Finite Group Databasebeta

← #29686933

Subgroups — #29686933

Order 15616.

906 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840023C211yesmaximal, characteristic
2C4×#14519807C211yesmaximal, characteristic, complemented
3C2×#16839862C211yesmaximal, characteristic, complemented
4#21454247C211yesmaximal, characteristic, complemented
5#21446725C211yesmaximal, characteristic, complemented
6#21446601C211yesmaximal, characteristic, complemented
7#21446484C211yesmaximal, characteristic, complemented
8#21446473C211yesmaximal, characteristic, complemented
9#21446332C211yesmaximal, characteristic, complemented
10#21446329C211yesmaximal, characteristic
11#21445881C211yesmaximal, characteristic, complemented
12#21445726C211yesmaximal, characteristic, complemented
13C4×#14519695C211yesmaximal, characteristic, complemented
14C4×#14519694C211yesmaximal, characteristic, complemented
15C4×AGL(1,ℤ/8)×C61C211yesmaximal, characteristic
16C2×#14519807C2211yescharacteristic
17C22×#11758200C2211yescharacteristic
18C2×#14519784C2211yescharacteristic
19C2×#14519784C2211yescharacteristic
20C2×C4×Dic122C2211yescharacteristic
21#16840023C418yescomplemented
22#16839867—21no—
23#16839862—21no—
24#16839862—21no—
25#16839862—21no—
26#16839862—21no—
27#16839862—21no—
28C4×#11758202—21no—
29C4×#11758200—21no—
30C2×#14519712C2211yescharacteristic
31C2×#14519712C2211yescharacteristic
32#16839805C2211yescharacteristic
33C2×#14519710C2211yescharacteristic
34#16839770C2211yescharacteristic
35#16839763C2211yescharacteristic
36C2×#14519700C2211yescharacteristic
37C2×#14519698C2211yescharacteristic
38C2×#14519695C2211yescharacteristic
39C2×#14519695C2211yescharacteristic
40C2×#14519694C2211yescharacteristic
41C2×#14519694C2211yescharacteristic
42#16839718—21no—
43#16839718—21no—
44#16839718C2211yescharacteristic
45#16839718C2211yescharacteristic
46C4×#11758194—21no—
47C4×#11758194C2211yescharacteristic
48C4×#11758194C2211yescharacteristic
49C2×#14519688—21no—
50C2×#14519688C2211yescharacteristic
51#16839531C2211yescharacteristic
52#16839531C2211yescharacteristic
53#16839530C2211yescharacteristic
54#16839530C2211yescharacteristic
55#16839526C2211yescharacteristic
56#16839526C2211yescharacteristic
57C2×AGL(1,ℤ/8)×C61C2211yescharacteristic
58C4×#41×C61C2211yescharacteristic
59C2×C4×D4×C61C2211yescharacteristic
60C61×#442—21no—
61C61×#440—21no—
62C4×SD16×C61—21no—
63C4×D8×C61—21no—
64C61×#428C2211yescharacteristic
65C61×#418C2211yescharacteristic
66C61×#417C2211yescharacteristic
67C4×M16×C61C2211yescharacteristic
68C22×Dic122C2311yescharacteristic
69#14519807—22no—
70#14519807C2×C414yescomplemented
71C2×#11758202—21no—
72C2×#11758200—21no—
73C2×#11758200—21no—
74C2×#11758200—21no—
75C2×#11758200—22no—
76C2×#11758200C2×C414yescomplemented
77#14519784—22no—
78#14519784—22no—
79#14519784—22no—
80#14519784C2×C414yescomplemented
81C2×#11758194—21no—
82C2×#11758194—21no—
83C2×#11758194C2311yescharacteristic
84C2×#11758194C2311yescharacteristic
85C2×#11758194C2311yescharacteristic
86C2×#11758194C2311yescharacteristic
87C22×#92418—21no—
88C22×#92418—21no—
89C22×#92418C2311yescharacteristic
90C22×#92418C2311yescharacteristic
91C4×Dic122—21no—
92C4×Dic122—21no—
93C4×Dic122—21no—
94C4×Dic122—21no—
95C2×C4⋊Dic61C2311yescharacteristic
96C2×#11758196—21no—
97C2×C4×Dic61—21no—
98#14519712—22no—
99#14519712C2×C414yescomplemented
100#14519710—21no—
101#14519710—21no—
102#14519710—21no—
103#14519708—21no—
104#14519701—21no—
105#14519700—21no—
106#14519700—21no—
107#14519700—21no—
108#14519700—21no—
109#14519698—21no—
110#14519698—21no—
111#14519698—21no—
112#14519695—22no—
113#14519695C2×C414yescomplemented
114#14519694—22no—
115#14519694C2×C414yescomplemented
116#14519688—21no—
117#14519688—21no—
118#14519688—21no—
119#14519688—21no—
120#14519688D411yescharacteristic
121#14519688D411yescharacteristic
122#14519688D411yescharacteristic
123#14519688D411yescharacteristic
124C4×#92418—21no—
125C4×#92418—21no—
126C4×#92418—21no—
127C4×#92418—21no—
128C2×#41×C61C2311yescharacteristic
129C22×D4×C61C2311yescharacteristic
130C23×C4×C61—21no—
131AGL(1,ℤ/8)×C61—22no—
132AGL(1,ℤ/8)×C61C2×C414yes—
133C2×SD16×C61—21no—
134C2×D8×C61—21no—
135C2×M16×C61C2311yescharacteristic
136C2×M16×C61C2311yescharacteristic
137C4×Q8×C61D411yescharacteristic
138C4×Q8×C61D411yescharacteristic
139C4×D4×C61—21no—
140C4×D4×C61—21no—
141C4×D4×C61—21no—
142C4×D4×C61—21no—
143C4×D4×C61—21no—
144C4×D4×C61—21no—
145C4×D4×C61D411yescharacteristic
146C4×D4×C61D411yescharacteristic
147#117×C61—21no—
148#117×C61C2311yescharacteristic
149C2×C4⋊C4×C61C2311yescharacteristic
150C2×C22⋊C4×C61—21no—
151C2×C42×C61—21no—
152C2×C42×C61C2311yescharacteristic
153#107×C61—21no—
154#106×C61—21no—
155#103×C61—21no—
156#102×C61—21no—
157#102×C61—21no—
158#102×C61—21no—
159#97×C61—21no—
160C4×C8×C61—21no—
161C24×C61—21no—
162#41×C61—21no—
163#41×C61—21no—
164#41×C61C22×C412yes—
165#41×C61#4112yes—
166C2×Q8×C61C2×D411yescharacteristic
167C2×D4×C61—21no—
168C2×D4×C61—21no—
169C2×D4×C61—21no—
170C2×D4×C61—21no—
171C2×D4×C61—21no—
172C2×D4×C61—21no—
173C2×D4×C61C22×C412yes—
174C2×D4×C61C2×D411yescharacteristic
175C22×C4×C61—21no—
176C22×C4×C61—21no—
177C22×C4×C61—21no—
178C22×C4×C61—21no—
179C22×C4×C61—41no—
180C22×C4×C61—41no—
181C22×C4×C61—82no—
182C22×C4×C61#4111yescharacteristic
183C22×C4×C61#4111yescharacteristic
184C22×C4×C61C2411yescharacteristic
185C22×Dic61—21no—
186C22×Dic61—21no—
187C2×Dic122—21no—
188C2×Dic122—21no—
189C2×Dic122—21no—
190C2×Dic122—21no—
191C2×Dic122—21no—
192C2×Dic122C22×C412yes—
193SD16×C61—24no—
194D8×C61—24no—
195M16×C61—21no—
196M16×C61—21no—
197M16×C61—21no—
198M16×C61C22×C412yes—
199C2×C8×C61—21no—
200C2×C8×C61—21no—
201C4⋊C4×C61—21no—
202C4⋊C4×C61—21no—
203C4⋊C4×C61—21no—
204C4⋊C4×C61—21no—
205C4⋊C4×C61C2×D411yescharacteristic
206C4⋊C4×C61C2×D411yescharacteristic
207C22⋊C4×C61—21no—
208C22⋊C4×C61—21no—
209C22⋊C4×C61—21no—
210C22⋊C4×C61—41no—
211C22⋊C4×C61—41no—
212C42×C61—21no—
213C42×C61—21no—
214C42×C61—21no—
215C42×C61C2×D411yescharacteristic
216C42×C61C2×D411yescharacteristic
217C42×C61C2×D411yescharacteristic
218C42×C61C2×D411yescharacteristic
219#11758202—24no—
220#11758200—24no—
221#11758200—24no—
222#11758200—24no—
223C4⋊Dic61—21no—
224C4⋊Dic61—21no—
225#11758196—41no—
226#11758196—41no—
227C4×Dic61—41no—
228C4×Dic61—41no—
229#11758194—24no—
230#11758194—21no—
231#11758194—21no—
232#11758194—21no—
233#11758194—24no—
234#11758194—21no—
235#11758194—21no—
236#11758194C22×C412yes—
237#11758194C22×C412yes—
238#11758194#4112yes—
239C2×#92418—24no—
240C2×#92418—21no—
241C2×#92418—21no—
242C2×#92418—21no—
243C2×#92418—21no—
244C2×#92418—21no—
245C2×#92418—24no—
246C2×#92418—21no—
247C2×#92418C22×C412yes—
248C2×#92418C2×D411yescharacteristic
249C2×#92418C2×D411yescharacteristic
250C2×#92418C2×D411yescharacteristic
251C2×#92418C2×D411yescharacteristic
252C2×#92418#4112yes—
253C23×C61—21no—
254C23×C61—22no—
255C23×C61—41no—
256C23×C61—41no—
257C23×C61—81no—
258C23×C61—81no—
259C23×C61C2×#4111yescharacteristic
260Q8×C61—21no—
261Q8×C61C4×D412yes—
262D4×C61—22no—
263D4×C61—22no—
264D4×C61—22no—
265D4×C61—22no—
266D4×C61—21no—
267D4×C61—41no—
268D4×C61—41no—
269D4×C61C4×D412yes—
270C2×C4×C61—21no—
271C2×C4×C61—22no—
272C2×C4×C61—22no—
273C2×C4×C61—21no—
274C2×C4×C61—21no—
275C2×C4×C61—41no—
276C2×C4×C61—41no—
277C2×C4×C61—41no—
278C2×C4×C61—81no—
279C2×C4×C61—81no—
280C2×C4×C61—82no—
281C2×C4×C61—82no—
282C2×C4×C61—81no—
283C2×C4×C61#12111yescharacteristic
284C2×C4×C61#12111yescharacteristic
285C2×C4×C61#12111yescharacteristic
286C2×C4×C61#12111yescharacteristic
287C2×C4×C61C23×C411yescharacteristic
288C2×C4×C61C22×D411yescharacteristic
289C2×C4×C61C22×D411yescharacteristic
290C2×C4×C61C2×#4111yescharacteristic
291C2×C4×C61C2×#4111yescharacteristic
292C2×C4×C61C2×#4111yescharacteristic
293C2×Dic61—22no—
294C2×Dic61—41no—
295C2×Dic61—41no—
296C2×Dic61—41no—
297C2×Dic61—81no—
298Dic122—22no—
299Dic122—22no—
300Dic122—41no—
301Dic122—41no—
302C8×C61—22no—
303C8×C61—41no—
304#92418—22no—
305#92418—22no—
306#92418—21no—
307#92418—22no—
308#92418—21no—
309#92418—41no—
310#92418C4×D412yes—
311#92418C4×D412yes—
312#21557—611nomaximal, Sylow
313C22×C61—21no—
314C22×C61—21no—
315C22×C61—22no—
316C22×C61—41no—
317C22×C61—41no—
318C22×C61—42no—
319C22×C61—82no—
320C22×C61—82no—
321C22×C61—81no—
322C22×C61—81no—
323C22×C61C4×#4111yescharacteristic
324C22×C61C2×#12111yescharacteristic
325C22×C61#52911yescharacteristic
326C4×C61—22no—
327C4×C61—21no—
328C4×C61—21no—
329C4×C61—21no—
330C4×C61—21no—
331C4×C61—41no—
332C4×C61—41no—
333C4×C61—81no—
334C4×C61C2×C4×D411yescharacteristic
335C4×C61C2×C4×D411yescharacteristic
336C4×C61#52511yescharacteristic
337C4×C61#53211yescharacteristic
338Dic61—42no—
339Dic61—81no—
340C2×#471—611no—
341C4×#137—611no—
342C4×AGL(1,ℤ/8)D6111yescharacteristic
343C2×#440—611no—
344#2875—611no—
345#1977—611no—
346#1952—611no—
347#1931—611no—
348#1930—611no—
349#1908—611no—
350#1907—611no—
351#1816—611no—
352#1778—611no—
353C4×#101—611no—
354C4×#100—611no—
355C2×C61—21no—
356C2×C61—21no—
357C2×C61—42no—
358C2×C61—82no—
359C2×C61C4×#12111yescharacteristic
360C2×C61#306811yescharacteristic
361C2×C61#306911yescharacteristic
362C2×#137—611no—
363C2×AGL(1,ℤ/8)C2×D6111yescharacteristic
364C22×SD16—611no—
365C2×#131—611no—
366C2×#131—611no—
367C4×#41C2×D6111yescharacteristic
368C2×C4×Q8—611no—
369C2×C4×D4C2×D6111yescharacteristic
370#471—618no—
371#442—21no—
372#441—1221no—
373#440—21no—
374#440—1221no—
375#440—1221no—
376#440—1221no—
377#440—1221no—
378#440—1221no—
379C4×Q16—1221no—
380C4×SD16—21no—
381C4×SD16—1221no—
382C4×D8—21no—
383C2×#108—611no—
384C2×#108—611no—
385#428—611no—
386#428C2×D6111yescharacteristic
387C2×#107—611no—
388#419—611no—
389#418C2×D6111yescharacteristic
390#417—611no—
391#417C2×D6111yescharacteristic
392C2×#103—611no—
393C2×#102—611no—
394C2×#101—611no—
395C2×#101—611no—
396C2×#100—611no—
397C2×#100—611no—
398#405—611no—
399#405—611no—
400#405—1221no—
401#405—1221no—
402C4×M16—611no—
403C4×M16—611no—
404C4×M16—1221no—
405C4×M16C2×D6111yescharacteristic
406C2×#97—611no—
407C2×#97—1221no—
408#343—611no—
409#343—611no—
410#343—611no—
411#343—611no—
412#341—611no—
413#341—611no—
414C61#2155711yescharacteristic, complemented, Sylow
415C2×#41C22×D6111yescharacteristic
416C22×Q8—611no—
417C22×D4C22×D6111yescharacteristic
418C23×C4—21no—
419#137—614no—
420#137—1222no—
421AGL(1,ℤ/8)—22no—
422AGL(1,ℤ/8)C4×D6114yes—
423C2×Q16—1221no—
424C2×SD16—21no—
425C2×SD16—614no—
426C2×SD16—1221no—
427C2×SD16—1221no—
428C2×SD16—1222no—
429C2×SD16—1221no—
430C2×D8—21no—
431#131—614no—
432#131—1222no—
433#131—1222no—
434#131—1222no—
435C2×M16—611no—
436C2×M16—611no—
437C2×M16—611no—
438C2×M16—611no—
439C2×M16—1221no—
440C2×M16—1221no—
441C2×M16C22×D6111yescharacteristic
442C2×M16C22×D6111yescharacteristic
443C22×C8—611no—
444C22×C8—611no—
445C22×C8—1221no—
446C22×C8—1221no—
447C4×Q8—1221no—
448C4×Q8—1221no—
449C4×Q8—1221no—
450C4×Q8—1221no—
451C4×Q8C22⋊D6111yescharacteristic
452C4×Q8C22⋊D6111yescharacteristic
453C4×D4—21no—
454C4×D4—21no—
455C4×D4—21no—
456C4×D4—21no—
457C4×D4—21no—
458C4×D4—21no—
459C4×D4C22⋊D6111yescharacteristic
460C4×D4C22⋊D6111yescharacteristic
461#117—21no—
462#117C22×D6111yescharacteristic
463C2×C4⋊C4—611no—
464C2×C4⋊C4—1221no—
465C2×C4⋊C4C22×D6111yescharacteristic
466C2×C22⋊C4—21no—
467C2×C42—21no—
468C2×C42—1221no—
469C2×C42C22×D6111yescharacteristic
470#108—614no—
471#108—1222no—
472#107—21no—
473#107—1221no—
474#107—1221no—
475#107—1221no—
476#106—21no—
477#106—1221no—
478#103—21no—
479#103—1221no—
480#103—1221no—
481#103—1221no—
482#103—1221no—
483#103—1221no—
484#102—21no—
485#102—21no—
486#102—21no—
487#102—1221no—
488#102—1221no—
489#102—1221no—
490#101—614no—
491#101—1222no—
492#100—614no—
493#100—1222no—
494#97—21no—
495#97—611no—
496#97—611no—
497#97—611no—
498#97—611no—
499#97—1221no—
500#97—1221no—
501#97—1221no—
502#97—1221no—
503C4×C8—21no—
504C4×C8—1221no—
505C4×C8—1221no—
506C4×C8—1221no—
507C4×C8—1221no—
508C24—21no—
509#41—21no—
510#41—21no—
511#41C2×C4×D6112yes—
512#41#1175822212yes—
513C2×Q8—612no—
514C2×Q8—1221no—
515C2×Q8—1221no—
516C2×Q8—1221no—
517C2×Q8—1221no—
518C2×Q8—1221no—
519C2×Q8C2×C22⋊D6111yescharacteristic
520C2×D4—21no—
521C2×D4—21no—
522C2×D4—21no—
523C2×D4—21no—
524C2×D4—21no—
525C2×D4—21no—
526C2×D4C2×C4×D6112yes—
527C2×D4C2×C22⋊D6111yescharacteristic
528C22×C4—21no—
529C22×C4—21no—
530C22×C4—21no—
531C22×C4—21no—
532C22×C4—41no—
533C22×C4—41no—
534C22×C4—82no—
535C22×C4—1221no—
536C22×C4—1221no—
537C22×C4#1175822411yescharacteristic
538C22×C4#1175822411yescharacteristic
539C22×C4C23×D6111yescharacteristic
540Q16—1224no—
541SD16—24no—
542SD16—1224no—
543SD16—1224no—
544SD16—1224no—
545D8—24no—
546M16—21no—
547M16—21no—
548M16—21no—
549M16—612no—
550M16—612no—
551M16—612no—
552M16—1224no—
553M16—1224no—
554M16—1221no—
555M16—1221no—
556M16—1221no—
557M16—1221no—
558M16—1221no—
559M16C2×C4×D6112yes—
560C2×C8—21no—
561C2×C8—21no—
562C2×C8—612no—
563C2×C8—611no—
564C2×C8—611no—
565C2×C8—612no—
566C2×C8—611no—
567C2×C8—611no—
568C2×C8—1224no—
569C2×C8—1224no—
570C2×C8—1221no—
571C2×C8—1221no—
572C2×C8—1221no—
573C2×C8—1221no—
574C2×C8—1221no—
575C2×C8—1221no—
576C4⋊C4—21no—
577C4⋊C4—21no—
578C4⋊C4—21no—
579C4⋊C4—21no—
580C4⋊C4—1221no—
581C4⋊C4—1221no—
582C4⋊C4—2441no—
583C4⋊C4—2441no—
584C4⋊C4C2×C22⋊D6111yescharacteristic
585C4⋊C4C2×C22⋊D6111yescharacteristic
586C22⋊C4—21no—
587C22⋊C4—21no—
588C22⋊C4—21no—
589C22⋊C4—41no—
590C22⋊C4—41no—
591C42—21no—
592C42—21no—
593C42—21no—
594C42—2441no—
595C42—2441no—
596C42D4×D6111yescharacteristic
597C42D4×D6111yescharacteristic
598C42C2×C22⋊D6111yescharacteristic
599C42C2×C22⋊D6111yescharacteristic
600C23—21no—
601C23—22no—
602C23—41no—
603C23—41no—
604C23—81no—
605C23—81no—
606C23C2×#1175822411yescharacteristic
607Q8—21no—
608Q8—1222no—
609Q8—1222no—
610Q8—2441no—
611Q8—2441no—
612Q8C4×C22⋊D6112yes—
613D4—22no—
614D4—22no—
615D4—22no—
616D4—22no—
617D4—21no—
618D4—41no—
619D4—41no—
620D4C4×C22⋊D6112yes—
621C2×C4—21no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—21no—
625C2×C4—21no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—81no—
630C2×C4—81no—
631C2×C4—82no—
632C2×C4—82no—
633C2×C4—81no—
634C2×C4—1222no—
635C2×C4—4881no—
636C2×C4—2441no—
637C2×C4—2441no—
638C2×C4—2441no—
639C2×C4#1451974911yescharacteristic
640C2×C4#1451974911yescharacteristic
641C2×C4#1451974911yescharacteristic
642C2×C4#1451974911yescharacteristic
643C2×C4C22×C4×D6111yescharacteristic
644C2×C4C2×D4×D6111yescharacteristic
645C2×C4C22×C22⋊D6111yescharacteristic
646C2×C4#41×D6111yescharacteristic
647C2×C4#41×D6111yescharacteristic
648C2×C4C2×#1175822211yescharacteristic
649C8—22no—
650C8—41no—
651C8—612no—
652C8—612no—
653C8—1222no—
654C8—1222no—
655C8—1221no—
656C8—1221no—
657C8—1222no—
658C8—2441no—
659C22—21no—
660C22—21no—
661C22—22no—
662C22—41no—
663C22—41no—
664C22—42no—
665C22—82no—
666C22—82no—
667C22—81no—
668C22—81no—
669C22C4×#1175822411yescharacteristic
670C22C2×#1451974911yescharacteristic
671C22#1684029211yescharacteristic
672C4—22no—
673C4—21no—
674C4—21no—
675C4—21no—
676C4—21no—
677C4—41no—
678C4—41no—
679C4—81no—
680C4—4881no—
681C4—2442no—
682C4C4×D4×D6111yescharacteristic
683C4C2×C4×C22⋊D6111yescharacteristic
684C4#1684027111yescharacteristic
685C4#1684030811yescharacteristic
686C2—21no—
687C2—21no—
688C2—42no—
689C2—82no—
690C2C4×#1451974911yescharacteristic
691C2#2145554411yescharacteristic
692C2#2145555811yescharacteristic
subgroupG/H
centerC22C2×#14519749
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC4×AGL(1,ℤ/8)×C61C2
socleC22×C61C2×#121
prime radicalC2×AGL(1,ℤ/8)×C61C22
cosocleC22×C4×C61C24
hypercenterC4×AGL(1,ℤ/8)D61
hypocenterC61#21557
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C4×AGL(1,ℤ/8)

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C4×#14519807C2semidirect
3. C2×#16839862C2semidirect
4. #21454247C2semidirect
5. #21446725C2semidirect
6. #21446601C2semidirect
7. #21446484C2semidirect
8. #21446473C2semidirect
9. #21446332C2semidirect
11. #21445881C2semidirect
12. #21445726C2semidirect
13. C4×#14519695C2semidirect
14. C4×#14519694C2semidirect
21. #16840023C4semidirect
70. #14519807C2×C4semidirect
76. C2×#11758200C2×C4semidirect
80. #14519784C2×C4semidirect
99. #14519712C2×C4semidirect
113. #14519695C2×C4semidirect
115. #14519694C2×C4semidirect
414. C61#21557semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#168400232
C4×#145198072
C2×#168398622
#214542472
#214467252
#214466012
#214464842
#214464732
#214463322
#214463292
#214458812
#214457262
C4×#145196952
C4×#145196942
C4×AGL(1,ℤ/8)×C612
#2155761