Finite Group Databasebeta

← #10793770

Subgroups — #10793770

Order 768.

1014 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5144C211yesmaximal, characteristic
2#83782C211yesmaximal, characteristic, complemented
3C2×#4823C211yesmaximal, characteristic
4#78819C211yesmaximal, characteristic
5#78550C211yesmaximal, characteristic, complemented
6C3×#1765C211yesmaximal, characteristic, complemented
7Dic3×#104C211yesmaximal, characteristic, complemented
8#70289C211yesmaximal, characteristic, complemented
9#69889C211yesmaximal, characteristic, complemented
10#69839C211yesmaximal, characteristic, complemented
11Dic3×#97C211yesmaximal, characteristic, complemented
12#68926C211yesmaximal, characteristic, complemented
13#68917C211yesmaximal, characteristic, complemented
14#68742C211yesmaximal, characteristic, complemented
15#68728C211yesmaximal, characteristic, complemented
16#20512—31nomaximal, Sylow
17C2×C3×#131C2211yescharacteristic
18Dic3×#41C2211yescharacteristic
19C2×#950C2211yescharacteristic
20C2×#948C2211yescharacteristic
21C2×#907C2211yescharacteristic
22C2×#901C2211yescharacteristic
23#5522C2211yescharacteristic
24C2×C3×#104C2211yescharacteristic
25C3×#416C2211yescharacteristic
26C3×#405C2211yescharacteristic
27C2×C3×#97C2211yescharacteristic
28C2×#837C2211yescharacteristic
29#5144C418yescomplemented
30C2×#825C2211yescharacteristic
31C2×#822C2211yescharacteristic
32C2×#822C2211yescharacteristic
33#5122—21no—
34#5122C2211yescharacteristic
35Dic3×M16—21no—
36Dic3×M16C2211yescharacteristic
37Dic3×M16C2211yescharacteristic
38#5113C2211yescharacteristic
39#5101—21no—
40#5101C2211yescharacteristic
41C2×#815—21no—
42C2×#815C2211yescharacteristic
43C2×#815C2211yescharacteristic
44C2×C8×Dic3C2211yescharacteristic
45C2×C8×Dic3C2211yescharacteristic
46#4998C2211yescharacteristic
47#4997C2211yescharacteristic
48C42×Dic3C2211yescharacteristic
49C2×#805C2211yescharacteristic
50C2×#803C2211yescharacteristic
51#4823C418yescomplemented
52#4817C412yescomplemented
53#4789C412yescomplemented
54#4785C412yescomplemented
55#4757C412yescomplemented
56#4708—21no—
57D3×#97—21no—
58#4689—21no—
59C4×ΓL(1,𝔽25)—21no—
60C3×#341C2211yescharacteristic
61C3×#339C2211yescharacteristic
62#4555C2211yescharacteristic
63#4552C2211yescharacteristic
64#4532C2211yescharacteristic
65#4523C2211yescharacteristic
66C2×#444—31no—
67C2×#443—31no—
68#2943—31no—
69#2875—31no—
70#2875—31no—
71#1778—31no—
72#1773—31no—
73#1772—31no—
74#1765—31no—
75#1765D311yescharacteristic, complemented
76#1764—31no—
77#1764—31no—
78C4×#104—31no—
79C4×#104—31no—
80C4×#97—31no—
81C2×C3×#41C2311yescharacteristic
82C2×#235C2311yescharacteristic
83C3×#131—22no—
84C3×#131C2×C414yescomplemented
85C2×C3×M16—21no—
86C2×C3×M16C2311yescharacteristic
87C2×C3×M16C2311yescharacteristic
88C22×C3×C8—21no—
89C22×C3×C8C2311yescharacteristic
90C22×C3×C8C2311yescharacteristic
91C3×#117C2311yescharacteristic
92C2×C3×C42C2311yescharacteristic
93#950C2×C414yes—
94#950C2×C414yes—
95#948—22no—
96#948C2×C414yescomplemented
97Q8×Dic3C2×C412yes—
98C2×#216C2×C411yescharacteristic
99C2×#215C2×C411yescharacteristic
100D4×Dic3—22no—
101D4×Dic3C2×C412yes—
102C2×#214C2×C411yescharacteristic
103C2×#213C2×C411yescharacteristic
104#926—21no—
105#926C2311yescharacteristic
106C2×C4×Dic3—21no—
107C2×C4×Dic3—22no—
108C2×C4×Dic3C2311yescharacteristic
109C2×C4×Dic3C2311yescharacteristic
110C2×#208—21no—
111C2×#208C2311yescharacteristic
112C2×#208C2311yescharacteristic
113C22×C3⋊C8—21no—
114C22×C3⋊C8C2311yescharacteristic
115C22×C3⋊C8C2311yescharacteristic
116#907—22no—
117#907C2×C414yescomplemented
118#901—22no—
119#901C2×C414yescomplemented
120C2×ΓL(1,𝔽25)—21no—
121C2×ΓL(1,𝔽25)—21no—
122C2×D3×C8—21no—
123C2×D3×C8—21no—
124#893C2×C412yes—
125#887C2×C412yes—
126#67⋊C4—22no—
127#872—21no—
128C42×D3—21no—
129C3×#104C2×C414yescomplemented
130C3×#103C2×C412yes—
131C3×#102C2×C412yes—
132C3×#97—21no—
133C3×#97—22no—
134C3×C4×C8—21no—
135#837C2×C414yescomplemented
136#825C2×C414yescomplemented
137#822C2×C414yescomplemented
138#822C2×C414yescomplemented
139#821C2×C412yes—
140#816C2×C412yes—
141#815—21no—
142#815—21no—
143#815—21no—
144#815—21no—
145#815C2×C412yes—
146#815C2×C412yes—
147#815D411yescharacteristic
148#815D411yescharacteristic
149#815D411yescharacteristic
150#815D411yescharacteristic
151C8×Dic3—21no—
152C8×Dic3—21no—
153C8×Dic3—21no—
154C8×Dic3—21no—
155C8×Dic3—22no—
156C8×Dic3C2×C414yescomplemented
157#808C2×C412yes—
158#807C2×C412yes—
159#805C2×C414yescomplemented
160#803—21no—
161#803C2×C414yescomplemented
162C4×C3⋊C8—21no—
163C2×#135—31no—
164C2×#131—31no—
165C2×#131—31no—
166C2×#131—31no—
167C2×#131C2×D311yescharacteristic
168C4×#41—31no—
169C4×#41—31no—
170#444—38no—
171#443—38no—
172#442—32no—
173#441—32no—
174#440—32no—
175#440—32no—
176C2×#108—31no—
177C2×#108—31no—
178#427—31no—
179C2×#104—31no—
180C2×#104—31no—
181C2×#104—31no—
182C2×#104C2×D311yescharacteristic
183#416—31no—
184#416C2×D311yescharacteristic
185#405—31no—
186#405—31no—
187#405—31no—
188#405—61no—
189#405—61no—
190#405—61no—
191#405—61no—
192#405C2×D311yescharacteristic
193C4×M16—31no—
194C4×M16—31no—
195C4×M16—61no—
196C4×M16—61no—
197C2×#97—31no—
198C2×#97—31no—
199C2×#97—31no—
200C2×#97—61no—
201C2×#97—61no—
202C2×#97C2×D311yescharacteristic
203C2×C4×C8—31no—
204C2×C4×C8—31no—
205C43—31no—
206#341—31no—
207#341C2×D311yescharacteristic
208#339—31no—
209#339—31no—
210#339—31no—
211#339C2×D311yescharacteristic
212C3×#41—21no—
213C3×#41—21no—
214C3×#41C22×C412yes—
215C3×#41C22×C412yes—
216C2×C3×Q8C22×C411yescharacteristic
217C2×C3×D4—21no—
218C2×C3×D4C22×C411yescharacteristic
219C22×C3×C4—21no—
220C22×C3×C4—21no—
221C22×C3×C4C2411yescharacteristic
222C2×#67—21no—
223C22×Dic3—21no—
224C22×Dic3—22no—
225C22×Dic3#4112yes—
226#235—21no—
227#235—21no—
228#235C22×C412yes—
229#235C22×C412yes—
230C2×D12C22×C411yescharacteristic
231C2×C4×D3—21no—
232C2×C4×D3—22no—
233C2×Dic6C22×C411yescharacteristic
234C3×M16—21no—
235C3×M16—21no—
236C3×M16—24no—
237C3×M16C22×C412yes—
238C3×M16C22×C412yes—
239C2×C3×C8—21no—
240C2×C3×C8—21no—
241C2×C3×C8—21no—
242C2×C3×C8—21no—
243C2×C3×C8—24no—
244C2×C3×C8C22×C412yes—
245C2×C3×C8C22×C412yes—
246C2×C3×C8C2×D411yescharacteristic
247C2×C3×C8C2×D411yescharacteristic
248C2×C3×C8#4112yes—
249C3×C4⋊C4C22×C412yes—
250C3×C22⋊C4—21no—
251C3×C42—21no—
252C3×C42—21no—
253C3×C42C22×C412yes—
254#217—21no—
255#217—22no—
256#216C4214yescomplemented
257#215C4214yescomplemented
258#214C4214yescomplemented
259#213C4214yescomplemented
260(C2×D3)⋊C4—22no—
261C4⋊Dic3—22no—
262C4⋊Dic3C22×C412yes—
263Dic3⋊C4—22no—
264C4×Dic3—21no—
265C4×Dic3—21no—
266C4×Dic3—22no—
267C4×Dic3—22no—
268C4×Dic3—44no—
269C4×Dic3C22×C412yes—
270C4×Dic3C22×C412yes—
271C4×Dic3C2×D411yescharacteristic
272C4×Dic3C2×D411yescharacteristic
273#208—21no—
274#208—21no—
275#208—24no—
276#208C22×C412yes—
277#208C22×C412yes—
278C2×C3⋊C8—21no—
279C2×C3⋊C8—21no—
280C2×C3⋊C8—24no—
281C2×C3⋊C8C22×C412yes—
282C2×C3⋊C8C22×C411yescharacteristic
283C2×C3⋊C8C22×C411yescharacteristic
284C2×C3⋊C8C22×C412yes—
285C2×C3⋊C8C22×C412yes—
286C2×C3⋊C8C2×D411yescharacteristic
287C2×C3⋊C8C2×D411yescharacteristic
288C2×C3⋊C8#4112yes—
289ΓL(1,𝔽25)—24no—
290ΓL(1,𝔽25)—24no—
291D3×C8—24no—
292D3×C8—24no—
293C2×#41—31no—
294C2×#41C22×D311yescharacteristic
295#135—34no—
296#135—34no—
297C2×Q16—31no—
298C2×SD16—31no—
299C2×SD16—31no—
300C2×D8—31no—
301#131—22no—
302#131—34no—
303#131—34no—
304#131—34no—
305#131—62no—
306#131—62no—
307#131—62no—
308#131C4×D314yescomplemented
309C2×M16—21no—
310C2×M16—31no—
311C2×M16—31no—
312C2×M16—61no—
313C2×M16—61no—
314C2×M16—61no—
315C2×M16C22×D311yescharacteristic
316C2×M16C22×D311yescharacteristic
317C22×C8—21no—
318C22×C8—31no—
319C22×C8—31no—
320C22×C8—61no—
321C22×C8—61no—
322C22×C8—61no—
323C22×C8C22×D311yescharacteristic
324C22×C8C22×D311yescharacteristic
325C4×Q8—32no—
326C4×Q8—32no—
327C4×D4—32no—
328C4×D4—32no—
329C4×D4—62no—
330C4×D4—62no—
331#117—31no—
332#117—61no—
333#117—61no—
334#117C22×D311yescharacteristic
335C2×C42—31no—
336C2×C42—31no—
337C2×C42—61no—
338C2×C42—61no—
339C2×C42—62no—
340C2×C42C22×D311yescharacteristic
341#108—34no—
342#108—34no—
343#107—32no—
344#106—32no—
345#104—34no—
346#104—34no—
347#104—34no—
348#104C4×D314yescomplemented
349#103—32no—
350#103C4×D312yes—
351#102—32no—
352#102C4×D312yes—
353#97—21no—
354#97—22no—
355#97—32no—
356#97—32no—
357#97—31no—
358#97—31no—
359#97—31no—
360#97—31no—
361#97—34no—
362#97—61no—
363#97—61no—
364#97—61no—
365#97—61no—
366#97—61no—
367C4×C8—21no—
368C4×C8—34no—
369C4×C8—61no—
370C4×C8—61no—
371C4×C8—61no—
372C4×C8—62no—
373C4×C8—61no—
374C4×C8—61no—
375C23×C3—21no—
376C23×C3C2×#4111yescharacteristic
377C22×D3—21no—
378C3×Q8—21no—
379C3×Q8C2×C4212yes—
380C3×D4—22no—
381C3×D4—22no—
382C3×D4—21no—
383C3×D4C2×C4212yes—
384C2×C3×C4—21no—
385C2×C3×C4—22no—
386C2×C3×C4—22no—
387C2×C3×C4—22no—
388C2×C3×C4—22no—
389C2×C3×C4—42no—
390C2×C3×C4C23×C411yescharacteristic
391C2×C3×C4C23×C411yescharacteristic
392C2×C3×C4C23×C411yescharacteristic
393C2×C3×C4C22×D411yescharacteristic
394C2×C3×C4C2×#4111yescharacteristic
395C2×C3×C4C2×#4111yescharacteristic
396#67—22no—
397#67—22no—
398C2×Dic3—24no—
399C2×Dic3—21no—
400C2×Dic3—22no—
401C2×Dic3—22no—
402C2×Dic3—42no—
403C2×Dic3—82no—
404C2×Dic3C4×D412yes—
405C2×Dic3C4×D412yes—
406D12—21no—
407D12C2×C4212yes—
408C4×D3—22no—
409C4×D3—22no—
410C4×D3—82no—
411Dic6—21no—
412Dic6C2×C4212yes—
413C3×C8—22no—
414C3×C8—22no—
415C3×C8—22no—
416C3×C8C4×D412yes—
417C3×C8C4×D412yes—
418C3⋊C8—22no—
419C3⋊C8—22no—
420C3⋊C8C2×C4212yes—
421C3⋊C8C2×C4212yes—
422C3⋊C8C4×D412yes—
423C3⋊C8C4×D412yes—
424#41—21no—
425#41—21no—
426#41—32no—
427#41—32no—
428#41—61no—
429#41—61no—
430#41C2×C4×D312yes—
431#41C2×C4×D312yes—
432C2×Q8—31no—
433C2×Q8C2×C4×D311yescharacteristic
434C2×D4—21no—
435C2×D4—31no—
436C2×D4—61no—
437C2×D4C2×C4×D311yescharacteristic
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—32no—
441C22×C4—62no—
442C22×C4—62no—
443C22×C4—61no—
444C22×C4—61no—
445C22×C4C23×D311yescharacteristic
446Q16—34no—
447SD16—34no—
448SD16—34no—
449D8—34no—
450M16—21no—
451M16—21no—
452M16—24no—
453M16—32no—
454M16—32no—
455M16—64no—
456M16—64no—
457M16—61no—
458M16—61no—
459M16—64no—
460M16C2×C4×D312yes—
461M16C2×C4×D312yes—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—24no—
467C2×C8—32no—
468C2×C8—31no—
469C2×C8—31no—
470C2×C8—32no—
471C2×C8—32no—
472C2×C8—31no—
473C2×C8—31no—
474C2×C8—32no—
475C2×C8—64no—
476C2×C8—64no—
477C2×C8—61no—
478C2×C8—61no—
479C2×C8—64no—
480C2×C8C2×C4×D312yes—
481C2×C8C2×C4×D312yes—
482C2×C8D3×D411yescharacteristic
483C2×C8D3×D411yescharacteristic
484C2×C8#23712yes—
485C4⋊C4—32no—
486C4⋊C4—62no—
487C4⋊C4—62no—
488C4⋊C4C2×C4×D312yes—
489C22⋊C4—21no—
490C22⋊C4—62no—
491C22⋊C4—62no—
492C22⋊C4—61no—
493C42—21no—
494C42—21no—
495C42—32no—
496C42—31no—
497C42—31no—
498C42—32no—
499C42—62no—
500C42—62no—
501C42—61no—
502C42—61no—
503C42—124no—
504C42C2×C4×D312yes—
505C22×C3—22no—
506C22×C3—22no—
507C22×C3—42no—
508C22×C3C2×C4×D411yescharacteristic
509C22×C3C4×#4111yescharacteristic
510C22×C3C4×#4111yescharacteristic
511C2×D3—22no—
512C2×D3—42no—
513C3×C4—22no—
514C3×C4—42no—
515C3×C4—42no—
516C3×C4C22×C4211yescharacteristic
517C3×C4C2×C4×D411yescharacteristic
518C3×C4C2×C4×D411yescharacteristic
519C3×C4C4×#4111yescharacteristic
520Dic3—24no—
521Dic3—22no—
522Dic3—22no—
523Dic3—22no—
524Dic3—42no—
525C23—21no—
526C23—61no—
527C23D3×#4111yescharacteristic
528Q8—21no—
529Q8—32no—
530Q8—61no—
531Q8C42×D312yes—
532D4—22no—
533D4—22no—
534D4—21no—
535D4—32no—
536D4—62no—
537D4—62no—
538D4—61no—
539D4C42×D312yes—
540C2×C4—21no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—32no—
546C2×C4—32no—
547C2×C4—42no—
548C2×C4—64no—
549C2×C4—62no—
550C2×C4—61no—
551C2×C4—62no—
552C2×C4—62no—
553C2×C4—62no—
554C2×C4—122no—
555C2×C4—242no—
556C2×C4—242no—
557C2×C4C22×C4×D311yescharacteristic
558C2×C4C22×C4×D311yescharacteristic
559C2×C4C22×C4×D311yescharacteristic
560C2×C4C2×D3×D411yescharacteristic
561C2×C4C2×#23711yescharacteristic
562C2×C4D3×#4111yescharacteristic
563C8—22no—
564C8—22no—
565C8—22no—
566C8—32no—
567C8—32no—
568C8—32no—
569C8—32no—
570C8—62no—
571C8—62no—
572C8#67⋊C412yes—
573C8#67⋊C412yes—
574C2×C3—21no—
575C2×C3—21no—
576C2×C3—42no—
577C2×C3C42×D411yescharacteristic
578C2×C3#295612yes—
579D3—42no—
580C22—22no—
581C22—22no—
582C22—42no—
583C22—62no—
584C22—122no—
585C22#552211yescharacteristic
586C22C4×#23711yescharacteristic
587C22C4×D3×D411yescharacteristic
588C4—22no—
589C4—42no—
590C4—42no—
591C4—64no—
592C4—62no—
593C4—62no—
594C4—62no—
595C4—122no—
596C4C2×C42×D311yescharacteristic
597C4C2×#67⋊C411yescharacteristic
598C4C4×#23711yescharacteristic
599C4C4×D3×D411yescharacteristic
600C3#2051211yescharacteristic, complemented, Sylow
601C2—21no—
602C2—21no—
603C2—42no—
604C2—122no—
605C2C4×#67⋊C411yescharacteristic
606C2#7989812yes—
subgroupG/H
centerC2×C4C2×D3×D4
commutatorC3×C4C22×C42
FrattiniC22×C4C23×D3
FittingC3×#1765C2
socleC22×C3C2×C4×D4
prime radicalC2×#950C22
cosocleC22×C3×C4C24
hypercenter#1765D3
hypocenterC3#20512
LCSG ▷ 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
2. #83782C2semidirect
5. #78550C2semidirect
6. C3×#1765C2semidirect
7. Dic3×#104C2semidirect
8. #70289C2semidirect
9. #69889C2semidirect
10. #69839C2semidirect
11. Dic3×#97C2semidirect
12. #68926C2semidirect
13. #68917C2semidirect
14. #68742C2semidirect
15. #68728C2semidirect
29. #5144C4semidirect
51. #4823C4semidirect
52. #4817C4semidirect
53. #4789C4semidirect
54. #4785C4semidirect
55. #4757C4semidirect
75. #1765D3semidirect
84. C3×#131C2×C4semidirect
96. #948C2×C4semidirect
117. #907C2×C4semidirect
119. #901C2×C4semidirect
129. C3×#104C2×C4semidirect
135. #837C2×C4semidirect
136. #825C2×C4semidirect
137. #822C2×C4semidirect
138. #822C2×C4semidirect
156. C8×Dic3C2×C4semidirect
159. #805C2×C4semidirect
161. #803C2×C4semidirect
256. #216C42semidirect
257. #215C42semidirect
258. #214C42semidirect
259. #213C42semidirect
308. #131C4×D3semidirect
348. #104C4×D3semidirect
600. C3#20512semidirect

Maximal subgroups

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

subgroupindex
C2×#51442
#837822
C2×#48232
#788192
#785502
C3×#17652
Dic3×#1042
#702892
#698892
#698392
Dic3×#972
#689262
#689172
#687422
#687282
#205123