Finite Group Databasebeta

← #10818098

Subgroups — #10818098

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#79209C211yesmaximal, characteristic
2#78948C211yesmaximal, characteristic, complemented
3#78774C211yesmaximal, characteristic
4C2×#4755C211yesmaximal, characteristic
5#78571C211yesmaximal, characteristic
6C3×#1930C211yesmaximal, characteristic, complemented
7#70677C211yesmaximal, characteristic, complemented
8#70348C211yesmaximal, characteristic
9#70259C211yesmaximal, characteristic, complemented
10#70102C211yesmaximal, characteristic, complemented
11#69890C211yesmaximal, characteristic, complemented
12#69004C211yesmaximal, characteristic
13#68984C211yesmaximal, characteristic, complemented
14#68930C211yesmaximal, characteristic, complemented
15#68927C211yesmaximal, characteristic
16#23047—31nomaximal, Sylow
17#5812C2211yescharacteristic
18C22×#216C2211yescharacteristic
19C2×#942C2211yescharacteristic
20C2×#909C2211yescharacteristic
21C2×#907C2211yescharacteristic
22#5523C2211yescharacteristic
23#5516C2211yescharacteristic
24#5504C2211yescharacteristic
25C3×#428C2211yescharacteristic
26C3×#419C2211yescharacteristic
27C2×C3×#103C2211yescharacteristic
28C2×C3×#101C2211yescharacteristic
29C3×#405C2211yescharacteristic
30#5229C2211yescharacteristic
31C2×#825C2211yescharacteristic
32C2×#824C2211yescharacteristic
33C2×#822C2211yescharacteristic
34#5122C2211yescharacteristic
35#5121C2211yescharacteristic
36#5121C2211yescharacteristic
37#5120C2211yescharacteristic
38#5113C2211yescharacteristic
39C2×#816C2211yescharacteristic
40#5101C2211yescharacteristic
41C2×#815C2211yescharacteristic
42#4998C2211yescharacteristic
43#4973C2211yescharacteristic
44#4964C2211yescharacteristic
45C2×#808C2211yescharacteristic
46#4881—21no—
47#4880—21no—
48#4861—21no—
49D3×#106—21no—
50#4817—21no—
51#4790—21no—
52#4790—21no—
53#4790—21no—
54#4789—21no—
55#4789—21no—
56#4789—21no—
57#4788—21no—
58#4755C418yescomplemented
59#4716—21no—
60#4710—21no—
61C4×Dic12—21no—
62C4×#204—21no—
63C3×#343C2211yescharacteristic
64C3×#341C2211yescharacteristic
65#4555C2211yescharacteristic
66#4555C2211yescharacteristic
67#4554C2211yescharacteristic
68#4532C2211yescharacteristic
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#1930D311yescharacteristic, complemented
78#1882—31no—
79#1882—31no—
80#1857—31no—
81#1857—31no—
82#1773—31no—
83#1765—31no—
84C22×C3×Q8C2311yescharacteristic
85C2×#235C2311yescharacteristic
86C22×Dic6C2311yescharacteristic
87C2×C3×M16C2311yescharacteristic
88C2×C3×M16C2311yescharacteristic
89C2×C3×M16C2311yescharacteristic
90C22×C3×C8C2311yescharacteristic
91C3×#117C2311yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93Q8×Dic3—21no—
94Q8×Dic3—21no—
95Q8×Dic3—21no—
96Q8×Dic3—21no—
97C2×#216—21no—
98C2×#216—21no—
99C2×#216—21no—
100C2×#216—22no—
101C2×#216C2×C414yescomplemented
102#942—22no—
103#942C2×C414yes—
104C2×#215—21no—
105C4×#67—41no—
106#926—41no—
107#926C2311yescharacteristic
108#926C2311yescharacteristic
109C2×C4⋊Dic3C2311yescharacteristic
110C2×C4×Dic3C2311yescharacteristic
111C2×#208C2311yescharacteristic
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×#106—21no—
137C3×#103—21no—
138C3×#103—21no—
139C3×#103—21no—
140C3×#103—21no—
141C3×#101C2×C414yescomplemented
142C3×#97—21no—
143C3×C4×C8—21no—
144C3⋊#95—41no—
145C3⋊#95—41no—
146#825—22no—
147#824C2×C414yescomplemented
148#822C2×C414yescomplemented
149#821D411yescharacteristic
150#821D411yescharacteristic
151#818—21no—
152#818—21no—
153#817—21no—
154#817—21no—
155#816—21no—
156#816—21no—
157#816—21no—
158#816—21no—
159#816D411yescharacteristic
160#816D411yescharacteristic
161#815—21no—
162#815—21no—
163#815—22no—
164C8×Dic3—21no—
165C8×Dic3—21no—
166#808—21no—
167#808—21no—
168#808—21no—
169#808D411yescharacteristic
170#808D411yescharacteristic
171#807—21no—
172#807D411yescharacteristic
173#807D411yescharacteristic
174C2×#137—31no—
175C2×#137—31no—
176C22×Q16—31no—
177C2×#131—31no—
178#520—31no—
179#520—31no—
180#519—31no—
181#519—31no—
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—31no—
199#428—61no—
200#428—61no—
201#428C2×D311yescharacteristic
202#427—31no—
203#427—61no—
204C2×#106—31no—
205C2×#106—61no—
206C2×#104—31no—
207#419—31no—
208#419C2×D311yescharacteristic
209#416—31no—
210C2×#103—31no—
211C2×#103C2×D311yescharacteristic
212C2×#101—31no—
213C2×#101C2×D311yescharacteristic
214#405—31no—
215#405—31no—
216#405C2×D311yescharacteristic
217C2×#97—31no—
218#391—31no—
219#391—31no—
220#343—31no—
221#343C2×D311yescharacteristic
222#341C2×D311yescharacteristic
223#339—31no—
224#339—31no—
225#339—31no—
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×C4C2411yescharacteristic
235C2×#67—21no—
236C22×Dic3—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×M16—21no—
257C3×M16—21no—
258C3×M16C22×C412yes—
259C3×M16C22×C412yes—
260C2×C3×C8—21no—
261C2×C3×C8—21no—
262C2×C3×C8—21no—
263C2×C3×C8—21no—
264C2×C3×C8—21no—
265C2×C3×C8C2×D411yescharacteristic
266C2×C3×C8C2×D411yescharacteristic
267C3×C4⋊C4—21no—
268C3×C4⋊C4—21no—
269C3×C4⋊C4C2×D411yescharacteristic
270C3×C4⋊C4C2×D411yescharacteristic
271C3×C22⋊C4—21no—
272C3×C42—21no—
273#217—21no—
274#217—21no—
275#216—24no—
276#216—24no—
277#216—24no—
278#215—24no—
279(C2×D3)⋊C4—41no—
280(C2×D3)⋊C4—41no—
281C4⋊Dic3—21no—
282C4⋊Dic3—21no—
283C4⋊Dic3—41no—
284C4⋊Dic3—41no—
285C4⋊Dic3C2×D411yescharacteristic
286C4⋊Dic3C2×D411yescharacteristic
287C4⋊Dic3C2×D411yescharacteristic
288C4⋊Dic3C2×D411yescharacteristic
289Dic3⋊C4—41no—
290Dic3⋊C4—41no—
291Dic3⋊C4—41no—
292Dic3⋊C4—41no—
293C4×Dic3—21no—
294C4×Dic3—21no—
295C4×Dic3—21no—
296C4×Dic3—21no—
297C4×Dic3—41no—
298C4×Dic3—41no—
299C4×Dic3—41no—
300C4×Dic3—41no—
301C4×Dic3—41no—
302C4×Dic3—41no—
303#208—21no—
304#208C22×C412yes—
305C2×C3⋊C8—21no—
306C2×C3⋊C8C22×C412yes—
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—61no—
333C2×M16C22×D311yescharacteristic
334C2×M16C22×D311yescharacteristic
335C2×M16C22×D311yescharacteristic
336C22×C8—31no—
337C22×C8—61no—
338C22×C8C22×D311yescharacteristic
339C4×Q8—61no—
340C4×Q8—61no—
341C4×Q8—61no—
342C4×Q8—61no—
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—31no—
355#117—61no—
356#117—61no—
357#117—121no—
358#117—121no—
359#117C22×D311yescharacteristic
360C2×C4⋊C4—31no—
361C2×C4⋊C4—61no—
362C2×C4⋊C4—61no—
363C2×C4⋊C4C22×D311yescharacteristic
364C2×C42—31no—
365#108—34no—
366#107—21no—
367#107—31no—
368#107—31no—
369#107—61no—
370#107—61no—
371#107—61no—
372#106—21no—
373#106—31no—
374#106—31no—
375#106—61no—
376#106—61no—
377#106—61no—
378#106—61no—
379#106—61no—
380#104—62no—
381#103—21no—
382#103—21no—
383#103—21no—
384#103—21no—
385#103—31no—
386#103—31no—
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—21no—
396#97—62no—
397#97—61no—
398#97—61no—
399C4×C8—21no—
400C4×C8—61no—
401C4×C8—61no—
402#95—121no—
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—41no—
414C2×C3×C4—41no—
415C2×C3×C4—41no—
416C2×C3×C4—81no—
417C2×C3×C4C23×C411yescharacteristic
418C2×C3×C4C22×D411yescharacteristic
419C2×C3×C4C22×D411yescharacteristic
420C2×C3×C4C2×#4111yescharacteristic
421C2×C3×C4C2×#4111yescharacteristic
422C2×C3×C42−1+411yescharacteristic
423#67—22no—
424#67—22no—
425C2×Dic3—21no—
426C2×Dic3—22no—
427C2×Dic3—21no—
428C2×Dic3—21no—
429C2×Dic3—21no—
430C2×Dic3—21no—
431C2×Dic3—41no—
432C2×Dic3—41no—
433C2×Dic3—41no—
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—21no—
453C3×C8—21no—
454C3×C8—41no—
455C3⋊C8—22no—
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—61no—
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—21no—
495M16—21no—
496M16—32no—
497M16—64no—
498M16—61no—
499M16C2×C4×D312yes—
500M16C2×C4×D312yes—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—32no—
507C2×C8—31no—
508C2×C8—31no—
509C2×C8—32no—
510C2×C8—64no—
511C2×C8—61no—
512C2×C8D3×D411yescharacteristic
513C2×C8D3×D411yescharacteristic
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—31no—
517C4⋊C4—31no—
518C4⋊C4—31no—
519C4⋊C4—31no—
520C4⋊C4—61no—
521C4⋊C4—61no—
522C4⋊C4—121no—
523C4⋊C4—121no—
524C4⋊C4—121no—
525C4⋊C4—121no—
526C4⋊C4—121no—
527C4⋊C4—121no—
528C4⋊C4D3×D411yescharacteristic
529C4⋊C4D3×D411yescharacteristic
530C22⋊C4—21no—
531C22⋊C4—61no—
532C22⋊C4—61no—
533C22⋊C4—121no—
534C22⋊C4—121no—
535C42—21no—
536C42—61no—
537C42—61no—
538C42—61no—
539C42—61no—
540C42—121no—
541C42—121no—
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—81no—
558C3×C4C2×C4×D411yescharacteristic
559C3×C4C2×C4×D411yescharacteristic
560C3×C4#54711yescharacteristic
561C3×C4#54811yescharacteristic
562Dic3—22no—
563Dic3—42no—
564Dic3—41no—
565Dic3—41no—
566Dic3—41no—
567Dic3—41no—
568Dic3—81no—
569Dic3—81no—
570C23—61no—
571C23#100911yescharacteristic
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—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—61no—
593C2×C4—62no—
594C2×C4—62no—
595C2×C4—62no—
596C2×C4—61no—
597C2×C4—61no—
598C2×C4—61no—
599C2×C4—61no—
600C2×C4—81no—
601C2×C4—121no—
602C2×C4—121no—
603C2×C4—121no—
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×C4D3×#4111yescharacteristic
614C2×C4D3×#4111yescharacteristic
615C2×C4#101011yescharacteristic
616C8—21no—
617C8—21no—
618C8—22no—
619C8—22no—
620C8—32no—
621C8—32no—
622C8—41no—
623C8—62no—
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#552311yescharacteristic
636C22#560911yescharacteristic
637C22#570911yescharacteristic
638C4—41no—
639C4—41no—
640C4—42no—
641C4—62no—
642C4—81no—
643C4—121no—
644C4—121no—
645C4—121no—
646C4—121no—
647C4—122no—
648C4—241no—
649C4—241no—
650C4C4×D3×D411yescharacteristic
651C4C4×D3×D411yescharacteristic
652C4#565511yescharacteristic
653C4#567711yescharacteristic
654C3#2304711yescharacteristic, complemented, Sylow
655C2—21no—
656C2—21no—
657C2—122no—
658C2#7547711yescharacteristic
659C2#8093311yescharacteristic
660C2#8094511yescharacteristic
subgroupG/H
centerC22#5609
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1930C2
socleC22×C3D42
prime radicalC2×#235C23
cosocleC22×C3×C4C24
hypercenter#1930D3
hypocenterC3#23047
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1930

Complements

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

NH ≅ G/Ntype
2. #78948C2semidirect
6. C3×#1930C2semidirect
7. #70677C2semidirect
9. #70259C2semidirect
10. #70102C2semidirect
11. #69890C2semidirect
13. #68984C2semidirect
14. #68930C2semidirect
58. #4755C4semidirect
77. #1930D3semidirect
101. C2×#216C2×C4semidirect
141. C3×#101C2×C4semidirect
147. #824C2×C4semidirect
148. #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
#792092
#789482
#787742
C2×#47552
#785712
C3×#19302
#706772
#703482
#702592
#701022
#698902
#690042
#689842
#689302
#689272
#230473