Finite Group Databasebeta

← #10801973

Subgroups — #10801973

Order 768.

906 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83795C211yesmaximal, characteristic, complemented
2C2×#5144C211yesmaximal, characteristic, complemented
3#78838C211yesmaximal, characteristic
4#78554C211yesmaximal, characteristic, complemented
5C2×#4708C211yesmaximal, characteristic, complemented
6C3×#1765C211yesmaximal, characteristic, complemented
7#70383C211yesmaximal, characteristic, complemented
8#70373C211yesmaximal, characteristic, complemented
9#70288C211yesmaximal, characteristic, complemented
10#69889C211yesmaximal, characteristic, complemented
11#69839C211yesmaximal, characteristic, complemented
12#68939C211yesmaximal, characteristic, complemented
13#68919C211yesmaximal, characteristic, complemented
14#68757C211yesmaximal, characteristic, complemented
15#68747C211yesmaximal, characteristic, complemented
16#21545—31nomaximal, Sylow
17C2×C3×#131C2211yescharacteristic
18#5828C2211yescharacteristic
19C2×#950C2211yescharacteristic
20C2×#901C2211yescharacteristic
21C22×ΓL(1,𝔽25)C2211yescharacteristic
22C22×D3×C8C2211yescharacteristic
23#5524C2211yescharacteristic
24C2×#872C2211yescharacteristic
25C2×C3×#104C2211yescharacteristic
26C3×#416C2211yescharacteristic
27C3×#405C2211yescharacteristic
28C2×C3×#97C2211yescharacteristic
29C2×#837C2211yescharacteristic
30#5234C2211yescharacteristic
31#5144—24no—
32C2×#825C2211yescharacteristic
33#5129C2211yescharacteristic
34#5128—22no—
35#5126C2211yescharacteristic
36#5126C2211yescharacteristic
37C2×#822C2211yescharacteristic
38C2×#822C2211yescharacteristic
39#5119C2211yescharacteristic
40#5113C2211yescharacteristic
41#5111—22no—
42#5101C2211yescharacteristic
43C2×#815C2211yescharacteristic
44C2×C8×Dic3C2211yescharacteristic
45#4999C2211yescharacteristic
46#4998C2211yescharacteristic
47C2×#805C2211yescharacteristic
48C2×C4×C3⋊C8C2211yescharacteristic
49#4813C412yescomplemented
50#4807C412yescomplemented
51#4778C412yescomplemented
52#4775C412yescomplemented
53#4708—24no—
54#4706—22no—
55#4688—22no—
56C3×#341C2211yescharacteristic
57C3×#339C2211yescharacteristic
58#4555C2211yescharacteristic
59#4552C2211yescharacteristic
60#4532C2211yescharacteristic
61#4523C2211yescharacteristic
62#3050—31no—
63C2×#443—31no—
64#2930—31no—
65#2930—31no—
66C2×#405—31no—
67#1974—31no—
68#1945—31no—
69#1938—31no—
70#1938—31no—
71#1897—31no—
72#1897—31no—
73#1810—31no—
74#1772—31no—
75#1765—31no—
76#1765D311yescharacteristic, complemented
77C2×C3×#41C2311yescharacteristic
78C2×#235C2311yescharacteristic
79C22×C4×D3C2311yescharacteristic
80C3×#131—22no—
81C3×#131—22no—
82C2×C3×M16—21no—
83C2×C3×M16C2311yescharacteristic
84C2×C3×M16C2311yescharacteristic
85C22×C3×C8—21no—
86C22×C3×C8C2311yescharacteristic
87C22×C3×C8C2311yescharacteristic
88C3×#117C2311yescharacteristic
89C2×C3×C42C2311yescharacteristic
90#950—22no—
91#950—22no—
92#946C2×C412yes—
93C2×#216C2×C411yescharacteristic
94C2×#215C2×C411yescharacteristic
95#938C2×C412yes—
96#937—21no—
97#935—21no—
98C2×#214C2×C411yescharacteristic
99C2×#213C2×C411yescharacteristic
100C4×#67—22no—
101C2×(C2×D3)⋊C4—21no—
102#926C2311yescharacteristic
103C2×Dic3⋊C4—21no—
104C2×C4×Dic3C2311yescharacteristic
105C2×#208C2311yescharacteristic
106C2×#208C2311yescharacteristic
107C22×C3⋊C8C2311yescharacteristic
108C22×C3⋊C8C2311yescharacteristic
109#901—22no—
110#901—22no—
111C2×ΓL(1,𝔽25)—22no—
112C2×ΓL(1,𝔽25)—22no—
113C2×ΓL(1,𝔽25)—21no—
114C2×ΓL(1,𝔽25)D412yes—
115C2×ΓL(1,𝔽25)D412yes—
116C2×D3×C8—22no—
117C2×D3×C8—22no—
118C2×D3×C8—21no—
119C2×D3×C8C2×C412yes—
120C2×D3×C8C2×C412yes—
121#897C2×C412yes—
122#895C2×C412yes—
123#886—21no—
124#882—21no—
125C4×D12—22no—
126#872—22no—
127#872—22no—
128C3×#104—22no—
129C3×#103C2×C412yes—
130C3×#102C2×C412yes—
131C3×#97—21no—
132C3×#97—22no—
133C3×C4×C8—21no—
134#837—22no—
135#830—22no—
136#825—22no—
137#822—22no—
138#822—22no—
139#821C2×C412yes—
140(C2×D3)⋊C8—22no—
141(C2×D3)⋊C8—22no—
142(C2×D3)⋊C8—22no—
143#816C2×C412yes—
144#815—21no—
145#815—22no—
146Dic3⋊C8—22no—
147C8×Dic3—21no—
148C8×Dic3D414yescomplemented
149#808C2×C412yes—
150#807C2×C412yes—
151#805—22no—
152C4⋊(C3⋊C8)—22no—
153C4×C3⋊C8—22no—
154C2×#135—31no—
155C2×#131—31no—
156C2×#131C2×D311yescharacteristic
157C22×M16—31no—
158C23×C8—31no—
159#525—31no—
160#525—31no—
161C2×#117—31no—
162#467—32no—
163#466—32no—
164#465—32no—
165#465—32no—
166#443—64no—
167#436—62no—
168#436—62no—
169#435—62no—
170#435—62no—
171C2×#108—31no—
172C2×#108—31no—
173#427—31no—
174#424—31no—
175#424—31no—
176C2×#104—31no—
177C2×#104—31no—
178C2×#104—31no—
179C2×#104C2×D311yescharacteristic
180#416—31no—
181#416C2×D311yescharacteristic
182#408—31no—
183#408—31no—
184#407—31no—
185#407—31no—
186#405—31no—
187#405—64no—
188#405C2×D311yescharacteristic
189C2×#97—31no—
190C2×#97C2×D311yescharacteristic
191C2×C4×C8—31no—
192C2×C4×C8—31no—
193#341—31no—
194#341C2×D311yescharacteristic
195#339—31no—
196#339—31no—
197#339—31no—
198#339C2×D311yescharacteristic
199C23×D3#4111yescharacteristic
200C3×#41—21no—
201C3×#41—21no—
202C3×#41—21no—
203C3×#41—21no—
204C2×C3×Q8C22×C411yescharacteristic
205C2×C3×D4—21no—
206C2×C3×D4C22×C411yescharacteristic
207C22×C3×C4—21no—
208C22×C3×C4—21no—
209C22×C3×C4C2411yescharacteristic
210C2×#67—21no—
211C2×#67—22no—
212C22×Dic3—21no—
213C22×Dic3#4111yescharacteristic
214#235—21no—
215#235—21no—
216#235—21no—
217#235—21no—
218C2×D12—22no—
219C2×D12C22×C411yescharacteristic
220C2×C4×D3—21no—
221C2×C4×D3—21no—
222C2×C4×D3—21no—
223C2×C4×D3—21no—
224C2×C4×D3—21no—
225C2×C4×D3C22×C412yes—
226C2×C4×D3C2×D412yes—
227C2×Dic6C22×C411yescharacteristic
228C3×M16—21no—
229C3×M16—21no—
230C3×M16—21no—
231C3×M16—21no—
232C3×M16—42no—
233C2×C3×C8—21no—
234C2×C3×C8—21no—
235C2×C3×C8—21no—
236C2×C3×C8—21no—
237C2×C3×C8—42no—
238C2×C3×C8C22×C412yes—
239C2×C3×C8C2×D411yescharacteristic
240C2×C3×C8C2×D412yes—
241C2×C3×C8C2×D411yescharacteristic
242C2×C3×C8#4112yes—
243C3×C4⋊C4C22×C412yes—
244C3×C22⋊C4—21no—
245C3×C42—21no—
246C3×C42—21no—
247C3×C42#4112yes—
248#217—21no—
249#217—22no—
250#216—22no—
251#215—22no—
252#214—22no—
253#213—22no—
254(C2×D3)⋊C4—24no—
255(C2×D3)⋊C4—22no—
256(C2×D3)⋊C4—22no—
257(C2×D3)⋊C4—22no—
258C4⋊Dic3—22no—
259C4⋊Dic3C22×C412yes—
260Dic3⋊C4—24no—
261Dic3⋊C4—22no—
262C4×Dic3—21no—
263C4×Dic3—21no—
264C4×Dic3C2×D412yes—
265#208—21no—
266#208—21no—
267#208—21no—
268#208—21no—
269C2×C3⋊C8—21no—
270C2×C3⋊C8—21no—
271C2×C3⋊C8—21no—
272C2×C3⋊C8—21no—
273C2×C3⋊C8C22×C411yescharacteristic
274C2×C3⋊C8C22×C411yescharacteristic
275C2×C3⋊C8C2×D412yes—
276C2×C3⋊C8C2×D412yes—
277C2×C3⋊C8#4112yes—
278ΓL(1,𝔽25)—24no—
279ΓL(1,𝔽25)—24no—
280ΓL(1,𝔽25)—42no—
281D3×C8—24no—
282D3×C8—24no—
283D3×C8—42no—
284C2×#41—31no—
285C2×#41C22×D311yescharacteristic
286C23×C4—31no—
287#135—62no—
288#135—62no—
289C2×Q16—31no—
290C2×SD16—31no—
291C2×SD16—31no—
292C2×D8—31no—
293#131—22no—
294#131—22no—
295#131—62no—
296#131—62no—
297C2×M16—21no—
298C2×M16—32no—
299C2×M16—32no—
300C2×M16—31no—
301C2×M16—31no—
302C2×M16—62no—
303C2×M16—62no—
304C2×M16—61no—
305C2×M16C22×D311yescharacteristic
306C2×M16C22×D311yescharacteristic
307C22×C8—21no—
308C22×C8—31no—
309C22×C8—31no—
310C22×C8—32no—
311C22×C8—32no—
312C22×C8—62no—
313C22×C8—62no—
314C22×C8—61no—
315C22×C8C22×D311yescharacteristic
316C22×C8C22×D311yescharacteristic
317#123—61no—
318#123—61no—
319C22⋊Q8—32no—
320C22⋊Q8—32no—
321#121—32no—
322#121—32no—
323#120—61no—
324#120—61no—
325C4×D4—62no—
326C4×D4—62no—
327#117—31no—
328#117—62no—
329#117—62no—
330#117C22×D311yescharacteristic
331C2×C4⋊C4—61no—
332C2×C22⋊C4—61no—
333C2×C42—31no—
334C2×C42C22×D311yescharacteristic
335#108—62no—
336#108—62no—
337#107—32no—
338#106—32no—
339C4⋊C8—62no—
340C4⋊C8—62no—
341#104—22no—
342#104—62no—
343#104—62no—
344#104—62no—
345#103—32no—
346#103C4×D312yes—
347#102—32no—
348#102C4×D312yes—
349C22⋊C8—62no—
350C22⋊C8—62no—
351C22⋊C8—62no—
352C22⋊C8—62no—
353#97—22no—
354#97—21no—
355#97—62no—
356#97—61no—
357C4×C8—21no—
358C4×C8—34no—
359C4×C8—62no—
360C4×C8—61no—
361C23×C3—21no—
362C23×C3C2×#4111yescharacteristic
363C22×D3—21no—
364C22×D3—21no—
365C22×D3—21no—
366C22×D3—42no—
367C22×D3C4×D412yes—
368C3×Q8—21no—
369C3×Q8—21no—
370C3×D4—21no—
371C3×D4—21no—
372C3×D4—41no—
373C3×D4—41no—
374C2×C3×C4—21no—
375C2×C3×C4—22no—
376C2×C3×C4—22no—
377C2×C3×C4—42no—
378C2×C3×C4—41no—
379C2×C3×C4—41no—
380C2×C3×C4C23×C411yescharacteristic
381C2×C3×C4C22×D411yescharacteristic
382C2×C3×C4C22×D411yescharacteristic
383C2×C3×C4C2×#4111yescharacteristic
384C2×C3×C4C2×#4111yescharacteristic
385C2×C3×C4C2×#4111yescharacteristic
386#67—41no—
387#67—41no—
388#67—82no—
389C2×Dic3—21no—
390C2×Dic3—21no—
391C2×Dic3—22no—
392C2×Dic3—21no—
393C2×Dic3—22no—
394C2×Dic3—42no—
395C2×Dic3C4×D412yes—
396D12—21no—
397D12—21no—
398D12—82no—
399C4×D3—22no—
400C4×D3—22no—
401C4×D3—22no—
402C4×D3—22no—
403C4×D3—41no—
404C4×D3—41no—
405Dic6—21no—
406Dic6—21no—
407C3×C8—22no—
408C3×C8—41no—
409C3×C8—41no—
410C3×C8#12112yes—
411C3×C8#12112yes—
412C3⋊C8—22no—
413C3⋊C8—41no—
414C3⋊C8—41no—
415C3⋊C8C4×D412yes—
416C3⋊C8C4×D412yes—
417C24—31no—
418#41—21no—
419#41—21no—
420#41—21no—
421#41—21no—
422#41—61no—
423#41—61no—
424#41—61no—
425#41—61no—
426C2×Q8—31no—
427C2×Q8C2×C4×D311yescharacteristic
428C2×D4—21no—
429C2×D4—31no—
430C2×D4—62no—
431C2×D4—61no—
432C2×D4—62no—
433C2×D4C2×C4×D311yescharacteristic
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—31no—
437C22×C4—32no—
438C22×C4—32no—
439C22×C4—61no—
440C22×C4—61no—
441C22×C4—61no—
442C22×C4—61no—
443C22×C4—61no—
444C22×C4—61no—
445C22×C4C23×D311yescharacteristic
446Q16—62no—
447SD16—62no—
448SD16—62no—
449D8—62no—
450M16—21no—
451M16—21no—
452M16—21no—
453M16—21no—
454M16—42no—
455M16—64no—
456M16—64no—
457M16—61no—
458M16—61no—
459M16—61no—
460M16—61no—
461M16—122no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—31no—
467C2×C8—31no—
468C2×C8—32no—
469C2×C8—32no—
470C2×C8—32no—
471C2×C8—42no—
472C2×C8—64no—
473C2×C8—64no—
474C2×C8—61no—
475C2×C8—61no—
476C2×C8—61no—
477C2×C8—61no—
478C2×C8—122no—
479C2×C8C2×C4×D312yes—
480C2×C8#23512yes—
481C2×C8D3×D412yes—
482C2×C8D3×D411yescharacteristic
483C2×C8D3×D411yescharacteristic
484C4⋊C4—32no—
485C4⋊C4—62no—
486C4⋊C4—62no—
487C4⋊C4—64no—
488C4⋊C4C2×C4×D312yes—
489C22⋊C4—21no—
490C22⋊C4—62no—
491C22⋊C4—62no—
492C22⋊C4—62no—
493C22⋊C4—64no—
494C22⋊C4—62no—
495C22⋊C4—61no—
496C42—21no—
497C42—21no—
498C42—32no—
499C42—61no—
500C42—61no—
501C42#23512yes—
502C22×C3—22no—
503C22×C3—42no—
504C22×C3—41no—
505C22×C3C2×C4×D411yescharacteristic
506C22×C3#52511yescharacteristic
507C22×C3#52911yescharacteristic
508C2×D3—22no—
509C2×D3—22no—
510C2×D3—22no—
511C2×D3—42no—
512C2×D3—42no—
513C2×D3—42no—
514C2×D3—41no—
515C3×C4—42no—
516C3×C4—42no—
517C3×C4—41no—
518C3×C4C2×C4×D411yescharacteristic
519C3×C4C4×#4111yescharacteristic
520C3×C4C2×#12111yescharacteristic
521C3×C4#53211yescharacteristic
522Dic3—22no—
523Dic3—42no—
524Dic3—42no—
525Dic3—41no—
526C23—21no—
527C23—32no—
528C23—61no—
529C23—61no—
530C23—61no—
531C23—122no—
532C23D3×#4111yescharacteristic
533Q8—21no—
534Q8—21no—
535Q8—61no—
536Q8—61no—
537D4—21no—
538D4—21no—
539D4—41no—
540D4—41no—
541D4—61no—
542D4—61no—
543D4—121no—
544D4—121no—
545D4—242no—
546D4—242no—
547C2×C4—22no—
548C2×C4—21no—
549C2×C4—22no—
550C2×C4—32no—
551C2×C4—42no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—61no—
555C2×C4—61no—
556C2×C4—62no—
557C2×C4—62no—
558C2×C4—62no—
559C2×C4—62no—
560C2×C4—62no—
561C2×C4—61no—
562C2×C4—62no—
563C2×C4—122no—
564C2×C4—121no—
565C2×C4—121no—
566C2×C4C22×C4×D311yescharacteristic
567C2×C4C2×#23511yescharacteristic
568C2×C4C2×#23511yescharacteristic
569C2×C4C2×D3×D411yescharacteristic
570C2×C4C2×D3×D411yescharacteristic
571C2×C4D3×#4111yescharacteristic
572C8—22no—
573C8—32no—
574C8—32no—
575C8—41no—
576C8—41no—
577C8—62no—
578C8—121no—
579C8—121no—
580C8#88412yes—
581C8#88412yes—
582C2×C3—21no—
583C2×C3—21no—
584C2×C3—81no—
585C2×C3C4×#12111yescharacteristic
586C2×C3#295712yes—
587D3—42no—
588D3—81no—
589C22—22no—
590C22—42no—
591C22—41no—
592C22—62no—
593C22—62no—
594C22—62no—
595C22—122no—
596C22—122no—
597C22—122no—
598C22—121no—
599C22#552811yescharacteristic
600C22C4×D3×D411yescharacteristic
601C22#554711yescharacteristic
602C4—42no—
603C4—42no—
604C4—41no—
605C4—62no—
606C4—122no—
607C4—122no—
608C4—121no—
609C4C4×#23511yescharacteristic
610C4C2×#88411yescharacteristic
611C4C4×D3×D411yescharacteristic
612C4#554811yescharacteristic
613C3#2154511yescharacteristic, complemented, Sylow
614C2—21no—
615C2—21no—
616C2—81no—
617C2—122no—
618C2—241no—
619C2C4×#88411yescharacteristic
620C2#7989912yes—
subgroupG/H
centerC2×C4C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1765C2
socleC22×C3C2×C4×D4
prime radical#78838C2
cosocleC22×C3×C4C24
hypercenter#1765D3
hypocenterC3#21545
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1765

Complements

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

NH ≅ G/Ntype
1. #83795C2semidirect
2. C2×#5144C2semidirect
4. #78554C2semidirect
5. C2×#4708C2semidirect
6. C3×#1765C2semidirect
7. #70383C2semidirect
8. #70373C2semidirect
9. #70288C2semidirect
10. #69889C2semidirect
11. #69839C2semidirect
12. #68939C2semidirect
13. #68919C2semidirect
14. #68757C2semidirect
15. #68747C2semidirect
49. #4813C4semidirect
50. #4807C4semidirect
51. #4778C4semidirect
52. #4775C4semidirect
76. #1765D3semidirect
148. C8×Dic3D4semidirect
613. C3#21545semidirect

Maximal subgroups

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

subgroupindex
#837952
C2×#51442
#788382
#785542
C2×#47082
C3×#17652
#703832
#703732
#702882
#698892
#698392
#689392
#689192
#687572
#687472
#215453