Finite Group Databasebeta

← #29686940

Subgroups — #29686940

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×#16840022C211yesmaximal, characteristic
2#21454634C211yesmaximal, characteristic, complemented
3#21454623C211yesmaximal, characteristic
4C2×#16839863C211yesmaximal, characteristic, complemented
5#21454246C211yesmaximal, characteristic, complemented
6#21446604C211yesmaximal, characteristic, complemented
7#21446484C211yesmaximal, characteristic, complemented
8#21446477C211yesmaximal, characteristic, complemented
9#21446331C211yesmaximal, characteristic, complemented
10#21446324C211yesmaximal, characteristic
11#21445880C211yesmaximal, characteristic, complemented
12#21445727C211yesmaximal, characteristic, complemented
13#101×Dic61C211yesmaximal, characteristic, complemented
14#100×Dic61C211yesmaximal, characteristic, complemented
15C61×#1977C211yesmaximal, characteristic, complemented
16C2×#14519808C2211yescharacteristic
17C2×#14519803C2211yescharacteristic
18C2×#14519785C2211yescharacteristic
19C2×#14519783C2211yescharacteristic
20#16840230C2211yescharacteristic
21C2×Q8×Dic61C2211yescharacteristic
22C2×D4×Dic61C2211yescharacteristic
23#16840022C418yescomplemented
24#16839871—21no—
25#16839866—21no—
26#16839865—21no—
27#16839863—21no—
28#16839863—21no—
29#16839863—21no—
30#16839863—21no—
31#16839859—21no—
32#16839854—21no—
33#16839853—21no—
34#16839847—21no—
35#16839842—21no—
36C2×#14519713C2211yescharacteristic
37#16839805C2211yescharacteristic
38#16839805C2211yescharacteristic
39C2×#14519711C2211yescharacteristic
40#16839764C2211yescharacteristic
41#16839764C2211yescharacteristic
42C2×#14519701C2211yescharacteristic
43C2×#14519698C2211yescharacteristic
44C2×#14519696C2211yescharacteristic
45C2×#14519694C2211yescharacteristic
46#16839719—21no—
47#16839718C2211yescharacteristic
48#16839716—21no—
49M16×Dic61C2211yescharacteristic
50M16×Dic61C2211yescharacteristic
51M16×Dic61C2211yescharacteristic
52C4×#11758189—21no—
53C2×#14519687C2211yescharacteristic
54#97×D61—21no—
55#16839530C2211yescharacteristic
56#16839530C2211yescharacteristic
57#16839530C2211yescharacteristic
58#16839530C2211yescharacteristic
59#16839526C2211yescharacteristic
60#16839526C2211yescharacteristic
61C22×SD16×C61C2211yescharacteristic
62C2×#108×C61C2211yescharacteristic
63C61×#419C2211yescharacteristic
64C61×#418C2211yescharacteristic
65C2×#101×C61C2211yescharacteristic
66C2×#100×C61C2211yescharacteristic
67C61×#405C2211yescharacteristic
68C2×#11758222C2311yescharacteristic
69C23×Dic61—21no—
70#14519808—22no—
71#14519808C2×C414yescomplemented
72#14519803—22no—
73#14519803C2×C414yes—
74C2×#11758202—21no—
75C2×#11758200—21no—
76C2×#11758201—21no—
77C2×#11758199—21no—
78#14519785—22no—
79#14519785—22no—
80#14519783—22no—
81#14519783C2×C414yescomplemented
82C2×#11758194C2311yescharacteristic
83C2×#11758194C2311yescharacteristic
84C2×#11758194C2311yescharacteristic
85C2×#11758189—21no—
86C2×#11758189—21no—
87C22×#92418C2311yescharacteristic
88C2×C8×D61—21no—
89C2×C8×D61—21no—
90Q8×Dic61—21no—
91Q8×Dic61—21no—
92Q8×Dic61—21no—
93Q8×Dic61—21no—
94#14519742D411yescharacteristic
95#14519742D411yescharacteristic
96D4×Dic61—21no—
97D4×Dic61—21no—
98D4×Dic61—21no—
99D4×Dic61—21no—
100#14519739D411yescharacteristic
101#14519739D411yescharacteristic
102#14519737—21no—
103#14519737—21no—
104#14519734—21no—
105C2×C4⋊Dic61—21no—
106C2×C4⋊Dic61C2311yescharacteristic
107C2×C4⋊Dic61C2311yescharacteristic
108C2×C22⋊Dic61—21no—
109C2×C4×Dic61—21no—
110C2×C4×Dic61C2311yescharacteristic
111C42×D61—21no—
112#14519713C2×C414yescomplemented
113#14519711—21no—
114#14519711—21no—
115#14519710—21no—
116#14519710—21no—
117#14519708—21no—
118#14519708—21no—
119#14519702—21no—
120#14519702—21no—
121#14519701—21no—
122#14519701—21no—
123#14519699—21no—
124#14519699—21no—
125#14519698—21no—
126#14519698—21no—
127#14519696C2×C414yescomplemented
128#14519694C2×C414yescomplemented
129#14519688—21no—
130#14519687—21no—
131#14519687—21no—
132#14519687—21no—
133#14519687D411yescharacteristic
134#14519687D411yescharacteristic
135#14519687D411yescharacteristic
136#14519687D411yescharacteristic
137C4×#92418—21no—
138C8×Dic61—21no—
139C8×Dic61—21no—
140C8×Dic61—21no—
141C22×Q8×C61C2311yescharacteristic
142C22×D4×C61C2311yescharacteristic
143C2×SD16×C61—21no—
144C2×SD16×C61—21no—
145C2×SD16×C61—22no—
146C2×SD16×C61C2×C414yescomplemented
147C2×M16×C61C2311yescharacteristic
148C2×M16×C61C2311yescharacteristic
149C2×M16×C61C2311yescharacteristic
150C22×C8×C61C2311yescharacteristic
151#117×C61C2311yescharacteristic
152#108×C61—22no—
153#103×C61—21no—
154#103×C61—21no—
155#102×C61—21no—
156#102×C61—21no—
157#101×C61—22no—
158#100×C61—22no—
159#97×C61—21no—
160C4×C8×C61—21no—
161C24×C61—21no—
162C2×Q8×C61—21no—
163C2×Q8×C61—21no—
164C2×Q8×C61—21no—
165C2×Q8×C61—21no—
166C2×Q8×C61—21no—
167C2×Q8×C61C22×C412yes—
168C2×D4×C61—21no—
169C2×D4×C61—21no—
170C2×D4×C61—21no—
171C2×D4×C61—21no—
172C2×D4×C61—21no—
173C2×D4×C61C22×C412yes—
174C22×C4×C61—21no—
175C22×C4×C61C2411yescharacteristic
176C2×C22⋊D61—21no—
177C22×Dic61—21no—
178C22×Dic61—21no—
179C22×Dic61—41no—
180C22×Dic61—41no—
181C22×Dic61—82no—
182C22×Dic61#4111yescharacteristic
183C22×Dic61#4111yescharacteristic
184#11758222—21no—
185#11758222—21no—
186#11758222C22×C412yes—
187#11758222#4112yes—
188C2×D244C2×D411yescharacteristic
189C2×C4×D61—21no—
190C2×C4×D61—21no—
191C2×C4×D61—21no—
192C2×Dic122C2×D411yescharacteristic
193SD16×C61—24no—
194SD16×C61—24no—
195M16×C61—21no—
196M16×C61—21no—
197M16×C61—21no—
198M16×C61—21no—
199M16×C61—21no—
200M16×C61#4112yes—
201C2×C8×C61—21no—
202C2×C8×C61—21no—
203C2×C8×C61—21no—
204C2×C8×C61C22×C412yes—
205C2×C8×C61C2×D411yescharacteristic
206C2×C8×C61C2×D411yescharacteristic
207C2×C8×C61#4112yes—
208C4⋊C4×C61C2×D411yescharacteristic
209C4⋊C4×C61C2×D411yescharacteristic
210C22⋊C4×C61—21no—
211C42×C61—21no—
212C22⋊Dic61—41no—
213C22⋊Dic61—41no—
214#11758202—24no—
215#11758201—24no—
216#11758200—24no—
217#11758199—24no—
218#11758198—21no—
219#11758198—21no—
220C4⋊Dic61—21no—
221C4⋊Dic61—21no—
222C4⋊Dic61—21no—
223C4⋊Dic61—21no—
224C4⋊Dic61—41no—
225C4⋊Dic61—41no—
226#11758196—21no—
227#11758196—21no—
228C4×Dic61—21no—
229C4×Dic61—21no—
230C4×Dic61—41no—
231C4×Dic61—41no—
232C4×Dic61C2×D411yescharacteristic
233C4×Dic61C2×D411yescharacteristic
234C4×Dic61C2×D411yescharacteristic
235C4×Dic61C2×D411yescharacteristic
236#11758194—21no—
237#11758194—21no—
238#11758194—21no—
239#11758194C22×C412yes—
240#11758194C22×C412yes—
241#11758194C22×C412yes—
242C2×#92418—21no—
243C2×#92418—21no—
244C2×#92418—21no—
245C2×#92418—21no—
246C2×#92418—21no—
247C2×#92418C2×D411yescharacteristic
248C2×#92418C2×D411yescharacteristic
249#11758189—24no—
250#11758189—24no—
251C8×D61—24no—
252C8×D61—24no—
253C23×C61—22no—
254C23×C61—41no—
255C23×C61—41no—
256C23×C61—81no—
257C23×C61—81no—
258C23×C61C2×#4111yescharacteristic
259C22×D61—21no—
260Q8×C61—22no—
261Q8×C61—22no—
262Q8×C61—41no—
263Q8×C61—41no—
264D4×C61—22no—
265D4×C61—22no—
266D4×C61—41no—
267D4×C61—41no—
268C2×C4×C61—22no—
269C2×C4×C61—21no—
270C2×C4×C61—21no—
271C2×C4×C61—41no—
272C2×C4×C61—41no—
273C2×C4×C61C23×C411yescharacteristic
274C2×C4×C61C22×D411yescharacteristic
275C2×C4×C61C22×D411yescharacteristic
276C2×C4×C61C2×#4111yescharacteristic
277C2×C4×C61C2×#4111yescharacteristic
278C2×C4×C61C2×#4111yescharacteristic
279C22⋊D61—22no—
280C22⋊D61—22no—
281C2×Dic61—21no—
282C2×Dic61—41no—
283C2×Dic61—41no—
284C2×Dic61—41no—
285C2×Dic61—41no—
286C2×Dic61—82no—
287C2×Dic61—82no—
288C2×Dic61—81no—
289C2×Dic61—81no—
290C2×Dic61#12111yescharacteristic
291C2×Dic61#12111yescharacteristic
292C2×Dic61#12111yescharacteristic
293C2×Dic61#12111yescharacteristic
294D244—21no—
295D244C4×D412yes—
296C4×D61—22no—
297C4×D61—22no—
298C4×D61—81no—
299C4×D61—81no—
300Dic122—21no—
301Dic122C4×D412yes—
302C8×C61—22no—
303C8×C61—41no—
304C8×C61—41no—
305C8×C61C4×D412yes—
306C8×C61C4×D412yes—
307#92418—22no—
308#92418—22no—
309#92418—22no—
310#92418—21no—
311#92418—21no—
312#21557—611nomaximal, Sylow
313C22×C61—21no—
314C22×C61—21no—
315C22×C61—42no—
316C22×C61—82no—
317C22×C61—82no—
318C22×C61—81no—
319C22×C61—81no—
320C22×C61C4×#4111yescharacteristic
321C22×C61C2×#12111yescharacteristic
322C22×C61#52911yescharacteristic
323C2×D61—22no—
324C2×D61—41no—
325C2×D61—41no—
326C4×C61—42no—
327C4×C61—41no—
328C4×C61—41no—
329C4×C61C2×C4×D411yescharacteristic
330C4×C61C2×C4×D411yescharacteristic
331C4×C61#52511yescharacteristic
332C4×C61#53211yescharacteristic
333Dic61—22no—
334Dic61—21no—
335Dic61—21no—
336Dic61—21no—
337Dic61—21no—
338Dic61—81no—
339Dic61—81no—
340C2×#471—611no—
341C4×#137—611no—
342C4×AGL(1,ℤ/8)—611no—
343C2×#440—611no—
344#2875—611no—
345#1977D6111yescharacteristic, complemented
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—82no—
358C2×C61C4×#12111yescharacteristic
359C2×C61#306811yescharacteristic
360C2×C61#306911yescharacteristic
361D61—42no—
362C2×#137—611no—
363C2×AGL(1,ℤ/8)—611no—
364C22×SD16C2×D6111yescharacteristic
365C2×#131—611no—
366C2×#131—611no—
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×#108C2×D6111yescharacteristic
385#428—611no—
386#428—611no—
387C2×#107—611no—
388#419C2×D6111yescharacteristic
389#418C2×D6111yescharacteristic
390#417—611no—
391#417—611no—
392C2×#103—611no—
393C2×#102—611no—
394C2×#101—611no—
395C2×#101C2×D6111yescharacteristic
396C2×#100—611no—
397C2×#100C2×D6111yescharacteristic
398#405—611no—
399#405—1221no—
400#405—1221no—
401#405C2×D6111yescharacteristic
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#343—611no—
412#341—611no—
413#341—611no—
414C61#2155711yescharacteristic, complemented, Sylow
415C2×#41—611no—
416C22×Q8C22×D6111yescharacteristic
417C22×D4C22×D6111yescharacteristic
418C23×C4—1221no—
419#137—614no—
420#137—1222no—
421AGL(1,ℤ/8)—614no—
422AGL(1,ℤ/8)—1222no—
423C2×Q16—1221no—
424C2×SD16—21no—
425C2×SD16—21no—
426C2×SD16—22no—
427C2×SD16—1221no—
428C2×SD16—1221no—
429C2×SD16C4×D6114yescomplemented
430C2×D8—1221no—
431#131—614no—
432#131—1222no—
433#131—1222no—
434#131—1222no—
435C2×M16—611no—
436C2×M16—611no—
437C2×M16—611no—
438C2×M16—1221no—
439C2×M16—1221no—
440C2×M16C22×D6111yescharacteristic
441C2×M16C22×D6111yescharacteristic
442C2×M16C22×D6111yescharacteristic
443C22×C8—611no—
444C22×C8—1221no—
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—1221no—
462#117C22×D6111yescharacteristic
463C2×C4⋊C4—611no—
464C2×C4⋊C4—611no—
465C2×C4⋊C4—1221no—
466C2×C22⋊C4—1221no—
467C2×C42—611no—
468C2×C42—1221no—
469C2×C42—1221no—
470#108—22no—
471#108—614no—
472#107—1221no—
473#107—1221no—
474#107—1221no—
475#107—1221no—
476#106—1221no—
477#106—1221no—
478#103—21no—
479#103—21no—
480#103—1221no—
481#103—1221no—
482#103—1221no—
483#103—1221no—
484#102—21no—
485#102—21no—
486#102—1221no—
487#102—1221no—
488#102—1221no—
489#102—1221no—
490#101—22no—
491#101—614no—
492#100—22no—
493#100—614no—
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—612no—
510#41—612no—
511#41—1221no—
512#41—1221no—
513C2×Q8—21no—
514C2×Q8—21no—
515C2×Q8—21no—
516C2×Q8—21no—
517C2×Q8—21no—
518C2×Q8—611no—
519C2×Q8C2×C4×D6112yes—
520C2×D4—21no—
521C2×D4—21no—
522C2×D4—21no—
523C2×D4—21no—
524C2×D4—21no—
525C2×D4—611no—
526C2×D4—1221no—
527C2×D4C2×C4×D6112yes—
528C22×C4—21no—
529C22×C4—611no—
530C22×C4—611no—
531C22×C4—1221no—
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—24no—
543SD16—1224no—
544SD16—1224no—
545D8—1224no—
546M16—21no—
547M16—21no—
548M16—21no—
549M16—21no—
550M16—21no—
551M16—612no—
552M16—612no—
553M16—612no—
554M16—1224no—
555M16—1221no—
556M16—1221no—
557M16—1221no—
558M16—1224no—
559M16#1175822412yes—
560C2×C8—21no—
561C2×C8—21no—
562C2×C8—21no—
563C2×C8—611no—
564C2×C8—611no—
565C2×C8—1224no—
566C2×C8—1221no—
567C2×C8—1221no—
568C2×C8—1221no—
569C2×C8—1221no—
570C2×C8—1221no—
571C2×C8—1224no—
572C2×C8C2×C4×D6112yes—
573C2×C8D4×D6111yescharacteristic
574C2×C8D4×D6111yescharacteristic
575C2×C8#1175822412yes—
576C4⋊C4—1221no—
577C4⋊C4—1221no—
578C4⋊C4—1221no—
579C4⋊C4—1221no—
580C4⋊C4—1221no—
581C4⋊C4—1221no—
582C4⋊C4—2441no—
583C4⋊C4—2441no—
584C4⋊C4D4×D6111yescharacteristic
585C4⋊C4D4×D6111yescharacteristic
586C22⋊C4—21no—
587C22⋊C4—1221no—
588C22⋊C4—1221no—
589C22⋊C4—2441no—
590C22⋊C4—2441no—
591C42—21no—
592C42—611no—
593C42—611no—
594C42—611no—
595C42—611no—
596C42—1221no—
597C42—1221no—
598C42—2441no—
599C42—2441no—
600C23—22no—
601C23—41no—
602C23—41no—
603C23—81no—
604C23—81no—
605C23—1221no—
606C23#41×D6111yescharacteristic
607Q8—22no—
608Q8—22no—
609Q8—41no—
610Q8—41no—
611Q8—612no—
612Q8—1221no—
613D4—22no—
614D4—22no—
615D4—41no—
616D4—41no—
617D4—612no—
618D4—1222no—
619D4—1222no—
620D4—1221no—
621C2×C4—22no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—611no—
627C2×C4—611no—
628C2×C4—611no—
629C2×C4—611no—
630C2×C4—1221no—
631C2×C4—1222no—
632C2×C4—1222no—
633C2×C4—4881no—
634C2×C4—4881no—
635C2×C4—4882no—
636C2×C4—4882no—
637C2×C4—4881no—
638C2×C4—4881no—
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—41no—
651C8—41no—
652C8—1222no—
653C8—1222no—
654C8—1222no—
655C8—1221no—
656C8—1221no—
657C8#1451973712yes—
658C8#1451973712yes—
659C22—21no—
660C22—21no—
661C22—42no—
662C22—82no—
663C22—82no—
664C22—81no—
665C22—81no—
666C22—1222no—
667C22—2441no—
668C22—2441no—
669C22C4×#1175822411yescharacteristic
670C22#121×D6111yescharacteristic
671C22#1684029111yescharacteristic
672C4—42no—
673C4—41no—
674C4—41no—
675C4—1222no—
676C4—1221no—
677C4—1221no—
678C4—1221no—
679C4—1221no—
680C4—4881no—
681C4—4881no—
682C4C4×D4×D6111yescharacteristic
683C4C2×#1451973711yescharacteristic
684C4#1684027211yescharacteristic
685C4#1684031011yescharacteristic
686C2—21no—
687C2—21no—
688C2—82no—
689C2—2442no—
690C2#2144833511yescharacteristic
691C2#2145554211yescharacteristic
692C2#2145555111yescharacteristic
subgroupG/H
centerC22#121×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1977C2
socleC22×C61C2×#121
prime radicalC2×#14519803C22
cosocleC22×C4×C61C24
hypercenter#1977D61
hypocenterC61#21557
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1977

Complements

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

NH ≅ G/Ntype
2. #21454634C2semidirect
4. C2×#16839863C2semidirect
5. #21454246C2semidirect
6. #21446604C2semidirect
7. #21446484C2semidirect
8. #21446477C2semidirect
9. #21446331C2semidirect
11. #21445880C2semidirect
12. #21445727C2semidirect
13. #101×Dic61C2semidirect
14. #100×Dic61C2semidirect
15. C61×#1977C2semidirect
23. #16840022C4semidirect
71. #14519808C2×C4semidirect
81. #14519783C2×C4semidirect
112. #14519713C2×C4semidirect
127. #14519696C2×C4semidirect
128. #14519694C2×C4semidirect
146. C2×SD16×C61C2×C4semidirect
345. #1977D61semidirect
414. C61#21557semidirect
429. C2×SD16C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168400222
#214546342
#214546232
C2×#168398632
#214542462
#214466042
#214464842
#214464772
#214463312
#214463242
#214458802
#214457272
#101×Dic612
#100×Dic612
C61×#19772
#2155761