Finite Group Databasebeta

← #72226246

Subgroups — #72226246

Order 1536.

962 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10948598C211yesmaximal, characteristic
2#10948443C211yesmaximal, characteristic, complemented
3#10706946C214yesmaximal, complemented
4#10706941C214yesmaximal, complemented
5#10706782C211yesmaximal, characteristic, complemented
6C2×#67766C211yesmaximal, characteristic, complemented
7C2×#67568C211yesmaximal, characteristic, complemented
8C3⋊C8×M32C211yesmaximal, characteristic, complemented
9C3×#33443C211yesmaximal, characteristic, complemented
10#483512—31nomaximal, Sylow
11C3×#3407—21no—
12C2×C3×#443C2211yescharacteristic
13#83894C411yescharacteristic
14#82007—21no—
15C2×#4700C2211yescharacteristic
16C3×#2171—22no—
17C3×#2170C2214yescomplemented
18C2×C3×#364C2211yescharacteristic
19C3×#2117—21no—
20C3×C4×M32C2211yescharacteristic
21#72080C411yescharacteristic
22#71437—21no—
23#70922C414yes—
24#70921C2214yes—
25#70690C411yescharacteristic
26#70683C411yescharacteristic
27C2×#4463C2211yescharacteristic
28C4×#812C2211yescharacteristic
29C2×C8×C3⋊C8C2211yescharacteristic
30#69076—22no—
31#69056C2214yescomplemented
32#67766C2212yescomplemented
33#67766C2212yescomplemented
34#67568C2212yescomplemented
35#67568C2212yescomplemented
36#67566C2214yescomplemented
37#67498C2212yescomplemented
38C16×C3⋊C8C2212yescomplemented
39#33463—31no—
40#33443—31no—
41#33443D311yescharacteristic, complemented
42#12047—34no—
43#12046—34no—
44#12011—31no—
45C2×#1335—31no—
46C2×#1335—31no—
47C8×M32—31no—
48C2×C3×#135C2×C411yescharacteristic
49C2×C3×#131—21no—
50C2×#904C2×C411yescharacteristic
51C2×#901—21no—
52C3×#504—24no—
53C3×#504—42no—
54C2×C3×M32—21no—
55C2×C3×M32—42no—
56C2×C3×M32D411yescharacteristic
57C2×C3×M32D411yescharacteristic
58C3×#443—21no—
59C3×#443—21no—
60C3×#443C2×C412yes—
61C3×#443C2312yes—
62C2×C3×#108C2×C411yescharacteristic
63C2×C3×#104—21no—
64C2×C3×C4×C8C2311yescharacteristic
65#5195C812yescomplemented
66#5194C2×C414yes—
67#5193C812yescomplemented
68#5179C812yescomplemented
69#5147C812yescomplemented
70C2×#819C2×C411yescharacteristic
71C2×#812—21no—
72C2×#812C2×C411yescharacteristic
73C2×#812C2×C411yescharacteristic
74C2×#805—21no—
75C2×C4×C3⋊C8—21no—
76#4907—42no—
77D3×M32—42no—
78#4901—24no—
79#4700—21no—
80#4700—21no—
81#4700C2×C412yes—
82#4700C2312yes—
83C3×#364—21no—
84C3×#364C2312yes—
85C3×#350—24no—
86C3×#346D412yes—
87C3×C4×C16C2312yes—
88#4567C2×C414yes—
89#4508—24no—
90#4497C2×C414yes—
91#4493C2×C414yes—
92#4463—21no—
93#4463C2312yes—
94#4461C2×C412yes—
95C4×C3⋊C16C2312yes—
96C8×C3⋊C8—21no—
97C8×C3⋊C8C2311yescharacteristic
98C8×C3⋊C8C2311yescharacteristic
99#3414—31no—
100#3407—21no—
101#3407—61no—
102C2×#443—31no—
103C2×#443C2×D311yescharacteristic
104#2179—34no—
105#2178—34no—
106#2171—22no—
107#2171—62no—
108#2170—34no—
109#2170C2×D314yescomplemented
110#2161—31no—
111C2×#364—31no—
112C2×#364C2×D311yescharacteristic
113#2143—31no—
114#2143—31no—
115#2117—21no—
116#2117—61no—
117C4×M32—31no—
118C4×M32C2×D311yescharacteristic
119C2×C82—31no—
120#1373—34no—
121#1335—32no—
122#1335—32no—
123#1335—32no—
124#1335—32no—
125#1311—32no—
126C8×C16—32no—
127C2×C3×#41—21no—
128C2×#235—21no—
129C3×#135—21no—
130C3×#135—21no—
131C3×#135C2×C812yes—
132C3×#135C22×C412yes—
133C2×C3×Q16C2×C811yescharacteristic
134C2×C3×SD16—21no—
135C2×C3×D8C2×C811yescharacteristic
136C3×#131—22no—
137C3×#131—22no—
138C3×#131—41no—
139C3×#131—41no—
140C2×C3×M16—21no—
141C2×C3×M16—41no—
142C22×C3×C8—41no—
143C22×C3×C8C22×C411yescharacteristic
144C22×C3×C8C2×D411yescharacteristic
145C2×C3×C42#4111yescharacteristic
146C2×#208—21no—
147C22×C3⋊C8—22no—
148C2×Dic12C2×C811yescharacteristic
149#904—21no—
150#904—21no—
151#904C2×C812yes—
152#904C22×C412yes—
153C2×D24C2×C811yescharacteristic
154C2×#204—21no—
155#901—22no—
156#901—22no—
157#901—41no—
158#901—41no—
159C2×ΓL(1,𝔽25)—41no—
160C2×D3×C8—41no—
161C3×M32—21no—
162C3×M32—21no—
163C3×M32—22no—
164C3×M32—41no—
165C3×M32—44no—
166C3×M32D812yes—
167C3×M32D812yes—
168C2×C3×C16—41no—
169C2×C3×C16—44no—
170C2×C3×C16C2×D412yes—
171C2×C3×C16C2×D412yes—
172C3×#108—21no—
173C3×#108C22×C412yes—
174C3×#104—22no—
175C3×#104—22no—
176C3×C4×C8C22×C411yescharacteristic
177C3×C4×C8C2×D411yescharacteristic
178C3×C4×C8#4111yescharacteristic
179C3×C4×C8C2411yescharacteristic
180#829C2×C814yescomplemented
181#828C2×C814yescomplemented
182#827C2×C814yescomplemented
183#826C2×C814yescomplemented
184#819—21no—
185#819C22×C412yes—
186#812—21no—
187#812—21no—
188#812—22no—
189#812—41no—
190#812C2×C812yes—
191#812C2×C812yes—
192C2×C3⋊C16—41no—
193C2×C3⋊C16C22×C412yes—
194C2×C3⋊C16C22×C412yes—
195#805—22no—
196#805—22no—
197C4×C3⋊C8—22no—
198C4×C3⋊C8—21no—
199C4×C3⋊C8—21no—
200#798—44no—
201D3×C16—44no—
202C2×#135—31no—
203C2×#135C4×D311yescharacteristic
204C2×#131—21no—
205C2×#131—61no—
206#510—32no—
207#510—32no—
208#509—32no—
209#509—32no—
210#508—34no—
211#504—24no—
212#504—42no—
213#504—64no—
214#504—122no—
215C2×M32—21no—
216C2×M32—31no—
217C2×M32—31no—
218C2×M32—42no—
219C2×M32—61no—
220C2×M32—122no—
221C2×M32#6711yescharacteristic
222C2×M32#6711yescharacteristic
223#443—21no—
224#443—21no—
225#443—32no—
226#443—32no—
227#443—61no—
228#443—61no—
229#443C4×D312yes—
230#443C22×D312yes—
231C2×#108—31no—
232C2×#108C4×D311yescharacteristic
233C2×#104—21no—
234C2×#104—61no—
235C2×C4×C8—61no—
236C2×C4×C8C22×D311yescharacteristic
237#368—34no—
238#364—21no—
239#364—32no—
240#364—61no—
241#364C22×D312yes—
242#359—34no—
243#359—34no—
244#350—24no—
245#350—64no—
246#346—32no—
247#346#6712yes—
248C4×C16—32no—
249C4×C16C22×D312yes—
250C82—31no—
251C82—31no—
252C82—61no—
253C3×#41—22no—
254C3×#41—22no—
255C3×#41—41no—
256C3×#41—41no—
257C2×C3×Q8—21no—
258C2×C3×D4—21no—
259C2×C3×D4—41no—
260C22×C3×C4—21no—
261C22×C3×C4—41no—
262C22×C3×C4C4×D411yescharacteristic
263C2×#67—41no—
264#235—22no—
265#235—22no—
266#235—41no—
267#235—41no—
268C2×D12—21no—
269C2×C4×D3—41no—
270C2×Dic6—21no—
271C3×Q16—21no—
272C3×Q16C22×C812yes—
273C3×SD16—22no—
274C3×SD16—22no—
275C3×D8—21no—
276C3×D8C22×C812yes—
277C3×M16—22no—
278C3×M16—41no—
279C3×M16—42no—
280C3×M16—42no—
281C2×C3×C8—41no—
282C2×C3×C8—42no—
283C2×C3×C8—42no—
284C2×C3×C8C4×D411yescharacteristic
285C2×C3×C8C4×D411yescharacteristic
286C2×C3×C8C22×C811yescharacteristic
287C2×C3×C8C22×C811yescharacteristic
288C2×C3×C8#13111yescharacteristic
289C2×C3×C8#13111yescharacteristic
290C2×C3×C8C2×D811yescharacteristic
291C2×C3×C8C2×D811yescharacteristic
292C2×C3×C8#13511yescharacteristic
293C2×C3×C8#13511yescharacteristic
294C2×C3×C8C23×C411yescharacteristic
295C2×C3×C8C22×D411yescharacteristic
296C3×C42—21no—
297C3×C42C4×D411yescharacteristic
298C3×C42C2×#4111yescharacteristic
299#208—22no—
300#208—41no—
301C2×C3⋊C8—22no—
302C2×C3⋊C8—22no—
303C2×C3⋊C8—44no—
304C2×C3⋊C8—41no—
305Dic12—21no—
306Dic12C22×C812yes—
307D24—21no—
308D24C22×C812yes—
309#204—22no—
310#204—22no—
311ΓL(1,𝔽25)—42no—
312ΓL(1,𝔽25)—42no—
313D3×C8—42no—
314D3×C8—42no—
315C3×C16—22no—
316C3×C16—42no—
317C3×C16C2×D814yes—
318C3⋊C16—22no—
319C3⋊C16—42no—
320C3⋊C16C22×C814yes—
321C2×#41—21no—
322C2×#41—61no—
323#135—21no—
324#135—21no—
325#135—32no—
326#135—32no—
327#135—61no—
328#135—61no—
329#135D3×C812yes—
330#135C2×C4×D312yes—
331C2×Q16—31no—
332C2×Q16D3×C811yescharacteristic
333C2×SD16—21no—
334C2×SD16—61no—
335C2×D8—31no—
336C2×D8D3×C811yescharacteristic
337#131—22no—
338#131—22no—
339#131—41no—
340#131—41no—
341#131—62no—
342#131—62no—
343#131—121no—
344#131—121no—
345C2×M16—21no—
346C2×M16—41no—
347C2×M16—61no—
348C2×M16—121no—
349C22×C8—41no—
350C22×C8—62no—
351C22×C8—121no—
352C22×C8C2×C4×D311yescharacteristic
353C22×C8C2×#6711yescharacteristic
354C2×C42#23511yescharacteristic
355Q32—34no—
356SD32—34no—
357SD32—34no—
358D16—34no—
359M32—21no—
360M32—21no—
361M32—22no—
362M32—32no—
363M32—32no—
364M32—41no—
365M32—44no—
366M32—61no—
367M32—61no—
368M32—62no—
369M32—121no—
370M32—124no—
371M32#21312yes—
372M32#21312yes—
373C2×C16—32no—
374C2×C16—32no—
375C2×C16—41no—
376C2×C16—44no—
377C2×C16—121no—
378C2×C16—124no—
379C2×C16C2×#6712yes—
380C2×C16C2×#6712yes—
381#108—21no—
382#108—32no—
383#108—61no—
384#108C2×C4×D312yes—
385#104—22no—
386#104—22no—
387#104—62no—
388#104—62no—
389C4×C8—62no—
390C4×C8—61no—
391C4×C8—61no—
392C4×C8C2×C4×D311yescharacteristic
393C4×C8#23511yescharacteristic
394C4×C8C2×#6711yescharacteristic
395C4×C8C23×D311yescharacteristic
396C23×C3—41no—
397C23×C3#44311yescharacteristic
398C22×D3—41no—
399C3×Q8—22no—
400C3×Q8—41no—
401C3×D4—22no—
402C3×D4—42no—
403C3×D4—42no—
404C3×D4—41no—
405C2×C3×C4—21no—
406C2×C3×C4—21no—
407C2×C3×C4—42no—
408C2×C3×C4—41no—
409C2×C3×C4—42no—
410C2×C3×C4—42no—
411C2×C3×C4C8×D411yescharacteristic
412C2×C3×C4C8×D411yescharacteristic
413C2×C3×C4C4×D811yescharacteristic
414C2×C3×C4C4×D811yescharacteristic
415C2×C3×C4#44311yescharacteristic
416C2×C3×C4C2×C4×D411yescharacteristic
417#67—42no—
418#67—42no—
419C2×Dic3—41no—
420D12—22no—
421D12—41no—
422C4×D3—42no—
423C4×D3—42no—
424Dic6—22no—
425Dic6—41no—
426C3×C8—42no—
427C3×C8C8×D411yescharacteristic
428C3×C8C8×D411yescharacteristic
429C3×C8C4×D811yescharacteristic
430C3×C8C4×D811yescharacteristic
431C3×C8C23×C811yescharacteristic
432C3×C8C2×#13111yescharacteristic
433C3×C8C22×D811yescharacteristic
434C3×C8C2×#13511yescharacteristic
435C3⋊C8—24no—
436C3⋊C8—42no—
437C3⋊C8—42no—
438#41—22no—
439#41—22no—
440#41—41no—
441#41—41no—
442#41—62no—
443#41—62no—
444#41—121no—
445#41—121no—
446C2×Q8—21no—
447C2×Q8—61no—
448C2×D4—21no—
449C2×D4—41no—
450C2×D4—61no—
451C2×D4—121no—
452C22×C4—21no—
453C22×C4—41no—
454C22×C4—121no—
455C22×C4C4×#6711yescharacteristic
456Q16—21no—
457Q16—32no—
458Q16—61no—
459Q16C2×D3×C812yes—
460SD16—22no—
461SD16—22no—
462SD16—62no—
463SD16—62no—
464D8—21no—
465D8—32no—
466D8—61no—
467D8C2×D3×C812yes—
468M16—22no—
469M16—41no—
470M16—42no—
471M16—42no—
472M16—62no—
473M16—121no—
474M16—122no—
475M16—122no—
476C2×C8—41no—
477C2×C8—42no—
478C2×C8—42no—
479C2×C8—62no—
480C2×C8—62no—
481C2×C8—124no—
482C2×C8—121no—
483C2×C8—122no—
484C2×C8—122no—
485C2×C8C2×D3×C811yescharacteristic
486C2×C8C2×D3×C811yescharacteristic
487C2×C8#90111yescharacteristic
488C2×C8#90111yescharacteristic
489C2×C8C4×#6711yescharacteristic
490C2×C8C4×#6711yescharacteristic
491C2×C8C2×#21311yescharacteristic
492C2×C8C2×#21311yescharacteristic
493C2×C8#95011yescharacteristic
494C2×C8#95011yescharacteristic
495C2×C8C22×C4×D311yescharacteristic
496C2×C8C22×#6711yescharacteristic
497C42—21no—
498C42C4×#6711yescharacteristic
499C42C2×#23511yescharacteristic
500C16—22no—
501C16—34no—
502C16—42no—
503C16—62no—
504C16—122no—
505C16C2×#21314yes—
506C22×C3—41no—
507C22×C3—42no—
508C22×C3—161no—
509C22×C3C8×D811yescharacteristic
510C22×C3C8×D811yescharacteristic
511C22×C3C2×#44311yescharacteristic
512C2×D3—42no—
513C2×D3—161no—
514C3×C4—22no—
515C3×C4—41no—
516C3×C4—42no—
517C3×C4C8×D811yescharacteristic
518C3×C4C8×D811yescharacteristic
519C3×C4C2×C8×D411yescharacteristic
520C3×C4C2×C4×D811yescharacteristic
521Dic3—42no—
522C23—41no—
523C23—121no—
524C23#514411yescharacteristic
525Q8—22no—
526Q8—41no—
527Q8—62no—
528Q8—121no—
529D4—22no—
530D4—42no—
531D4—42no—
532D4—41no—
533D4—62no—
534D4—122no—
535D4—122no—
536D4—121no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—42no—
540C2×C4—41no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—121no—
544C2×C4—122no—
545C2×C4—122no—
546C2×C4C4×#21311yescharacteristic
547C2×C4C4×#21311yescharacteristic
548C2×C4C8×#6711yescharacteristic
549C2×C4C8×#6711yescharacteristic
550C2×C4#514411yescharacteristic
551C2×C4C2×C4×#6711yescharacteristic
552C8—42no—
553C8—64no—
554C8—122no—
555C8—122no—
556C8C4×#21311yescharacteristic
557C8C4×#21311yescharacteristic
558C8C8×#6711yescharacteristic
559C8C8×#6711yescharacteristic
560C8C22×D3×C811yescharacteristic
561C8C2×#90111yescharacteristic
562C8C22×#21311yescharacteristic
563C8C2×#95011yescharacteristic
564C2×C3—21no—
565C2×C3—21no—
566C2×C3—21no—
567C2×C3—82no—
568C2×C3C2×C8×D811yescharacteristic
569D3—82no—
570C22—41no—
571C22—42no—
572C22—122no—
573C22—161no—
574C22—481no—
575C22C8×#21311yescharacteristic
576C22C8×#21311yescharacteristic
577C22C2×#514411yescharacteristic
578C4—22no—
579C4—41no—
580C4—42no—
581C4—122no—
582C4C8×#21311yescharacteristic
583C4C8×#21311yescharacteristic
584C4C2×C4×#21311yescharacteristic
585C4C2×C8×#6711yescharacteristic
586C3#48351211yescharacteristic, complemented, Sylow
587C2—21no—
588C2—21no—
589C2—21no—
590C2—82no—
591C2—242no—
592C2C2×C8×#21311yescharacteristic
subgroupG/H
centerC8C22×#213
commutatorC3×C8C23×C8
FrattiniC4×C8C23×D3
FittingC3×#33443C2
socleC2×C3C2×C8×D8
prime radical#83894C4
cosocleC3×C4×C8C24
hypercenter#33443D3
hypocenterC3#483512
LCSG ▷ C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C8 ◁ C2×M32 ◁ C4×M32 ◁ #33443

Complements

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

NH ≅ G/Ntype
2. #10948443C2semidirect
3. #10706946C2semidirect
4. #10706941C2semidirect
5. #10706782C2semidirect
6. C2×#67766C2semidirect
7. C2×#67568C2semidirect
8. C3⋊C8×M32C2semidirect
9. C3×#33443C2semidirect
17. C3×#2170C22semidirect
31. #69056C22semidirect
32. #67766C22semidirect
33. #67766C22semidirect
34. #67568C22semidirect
35. #67568C22semidirect
36. #67566C22semidirect
37. #67498C22semidirect
38. C16×C3⋊C8C22semidirect
41. #33443D3semidirect
65. #5195C8semidirect
67. #5193C8semidirect
68. #5179C8semidirect
69. #5147C8semidirect
109. #2170C2×D3semidirect
180. #829C2×C8semidirect
181. #828C2×C8semidirect
182. #827C2×C8semidirect
183. #826C2×C8semidirect
586. C3#483512semidirect

Maximal subgroups

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

subgroupindex
#109485982
#109484432
#107069462
#107069412
#107067822
C2×#677662
C2×#675682
C3⋊C8×M322
C3×#334432
#4835123