Finite Group Databasebeta

← #11904611

Subgroups — #11904611

Order 1152.

4974 conjugacy classes of subgroups.

The subgroup lattice has 483 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C32×#41C211yesmaximal, characteristic, complemented
2C2×#64366C211yesmaximal, characteristic, complemented
3#10596305C212yesmaximal, complemented
4#10596304C213yesmaximal, complemented
5#10596301C213yesmaximal, complemented
6#10596300C216yesmaximal, complemented
7#10596298C211yesmaximal, characteristic, complemented
8#10596288C212yesmaximal, complemented
9D4×#638C216yesmaximal, complemented
10#10596262C213yesmaximal, complemented
11C2×#64139C213yesmaximal, complemented
12#87275—34nomaximal
13C2×C32×#41—28no—
14C2×C32×#41C2211yescharacteristic
15C2×C32×#41C2213yescomplemented
16C2×C32×#41C2213yescomplemented
17C2×C32×#41C2211yescharacteristic, complemented
18C22×C32×Q8C2211yescharacteristic, complemented
19C22×C32×D4C2213yescomplemented
20C23×C32×C4C2213yescomplemented
21C22×#638C2216yescomplemented
22#64366—28no—
23C2×#3832C2212yescomplemented
24C2×D4×C3⋊D3C2216yescomplemented
25C2×#3828C2213yescomplemented
26C22×C4⋊(C3⋊D3)C2213yescomplemented
27#64162C2216yescomplemented
28#64156C2216yescomplemented
29#64155C2216yescomplemented
30Q8×C3⋊Dic3C2212yescomplemented
31#64149C2216yescomplemented
32#64148C22112yescomplemented
33#64147C2216yescomplemented
34#64146C22112yescomplemented
35D4×C3⋊Dic3C2216yescomplemented
36#64139C2213yescomplemented
37#64139C2216yescomplemented
38#64139C2216yescomplemented
39#64139C2216yescomplemented
40#64139C2213yescomplemented
41#64138C2216yescomplemented
42C4×#638C2216yescomplemented
43C4×#638C2216yescomplemented
44C2×#3752C2216yescomplemented
45#64135C2213yescomplemented
46C2×C4⋊(C3⋊Dic3)C2213yescomplemented
47#64133C2211yescharacteristic, complemented
48#64133C2213yescomplemented
49C22×C3×#41D314yescomplemented
50C2×#1009—34no—
51#5834—38no—
52#5833—312no—
53#5830—312no—
54#5829—324no—
55#5827—34no—
56#5817—38no—
57D4×#67—324no—
58#5791—312no—
59C2×#930—312no—
60C24×C32C2313yes—
61C23×C3⋊D3C2316yes—
62C32×#41—24no—
63C32×#41—212no—
64C32×#41—212no—
65C32×#41D418yescomplemented
66C2×C32×Q8—24no—
67C2×C32×Q8C2313yes—
68C2×C32×Q8C2311yescharacteristic
69C2×C32×D4—212no—
70C2×C32×D4C2316yes—
71C2×C32×D4C2313yes—
72C2×C32×D4C2313yes—
73C22×C32×C4—212no—
74C22×C32×C4C2311yescharacteristic
75C22×C32×C4C2313yes—
76C22×C32×C4C2313yes—
77C22×C32×C4C2313yes—
78C22×C32×C4C2313yes—
79C22×C32×C4C2313yes—
80C2×#638—412no—
81C2×#638C2316yes—
82C2×#638C2316yes—
83C2×#638C23112yes—
84C2×#638C2316yes—
85C22×C3⋊Dic3C2316yes—
86#3832—28no—
87D4×C3⋊D3—224no—
88#3828—212no—
89C2×C4⋊(C3⋊D3)—212no—
90C2×C4⋊(C3⋊D3)C2313yes—
91C2×C4⋊(C3⋊D3)C2316yes—
92C2×C4×C3⋊D3C2316yes—
93C2×#634C2311yescharacteristic
94#3757C2312yes—
95#3757C2316yes—
96#3757C2316yes—
97#3752C2316yes—
98#3752C23112yes—
99#3752C2316yes—
100C4⋊(C3⋊Dic3)C2313yes—
101C4⋊(C3⋊Dic3)C2311yescharacteristic
102C4⋊(C3⋊Dic3)C2316yes—
103#3750C2312yes—
104#3750C2316yes—
105C4×C3⋊Dic3C2316yes—
106#3488—91noSylow
107C2×C3×#41—232no—
108C2×C3×#41C2×D314yes—
109C2×C3×#41C2×D3112yescomplemented
110C2×C3×#41C2×D3112yescomplemented
111C2×C3×#41C2×D314yescomplemented
112C22×C3×Q8C2×D314yescomplemented
113C22×C3×D4C2×D3112yescomplemented
114C23×C3×C4C2×D3112yescomplemented
115C22×#67—324no—
116#1009—632no—
117C2×#239—38no—
118C2×D3×D4—324no—
119C2×#235—312no—
120C22×D12—312no—
121#953—324no—
122#947—324no—
123#946—324no—
124Q8×Dic3—38no—
125#940—324no—
126#939—348no—
127#938—324no—
128#937—348no—
129D4×Dic3—324no—
130#930—312no—
131#930—324no—
132#930—324no—
133#930—324no—
134#930—312no—
135#929—324no—
136C4×#67—324no—
137C4×#67—324no—
138C2×(C2×D3)⋊C4—324no—
139#926—312no—
140C2×C4⋊Dic3—312no—
141#924—34no—
142#924—312no—
143C23×C32—26no—
144C23×C32—26no—
145C23×C32—43no—
146C23×C32C2413yes—
147C23×C32C2413yes—
148C23×C32C2411yescharacteristic
149C22×C3⋊D3—212no—
150C22×C3⋊D3—46no—
151C22×C3⋊D3—46no—
152C22×C3⋊D3C2416yes—
153C32×Q8—26no—
154C32×Q8C2×D414yescomplemented
155C32×D4—212no—
156C32×D4—26no—
157C32×D4C2×D4112yescomplemented
158C2×C32×C4—26no—
159C2×C32×C4—22no—
160C2×C32×C4—26no—
161C2×C32×C4—26no—
162C2×C32×C4—26no—
163C2×C32×C4C2×D4112yescomplemented
164C2×C32×C4C2411yescharacteristic
165C2×C32×C4C2413yes—
166C2×C32×C4C2413yes—
167C2×C32×C4C2411yescharacteristic
168#638—212no—
169#638—212no—
170#638—412no—
171#638—46no—
172C2×C3⋊Dic3—46no—
173C2×C3⋊Dic3C2412yes—
174C2×C3⋊Dic3C2416yes—
175C2×C3⋊Dic3C2412yes—
176C4⋊(C3⋊D3)—26no—
177C4⋊(C3⋊D3)—212no—
178C4×C3⋊D3—212no—
179#634—22no—
180C2×H(5,𝔽2)—91no—
181C22×#41C3⋊D311yescharacteristic, complemented
182#550—92no—
183#548—92no—
184#546—96no—
185D42—96no—
186#537—93no—
187#535—93no—
188#525—93no—
189C2×#121—93no—
190#520—91no—
191C24×C3C22×D3112yes—
192C23×D3—324no—
193C3×#41—216no—
194C3×#41—248no—
195C3×#41—248no—
196C3×#41#67132yescomplemented
197C2×C3×Q8—216no—
198C2×C3×Q8C22×D3112yes—
199C2×C3×Q8C22×D314yes—
200C2×C3×D4—248no—
201C2×C3×D4C22×D3124yes—
202C2×C3×D4C22×D3112yes—
203C2×C3×D4C22×D3112yes—
204C22×C3×C4—248no—
205C22×C3×C4C22×D314yes—
206C22×C3×C4C22×D3112yes—
207C22×C3×C4C22×D3112yes—
208C22×C3×C4C22×D3112yes—
209C22×C3×C4C22×D3112yes—
210C22×C3×C4C22×D3112yes—
211C2×#67—324no—
212C2×#67—324no—
213C2×#67—324no—
214C2×#67—348no—
215C2×#67—1248no—
216C22×Dic3—324no—
217#239—632no—
218D3×D4—696no—
219#235—648no—
220C2×D12—312no—
221C2×D12—324no—
222C2×D12—648no—
223C2×C4×D3—324no—
224C2×Dic6—34no—
225#217—38no—
226#217—324no—
227#217—324no—
228(C2×D3)⋊C4—324no—
229(C2×D3)⋊C4—348no—
230(C2×D3)⋊C4—324no—
231C4⋊Dic3—312no—
232C4⋊Dic3—34no—
233C4⋊Dic3—324no—
234Dic3⋊C4—38no—
235Dic3⋊C4—324no—
236C4×Dic3—324no—
237C22×C32—21no—
238C22×C32—23no—
239C22×C32—23no—
240C22×C32—21no—
241C22×C32—23no—
242C22×C32—23no—
243C22×C32—23no—
244C22×C32—412no—
245C22×C32C22×D416yes—
246C22×C32H(5,𝔽2)12yescomplemented
247C22×C32C2511yescharacteristic
248C2×C3⋊D3—26no—
249C2×C3⋊D3—26no—
250C2×C3⋊D3—26no—
251C2×C3⋊D3—812no—
252C32×C4—21no—
253C32×C4—23no—
254C32×C4C22×D412yes—
255C32×C4C22×D416yes—
256C3⋊Dic3—22no—
257C3⋊Dic3—26no—
258C3⋊Dic3—22no—
259H(5,𝔽2)—188no—
260C2×#41—28no—
261C2×#41—92no—
262C2×#41—93no—
263C2×#41C2×C3⋊D311yescharacteristic
264C2×#41C2×C3⋊D313yescomplemented
265C2×#41C2×C3⋊D313yescomplemented
266C2×#41C2×C3⋊D311yescharacteristic, complemented
267C22×Q8C2×C3⋊D311yescharacteristic, complemented
268C22×D4—96no—
269C22×D4—96no—
270C22×D4—93no—
271C22×D4C2×C3⋊D313yescomplemented
272C23×C4C2×C3⋊D313yescomplemented
273#127—96no—
274#124—96no—
275#123—96no—
276C22⋊Q8—96no—
277C22⋊Q8—91no—
278C22⋊Q8—93no—
279#121—912no—
280#121—96no—
281#121—93no—
282#121—96no—
283#121—96no—
284#121—96no—
285#121—93no—
286#120—96no—
287#120—912no—
288C4×Q8—92no—
289C4×D4—96no—
290C4×D4—96no—
291C4×D4—96no—
292#117—93no—
293C2×C4⋊C4—93no—
294C2×C22⋊C4—96no—
295C23×C3—224no—
296C23×C3—224no—
297C23×C3—412no—
298C23×C3C23×D3112yes—
299C23×C3C23×D314yes—
300C23×C3C23×D3112yes—
301C22×D3—324no—
302C22×D3—648no—
303C22×D3—1224no—
304C22×D3—1224no—
305C3×Q8—224no—
306C3×Q8C2×#67116yescomplemented
307C3×D4—248no—
308C3×D4—224no—
309C3×D4C2×#67148yescomplemented
310C2×C3×C4—224no—
311C2×C3×C4—28no—
312C2×C3×C4—224no—
313C2×C3×C4—224no—
314C2×C3×C4—224no—
315C2×C3×C4C2×#67148yescomplemented
316C2×C3×C4C23×D314yes—
317C2×C3×C4C23×D3112yes—
318C2×C3×C4C23×D3112yes—
319C2×C3×C4C23×D314yes—
320#67—648no—
321#67—648no—
322#67—1224no—
323#67—1248no—
324C2×Dic3—38no—
325C2×Dic3—324no—
326C2×Dic3—38no—
327C2×Dic3—1224no—
328D12—624no—
329D12—648no—
330C4×D3—648no—
331Dic6—68no—
332C2×C32—26no—
333C2×C32—22no—
334C2×C32—43no—
335C2×C32C23×D411yescharacteristic
336C2×C32C2×H(5,𝔽2)11yescharacteristic
337C2×C32#58511yescharacteristic
338C3⋊D3—46no—
339C24—96no—
340C24C22×C3⋊D313yes—
341#41—24no—
342#41—212no—
343#41—212no—
344#41—188no—
345#41—1812no—
346#41#63818yescomplemented
347C2×Q8—24no—
348C2×Q8—91no—
349C2×Q8C22×C3⋊D313yes—
350C2×Q8C22×C3⋊D311yescharacteristic
351C2×D4—212no—
352C2×D4—96no—
353C2×D4—96no—
354C2×D4—96no—
355C2×D4—912no—
356C2×D4—93no—
357C2×D4—96no—
358C2×D4—1824no—
359C2×D4—1812no—
360C2×D4—3612no—
361C2×D4C22×C3⋊D316yes—
362C2×D4C22×C3⋊D313yes—
363C2×D4C22×C3⋊D313yes—
364C22×C4—212no—
365C22×C4—96no—
366C22×C4—96no—
367C22×C4C22×C3⋊D313yes—
368C22×C4C22×C3⋊D313yes—
369C22×C4C22×C3⋊D313yes—
370C22×C4C22×C3⋊D311yescharacteristic
371C22×C4C22×C3⋊D313yes—
372C22×C4C22×C3⋊D313yes—
373C4⋊C4—93no—
374C4⋊C4—96no—
375C4⋊C4—91no—
376C4⋊C4—92no—
377C4⋊C4—96no—
378C22⋊C4—96no—
379C22⋊C4—92no—
380C22⋊C4—96no—
381C22⋊C4—96no—
382C22⋊C4—912no—
383C22⋊C4—96no—
384C42—96no—
385C22×C3—212no—
386C22×C3—212no—
387C22×C3—24no—
388C22×C3—24no—
389C22×C3—212no—
390C22×C3—212no—
391C22×C3—212no—
392C22×C3—448no—
393C22×C3#100918yescomplemented
394C22×C3C22×#67124yes—
395C22×C3C24×D314yes—
396C2×D3—624no—
397C2×D3—624no—
398C2×D3—624no—
399C2×D3—2448no—
400C3×C4—24no—
401C3×C4—212no—
402C3×C4C22×#6718yes—
403C3×C4C22×#67124yes—
404Dic3—624no—
405Dic3—68no—
406Dic3—68no—
407C32#348811yescharacteristic, complemented, Sylow
408C23—26no—
409C23—26no—
410C23—43no—
411C23—96no—
412C23—1812no—
413C23—366no—
414C23—366no—
415C23C23×C3⋊D313yes—
416C23C23×C3⋊D311yescharacteristic
417C23C23×C3⋊D313yes—
418Q8—26no—
419Q8—182no—
420Q8C2×#63814yescomplemented
421D4—212no—
422D4—26no—
423D4—1812no—
424D4—1812no—
425D4—186no—
426D4—1812no—
427D4—3612no—
428D4—366no—
429D4C2×#638112yescomplemented
430C2×C4—26no—
431C2×C4—22no—
432C2×C4—26no—
433C2×C4—26no—
434C2×C4—26no—
435C2×C4—92no—
436C2×C4—96no—
437C2×C4—92no—
438C2×C4—1812no—
439C2×C4—366no—
440C2×C4C2×#638112yescomplemented
441C2×C4C23×C3⋊D311yescharacteristic
442C2×C4C23×C3⋊D313yes—
443C2×C4C23×C3⋊D313yes—
444C2×C4C23×C3⋊D311yescharacteristic
445C2×C3—224no—
446C2×C3—28no—
447C2×C3—412no—
448C2×C3C2×#100914yes—
449C2×C3#596414yes—
450C2×C3C23×#6714yes—
451D3—1224no—
452C22—23no—
453C22—21no—
454C22—23no—
455C22—21no—
456C22—23no—
457C22—23no—
458C22—23no—
459C22—412no—
460C22—186no—
461C22—186no—
462C22—186no—
463C22—7212no—
464C22#6436612yescomplemented
465C22C22×#63816yes—
466C22C24×C3⋊D311yescharacteristic
467C4—21no—
468C4—23no—
469C4—186no—
470C4—182no—
471C4—182no—
472C4C22×#63812yes—
473C4C22×#63816yes—
474C3#8727514yescomplemented
475C2—22no—
476C2—26no—
477C2—43no—
478C2—366no—
479C2C2×#6436611yescharacteristic
480C2#1059675611yescharacteristic
481C2C23×#63811yescharacteristic
subgroupG/H
center, FrattiniC22C24×C3⋊D3
commutator, socle, cosocleC22×C32C25
FittingC22×C32×#41C2
hypercenterC22×#41C3⋊D3
hypocenterC32#3488
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×#41

Complements

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

NH ≅ G/Ntype
1. C22×C32×#41C2semidirect
2. C2×#64366C2semidirect
3. #10596305C2semidirect
4. #10596304C2semidirect
5. #10596301C2semidirect
6. #10596300C2semidirect
7. #10596298C2semidirect
8. #10596288C2semidirect
9. D4×#638C2semidirect
10. #10596262C2semidirect
11. C2×#64139C2semidirect
15. C2×C32×#41C22semidirect
16. C2×C32×#41C22semidirect
17. C2×C32×#41C22semidirect
18. C22×C32×Q8C22semidirect
19. C22×C32×D4C22semidirect
20. C23×C32×C4C22semidirect
21. C22×#638C22semidirect
23. C2×#3832C22semidirect
24. C2×D4×C3⋊D3C22semidirect
25. C2×#3828C22semidirect
26. C22×C4⋊(C3⋊D3)C22semidirect
27. #64162C22semidirect
28. #64156C22semidirect
29. #64155C22semidirect
30. Q8×C3⋊Dic3C22semidirect
31. #64149C22semidirect
32. #64148C22semidirect
33. #64147C22semidirect
34. #64146C22semidirect
35. D4×C3⋊Dic3C22semidirect
36. #64139C22semidirect
37. #64139C22semidirect
38. #64139C22semidirect
39. #64139C22semidirect
40. #64139C22semidirect
41. #64138C22semidirect
42. C4×#638C22semidirect
43. C4×#638C22semidirect
44. C2×#3752C22semidirect
45. #64135C22semidirect
46. C2×C4⋊(C3⋊Dic3)C22semidirect
47. #64133C22semidirect
48. #64133C22semidirect
49. C22×C3×#41D3semidirect
65. C32×#41D4semidirect
109. C2×C3×#41C2×D3semidirect
110. C2×C3×#41C2×D3semidirect
111. C2×C3×#41C2×D3semidirect
112. C22×C3×Q8C2×D3semidirect
113. C22×C3×D4C2×D3semidirect
114. C23×C3×C4C2×D3semidirect
154. C32×Q8C2×D4semidirect
157. C32×D4C2×D4semidirect
163. C2×C32×C4C2×D4semidirect
181. C22×#41C3⋊D3semidirect
196. C3×#41#67semidirect
246. C22×C32H(5,𝔽2)semidirect
264. C2×#41C2×C3⋊D3semidirect
265. C2×#41C2×C3⋊D3semidirect
266. C2×#41C2×C3⋊D3semidirect
267. C22×Q8C2×C3⋊D3semidirect
271. C22×D4C2×C3⋊D3semidirect
272. C23×C4C2×C3⋊D3semidirect
306. C3×Q8C2×#67semidirect
309. C3×D4C2×#67semidirect
315. C2×C3×C4C2×#67semidirect
346. #41#638semidirect
393. C22×C3#1009semidirect
407. C32#3488semidirect
420. Q8C2×#638semidirect
429. D4C2×#638semidirect
440. C2×C4C2×#638semidirect
464. C22#64366semidirect
474. C3#87275semidirect

Maximal subgroups

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

subgroupindex
C22×C32×#412
C2×#643662
#105963052
#105963042
#105963012
#105963002
#105962982
#105962882
D4×#6382
#105962622
C2×#641392
#872753