Finite Group Databasebeta

← #11899684

Subgroups — #11899684

Order 1152.

785 conjugacy classes of subgroups.

The subgroup lattice has 557 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5757C211yesmaximal, characteristic, complemented
2#10594678C211yesmaximal, characteristic, complemented
3#10592846C211yesmaximal, characteristic, complemented
4C2×C3×#826C211yesmaximal, characteristic, complemented
5#10591032C212yesmaximal, complemented
6#10591031C212yesmaximal, complemented
7#10591024C212yesmaximal, complemented
8D3×#826C212yesmaximal, complemented
9#10590632C211yesmaximal, characteristic, complemented
10#10590630C211yesmaximal, characteristic, complemented
11#10590621C211yesmaximal, characteristic, complemented
12#84007—31nomaximal
13#82019—31nomaximal
14C3×#1004—21no—
15#64307—21no—
16C2×C32×D8C2211yescharacteristic, complemented
17#64113C2211yescharacteristic, complemented
18C3×#932—21no—
19C3×#912C2212yescomplemented
20C3×#911—22no—
21C3×D3×D8C2212yescomplemented
22C3×#904C2211yescharacteristic, complemented
23C2×C3×D24C2211yescharacteristic, complemented
24C3×#901C2211yescharacteristic
25#63830—21no—
26#63807C2212yescomplemented
27#63795—22no—
28D3×D24C2212yescomplemented
29#63654C2212yescomplemented
30#63653C2212yescomplemented
31#63639C2211yescharacteristic, complemented
32C3×#826C2212yescomplemented
33C3×#826C2212yescomplemented
34C2×C3×C3⋊C16C2211yescharacteristic, complemented
35#63551C2211yescharacteristic, complemented
36#63549C2211yescharacteristic, complemented
37#63548C2211yescharacteristic, complemented
38#63547C2212yescomplemented
39#63546C2211yescharacteristic, complemented
40#63545C2212yescomplemented
41#63543C2212yescomplemented
42D3×C3⋊C16C2212yescomplemented
43C3×#576—31no—
44#5757—31no—
45#5757D311yescharacteristic, complemented
46#5752—31no—
47C2×C3×D16—31no—
48#5194—32no—
49#5194—31no—
50#5193—31no—
51#5193—32no—
52#5147—31no—
53#5147—31no—
54#5147—31no—
55#5147—32no—
56#5147—61no—
57C2×#826—32no—
58C2×#826D311yescharacteristic, complemented
59#5140—31no—
60#4912—32no—
61#4911—32no—
62#4911—32no—
63D3×D16—32no—
64#4904—31no—
65#4901—31no—
66C2×C32×D4—21no—
67#3828—21no—
68C2×C3×#67—42no—
69C3×#237—22no—
70C3×#237—41no—
71C3×D3×D4—22no—
72C3×D3×D4—41no—
73C3×#235—21no—
74C3×#235D411yescharacteristic
75C2×C3×D12—21no—
76D3×#67—42no—
77#3802—41no—
78D3×D12—22no—
79#3796—41no—
80#3795—22no—
81C32×D8—21no—
82C32×D8C2312yes—
83C2×C32×C8C2311yescharacteristic
84#3746C2311yescharacteristic
85#3745C2311yescharacteristic
86#3744—21no—
87C3×#214—22no—
88C3×#213—22no—
89C3×#208D411yescharacteristic
90C3×Dic12C2311yescharacteristic
91C3×D24—21no—
92C3×D24C2312yes—
93C3×D24C2311yescharacteristic
94C3×#204—21no—
95C3×ΓL(1,𝔽25)D412yes—
96C3×D3×C8C2312yes—
97#3716—22no—
98#3714—22no—
99C3⋊(C3⋊C16)C2312yes—
100C3×C3⋊C16C2311yescharacteristic
101C3×C3⋊C16C2311yescharacteristic
102#3415—91noSylow
103C3×H(5,𝔽2)—61no—
104#1009—61no—
105#1004—21no—
106#1004—61no—
107C3×AGL(1,ℤ/8)—62no—
108C3×AGL(1,ℤ/8)—61no—
109C3×#135—32no—
110C3×#135—31no—
111C2×C3×D8—21no—
112C2×C3×D8—32no—
113C2×C3×D8—31no—
114C2×C3×D8C2×D311yescharacteristic, complemented
115C2×C3×D8C2×D311yescharacteristic, complemented
116C3×#131—31no—
117#932—21no—
118#932—61no—
119#912—32no—
120#912C2×D312yescomplemented
121#911—22no—
122#911—62no—
123D3×D8—32no—
124D3×D8C2×D312yescomplemented
125#908—62no—
126#908—61no—
127#904—31no—
128#904—31no—
129#904—32no—
130#904—61no—
131#904C2×D311yescharacteristic, complemented
132C2×D24—32no—
133C2×D24C2×D311yescharacteristic, complemented
134#901C2×D311yescharacteristic
135C3×D16—32no—
136C3×D16—32no—
137C2×C3×C16—31no—
138#829—31no—
139#828—32no—
140#828—31no—
141#827—32no—
142#827—32no—
143#827—32no—
144#827—31no—
145#827—62no—
146#826—32no—
147#826—32no—
148#826—32no—
149#826—32no—
150#826—31no—
151#826—62no—
152#826C2×D312yescomplemented
153#826C2×D312yescomplemented
154#812—31no—
155#812—31no—
156#812—32no—
157#812—61no—
158C2×C3⋊C16—32no—
159C2×C3⋊C16C2×D311yescharacteristic, complemented
160Dic24—31no—
161#800—31no—
162#800—31no—
163D48—31no—
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
173#638—41no—
174C4⋊(C3⋊D3)—21no—
175C4×C3⋊D3—41no—
176#634—21no—
177C3×#67—41no—
178C3×#67—42no—
179C3×#67—42no—
180C3×#67D812yes—
181C2×C3×Dic3—42no—
182C3×D12—22no—
183C3×D12—21no—
184C3×D12—41no—
185C3×D12D811yescharacteristic
186C3×C4×D3—41no—
187C3×C4×D3C2×D412yes—
188C3×Dic6—21no—
189C3×Dic6D811yescharacteristic
190#626—42no—
191#625—42no—
192D3×Dic3—42no—
193C32×C8C2×D411yescharacteristic
194C32×C8C2411yescharacteristic
195C3×C3⋊C8C2×D412yes—
196#576—31no—
197#576—91no—
198#509—92no—
199#509—92no—
200#509—91no—
201#509—91no—
202#508—91no—
203#508—92no—
204C2×D16—31no—
205C2×D16—92no—
206#504—91no—
207C3×#41—31no—
208C3×#41—62no—
209C3×#41—61no—
210C3×#41—121no—
211C2×C3×D4—21no—
212C2×C3×D4—21no—
213C2×C3×D4—41no—
214C2×C3×D4—62no—
215C2×C3×D4—61no—
216C2×C3×D4—121no—
217C2×C3×D4—122no—
218C2×#67—42no—
219C2×#67—122no—
220#239—121no—
221#237—22no—
222#237—41no—
223#237—62no—
224#237—121no—
225D3×D4—22no—
226D3×D4—41no—
227D3×D4—62no—
228D3×D4—121no—
229D3×D4—122no—
230D3×D4—121no—
231#235—21no—
232#235—62no—
233#235—61no—
234#235—61no—
235#235—121no—
236#235#6711yescharacteristic
237C2×D12—21no—
238C2×D12—62no—
239C3×Q16—31no—
240C3×SD16—62no—
241C3×SD16—61no—
242C3×D8—21no—
243C3×D8—21no—
244C3×D8—22no—
245C3×D8—31no—
246C3×D8—32no—
247C3×D8—41no—
248C3×D8—62no—
249C3×D8—61no—
250C3×D8C22×D312yes—
251C3×D8C22×D312yes—
252C3×M16—31no—
253C3×M16—32no—
254C2×C3×C8—21no—
255C2×C3×C8—32no—
256C2×C3×C8C22×D311yescharacteristic
257C2×C3×C8C22×D311yescharacteristic
258#214—22no—
259#214—62no—
260#213—22no—
261#213—62no—
262#208#6711yescharacteristic
263Dic12—31no—
264Dic12—31no—
265Dic12—61no—
266Dic12C22×D311yescharacteristic
267D24—21no—
268D24—31no—
269D24—31no—
270D24—61no—
271D24—62no—
272D24C22×D311yescharacteristic
273D24C22×D312yes—
274#204—21no—
275#204—62no—
276#204—61no—
277#204—61no—
278#204—121no—
279ΓL(1,𝔽25)#6712yes—
280D3×C8C22×D312yes—
281C3×C16—31no—
282C3×C16—31no—
283C3⋊C16—32no—
284C3⋊C16—32no—
285C3⋊C16—62no—
286C3⋊C16C22×D311yescharacteristic
287C3⋊C16C22×D311yescharacteristic
288C22×C32—42no—
289C22×C32—161no—
290C22×C32C2×D811yescharacteristic
291C2×C3⋊D3—41no—
292C2×C3×D3—42no—
293C2×C3×D3—41no—
294C2×C3×D3—162no—
295C2×C3×D3—161no—
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—92no—
312#135—91no—
313C2×D8—32no—
314C2×D8—31no—
315C2×D8—92no—
316C2×D8D3211yescharacteristic, complemented
317#131—31no—
318Q32—91no—
319SD32—91no—
320SD32—91no—
321SD32—92no—
322SD32—92no—
323D16—32no—
324D16—32no—
325D16—91no—
326D16—92no—
327D16—92no—
328M32—92no—
329M32—91no—
330C2×C16—31no—
331C2×C16—92no—
332C23×C3—41no—
333C23×C3—41no—
334C23×C3—81no—
335C23×C3—122no—
336C23×C3—121no—
337C22×D3—41no—
338C22×D3—42no—
339C22×D3—122no—
340C22×D3—122no—
341C3×Q8—31no—
342C3×Q8—61no—
343C3×D4—22no—
344C3×D4—22no—
345C3×D4—32no—
346C3×D4—31no—
347C3×D4—42no—
348C3×D4—41no—
349C3×D4—41no—
350C3×D4—61no—
351C3×D4—62no—
352C3×D4—81no—
353C3×D4—122no—
354C3×D4—122no—
355C3×D4—121no—
356C3×D4—121no—
357C2×C3×C4—21no—
358C2×C3×C4—32no—
359C2×C3×C4—122no—
360C2×C3×C4—121no—
361C2×C3×C4D3×D411yescharacteristic
362C2×C3×C4C2×#6711yescharacteristic
363#67—41no—
364#67—42no—
365#67—42no—
366#67—122no—
367#67—121no—
368#67—121no—
369#67—122no—
370#67—122no—
371#67—241no—
372#67#21312yes—
373C2×Dic3—42no—
374C2×Dic3—122no—
375D12—22no—
376D12—21no—
377D12—41no—
378D12—61no—
379D12—61no—
380D12—121no—
381D12—122no—
382D12#21311yescharacteristic
383C4×D3—41no—
384C4×D3—122no—
385C4×D3—121no—
386C4×D3—121no—
387C4×D3—241no—
388C4×D3C2×#6712yes—
389Dic6—21no—
390Dic6—61no—
391Dic6—61no—
392Dic6—121no—
393Dic6#21311yescharacteristic
394C3×C8—21no—
395C3×C8—21no—
396C3×C8—32no—
397C3×C8D3×D411yescharacteristic
398C3×C8C2×#6711yescharacteristic
399C3×C8C23×D311yescharacteristic
400C3×C8C23×D311yescharacteristic
401C3⋊C8C2×#6712yes—
402C2×C32—21no—
403C2×C32—82no—
404C2×C32C22×D811yescharacteristic
405C3⋊D3—81no—
406C3×D3—22no—
407C3×D3—82no—
408C3×D3—81no—
409#41—31no—
410#41—62no—
411#41—61no—
412#41—121no—
413#41—182no—
414#41—181no—
415#41—361no—
416C2×D4—21no—
417C2×D4—62no—
418C2×D4—61no—
419C2×D4—121no—
420C2×D4—122no—
421C2×D4—182no—
422C2×D4—361no—
423C2×D4—362no—
424Q16—31no—
425Q16—91no—
426SD16—62no—
427SD16—61no—
428SD16—182no—
429SD16—181no—
430D8—21no—
431D8—32no—
432D8—31no—
433D8—62no—
434D8—61no—
435D8—91no—
436D8—182no—
437D8C2×D3212yes—
438M16—31no—
439M16—32no—
440C2×C8—32no—
441C2×C8C2×D3211yescharacteristic
442C16—31no—
443C16—31no—
444C16—92no—
445C22×C3—21no—
446C22×C3—32no—
447C22×C3—42no—
448C22×C3—42no—
449C22×C3—82no—
450C22×C3—121no—
451C22×C3—122no—
452C22×C3—161no—
453C22×C3—161no—
454C22×C3—162no—
455C22×C3—481no—
456C22×C3—482no—
457C22×C3D3×D811yescharacteristic
458C22×C3C2×#21311yescharacteristic
459C2×D3—42no—
460C2×D3—41no—
461C2×D3—121no—
462C2×D3—121no—
463C2×D3—162no—
464C2×D3—161no—
465C2×D3—241no—
466C2×D3—482no—
467C2×D3—482no—
468C2×D3C2×#21312yes—
469C3×C4—21no—
470C3×C4—21no—
471C3×C4—32no—
472C3×C4—121no—
473C3×C4D3×D811yescharacteristic
474C3×C4C2×#21311yescharacteristic
475C3×C4C2×D3×D411yescharacteristic
476C3×C4C22×#6711yescharacteristic
477Dic3—41no—
478Dic3—121no—
479Dic3—121no—
480Dic3—241no—
481Dic3C2×#21312yes—
482C32#341511yescharacteristic, complemented, Sylow
483C23—41no—
484C23—122no—
485C23—121no—
486C23—362no—
487Q8—31no—
488Q8—61no—
489Q8—181no—
490D4—22no—
491D4—32no—
492D4—31no—
493D4—41no—
494D4—61no—
495D4—62no—
496D4—122no—
497D4—122no—
498D4—121no—
499D4—121no—
500D4—181no—
501D4—362no—
502D4—362no—
503D4—361no—
504C2×C4—32no—
505C2×C4—122no—
506C2×C4—121no—
507C2×C4—362no—
508C2×C4—361no—
509C2×C4D3×#6711yescharacteristic
510C8—32no—
511C8D3×#6711yescharacteristic
512C8C22×D3211yescharacteristic
513C2×C3—21no—
514C2×C3—21no—
515C2×C3—21no—
516C2×C3—41no—
517C2×C3—62no—
518C2×C3—82no—
519C2×C3—82no—
520C2×C3—162no—
521C2×C3—242no—
522C2×C3—241no—
523C2×C3C2×D3×D811yescharacteristic
524C2×C3C22×#21311yescharacteristic
525D3—22no—
526D3—81no—
527D3—82no—
528D3—241no—
529D3—241no—
530D3—481no—
531C22—32no—
532C22—42no—
533C22—121no—
534C22—122no—
535C22—161no—
536C22—361no—
537C22—481no—
538C22—482no—
539C22—1442no—
540C22D3×#21311yescharacteristic
541C4—32no—
542C4—121no—
543C4—361no—
544C4D3×#21311yescharacteristic
545C4C2×D3×#6711yescharacteristic
546C3—21no—
547C3#8201911yescharacteristic, complemented
548C3#8400711yescharacteristic, complemented
549C2—21no—
550C2—62no—
551C2—82no—
552C2—241no—
553C2—242no—
554C2—721no—
555C2C2×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. #10594678C2semidirect
3. #10592846C2semidirect
4. C2×C3×#826C2semidirect
5. #10591032C2semidirect
6. #10591031C2semidirect
7. #10591024C2semidirect
8. D3×#826C2semidirect
9. #10590632C2semidirect
10. #10590630C2semidirect
11. #10590621C2semidirect
16. C2×C32×D8C22semidirect
17. #64113C22semidirect
19. C3×#912C22semidirect
21. C3×D3×D8C22semidirect
22. C3×#904C22semidirect
23. C2×C3×D24C22semidirect
26. #63807C22semidirect
28. D3×D24C22semidirect
29. #63654C22semidirect
30. #63653C22semidirect
31. #63639C22semidirect
32. C3×#826C22semidirect
33. C3×#826C22semidirect
34. C2×C3×C3⋊C16C22semidirect
35. #63551C22semidirect
36. #63549C22semidirect
37. #63548C22semidirect
38. #63547C22semidirect
39. #63546C22semidirect
40. #63545C22semidirect
41. #63543C22semidirect
42. D3×C3⋊C16C22semidirect
45. #5757D3semidirect
58. C2×#826D3semidirect
114. C2×C3×D8C2×D3semidirect
115. C2×C3×D8C2×D3semidirect
120. #912C2×D3semidirect
124. D3×D8C2×D3semidirect
131. #904C2×D3semidirect
133. C2×D24C2×D3semidirect
152. #826C2×D3semidirect
153. #826C2×D3semidirect
159. C2×C3⋊C16C2×D3semidirect
316. C2×D8D32semidirect
482. C32#3415semidirect
547. C3#82019semidirect
548. C3#84007semidirect

Maximal subgroups

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

subgroupindex
C3×#57572
#105946782
#105928462
C2×C3×#8262
#105910322
#105910312
#105910242
D3×#8262
#105906322
#105906302
#105906212
#840073
#820193