Finite Group Databasebeta

← #10802136

Subgroups — #10802136

Order 768.

868 conjugacy classes of subgroups.

The subgroup lattice has 714 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#79257C211yesmaximal, characteristic, complemented
2#78989C211yesmaximal, characteristic, complemented
3C2×#4823C211yesmaximal, characteristic, complemented
4#78719C211yesmaximal, characteristic
5#78516C211yesmaximal, characteristic, complemented
6C3×#1932C211yesmaximal, characteristic, complemented
7#70725C211yesmaximal, characteristic, complemented
8#70397C211yesmaximal, characteristic, complemented
9#70391C211yesmaximal, characteristic, complemented
10#70133C211yesmaximal, characteristic, complemented
11#69874C211yesmaximal, characteristic, complemented
12#69032C211yesmaximal, characteristic, complemented
13#69014C211yesmaximal, characteristic, complemented
14#68778C211yesmaximal, characteristic, complemented
15#68742C211yesmaximal, characteristic, complemented
16#21559—31nomaximal, Sylow
17#5822C2211yescharacteristic
18C2×#950C2211yescharacteristic
19C2×#948C2211yescharacteristic
20#5748C2211yescharacteristic
21C2×#907C2211yescharacteristic
22C2×#907C2211yescharacteristic
23C2×#907C2211yescharacteristic
24C4×#235C2211yescharacteristic
25C3×#430C2211yescharacteristic
26C3×#421C2211yescharacteristic
27C2×C3×#104C2211yescharacteristic
28C3×#413C2211yescharacteristic
29C3×C4×M16C2211yescharacteristic
30#5240C2211yescharacteristic
31#5139C2211yescharacteristic
32#5139C2211yescharacteristic
33#5132C2211yescharacteristic
34#5124C2211yescharacteristic
35C2×#822C2211yescharacteristic
36C2×#822C2211yescharacteristic
37#5122C2211yescharacteristic
38#5122C2211yescharacteristic
39#5122C2211yescharacteristic
40#5006C2211yescharacteristic
41#5005C2211yescharacteristic
42#4998C2211yescharacteristic
43#4997C2211yescharacteristic
44C2×#805C2211yescharacteristic
45C2×#803C2211yescharacteristic
46#4895—22no—
47#4894—22no—
48#4824—22no—
49#4823—22no—
50#4823—21no—
51#4823—21no—
52#4823—21no—
53#4823—21no—
54#4755C412yescomplemented
55#4754C412yescomplemented
56#4749C412yescomplemented
57#4748C412yescomplemented
58#4707—21no—
59D3×#97—21no—
60#4689—21no—
61#4689—21no—
62C3×#344C2211yescharacteristic
63C3×#339C2211yescharacteristic
64#4559C2211yescharacteristic
65#4559C2211yescharacteristic
66#4556C2211yescharacteristic
67#4523C2211yescharacteristic
68#3067—31no—
69#2955—31no—
70#2955—31no—
71C2×#444—31no—
72#2875—31no—
73#1976—31no—
74#1953—31no—
75#1932—31no—
76#1932D311yescharacteristic, complemented
77#1906—31no—
78#1906—31no—
79#1814—31no—
80#1773—31no—
81#1757—31no—
82#1757—31no—
83C2×C3×#41C2311yescharacteristic
84C2×#235C2311yescharacteristic
85C2×#235C2311yescharacteristic
86C2×C3×M16C2311yescharacteristic
87C2×C3×M16C2311yescharacteristic
88C2×C3×M16C2311yescharacteristic
89C2×C3×M16C2311yescharacteristic
90C3×#117C2311yescharacteristic
91C2×C3×C42C2311yescharacteristic
92#951—21no—
93#950—22no—
94#950—21no—
95#950—21no—
96#950—21no—
97#950—21no—
98#949—21no—
99#948—21no—
100#948—21no—
101#948—21no—
102#948—21no—
103C2×#216C2×C411yescharacteristic, complemented
104#942C2×C412yes—
105C2×#215C2×C411yescharacteristic, complemented
106C2×#214C2×C411yescharacteristic, complemented
107#932C2×C412yes—
108C2×#213C2×C411yescharacteristic, complemented
109C4×#67—22no—
110#926C2311yescharacteristic
111#926C2311yescharacteristic
112C2×#208—21no—
113C2×#208C2311yescharacteristic
114C2×#208C2311yescharacteristic
115C22×C3⋊C8—21no—
116C22×C3⋊C8C2311yescharacteristic
117C22×C3⋊C8C2311yescharacteristic
118#909—21no—
119#909C2×C412yes—
120#908—21no—
121#908C2×C412yes—
122#907—21no—
123#907—21no—
124#907—21no—
125#907—21no—
126#907—21no—
127#907—21no—
128#907—21no—
129#907—21no—
130#907C2×C412yes—
131#907C2×C412yes—
132#907D412yes—
133#907D412yes—
134#904—22no—
135C2×ΓL(1,𝔽25)—21no—
136C2×ΓL(1,𝔽25)—21no—
137C2×ΓL(1,𝔽25)—21no—
138C2×D3×C8—21no—
139C2×D3×C8—21no—
140C2×D3×C8—21no—
141C4×D12—21no—
142#872—21no—
143C42×D3—21no—
144C4×Dic6—21no—
145C3×#108—22no—
146C3×#104—22no—
147C3×#104—21no—
148C3×#104—21no—
149C3×#101C2×C412yescomplemented
150C3×#100C2×C412yescomplemented
151C3×#97—21no—
152C3×C4×C8—21no—
153#834—22no—
154#825—22no—
155#825—22no—
156#824C2×C412yescomplemented
157#823C2×C412yescomplemented
158#822—22no—
159#822—21no—
160#822—21no—
161#822C2×C412yescomplemented
162#822C2×C412yescomplemented
163#815—21no—
164#815—21no—
165#815—21no—
166C8×Dic3—21no—
167C8×Dic3—21no—
168C8×Dic3—21no—
169#808—21no—
170#807—21no—
171#806—22no—
172#805—22no—
173#805—21no—
174#805—21no—
175#803—21no—
176#803D411yescharacteristic
177#803D411yescharacteristic
178#803D411yescharacteristic
179#803D411yescharacteristic
180C4×C3⋊C8—21no—
181#575—31no—
182#575—31no—
183C2×#135—31no—
184C2×#131—31no—
185C2×#131—31no—
186C2×#131—31no—
187C2×#131—31no—
188C4×#41—31no—
189#473—32no—
190#472—32no—
191#471—32no—
192#471—32no—
193#444—61no—
194#444—61no—
195#444—61no—
196#444—61no—
197#444—62no—
198#444—62no—
199#443—62no—
200#443—62no—
201#430—31no—
202#430C2×D311yescharacteristic
203C2×#108—31no—
204C2×#108—31no—
205#427—31no—
206#421—31no—
207#421—31no—
208#421—31no—
209#421C2×D311yescharacteristic
210C2×#104—31no—
211C2×#104C2×D311yescharacteristic
212#413—31no—
213#413C2×D311yescharacteristic
214#410—31no—
215#410—31no—
216#405—31no—
217#405—31no—
218#405—31no—
219#405—31no—
220#405—61no—
221#405—61no—
222C4×M16—61no—
223C4×M16C2×D311yescharacteristic
224C2×#97—31no—
225C2×#97—61no—
226#344—31no—
227#344—31no—
228#344—31no—
229#344C2×D311yescharacteristic
230#339—31no—
231#339C2×D311yescharacteristic
232C3×#41—21no—
233C3×#41—21no—
234C3×#41—21no—
235C3×#41—21no—
236C2×C3×Q8C22×C411yescharacteristic
237C2×C3×D4—21no—
238C2×C3×D4C22×C411yescharacteristic
239C22×C3×C4—21no—
240C22×C3×C4—21no—
241C22×C3×C4C2411yescharacteristic
242C2×#67—21no—
243C2×#67—21no—
244#235—21no—
245#235—21no—
246#235—21no—
247#235—21no—
248#235—21no—
249#235—21no—
250#235C22×C412yes—
251#235C2×D412yes—
252C2×D12C22×C411yescharacteristic
253C2×D12#4111yescharacteristic
254C2×C4×D3—21no—
255C2×C4×D3—21no—
256C2×C4×D3—22no—
257C2×Dic6C22×C411yescharacteristic
258C2×Dic6#4111yescharacteristic
259C3×M16—21no—
260C3×M16—21no—
261C3×M16—21no—
262C3×M16—21no—
263C3×M16—21no—
264C3×M16C22×C412yes—
265C3×M16C22×C412yes—
266C3×M16C2×D412yes—
267C2×C3×C8—21no—
268C2×C3×C8—21no—
269C2×C3×C8—21no—
270C2×C3×C8—21no—
271C3×C4⋊C4—21no—
272C3×C22⋊C4—21no—
273C3×C42—21no—
274C3×C42—21no—
275C3×C42C2×D411yescharacteristic
276C3×C42C2×D411yescharacteristic
277#217—21no—
278#217#4112yes—
279#216—22no—
280#216—21no—
281#216—21no—
282#215—22no—
283#215—21no—
284#215—21no—
285#214—22no—
286#214—21no—
287#214—21no—
288#213—22no—
289#213—21no—
290#213—21no—
291(C2×D3)⋊C4—22no—
292C4⋊Dic3—21no—
293C4⋊Dic3—21no—
294Dic3⋊C4—22no—
295C4×Dic3—22no—
296C4×Dic3—21no—
297C4×Dic3—21no—
298#208—21no—
299#208—21no—
300#208—21no—
301#208—41no—
302#208—41no—
303#208C22×C412yes—
304C2×C3⋊C8—21no—
305C2×C3⋊C8—21no—
306C2×C3⋊C8—41no—
307C2×C3⋊C8—41no—
308C2×C3⋊C8C22×C411yescharacteristic
309C2×C3⋊C8C22×C411yescharacteristic
310C2×C3⋊C8C2×D411yescharacteristic
311C2×C3⋊C8C2×D411yescharacteristic
312C2×C3⋊C8C2×D411yescharacteristic
313C2×C3⋊C8C2×D411yescharacteristic
314C2×C3⋊C8C2×D411yescharacteristic
315C2×C3⋊C8C2×D411yescharacteristic
316C2×C3⋊C8#4111yescharacteristic
317C2×C3⋊C8#4111yescharacteristic
318C2×C3⋊C8#4111yescharacteristic
319C2×C3⋊C8#4111yescharacteristic
320Dic12—22no—
321D24—22no—
322#204—22no—
323#204—22no—
324ΓL(1,𝔽25)—22no—
325ΓL(1,𝔽25)—22no—
326ΓL(1,𝔽25)—22no—
327ΓL(1,𝔽25)—22no—
328ΓL(1,𝔽25)—41no—
329ΓL(1,𝔽25)—41no—
330D3×C8—22no—
331D3×C8—22no—
332D3×C8—22no—
333D3×C8—22no—
334D3×C8—41no—
335D3×C8—41no—
336C2×#41—31no—
337C2×#41—31no—
338C2×#41C22×D311yescharacteristic
339#137—32no—
340#137—32no—
341#137—61no—
342#137—61no—
343AGL(1,ℤ/8)—32no—
344AGL(1,ℤ/8)—32no—
345AGL(1,ℤ/8)—61no—
346AGL(1,ℤ/8)—61no—
347#135—62no—
348#135—61no—
349#135—61no—
350#135—61no—
351#135—61no—
352#135—62no—
353C2×Q16—31no—
354C2×SD16—31no—
355C2×SD16—31no—
356C2×D8—31no—
357#131—32no—
358#131—32no—
359#131—32no—
360#131—32no—
361#131—61no—
362#131—61no—
363#131—61no—
364#131—61no—
365#131—61no—
366#131—61no—
367#131—61no—
368#131—61no—
369#131—61no—
370#131—61no—
371#131—61no—
372#131—61no—
373C2×M16—31no—
374C2×M16—31no—
375C2×M16—61no—
376C2×M16—61no—
377C2×M16—61no—
378C2×M16—61no—
379C2×M16C22×D311yescharacteristic
380C2×M16C22×D311yescharacteristic
381C2×M16C22×D311yescharacteristic
382C2×M16C22×D311yescharacteristic
383C22×C8—31no—
384C22×C8—31no—
385C22×C8—61no—
386C22×C8—61no—
387C22×C8—61no—
388C22×C8—61no—
389C4×Q8—61no—
390C4×D4—61no—
391C4×D4—62no—
392#117—31no—
393#117—31no—
394#117—61no—
395#117C22×D311yescharacteristic
396C2×C42—61no—
397C2×C42C22×D311yescharacteristic
398#108—22no—
399#108—32no—
400#108—32no—
401#108—61no—
402#108—61no—
403#108—62no—
404#107—61no—
405#106—61no—
406#104—22no—
407#104—21no—
408#104—21no—
409#104—62no—
410#104—62no—
411#104—62no—
412#104—61no—
413#104—61no—
414#101—32no—
415#101C4×D312yescomplemented
416#100—32no—
417#100C4×D312yescomplemented
418#99—62no—
419#99—62no—
420#97—21no—
421#97—31no—
422#97—31no—
423#97—31no—
424#97—31no—
425#97—61no—
426#97—61no—
427#97—61no—
428#97—61no—
429C4×C8—21no—
430C4×C8—61no—
431C4×C8—61no—
432C4×C8—61no—
433C4×C8—61no—
434C23×C3—21no—
435C23×C3C2×#4111yescharacteristic
436C22×D3—21no—
437C22×D3—21no—
438C3×Q8—21no—
439C3×Q8—21no—
440C3×D4—21no—
441C3×D4—21no—
442C3×D4—41no—
443C3×D4—41no—
444C2×C3×C4—21no—
445C2×C3×C4—21no—
446C2×C3×C4—21no—
447C2×C3×C4—41no—
448C2×C3×C4—41no—
449C2×C3×C4—41no—
450C2×C3×C4—81no—
451C2×C3×C4C23×C411yescharacteristic
452C2×C3×C4C22×D411yescharacteristic
453C2×C3×C4C22×D411yescharacteristic
454C2×C3×C4C2×#4111yescharacteristic
455C2×C3×C4C2×#4111yescharacteristic
456C2×C3×C4C2×#4111yescharacteristic
457#67—22no—
458#67—22no—
459#67—41no—
460#67—41no—
461C2×Dic3—21no—
462C2×Dic3—21no—
463C2×Dic3—22no—
464C2×Dic3—41no—
465D12—21no—
466D12—21no—
467D12—21no—
468D12C4×D412yes—
469C4×D3—22no—
470C4×D3—22no—
471C4×D3—41no—
472C4×D3—41no—
473C4×D3—82no—
474Dic6—21no—
475Dic6—21no—
476Dic6—21no—
477Dic6C4×D412yes—
478C3×C8—22no—
479C3×C8—22no—
480C3×C8—22no—
481C3×C8—41no—
482C3⋊C8—22no—
483C3⋊C8—41no—
484C3⋊C8C4×D411yescharacteristic
485C3⋊C8C4×D411yescharacteristic
486C3⋊C8C4×D411yescharacteristic
487C3⋊C8C4×D411yescharacteristic
488C3⋊C8#12111yescharacteristic
489C3⋊C8#12111yescharacteristic
490C3⋊C8#12111yescharacteristic
491C3⋊C8#12111yescharacteristic
492#41—21no—
493#41—21no—
494#41—21no—
495#41—21no—
496#41—32no—
497#41—32no—
498#41—61no—
499#41—61no—
500#41—61no—
501#41—61no—
502#41—61no—
503#41—61no—
504C2×Q8—31no—
505C2×Q8—31no—
506C2×Q8C2×C4×D311yescharacteristic
507C2×D4—21no—
508C2×D4—31no—
509C2×D4—31no—
510C2×D4—61no—
511C2×D4—61no—
512C2×D4C2×C4×D311yescharacteristic
513C22×C4—21no—
514C22×C4—21no—
515C22×C4—62no—
516C22×C4—61no—
517C22×C4—61no—
518C22×C4C23×D311yescharacteristic
519Q16—62no—
520Q16—61no—
521Q16—61no—
522Q16—62no—
523SD16—62no—
524SD16—62no—
525SD16—61no—
526SD16—61no—
527SD16—61no—
528SD16—61no—
529SD16—62no—
530SD16—62no—
531D8—62no—
532D8—61no—
533D8—61no—
534D8—62no—
535M16—21no—
536M16—21no—
537M16—21no—
538M16—21no—
539M16—21no—
540M16—32no—
541M16—62no—
542M16—62no—
543M16—62no—
544M16—62no—
545M16—61no—
546M16—61no—
547M16—61no—
548M16—121no—
549M16—121no—
550M16—121no—
551M16—121no—
552M16C2×C4×D312yes—
553M16C2×C4×D312yes—
554M16D3×D412yes—
555C2×C8—21no—
556C2×C8—21no—
557C2×C8—21no—
558C2×C8—21no—
559C2×C8—31no—
560C2×C8—31no—
561C2×C8—31no—
562C2×C8—31no—
563C2×C8—31no—
564C2×C8—31no—
565C2×C8—31no—
566C2×C8—31no—
567C2×C8—31no—
568C2×C8—31no—
569C2×C8—31no—
570C2×C8—31no—
571C2×C8—62no—
572C2×C8—62no—
573C2×C8—62no—
574C2×C8—62no—
575C2×C8—61no—
576C2×C8—61no—
577C2×C8—121no—
578C2×C8—121no—
579C2×C8—121no—
580C2×C8—121no—
581C4⋊C4—21no—
582C4⋊C4—62no—
583C4⋊C4—61no—
584C4⋊C4—61no—
585C22⋊C4—21no—
586C22⋊C4—32no—
587C22⋊C4—62no—
588C22⋊C4—61no—
589C42—21no—
590C42—21no—
591C42—62no—
592C42—61no—
593C42—61no—
594C42D3×D411yescharacteristic
595C42D3×D411yescharacteristic
596C22×C3—41no—
597C22×C3—41no—
598C22×C3—81no—
599C22×C3C4×#4111yescharacteristic
600C22×C3#52511yescharacteristic
601C22×C3#53211yescharacteristic
602C2×D3—22no—
603C2×D3—42no—
604C2×D3—41no—
605C2×D3—81no—
606C3×C4—41no—
607C3×C4—41no—
608C3×C4—41no—
609C3×C4—81no—
610C3×C4C2×C4×D411yescharacteristic
611C3×C4C2×C4×D411yescharacteristic
612C3×C4C2×#12111yescharacteristic
613C3×C4#52911yescharacteristic
614Dic3—22no—
615Dic3—42no—
616Dic3—41no—
617Dic3—81no—
618C23—21no—
619C23—61no—
620C23—61no—
621C23D3×#4111yescharacteristic
622Q8—21no—
623Q8—21no—
624Q8—32no—
625Q8—61no—
626Q8—61no—
627Q8—61no—
628D4—21no—
629D4—21no—
630D4—32no—
631D4—41no—
632D4—41no—
633D4—62no—
634D4—62no—
635D4—61no—
636D4—61no—
637D4—61no—
638D4—121no—
639D4—121no—
640C2×C4—21no—
641C2×C4—21no—
642C2×C4—21no—
643C2×C4—41no—
644C2×C4—41no—
645C2×C4—41no—
646C2×C4—61no—
647C2×C4—62no—
648C2×C4—62no—
649C2×C4—62no—
650C2×C4—61no—
651C2×C4—81no—
652C2×C4—121no—
653C2×C4—121no—
654C2×C4—121no—
655C2×C4—242no—
656C2×C4C22×C4×D311yescharacteristic
657C2×C4C2×D3×D411yescharacteristic
658C2×C4C2×D3×D411yescharacteristic
659C2×C4D3×#4111yescharacteristic
660C2×C4D3×#4111yescharacteristic
661C2×C4D3×#4111yescharacteristic
662C8—22no—
663C8—22no—
664C8—22no—
665C8—31no—
666C8—31no—
667C8—31no—
668C8—31no—
669C8—31no—
670C8—31no—
671C8—31no—
672C8—31no—
673C8—41no—
674C8—62no—
675C8—121no—
676C2×C3—21no—
677C2×C3—21no—
678C2×C3—21no—
679C2×C3—81no—
680C2×C3C4×#12111yescharacteristic
681D3—42no—
682D3—81no—
683C22—41no—
684C22—41no—
685C22—62no—
686C22—81no—
687C22—122no—
688C22—121no—
689C22—241no—
690C22#552211yescharacteristic
691C22#565611yescharacteristic
692C22#566811yescharacteristic
693C4—41no—
694C4—41no—
695C4—41no—
696C4—62no—
697C4—81no—
698C4—122no—
699C4—121no—
700C4—241no—
701C4C4×D3×D411yescharacteristic
702C4C4×D3×D411yescharacteristic
703C4D3×#12111yescharacteristic
704C4#570611yescharacteristic
705C3#2155911yescharacteristic, complemented, Sylow
706C2—21no—
707C2—21no—
708C2—21no—
709C2—81no—
710C2—122no—
711C2—241no—
712C2#7545211yescharacteristic
subgroupG/H
centerC4D3×#121
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1932C2
socleC2×C3C4×#121
prime radical#78719C2
cosocleC22×C3×C4C24
hypercenter#1932D3
hypocenterC3#21559
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C22×C4 ◁ #1932

Complements

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

NH ≅ G/Ntype
1. #79257C2semidirect
2. #78989C2semidirect
3. C2×#4823C2semidirect
5. #78516C2semidirect
6. C3×#1932C2semidirect
7. #70725C2semidirect
8. #70397C2semidirect
9. #70391C2semidirect
10. #70133C2semidirect
11. #69874C2semidirect
12. #69032C2semidirect
13. #69014C2semidirect
14. #68778C2semidirect
15. #68742C2semidirect
54. #4755C4semidirect
55. #4754C4semidirect
56. #4749C4semidirect
57. #4748C4semidirect
76. #1932D3semidirect
103. C2×#216C2×C4semidirect
105. C2×#215C2×C4semidirect
106. C2×#214C2×C4semidirect
108. C2×#213C2×C4semidirect
149. C3×#101C2×C4semidirect
150. C3×#100C2×C4semidirect
156. #824C2×C4semidirect
157. #823C2×C4semidirect
161. #822C2×C4semidirect
162. #822C2×C4semidirect
415. #101C4×D3semidirect
417. #100C4×D3semidirect
705. C3#21559semidirect

Maximal subgroups

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

subgroupindex
#792572
#789892
C2×#48232
#787192
#785162
C3×#19322
#707252
#703972
#703912
#701332
#698742
#690322
#690142
#687782
#687422
#215593