Finite Group Databasebeta

← #10818104

Subgroups — #10818104

Order 768.

840 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82549C211yesmaximal, characteristic, complemented
2#82547C211yesmaximal, characteristic
3#79209C211yesmaximal, characteristic
4#79202C211yesmaximal, characteristic
5C2×#4755C211yesmaximal, characteristic
6C3×#1857C211yesmaximal, characteristic, complemented
7#70672C211yesmaximal, characteristic, complemented
8#70668C211yesmaximal, characteristic, complemented
9#70106C211yesmaximal, characteristic, complemented
10#69693C211yesmaximal, characteristic
11#69678C211yesmaximal, characteristic, complemented
12#68775C211yesmaximal, characteristic
13#68764C211yesmaximal, characteristic, complemented
14#68742C211yesmaximal, characteristic, complemented
15#68740C211yesmaximal, characteristic
16#23047—31nomaximal, Sylow
17C3×#519C2211yescharacteristic
18C22×#216C2211yescharacteristic
19C2×#942C2211yescharacteristic
20C2×#909C2211yescharacteristic
21C2×#907C2211yescharacteristic
22#5523C2211yescharacteristic
23#5516C2211yescharacteristic
24#5504C2211yescharacteristic
25C3×#428C2211yescharacteristic
26C3×#428C2211yescharacteristic
27C2×C3×#101C2211yescharacteristic
28C3×#391C2211yescharacteristic
29#5225C2211yescharacteristic
30C2×#835C2211yescharacteristic
31C2×#824C2211yescharacteristic
32C2×#822C2211yescharacteristic
33#5122C2211yescharacteristic
34#5121C2211yescharacteristic
35#5030—21no—
36#5030—21no—
37#5030—21no—
38C4×#216—21no—
39C4×#216—21no—
40C4×#216—21no—
41#5026—21no—
42C4×#215—21no—
43#5004C2211yescharacteristic
44#5002C2211yescharacteristic
45#4998C2211yescharacteristic
46#4997C2211yescharacteristic
47#4997C2211yescharacteristic
48#4995C2211yescharacteristic
49C2×#810C2211yescharacteristic
50C2×#808C2211yescharacteristic
51#4929C2211yescharacteristic
52C2×#805C2211yescharacteristic
53C2×#803C2211yescharacteristic
54#4889—21no—
55#4881—21no—
56#4880—21no—
57#4873—21no—
58#4861—21no—
59D3×#106—21no—
60#4853—21no—
61#4852—21no—
62#4755C418yescomplemented
63C3×#339C2211yescharacteristic
64C3×#339C2211yescharacteristic
65#4558C2211yescharacteristic
66#4554C2211yescharacteristic
67#4530C2211yescharacteristic
68#4523C2211yescharacteristic
69C2×#473—31no—
70#2950—31no—
71#2947—31no—
72#2947—31no—
73#2912—31no—
74#1971—31no—
75#1951—31no—
76#1930—31no—
77#1930—31no—
78#1882—31no—
79#1882—31no—
80#1857—31no—
81#1857D311yescharacteristic, complemented
82#1773—31no—
83#1765—31no—
84C22×C3×Q8C2311yescharacteristic
85C2×#235C2311yescharacteristic
86C22×Dic6C2311yescharacteristic
87C2×C3×M16C2311yescharacteristic
88C2×C3×M16C2311yescharacteristic
89C3×C4×Q8—21no—
90C3×C4×Q8—21no—
91C3×C4×Q8—21no—
92C3×C4×Q8—21no—
93C3×#117—41no—
94C3×#117C2311yescharacteristic
95C3×#117C2311yescharacteristic
96C2×C3×C4⋊C4C2311yescharacteristic
97C2×C3×C42C2311yescharacteristic
98C2×#216—21no—
99C2×#216—21no—
100C2×#216—21no—
101C2×#216—22no—
102C2×#216C2×C414yescomplemented
103#942—22no—
104#942C2×C414yes—
105C2×#215—21no—
106C4×#67—41no—
107#926C2311yescharacteristic
108C2×C4⋊Dic3C2311yescharacteristic
109C2×#208C2311yescharacteristic
110C2×#208C2311yescharacteristic
111C22×C3⋊C8C2311yescharacteristic
112C22×C3⋊C8C2311yescharacteristic
113#909—22no—
114#909C2×C414yes—
115#907—22no—
116#907C2×C414yes—
117C2×Dic12—21no—
118C2×#204—21no—
119C2×ΓL(1,𝔽25)—21no—
120C2×D3×C8—21no—
121#893—21no—
122#892—21no—
123#892—21no—
124D3×C4⋊C4—21no—
125D3×C4⋊C4—21no—
126#887—21no—
127#887—21no—
128#887—21no—
129#67⋊C4—41no—
130#877—41no—
131C4×D12—21no—
132C4×Dic6—21no—
133C4×Dic6—21no—
134C4×Dic6—21no—
135C3×#107—21no—
136C3×#107—21no—
137C3×#106—21no—
138C3×#106—21no—
139C3×#101C2×C414yescomplemented
140C3×#95—41no—
141#835—21no—
142#835—21no—
143#835—21no—
144#835—21no—
145C3⋊#95—41no—
146#824C2×C414yescomplemented
147#822C2×C414yescomplemented
148#818—21no—
149#817—21no—
150#815—21no—
151C8×Dic3—21no—
152#810—21no—
153#810—21no—
154#810—21no—
155#810—21no—
156#810D411yescharacteristic
157#810D411yescharacteristic
158#809D411yescharacteristic
159#809D411yescharacteristic
160#808—21no—
161#808—21no—
162#808—21no—
163#808D411yescharacteristic
164#808D411yescharacteristic
165#807—21no—
166#807D411yescharacteristic
167#807D411yescharacteristic
168#805—22no—
169#803—21no—
170#803—21no—
171#803—22no—
172C4×C3⋊C8—21no—
173C4×C3⋊C8—21no—
174C2×#137—31no—
175C2×#137—31no—
176C22×Q16—31no—
177C2×#131—31no—
178#520—31no—
179#520—31no—
180#519—31no—
181#519C2×D311yescharacteristic
182#473—38no—
183#441—61no—
184#441—61no—
185#441—61no—
186#441—61no—
187#440—61no—
188#440—61no—
189C4×Q16—61no—
190C4×Q16—61no—
191C4×Q16—61no—
192C4×Q16—61no—
193C4×SD16—61no—
194C4×SD16—61no—
195C2×#108—31no—
196#428—31no—
197#428—31no—
198#428—61no—
199#428—61no—
200#428C2×D311yescharacteristic
201#428C2×D311yescharacteristic
202#427—31no—
203#427—61no—
204C2×#106—31no—
205C2×#106—61no—
206C2×#104—31no—
207#419—31no—
208#419—31no—
209#416—31no—
210C2×#103—31no—
211C2×#103—31no—
212C2×#101—31no—
213C2×#101C2×D311yescharacteristic
214#405—31no—
215#405—31no—
216#405—31no—
217C2×#97—31no—
218#391—31no—
219#391C2×D311yescharacteristic
220#343—31no—
221#343—31no—
222#341—31no—
223#339—31no—
224#339C2×D311yescharacteristic
225#339C2×D311yescharacteristic
226C2×C3×Q8—21no—
227C2×C3×Q8—21no—
228C2×C3×Q8—21no—
229C2×C3×Q8—21no—
230C2×C3×Q8—21no—
231C2×C3×Q8C22×C412yes—
232C22×C3×C4—21no—
233C22×C3×C4—21no—
234C22×C3×C4—21no—
235C22×C3×C4C2411yescharacteristic
236C2×#67—21no—
237C22×Dic3—21no—
238C22×Dic3—21no—
239#235—21no—
240#235—21no—
241#235C22×C412yes—
242#235#4112yes—
243C2×D12C2×D411yescharacteristic
244C2×C4×D3—21no—
245C2×C4×D3—41no—
246C2×C4×D3—41no—
247C2×Dic6—21no—
248C2×Dic6—21no—
249C2×Dic6—21no—
250C2×Dic6—21no—
251C2×Dic6—21no—
252C2×Dic6C22×C412yes—
253C2×Dic6C2×D411yescharacteristic
254C3×M16—21no—
255C3×M16—21no—
256C3×M16C22×C412yes—
257C3×M16C22×C412yes—
258C2×C3×C8—21no—
259C2×C3×C8—21no—
260C3×C4⋊C4—21no—
261C3×C4⋊C4—21no—
262C3×C4⋊C4—41no—
263C3×C4⋊C4—41no—
264C3×C4⋊C4C2×D411yescharacteristic
265C3×C4⋊C4C2×D411yescharacteristic
266C3×C4⋊C4C2×D411yescharacteristic
267C3×C4⋊C4C2×D411yescharacteristic
268C3×C22⋊C4—21no—
269C3×C22⋊C4—21no—
270C3×C42—21no—
271C3×C42—21no—
272C3×C42—21no—
273C3×C42—21no—
274C3×C42—41no—
275C3×C42—41no—
276#217—21no—
277#216—24no—
278#216—24no—
279#216—24no—
280#215—24no—
281(C2×D3)⋊C4—41no—
282(C2×D3)⋊C4—41no—
283C4⋊Dic3—21no—
284C4⋊Dic3—21no—
285C4⋊Dic3C2×D411yescharacteristic
286C4⋊Dic3C2×D411yescharacteristic
287Dic3⋊C4—41no—
288Dic3⋊C4—41no—
289Dic3⋊C4—41no—
290Dic3⋊C4—41no—
291C4×Dic3—21no—
292C4×Dic3—41no—
293C4×Dic3—41no—
294C4×Dic3—41no—
295C4×Dic3—41no—
296#208—21no—
297#208—21no—
298#208—21no—
299#208C22×C412yes—
300C2×C3⋊C8—21no—
301C2×C3⋊C8—21no—
302C2×C3⋊C8—21no—
303C2×C3⋊C8—21no—
304C2×C3⋊C8C22×C412yes—
305C2×C3⋊C8C2×D411yescharacteristic
306C2×C3⋊C8C2×D411yescharacteristic
307C2×C3⋊C8C2×D411yescharacteristic
308C2×C3⋊C8C2×D411yescharacteristic
309C2×C3⋊C8#4112yes—
310Dic12—24no—
311#204—24no—
312ΓL(1,𝔽25)—24no—
313D3×C8—24no—
314C2×#41—31no—
315C22×Q8—31no—
316C22×Q8C22×D311yescharacteristic
317#137—34no—
318#137—34no—
319#137—62no—
320#137—62no—
321C2×Q16—34no—
322C2×Q16—61no—
323C2×Q16—62no—
324C2×Q16—61no—
325C2×Q16—61no—
326C2×Q16—61no—
327C2×SD16—61no—
328C2×SD16—61no—
329#131—34no—
330#131—62no—
331C2×M16—31no—
332C2×M16—31no—
333C2×M16—61no—
334C2×M16C22×D311yescharacteristic
335C2×M16C22×D311yescharacteristic
336C22×C8—31no—
337C22×C8—31no—
338C22×C8—61no—
339C4×Q8—21no—
340C4×Q8—21no—
341C4×Q8—21no—
342C4×Q8—21no—
343C4×Q8—61no—
344C4×Q8—61no—
345C4×Q8—61no—
346C4×Q8—61no—
347C4×Q8—61no—
348C4×Q8—61no—
349C4×D4—61no—
350C4×D4—61no—
351C4×D4—121no—
352C4×D4—121no—
353#117—31no—
354#117—41no—
355#117—61no—
356#117—61no—
357#117—121no—
358#117C22×D311yescharacteristic
359#117C22×D311yescharacteristic
360C2×C4⋊C4—31no—
361C2×C4⋊C4—61no—
362C2×C4⋊C4—61no—
363C2×C4⋊C4C22×D311yescharacteristic
364C2×C42C22×D311yescharacteristic
365#108—34no—
366#107—21no—
367#107—21no—
368#107—31no—
369#107—31no—
370#107—61no—
371#107—61no—
372#106—21no—
373#106—21no—
374#106—31no—
375#106—31no—
376#106—61no—
377#106—61no—
378#106—61no—
379#106—61no—
380#104—62no—
381#103—31no—
382#103—31no—
383#103—61no—
384#103—61no—
385#103—61no—
386#103—61no—
387#103—61no—
388#103—61no—
389#103—61no—
390#103—61no—
391#102—31no—
392#102—31no—
393#101—34no—
394#101C4×D314yescomplemented
395#97—62no—
396#97—61no—
397#97—61no—
398#97—61no—
399C4×C8—61no—
400C4×C8—61no—
401C4×C8—61no—
402#95—41no—
403#95—121no—
404C23×C3H(5,𝔽2)11yescharacteristic
405C22×D3—21no—
406C3×Q8—22no—
407C3×Q8—22no—
408C3×Q8—41no—
409C3×Q8—41no—
410C2×C3×C4—22no—
411C2×C3×C4—21no—
412C2×C3×C4—21no—
413C2×C3×C4—21no—
414C2×C3×C4—21no—
415C2×C3×C4—41no—
416C2×C3×C4—41no—
417C2×C3×C4—41no—
418C2×C3×C4—41no—
419C2×C3×C4—41no—
420C2×C3×C4—41no—
421C2×C3×C4—81no—
422C2×C3×C4C23×C411yescharacteristic
423C2×C3×C4C22×D411yescharacteristic
424C2×C3×C4C22×D411yescharacteristic
425C2×C3×C4C2×#4111yescharacteristic
426C2×C3×C4C2×#4111yescharacteristic
427C2×C3×C42−1+411yescharacteristic
428#67—22no—
429#67—22no—
430C2×Dic3—21no—
431C2×Dic3—22no—
432C2×Dic3—21no—
433C2×Dic3—21no—
434C2×Dic3—41no—
435C2×Dic3—41no—
436C2×Dic3—41no—
437C2×Dic3—81no—
438D12—21no—
439D12C4×D412yes—
440C4×D3—22no—
441C4×D3—22no—
442C4×D3—161no—
443C4×D3—161no—
444Dic6—21no—
445Dic6—22no—
446Dic6—22no—
447Dic6—41no—
448Dic6—41no—
449Dic6C4×D412yes—
450C3×C8—22no—
451C3×C8—22no—
452C3⋊C8—22no—
453C3⋊C8—21no—
454C3⋊C8—21no—
455C3⋊C8—41no—
456C3⋊C8C4×D412yes—
457C3⋊C8C4×D412yes—
458#41—32no—
459#41—32no—
460#41—61no—
461#41—61no—
462C2×Q8—21no—
463C2×Q8—21no—
464C2×Q8—21no—
465C2×Q8—21no—
466C2×Q8—21no—
467C2×Q8—32no—
468C2×Q8—31no—
469C2×Q8—61no—
470C2×Q8—61no—
471C2×Q8—61no—
472C2×Q8—61no—
473C2×Q8—61no—
474C2×Q8C2×C4×D312yes—
475C2×D4—31no—
476C2×D4—61no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—61no—
481C22×C4—61no—
482C22×C4—61no—
483C22×C4—121no—
484C22×C4—121no—
485C22×C4C23×D311yescharacteristic
486Q16—64no—
487Q16—64no—
488Q16—64no—
489Q16—64no—
490SD16—64no—
491SD16—64no—
492M16—21no—
493M16—21no—
494M16—32no—
495M16—64no—
496M16—61no—
497M16—61no—
498M16—61no—
499M16C2×C4×D312yes—
500M16C2×C4×D312yes—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—32no—
504C2×C8—31no—
505C2×C8—31no—
506C2×C8—32no—
507C2×C8—31no—
508C2×C8—31no—
509C2×C8—64no—
510C2×C8—61no—
511C2×C8—61no—
512C2×C8—61no—
513C2×C8—61no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—31no—
517C4⋊C4—31no—
518C4⋊C4—41no—
519C4⋊C4—41no—
520C4⋊C4—61no—
521C4⋊C4—61no—
522C4⋊C4—121no—
523C4⋊C4—121no—
524C4⋊C4—121no—
525C4⋊C4—121no—
526C4⋊C4D3×D411yescharacteristic
527C4⋊C4D3×D411yescharacteristic
528C4⋊C4D3×D411yescharacteristic
529C4⋊C4D3×D411yescharacteristic
530C22⋊C4—21no—
531C22⋊C4—21no—
532C22⋊C4—61no—
533C22⋊C4—121no—
534C22⋊C4—121no—
535C42—21no—
536C42—21no—
537C42—21no—
538C42—21no—
539C42—41no—
540C42—41no—
541C42—61no—
542C42—121no—
543C42—121no—
544C42—121no—
545C42—121no—
546C22×C3—21no—
547C22×C3—21no—
548C22×C3#52011yescharacteristic
549C22×C3D4211yescharacteristic
550C22×C3#55311yescharacteristic
551C2×D3—22no—
552C2×D3—41no—
553C2×D3—41no—
554C3×C4—42no—
555C3×C4—41no—
556C3×C4—41no—
557C3×C4—41no—
558C3×C4—41no—
559C3×C4—81no—
560C3×C4—81no—
561C3×C4C2×C4×D411yescharacteristic
562C3×C4C2×C4×D411yescharacteristic
563C3×C4#54711yescharacteristic
564C3×C4#54811yescharacteristic
565Dic3—22no—
566Dic3—42no—
567Dic3—41no—
568Dic3—41no—
569Dic3—81no—
570C23—61no—
571C23#100411yescharacteristic
572Q8—22no—
573Q8—22no—
574Q8—32no—
575Q8—41no—
576Q8—41no—
577Q8—61no—
578Q8—62no—
579Q8—62no—
580Q8—121no—
581Q8—121no—
582D4—32no—
583D4—62no—
584D4—62no—
585D4—61no—
586C2×C4—22no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—61no—
598C2×C4—62no—
599C2×C4—62no—
600C2×C4—62no—
601C2×C4—61no—
602C2×C4—61no—
603C2×C4—81no—
604C2×C4—121no—
605C2×C4—121no—
606C2×C4—121no—
607C2×C4—241no—
608C2×C4—481no—
609C2×C4—481no—
610C2×C4C22×C4×D311yescharacteristic
611C2×C4C2×D3×D411yescharacteristic
612C2×C4C2×D3×D411yescharacteristic
613C2×C4#100711yescharacteristic
614C2×C4D3×#4111yescharacteristic
615C2×C4D3×#4111yescharacteristic
616C8—22no—
617C8—22no—
618C8—32no—
619C8—32no—
620C8—62no—
621C8—61no—
622C8—61no—
623C8—121no—
624C2×C3—21no—
625C2×C3—21no—
626C2×C3#236111yescharacteristic
627C2×C3#328911yescharacteristic
628C2×C3#328911yescharacteristic
629D3—42no—
630C22—21no—
631C22—21no—
632C22—62no—
633C22—121no—
634C22—121no—
635C22#550411yescharacteristic
636C22#559011yescharacteristic
637C22#566111yescharacteristic
638C4—42no—
639C4—41no—
640C4—41no—
641C4—41no—
642C4—41no—
643C4—62no—
644C4—81no—
645C4—81no—
646C4—122no—
647C4—121no—
648C4—121no—
649C4—241no—
650C4C4×D3×D411yescharacteristic
651C4C4×D3×D411yescharacteristic
652C4#563111yescharacteristic
653C4#563311yescharacteristic
654C3#2304711yescharacteristic, complemented, Sylow
655C2—21no—
656C2—21no—
657C2—122no—
658C2#7439711yescharacteristic
659C2#8028811yescharacteristic
660C2#8030111yescharacteristic
subgroupG/H
centerC22#5590
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1857C2
socleC22×C3D42
prime radicalC2×#235C23
cosocleC22×C3×C4C24
hypercenter#1857D3
hypocenterC3#23047
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1857

Complements

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

NH ≅ G/Ntype
1. #82549C2semidirect
6. C3×#1857C2semidirect
7. #70672C2semidirect
8. #70668C2semidirect
9. #70106C2semidirect
11. #69678C2semidirect
13. #68764C2semidirect
14. #68742C2semidirect
62. #4755C4semidirect
81. #1857D3semidirect
102. C2×#216C2×C4semidirect
139. C3×#101C2×C4semidirect
146. #824C2×C4semidirect
147. #822C2×C4semidirect
394. #101C4×D3semidirect
654. C3#23047semidirect

Maximal subgroups

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

subgroupindex
#825492
#825472
#792092
#792022
C2×#47552
C3×#18572
#706722
#706682
#701062
#696932
#696782
#687752
#687642
#687422
#687402
#230473