Finite Group Databasebeta

← #11899682

Subgroups — #11899682

Order 1152.

789 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5757C211yesmaximal, characteristic, complemented
2#10594680C211yesmaximal, characteristic, complemented
3C2×#63653C211yesmaximal, characteristic, complemented
4C3×#5147C211yesmaximal, characteristic, complemented
5#10591034C212yesmaximal, complemented
6#10591033C212yesmaximal, complemented
7#10591024C212yesmaximal, complemented
8D3×#826C212yesmaximal, complemented
9#10590634C211yesmaximal, characteristic, complemented
10#10590629C211yesmaximal, characteristic, complemented
11#10590622C211yesmaximal, characteristic, complemented
12#84007—31nomaximal
13#82036—31nomaximal
14C3×#1004—21no—
15#64308—21no—
16C2×C32×D8C2211yescharacteristic, complemented
17C2×#3745C2211yescharacteristic, complemented
18C3×#932—21no—
19C3×#912C2212yescomplemented
20C3×#911—22no—
21C3×D3×D8C2212yescomplemented
22C3×#904C2211yescharacteristic, complemented
23C3×#904C2211yescharacteristic, complemented
24C3×#901C2211yescharacteristic
25#63825—21no—
26#63808C2212yescomplemented
27#63794—22no—
28D3×D24C2212yescomplemented
29#63653C2212yescomplemented
30#63653C2212yescomplemented
31C2×C3⋊(C3⋊C16)C2211yescharacteristic, complemented
32C3×#827C2212yescomplemented
33C3×#826C2212yescomplemented
34C3×#812C2211yescharacteristic, complemented
35#63550C2211yescharacteristic, complemented
36#63549C2212yescomplemented
37#63547C2211yescharacteristic, complemented
38#63547C2211yescharacteristic, complemented
39#63546C2212yescomplemented
40#63545C2211yescharacteristic, complemented
41#63543C2212yescomplemented
42D3×C3⋊C16C2212yescomplemented
43C3×#576—31no—
44#5769—31no—
45#5757D311yescharacteristic, complemented
46#5752—31no—
47C3×#509—31no—
48#5194—31no—
49#5194—31no—
50#5194—32no—
51#5193—32no—
52#5193—31no—
53#5147—32no—
54#5147D311yescharacteristic, complemented
55C2×#826—31no—
56C2×#826—31no—
57C2×#826—32no—
58C2×#826—61no—
59#5140—31no—
60#4916—32no—
61#4914—32no—
62#4911—32no—
63D3×D16—32no—
64#4908—31no—
65#4907—31no—
66C2×C32×D4—21no—
67C2×C4⋊(C3⋊D3)—21no—
68C2×C3×#67—42no—
69C3×#237—22no—
70C3×#237—41no—
71C3×D3×D4—22no—
72C3×D3×D4—41no—
73C3×#235—21no—
74C3×#235—21no—
75C3×#235D411yescharacteristic
76#3811—42no—
77D3×D12—22no—
78D3×D12—41no—
79#3799—22no—
80#3799—41no—
81C32×D8—21no—
82C32×D8C2312yes—
83C2×C32×C8C2311yescharacteristic
84#3745—21no—
85#3745C2312yes—
86C3×#214—22no—
87C3×#213—22no—
88C3×#208D411yescharacteristic
89C3×Dic12C2311yescharacteristic
90C3×Dic12C2311yescharacteristic
91C3×D24C2311yescharacteristic
92C3×D24C2311yescharacteristic
93C3×#204—21no—
94C3×#204—21no—
95C3×ΓL(1,𝔽25)D412yes—
96C3×D3×C8C2312yes—
97#3717—22no—
98#3714—22no—
99C3⋊(C3⋊C16)C2311yescharacteristic
100C3⋊(C3⋊C16)C2311yescharacteristic
101C3×C3⋊C16C2312yes—
102#3415—91noSylow
103C3×H(5,𝔽2)—61no—
104#1009—61no—
105#1009—61no—
106#1004—21no—
107C3×AGL(1,ℤ/8)—62no—
108C3×AGL(1,ℤ/8)—61no—
109C3×#135—32no—
110C3×#135—31no—
111C3×#135—31no—
112C2×C3×D8—21no—
113C2×C3×D8—32no—
114C2×C3×D8C2×D311yescharacteristic, complemented
115C2×C3×D8C2×D311yescharacteristic, complemented
116C3×#131—31no—
117#949—61no—
118#932—21no—
119#919—32no—
120#914—62no—
121#912C2×D312yescomplemented
122#911—22no—
123D3×D8—32no—
124D3×D8C2×D312yescomplemented
125#908—62no—
126#908—61no—
127#904—32no—
128#904C2×D311yescharacteristic, complemented
129#904C2×D311yescharacteristic, complemented
130C2×D24—31no—
131C2×D24—31no—
132C2×D24—32no—
133C2×D24—61no—
134#901C2×D311yescharacteristic
135C3×SD32—32no—
136C3×D16—32no—
137C3×M32—31no—
138#829—31no—
139#829—31no—
140#828—31no—
141#828—31no—
142#828—32no—
143#827—31no—
144#827—31no—
145#827C2×D312yescomplemented
146#826—32no—
147#826—32no—
148#826—32no—
149#826—32no—
150#826—31no—
151#826—31no—
152#826—32no—
153#826—62no—
154#826—62no—
155#826C2×D312yescomplemented
156#812—32no—
157#812C2×D311yescharacteristic, complemented
158C2×C3⋊C16—31no—
159C2×C3⋊C16—31no—
160C2×C3⋊C16—32no—
161C2×C3⋊C16—61no—
162#800—32no—
163D48—32no—
164#798—32no—
165D3×C16—32no—
166C23×C32—41no—
167C22×C3⋊D3—41no—
168C22×C3×D3—42no—
169C2×D32—42no—
170C32×D4—22no—
171C32×D4—41no—
172C2×C32×C4C2×D411yescharacteristic
173C4⋊(C3⋊D3)—22no—
174C4⋊(C3⋊D3)—41no—
175C3×#67—41no—
176C3×#67—41no—
177C3×#67—42no—
178C3×#67—42no—
179C3×#67D812yes—
180C2×C3×Dic3—42no—
181C3×D12—21no—
182C3×D12—21no—
183C3×D12D811yescharacteristic
184C3×C4×D3—41no—
185C3×C4×D3—41no—
186C3×C4×D3C2×D412yes—
187C3×Dic6—21no—
188C3×Dic6—21no—
189C3×Dic6D811yescharacteristic
190#626—42no—
191#626—42no—
192#624—42no—
193C32×C8C2×D411yescharacteristic
194C32×C8C2411yescharacteristic
195C3×C3⋊C8C2×D412yes—
196#576—31no—
197#576—91no—
198#509—31no—
199#509—91no—
200#509—92no—
201#509—92no—
202#508—92no—
203#508—91no—
204C2×D16—92no—
205C2×D16—91no—
206#504—91no—
207C3×#41—31no—
208C3×#41—62no—
209C3×#41—61no—
210C3×#41—61no—
211C3×#41—121no—
212C2×C3×D4—21no—
213C2×C3×D4—21no—
214C2×C3×D4—41no—
215C2×C3×D4—62no—
216C2×C3×D4—121no—
217C2×C3×D4—122no—
218C2×#67—42no—
219#239—62no—
220#239—121no—
221#237—22no—
222#237—41no—
223D3×D4—22no—
224D3×D4—41no—
225D3×D4—62no—
226D3×D4—121no—
227D3×D4—122no—
228D3×D4—122no—
229#235—21no—
230#235—21no—
231#235—62no—
232#235—121no—
233#235#6711yescharacteristic
234C2×D12—62no—
235C2×D12—61no—
236C2×D12—61no—
237C2×D12—121no—
238C2×D12—121no—
239C3×Q16—31no—
240C3×Q16—31no—
241C3×SD16—62no—
242C3×SD16—61no—
243C3×SD16—61no—
244C3×D8—21no—
245C3×D8—21no—
246C3×D8—22no—
247C3×D8—31no—
248C3×D8—31no—
249C3×D8—41no—
250C3×D8—62no—
251C3×D8C22×D312yes—
252C3×D8C22×D312yes—
253C3×M16—31no—
254C3×M16—32no—
255C2×C3×C8—21no—
256C2×C3×C8—32no—
257C2×C3×C8C22×D311yescharacteristic
258C2×C3×C8C22×D311yescharacteristic
259#215—62no—
260#214—22no—
261#213—22no—
262#213—62no—
263#208#6711yescharacteristic
264Dic12C22×D311yescharacteristic
265Dic12C22×D311yescharacteristic
266D24—32no—
267D24—32no—
268D24—62no—
269D24—62no—
270D24—61no—
271D24—61no—
272D24—121no—
273D24C22×D311yescharacteristic
274D24C22×D311yescharacteristic
275#204—21no—
276#204—21no—
277#204—62no—
278ΓL(1,𝔽25)#6712yes—
279D3×C8C22×D312yes—
280C3×C16—32no—
281C3⋊C16—31no—
282C3⋊C16—31no—
283C3⋊C16—31no—
284C3⋊C16—31no—
285C3⋊C16—61no—
286C3⋊C16—61no—
287C3⋊C16C22×D312yes—
288C22×C32—42no—
289C22×C32—161no—
290C22×C32C2×D811yescharacteristic
291C2×C3⋊D3—42no—
292C2×C3⋊D3—161no—
293C2×C3×D3—41no—
294C2×C3×D3—41no—
295C2×C3×D3—162no—
296C2×C3×D3C2×D812yes—
297D32—162no—
298C32×C4C2×D811yescharacteristic
299C32×C4C22×D411yescharacteristic
300C3×Dic3—41no—
301C3×Dic3—41no—
302C3×Dic3C2×D812yes—
303H(5,𝔽2)—61no—
304H(5,𝔽2)—181no—
305AGL(1,ℤ/8)—62no—
306AGL(1,ℤ/8)—61no—
307AGL(1,ℤ/8)—182no—
308AGL(1,ℤ/8)—181no—
309#135—32no—
310#135—31no—
311#135—31no—
312#135—92no—
313C2×D8—32no—
314C2×D8—92no—
315C2×D8—91no—
316C2×D8D3211yescharacteristic, complemented
317#131—31no—
318Q32—91no—
319SD32—32no—
320SD32—92no—
321SD32—91no—
322SD32—91no—
323D16—32no—
324D16—92no—
325D16—91no—
326D16—92no—
327D16—92no—
328M32—31no—
329M32—92no—
330C2×C16—92no—
331C2×C16—91no—
332C23×C3—41no—
333C23×C3—41no—
334C23×C3—81no—
335C23×C3—122no—
336C22×D3—42no—
337C22×D3—122no—
338C22×D3—121no—
339C22×D3—121no—
340C22×D3—122no—
341C22×D3—241no—
342C3×Q8—31no—
343C3×Q8—61no—
344C3×Q8—61no—
345C3×D4—22no—
346C3×D4—22no—
347C3×D4—32no—
348C3×D4—31no—
349C3×D4—42no—
350C3×D4—41no—
351C3×D4—41no—
352C3×D4—61no—
353C3×D4—61no—
354C3×D4—81no—
355C3×D4—122no—
356C3×D4—122no—
357C3×D4—121no—
358C3×D4—121no—
359C2×C3×C4—21no—
360C2×C3×C4—32no—
361C2×C3×C4—122no—
362C2×C3×C4—121no—
363C2×C3×C4—121no—
364C2×C3×C4D3×D411yescharacteristic
365C2×C3×C4C2×#6711yescharacteristic
366#67—41no—
367#67—41no—
368#67—42no—
369#67—42no—
370#67—122no—
371#67—122no—
372#67#21312yes—
373C2×Dic3—42no—
374D12—21no—
375D12—21no—
376D12—62no—
377D12—62no—
378D12—122no—
379D12—122no—
380D12—122no—
381D12—121no—
382D12—121no—
383D12—241no—
384D12#21311yescharacteristic
385C4×D3—41no—
386C4×D3—41no—
387C4×D3—122no—
388C4×D3—122no—
389C4×D3C2×#6712yes—
390Dic6—21no—
391Dic6—21no—
392Dic6#21311yescharacteristic
393C3×C8—21no—
394C3×C8—21no—
395C3×C8—32no—
396C3×C8D3×D411yescharacteristic
397C3×C8C2×#6711yescharacteristic
398C3×C8C23×D311yescharacteristic
399C3×C8C23×D311yescharacteristic
400C3⋊C8C2×#6712yes—
401C2×C32—21no—
402C2×C32—82no—
403C2×C32C22×D811yescharacteristic
404C3⋊D3—82no—
405C3×D3—22no—
406C3×D3—81no—
407C3×D3—81no—
408#41—31no—
409#41—62no—
410#41—61no—
411#41—61no—
412#41—121no—
413#41—182no—
414#41—361no—
415C2×D4—21no—
416C2×D4—62no—
417C2×D4—121no—
418C2×D4—122no—
419C2×D4—182no—
420C2×D4—181no—
421C2×D4—361no—
422C2×D4—362no—
423Q16—31no—
424Q16—31no—
425SD16—62no—
426SD16—61no—
427SD16—61no—
428SD16—182no—
429D8—21no—
430D8—31no—
431D8—31no—
432D8—62no—
433D8—92no—
434D8—182no—
435D8—181no—
436D8C2×D3212yes—
437M16—31no—
438M16—32no—
439C2×C8—32no—
440C2×C8C2×D3211yescharacteristic
441C16—32no—
442C16—91no—
443C16—91no—
444C22×C3—21no—
445C22×C3—32no—
446C22×C3—42no—
447C22×C3—42no—
448C22×C3—82no—
449C22×C3—121no—
450C22×C3—121no—
451C22×C3—161no—
452C22×C3—161no—
453C22×C3—162no—
454C22×C3—482no—
455C22×C3D3×D811yescharacteristic
456C22×C3C2×#21311yescharacteristic
457C2×D3—41no—
458C2×D3—41no—
459C2×D3—122no—
460C2×D3—122no—
461C2×D3—162no—
462C2×D3—242no—
463C2×D3—482no—
464C2×D3—482no—
465C2×D3—481no—
466C2×D3—481no—
467C2×D3—482no—
468C2×D3C2×#21312yes—
469C3×C4—21no—
470C3×C4—21no—
471C3×C4—32no—
472C3×C4—121no—
473C3×C4—121no—
474C3×C4D3×D811yescharacteristic
475C3×C4C2×#21311yescharacteristic
476C3×C4C2×D3×D411yescharacteristic
477C3×C4C22×#6711yescharacteristic
478Dic3—41no—
479Dic3—41no—
480Dic3C2×#21312yes—
481C32#341511yescharacteristic, complemented, Sylow
482C23—41no—
483C23—122no—
484C23—362no—
485C23—361no—
486Q8—31no—
487Q8—61no—
488Q8—61no—
489D4—22no—
490D4—32no—
491D4—31no—
492D4—41no—
493D4—61no—
494D4—61no—
495D4—122no—
496D4—122no—
497D4—121no—
498D4—121no—
499D4—182no—
500D4—362no—
501D4—362no—
502D4—361no—
503C2×C4—32no—
504C2×C4—122no—
505C2×C4—121no—
506C2×C4—121no—
507C2×C4—362no—
508C2×C4D3×#6711yescharacteristic
509C8—32no—
510C8D3×#6711yescharacteristic
511C8C22×D3211yescharacteristic
512C2×C3—22no—
513C2×C3—21no—
514C2×C3—21no—
515C2×C3—21no—
516C2×C3—62no—
517C2×C3—82no—
518C2×C3—82no—
519C2×C3—162no—
520C2×C3—241no—
521C2×C3—241no—
522C2×C3C2×D3×D811yescharacteristic
523C2×C3C22×#21311yescharacteristic
524D3—22no—
525D3—81no—
526D3—81no—
527D3—242no—
528D3—242no—
529D3—482no—
530C22—32no—
531C22—42no—
532C22—121no—
533C22—121no—
534C22—161no—
535C22—362no—
536C22—482no—
537C22—1441no—
538C22—1442no—
539C22D3×#21311yescharacteristic
540C4—32no—
541C4—121no—
542C4—121no—
543C4D3×#21311yescharacteristic
544C4C2×D3×#6711yescharacteristic
545C3—21no—
546C3#8203611yescharacteristic, complemented
547C3#8400711yescharacteristic, complemented
548C2—21no—
549C2—62no—
550C2—82no—
551C2—241no—
552C2—241no—
553C2—722no—
554C2C2×D3×#21311yescharacteristic
subgroupG/H
centerC2C2×D3×#213
commutator, cosocleC32×C8C24
FrattiniC8C22×D32
FittingC2×C32×D8C22
socleC2×C32C22×D8
hypercenterC2×D8D32
hypocenterC32#3415
LCSG ▷ C32×C8 ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×D8

Complements

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

NH ≅ G/Ntype
1. C3×#5757C2semidirect
2. #10594680C2semidirect
3. C2×#63653C2semidirect
4. C3×#5147C2semidirect
5. #10591034C2semidirect
6. #10591033C2semidirect
7. #10591024C2semidirect
8. D3×#826C2semidirect
9. #10590634C2semidirect
10. #10590629C2semidirect
11. #10590622C2semidirect
16. C2×C32×D8C22semidirect
17. C2×#3745C22semidirect
19. C3×#912C22semidirect
21. C3×D3×D8C22semidirect
22. C3×#904C22semidirect
23. C3×#904C22semidirect
26. #63808C22semidirect
28. D3×D24C22semidirect
29. #63653C22semidirect
30. #63653C22semidirect
31. C2×C3⋊(C3⋊C16)C22semidirect
32. C3×#827C22semidirect
33. C3×#826C22semidirect
34. C3×#812C22semidirect
35. #63550C22semidirect
36. #63549C22semidirect
37. #63547C22semidirect
38. #63547C22semidirect
39. #63546C22semidirect
40. #63545C22semidirect
41. #63543C22semidirect
42. D3×C3⋊C16C22semidirect
45. #5757D3semidirect
54. #5147D3semidirect
114. C2×C3×D8C2×D3semidirect
115. C2×C3×D8C2×D3semidirect
121. #912C2×D3semidirect
124. D3×D8C2×D3semidirect
128. #904C2×D3semidirect
129. #904C2×D3semidirect
145. #827C2×D3semidirect
155. #826C2×D3semidirect
157. #812C2×D3semidirect
316. C2×D8D32semidirect
481. C32#3415semidirect
546. C3#82036semidirect
547. C3#84007semidirect

Maximal subgroups

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

subgroupindex
C3×#57572
#105946802
C2×#636532
C3×#51472
#105910342
#105910332
#105910242
D3×#8262
#105906342
#105906292
#105906222
#840073
#820363