Finite Group Databasebeta

← #29686942

Subgroups — #29686942

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
2#21454643C211yesmaximal, characteristic, complemented
3AGL(1,ℤ/8)×Dic61C211yesmaximal, characteristic
4C2×#16839859C211yesmaximal, characteristic, complemented
5#21454245C211yesmaximal, characteristic, complemented
6#21446719C211yesmaximal, characteristic, complemented
7#21446603C211yesmaximal, characteristic, complemented
8#21446478C211yesmaximal, characteristic, complemented
9#21446475C211yesmaximal, characteristic, complemented
10#21446334C211yesmaximal, characteristic, complemented
11#21445881C211yesmaximal, characteristic, complemented
12#21445724C211yesmaximal, characteristic, complemented
13#21445614C211yesmaximal, characteristic, complemented
14#100×Dic61C211yesmaximal, characteristic, complemented
15C61×#1907C211yesmaximal, characteristic
16C2×#14519807C2211yescharacteristic
17C22×#11758200C2211yescharacteristic
18C2×#14519784C2211yescharacteristic
19#41×Dic61C2211yescharacteristic
20C2×#14519742C2211yescharacteristic
21C2×D4×Dic61C2211yescharacteristic
22#16840023C418yescomplemented
23#16839874—21no—
24#16839866—21no—
25#16839863—21no—
26#16839859—21no—
27#16839859—21no—
28#16839859—21no—
29#16839859—21no—
30#16839859—21no—
31#16839854—21no—
32#16839853—21no—
33SD16×Dic61—21no—
34D8×Dic61—21no—
35C2×#14519712C2211yescharacteristic
36C2×#14519712C2211yescharacteristic
37#16839804C2211yescharacteristic
38#16839802C2211yescharacteristic
39C2×#14519710C2211yescharacteristic
40#16839772C2211yescharacteristic
41#16839767C2211yescharacteristic
42#16839763C2211yescharacteristic
43C2×#14519702C2211yescharacteristic
44C2×#14519695C2211yescharacteristic
45C2×#14519695C2211yescharacteristic
46C2×#14519694C2211yescharacteristic
47#16839720—21no—
48#16839720C2211yescharacteristic
49#16839717—21no—
50#16839717C2211yescharacteristic
51M16×Dic61—21no—
52M16×Dic61C2211yescharacteristic
53M16×Dic61C2211yescharacteristic
54M16×Dic61C2211yescharacteristic
55C2×#14519687—21no—
56C2×#14519687C2211yescharacteristic
57#16839531C2211yescharacteristic
58#16839530C2211yescharacteristic
59#16839530C2211yescharacteristic
60#16839527C2211yescharacteristic
61C2×AGL(1,ℤ/8)×C61C2211yescharacteristic
62C2×#131×C61C2211yescharacteristic
63C61×#417C2211yescharacteristic
64C2×#102×C61C2211yescharacteristic
65C2×#100×C61C2211yescharacteristic
66C61×#343C2211yescharacteristic
67C61×#341C2211yescharacteristic
68C22×Dic122C2311yescharacteristic
69C23×Dic61—21no—
70#14519807—22no—
71#14519807C2×C414yescomplemented
72C2×#11758202—21no—
73C2×#11758200—21no—
74C2×#11758200—21no—
75C2×#11758200—21no—
76C2×#11758200—22no—
77C2×#11758200C2×C414yescomplemented
78#14519784—22no—
79#14519784C2×C414yescomplemented
80C2×#11758194—21no—
81C2×#11758194C2311yescharacteristic
82C2×#11758194C2311yescharacteristic
83C2×#11758194C2311yescharacteristic
84C22×#92418—21no—
85C22×#92418C2311yescharacteristic
86Q8×Dic61D411yescharacteristic
87Q8×Dic61D411yescharacteristic
88#14519742—21no—
89#14519742—21no—
90#14519742—21no—
91#14519742—21no—
92D4×Dic61—21no—
93D4×Dic61—21no—
94D4×Dic61—21no—
95D4×Dic61—21no—
96D4×Dic61—21no—
97D4×Dic61—21no—
98D4×Dic61D411yescharacteristic
99D4×Dic61D411yescharacteristic
100#14519736—21no—
101#14519736C2311yescharacteristic
102C2×C4⋊Dic61C2311yescharacteristic
103C2×#11758196—21no—
104C2×C22⋊Dic61—21no—
105C2×C4×Dic61—21no—
106C2×C4×Dic61—21no—
107C2×C4×Dic61C2311yescharacteristic
108#14519712—22no—
109#14519712C2×C414yescomplemented
110#14519711—21no—
111#14519710—21no—
112#14519710—21no—
113#14519710—21no—
114#14519709—21no—
115#14519708—21no—
116#14519702—21no—
117#14519702—21no—
118#14519702—21no—
119#14519702—21no—
120#14519701—21no—
121#14519698—21no—
122#14519698—21no—
123#14519698—21no—
124#14519695—22no—
125#14519695C2×C414yescomplemented
126#14519694C2×C414yescomplemented
127#14519687—21no—
128#14519687—21no—
129#14519687—21no—
130#14519687—21no—
131#14519687—21no—
132#14519687D411yescharacteristic
133#14519687D411yescharacteristic
134#14519687D411yescharacteristic
135#14519687D411yescharacteristic
136C8×Dic61—21no—
137C8×Dic61—21no—
138C8×Dic61—21no—
139C8×Dic61—21no—
140C8×Dic61—21no—
141C2×#41×C61C2311yescharacteristic
142C22×D4×C61C2311yescharacteristic
143AGL(1,ℤ/8)×C61—22no—
144AGL(1,ℤ/8)×C61C2×C414yes—
145C2×SD16×C61—21no—
146C2×D8×C61—21no—
147#131×C61—22no—
148#131×C61—22no—
149C2×M16×C61—21no—
150C2×M16×C61C2311yescharacteristic
151C2×M16×C61C2311yescharacteristic
152C2×M16×C61C2311yescharacteristic
153C22×C8×C61—21no—
154C22×C8×C61C2311yescharacteristic
155C2×C4⋊C4×C61C2311yescharacteristic
156#103×C61—21no—
157#102×C61—21no—
158#102×C61—21no—
159#102×C61—21no—
160#100×C61—22no—
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×C61C2411yescharacteristic
178C22×Dic61—21no—
179C22×Dic61—21no—
180C22×Dic61—21no—
181C22×Dic61—21no—
182C22×Dic61—41no—
183C22×Dic61—41no—
184C22×Dic61—82no—
185C22×Dic61#4111yescharacteristic
186C22×Dic61#4111yescharacteristic
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—24no—
196M16×C61—21no—
197M16×C61—21no—
198M16×C61—21no—
199M16×C61—21no—
200M16×C61C22×C412yes—
201M16×C61#4112yes—
202C2×C8×C61—24no—
203C2×C8×C61—21no—
204C2×C8×C61—21no—
205C2×C8×C61—21no—
206C2×C8×C61—21no—
207C2×C8×C61—21no—
208C2×C8×C61C2×D411yescharacteristic
209C2×C8×C61C2×D411yescharacteristic
210C4⋊C4×C61—21no—
211C4⋊C4×C61—21no—
212C22⋊Dic61—21no—
213C22⋊Dic61—21no—
214C22⋊Dic61—21no—
215C22⋊Dic61—41no—
216C22⋊Dic61—41no—
217#11758202—24no—
218#11758200—24no—
219#11758200—24no—
220#11758200—24no—
221C4⋊Dic61—21no—
222C4⋊Dic61—21no—
223C4⋊Dic61—21no—
224C4⋊Dic61—21no—
225C4⋊Dic61C2×D411yescharacteristic
226C4⋊Dic61C2×D411yescharacteristic
227#11758196—41no—
228#11758196—41no—
229C4×Dic61—21no—
230C4×Dic61—21no—
231C4×Dic61—21no—
232C4×Dic61—41no—
233C4×Dic61—41no—
234C4×Dic61C2×D411yescharacteristic
235C4×Dic61C2×D411yescharacteristic
236C4×Dic61C2×D411yescharacteristic
237C4×Dic61C2×D411yescharacteristic
238#11758194—21no—
239#11758194—21no—
240#11758194—24no—
241#11758194—21no—
242#11758194—21no—
243#11758194C22×C412yes—
244#11758194C22×C412yes—
245C2×#92418—21no—
246C2×#92418—21no—
247C2×#92418—24no—
248C2×#92418—21no—
249C2×#92418C22×C412yes—
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—41no—
274C2×C4×C61—81no—
275C2×C4×C61C23×C411yescharacteristic
276C2×C4×C61C22×D411yescharacteristic
277C2×C4×C61C22×D411yescharacteristic
278C2×C4×C61C2×#4111yescharacteristic
279C2×C4×C61C2×#4111yescharacteristic
280C2×C4×C61C2×#4111yescharacteristic
281C2×Dic61—22no—
282C2×Dic61—21no—
283C2×Dic61—21no—
284C2×Dic61—41no—
285C2×Dic61—41no—
286C2×Dic61—41no—
287C2×Dic61—41no—
288C2×Dic61—41no—
289C2×Dic61—81no—
290C2×Dic61—81no—
291C2×Dic61—82no—
292C2×Dic61—82no—
293C2×Dic61—81no—
294C2×Dic61#12111yescharacteristic
295C2×Dic61#12111yescharacteristic
296C2×Dic61#12111yescharacteristic
297C2×Dic61#12111yescharacteristic
298Dic122—22no—
299Dic122—22no—
300Dic122—41no—
301Dic122—41no—
302C8×C61—22no—
303C8×C61—21no—
304C8×C61—22no—
305C8×C61—21no—
306C8×C61—41no—
307#92418—22no—
308#92418—22no—
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—81no—
328C4×C61C2×C4×D411yescharacteristic
329C4×C61C2×C4×D411yescharacteristic
330C4×C61#52511yescharacteristic
331C4×C61#53211yescharacteristic
332Dic61—21no—
333Dic61—21no—
334Dic61—21no—
335Dic61—21no—
336Dic61—42no—
337Dic61—41no—
338Dic61—41no—
339Dic61—81no—
340C2×#471—611no—
341C4×#137—611no—
342C4×AGL(1,ℤ/8)—611no—
343C2×#440—611no—
344#2875—611no—
345#1977—611no—
346#1952—611no—
347#1931—611no—
348#1930—611no—
349#1908—611no—
350#1907D6111yescharacteristic
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×#131C2×D6111yescharacteristic
367C4×#41—611no—
368C2×C4×Q8—611no—
369C2×C4×D4—611no—
370#471—618no—
371#442—1221no—
372#441—1221no—
373#440—1221no—
374#440—1221no—
375#440—1221no—
376#440—1221no—
377#440—1221no—
378#440—1221no—
379C4×Q16—1221no—
380C4×SD16—1221no—
381C4×SD16—1221no—
382C4×D8—1221no—
383C2×#108—611no—
384C2×#108—611no—
385#428—611no—
386#428—611no—
387C2×#107—611no—
388#419—611no—
389#418—611no—
390#417—611no—
391#417C2×D6111yescharacteristic
392C2×#103—611no—
393C2×#102C2×D6111yescharacteristic
394C2×#101—611no—
395C2×#101—611no—
396C2×#100—611no—
397C2×#100C2×D6111yescharacteristic
398#405—611no—
399#405—611no—
400#405—1221no—
401#405—1221no—
402C4×M16—611no—
403C4×M16—611no—
404C4×M16—611no—
405C4×M16—1221no—
406C2×#97—611no—
407C2×#97—1221no—
408#343—611no—
409#343—611no—
410#343—611no—
411#343C2×D6111yescharacteristic
412#341—611no—
413#341C2×D6111yescharacteristic
414C61#2155711yescharacteristic, complemented, Sylow
415C2×#41C22×D6111yescharacteristic
416C22×Q8—611no—
417C22×D4C22×D6111yescharacteristic
418C23×C4—1221no—
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—22no—
432#131—22no—
433#131—614no—
434#131—1222no—
435C2×M16—21no—
436C2×M16—611no—
437C2×M16—611no—
438C2×M16—611no—
439C2×M16—1221no—
440C2×M16C22×D6111yescharacteristic
441C2×M16C22×D6111yescharacteristic
442C2×M16C22×D6111yescharacteristic
443C22×C8—21no—
444C22×C8—611no—
445C22×C8—1221no—
446C22×C8C22×D6111yescharacteristic
447C4×Q8—611no—
448C4×Q8—611no—
449C4×Q8—1221no—
450C4×Q8—1221no—
451C4×Q8—1221no—
452C4×Q8—1221no—
453C4×D4—611no—
454C4×D4—611no—
455C4×D4—1221no—
456C4×D4—1221no—
457C4×D4—1221no—
458C4×D4—1221no—
459C4×D4—1221no—
460C4×D4—1221no—
461#117—611no—
462#117—1221no—
463C2×C4⋊C4—611no—
464C2×C4⋊C4—1221no—
465C2×C4⋊C4C22×D6111yescharacteristic
466C2×C22⋊C4—1221no—
467C2×C42—611no—
468C2×C42—1221no—
469C2×C42—1221no—
470#108—614no—
471#108—1222no—
472#107—1221no—
473#107—1221no—
474#107—1221no—
475#107—1221no—
476#106—1221no—
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—22no—
493#100—614no—
494#97—611no—
495#97—611no—
496#97—611no—
497#97—611no—
498#97—1221no—
499#97—1221no—
500#97—1221no—
501#97—1221no—
502#97—1221no—
503C4×C8—1221no—
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#1175822412yes—
513C2×Q8—612no—
514C2×Q8—1221no—
515C2×Q8—1221no—
516C2×Q8—1221no—
517C2×Q8—1221no—
518C2×Q8—1221no—
519C2×Q8D4×D6111yescharacteristic
520C2×D4—21no—
521C2×D4—21no—
522C2×D4—21no—
523C2×D4—21no—
524C2×D4—21no—
525C2×D4—21no—
526C2×D4C2×C4×D6112yes—
527C2×D4D4×D6111yescharacteristic
528C22×C4—21no—
529C22×C4—21no—
530C22×C4—611no—
531C22×C4—611no—
532C22×C4—1221no—
533C22×C4—1221no—
534C22×C4—1221no—
535C22×C4—1221no—
536C22×C4—4882no—
537C22×C4—2441no—
538C22×C4—2441no—
539C22×C4C23×D6111yescharacteristic
540Q16—1224no—
541SD16—24no—
542SD16—1224no—
543SD16—1224no—
544SD16—1224no—
545D8—24no—
546M16—24no—
547M16—21no—
548M16—21no—
549M16—21no—
550M16—21no—
551M16—612no—
552M16—612no—
553M16—1224no—
554M16—1221no—
555M16—1221no—
556M16—1221no—
557M16—1221no—
558M16C2×C4×D6112yes—
559M16#1175822412yes—
560C2×C8—24no—
561C2×C8—21no—
562C2×C8—21no—
563C2×C8—21no—
564C2×C8—21no—
565C2×C8—21no—
566C2×C8—612no—
567C2×C8—611no—
568C2×C8—611no—
569C2×C8—612no—
570C2×C8—1224no—
571C2×C8—1221no—
572C2×C8—1221no—
573C2×C8—1221no—
574C2×C8D4×D6111yescharacteristic
575C2×C8D4×D6111yescharacteristic
576C4⋊C4—21no—
577C4⋊C4—21no—
578C4⋊C4—611no—
579C4⋊C4—611no—
580C4⋊C4—1221no—
581C4⋊C4—1221no—
582C4⋊C4—1221no—
583C4⋊C4—1221no—
584C4⋊C4—2441no—
585C4⋊C4—2441no—
586C22⋊C4—1221no—
587C22⋊C4—1221no—
588C22⋊C4—1221no—
589C22⋊C4—2441no—
590C22⋊C4—2441no—
591C42—611no—
592C42—611no—
593C42—611no—
594C42—611no—
595C42—1221no—
596C42—1221no—
597C42—1221no—
598C42—2441no—
599C42—2441no—
600C23—21no—
601C23—22no—
602C23—41no—
603C23—41no—
604C23—81no—
605C23—81no—
606C23#41×D6111yescharacteristic
607Q8—21no—
608Q8—1222no—
609Q8—1222no—
610Q8—2441no—
611Q8—2441no—
612Q8#1451973712yes—
613D4—22no—
614D4—22no—
615D4—22no—
616D4—22no—
617D4—21no—
618D4—41no—
619D4—41no—
620D4#1451973712yes—
621C2×C4—21no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—41no—
625C2×C4—81no—
626C2×C4—611no—
627C2×C4—611no—
628C2×C4—611no—
629C2×C4—611no—
630C2×C4—1222no—
631C2×C4—1221no—
632C2×C4—1221no—
633C2×C4—4882no—
634C2×C4—4882no—
635C2×C4—4881no—
636C2×C4—4881no—
637C2×C4—4881no—
638C2×C4—2441no—
639C2×C4—2441no—
640C2×C4—2441no—
641C2×C4—2441no—
642C2×C4—2441no—
643C2×C4C22×C4×D6111yescharacteristic
644C2×C4C2×D4×D6111yescharacteristic
645C2×C4C2×D4×D6111yescharacteristic
646C2×C4#41×D6111yescharacteristic
647C2×C4C2×#1175822411yescharacteristic
648C2×C4C2×#1175822411yescharacteristic
649C8—22no—
650C8—21no—
651C8—22no—
652C8—21no—
653C8—41no—
654C8—612no—
655C8—612no—
656C8—1222no—
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
670C22#121×D6111yescharacteristic
671C22#1684028911yescharacteristic
672C4—22no—
673C4—81no—
674C4—1221no—
675C4—1221no—
676C4—1221no—
677C4—1221no—
678C4—4881no—
679C4—2442no—
680C4—2441no—
681C4—2441no—
682C4C4×D4×D6111yescharacteristic
683C4C2×#1451973711yescharacteristic
684C4#1684026811yescharacteristic
685C4#1684030711yescharacteristic
686C2—21no—
687C2—21no—
688C2—42no—
689C2—82no—
690C2#2144832911yescharacteristic
691C2#2145554511yescharacteristic
692C2#2145555111yescharacteristic
subgroupG/H
centerC22#121×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1907C2
socleC22×C61C2×#121
prime radicalC2×AGL(1,ℤ/8)×C61C22
cosocleC22×C4×C61C24
hypercenter#1907D61
hypocenterC61#21557
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1907

Complements

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

NH ≅ G/Ntype
2. #21454643C2semidirect
4. C2×#16839859C2semidirect
5. #21454245C2semidirect
6. #21446719C2semidirect
7. #21446603C2semidirect
8. #21446478C2semidirect
9. #21446475C2semidirect
10. #21446334C2semidirect
11. #21445881C2semidirect
12. #21445724C2semidirect
13. #21445614C2semidirect
14. #100×Dic61C2semidirect
22. #16840023C4semidirect
71. #14519807C2×C4semidirect
77. C2×#11758200C2×C4semidirect
79. #14519784C2×C4semidirect
109. #14519712C2×C4semidirect
125. #14519695C2×C4semidirect
126. #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
#214546432
AGL(1,ℤ/8)×Dic612
C2×#168398592
#214542452
#214467192
#214466032
#214464782
#214464752
#214463342
#214458812
#214457242
#214456142
#100×Dic612
C61×#19072
#2155761