Finite Group Databasebeta

← #11900164

Subgroups — #11900164

Order 1152.

4980 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#64366C211yesmaximal, characteristic
2C32×#520C211yesmaximal, characteristic, complemented
3#41×C3⋊Dic3C211yesmaximal, characteristic, complemented
4#10596045C212yesmaximal, complemented
5C4×#3832C212yesmaximal, complemented
6#10596016C216yesmaximal, complemented
7C4×D4×C3⋊D3C216yesmaximal, complemented
8#10595993C213yesmaximal, complemented
9#10595992C213yesmaximal, complemented
10#10595975C213yesmaximal, complemented
11C2×#64102C213yesmaximal, complemented
12#85886—34nomaximal
13C2×C32×#41C2211yescharacteristic
14#64366C4116yescomplemented
15C2×#3832C2212yes—
16C2×D4×C3⋊D3C2216yes—
17C2×#3828C2213yes—
18C22×C4⋊(C3⋊D3)C2213yes—
19C22×C4×C3⋊D3C2216yescomplemented
20C32×C4×Q8C2212yescomplemented
21C32×C4×D4C2216yescomplemented
22C32×#117C2213yescomplemented
23C2×C32×C4⋊C4C2213yescomplemented
24Q8×C3⋊Dic3C2211yescharacteristic, complemented
25Q8×C3⋊Dic3C2211yescharacteristic, complemented
26D4×C3⋊Dic3C2213yescomplemented
27D4×C3⋊Dic3C2213yescomplemented
28C4×#638C22112yescomplemented
29C2×#3752C2216yescomplemented
30#64135C2213yescomplemented
31C2×C4×C3⋊Dic3C2213yescomplemented
32#64102C2213yescomplemented
33#64102C2216yescomplemented
34#64102C2216yescomplemented
35#64102C2216yescomplemented
36#64102C2213yescomplemented
37#64101C2216yescomplemented
38C4⋊C4×C3⋊D3C2216yescomplemented
39#64096C2213yescomplemented
40#64096C2211yescharacteristic, complemented
41#64090C2216yescomplemented
42#64090C2216yescomplemented
43C22⋊C4×C3⋊D3C22112yescomplemented
44C4×C4⋊(C3⋊D3)C2216yescomplemented
45C4×C4⋊(C3⋊D3)C2216yescomplemented
46#64081C2216yescomplemented
47C42×C3⋊D3C2216yescomplemented
48C2×#1009—34no—
49C3×#520D314yescomplemented
50Dic3×#41—34no—
51#5574—38no—
52C4×#239—38no—
53#5545—324no—
54C4×D3×D4—324no—
55#5522—312no—
56#5521—312no—
57#5504—312no—
58C2×#893—312no—
59C23×C3⋊D3C2316yes—
60C32×#41C2×C418yescomplemented
61C2×C32×Q8C2311yescharacteristic
62C2×C32×D4C2313yes—
63C22×C32×C4C2316yes—
64C22×C32×C4C2313yes—
65C2×#638C2316yes—
66C22×C3⋊Dic3C2313yes—
67C22×C3⋊Dic3C2313yes—
68#3832C2×C4116yescomplemented
69D4×C3⋊D3C2×C4148yescomplemented
70#3828C2×C4124yescomplemented
71C2×C4⋊(C3⋊D3)C2×C4124yescomplemented
72C2×C4⋊(C3⋊D3)C2316yes—
73C2×C4⋊(C3⋊D3)C2313yes—
74C2×C4×C3⋊D3—412no—
75C2×C4×C3⋊D3C2316yes—
76C2×C4×C3⋊D3C2316yes—
77C2×C4×C3⋊D3C23112yes—
78C2×C4×C3⋊D3C2316yes—
79C2×#634C2311yescharacteristic
80C32×C4⋊C4C2313yes—
81C32×C4⋊C4C2311yescharacteristic
82C32×C4⋊C4C2316yes—
83C32×C22⋊C4C2316yes—
84C32×C42C2316yes—
85#3757C2316yes—
86#3752C2316yes—
87#3752C2316yes—
88#3752C23112yes—
89C4⋊(C3⋊Dic3)C2313yes—
90C4⋊(C3⋊Dic3)C2313yes—
91#3750C2312yes—
92#3750C2316yes—
93C4×C3⋊Dic3C2312yes—
94C4×C3⋊Dic3C2313yes—
95C4×C3⋊Dic3C2316yes—
96C4×C3⋊Dic3C2313yes—
97C4×C3⋊Dic3C2311yescharacteristic
98C4×C3⋊Dic3C2313yes—
99C4×H(5,𝔽2)—91noSylow
100C2×C3×#41C2×D314yes—
101#1009—364no—
102C2×#239—38no—
103C2×D3×D4—324no—
104C2×#235—312no—
105C22×D12—312no—
106C22×C4×D3—324no—
107C3×C4×Q8C2×D318yescomplemented
108C3×C4×D4C2×D3124yescomplemented
109C3×#117C2×D3112yescomplemented
110C2×C3×C4⋊C4C2×D3112yescomplemented
111Q8×Dic3—34no—
112Q8×Dic3—34no—
113D4×Dic3—312no—
114D4×Dic3—312no—
115C4×#67—348no—
116C2×(C2×D3)⋊C4—324no—
117#926—312no—
118C2×C4×Dic3—312no—
119#893—312no—
120#893—324no—
121#893—324no—
122#893—324no—
123#893—312no—
124#892—324no—
125D3×C4⋊C4—324no—
126#887—312no—
127#887—34no—
128#67⋊C4—324no—
129#67⋊C4—324no—
130D3×C22⋊C4—348no—
131C4×D12—324no—
132C4×D12—324no—
133#872—324no—
134C42×D3—324no—
135C23×C32C2413yes—
136C22×C3⋊D3—46no—
137C22×C3⋊D3—46no—
138C22×C3⋊D3C22×C4124yescomplemented
139C22×C3⋊D3C2416yes—
140C32×Q8C22×C414yescomplemented
141C32×D4C22×C4112yescomplemented
142C2×C32×C4—46no—
143C2×C32×C4C22×C4112yescomplemented
144C2×C32×C4C2412yes—
145C2×C32×C4C2416yes—
146C2×C32×C4C2411yescharacteristic
147C2×C32×C4C2413yes—
148#638C22×C4124yescomplemented
149C2×C3⋊Dic3—26no—
150C2×C3⋊Dic3—43no—
151C2×C3⋊Dic3C2411yescharacteristic
152C2×C3⋊Dic3C2413yes—
153C2×C3⋊Dic3C2413yes—
154C2×C3⋊Dic3C2412yes—
155C2×C3⋊Dic3C2411yescharacteristic
156C4⋊(C3⋊D3)C22×C4112yescomplemented
157C4⋊(C3⋊D3)C22×C4124yescomplemented
158C4×C3⋊D3—212no—
159C4×C3⋊D3—412no—
160C4×C3⋊D3—46no—
161C4×C3⋊D3C22×C4124yescomplemented
162#634C22×C414yescomplemented
163C2×H(5,𝔽2)—91no—
164#520—92no—
165#520—93no—
166#520C3⋊D311yescharacteristic, complemented
167#518—96no—
168#518—93no—
169C4×#41—91no—
170C4×#41—92no—
171C4×#41—93no—
172C2×C4×D4—96no—
173C2×C4×D4—93no—
174C23×D3—324no—
175C3×#41C4×D3132yescomplemented
176C2×C3×Q8C22×D314yes—
177C2×C3×D4C22×D3112yes—
178C22×C3×C4C22×D3124yes—
179C22×C3×C4C22×D3112yes—
180C2×#67—324no—
181C22×Dic3—312no—
182C22×Dic3—312no—
183#239—364no—
184D3×D4—3192no—
185#235—396no—
186C2×D12—396no—
187C2×D12—324no—
188C2×D12—312no—
189C2×C4×D3—324no—
190C2×C4×D3—324no—
191C2×C4×D3—324no—
192C2×C4×D3—348no—
193C2×C4×D3—1248no—
194C2×Dic6—34no—
195C3×C4⋊C4C22×D3112yes—
196C3×C4⋊C4C22×D314yes—
197C3×C4⋊C4C22×D3124yes—
198C3×C22⋊C4C22×D3124yes—
199C3×C42C22×D3124yes—
200#217—324no—
201(C2×D3)⋊C4—324no—
202(C2×D3)⋊C4—324no—
203(C2×D3)⋊C4—348no—
204C4⋊Dic3—312no—
205C4⋊Dic3—312no—
206Dic3⋊C4—38no—
207Dic3⋊C4—324no—
208C4×Dic3—38no—
209C4×Dic3—312no—
210C4×Dic3—324no—
211C4×Dic3—312no—
212C4×Dic3—34no—
213C4×Dic3—312no—
214C22×C32—23no—
215C22×C32—23no—
216C22×C32C23×C416yes—
217C22×C32C2511yescharacteristic
218C2×C3⋊D3—26no—
219C2×C3⋊D3—26no—
220C2×C3⋊D3—424no—
221C2×C3⋊D3C23×C4112yes—
222C32×C4—26no—
223C32×C4—22no—
224C32×C4C23×C412yes—
225C32×C4C23×C416yes—
226C3⋊Dic3—23no—
227C3⋊Dic3—23no—
228C3⋊Dic3—21no—
229C3⋊Dic3C23×C414yes—
230C3⋊Dic3H(5,𝔽2)12yescomplemented
231H(5,𝔽2)—916no—
232C2×#41—92no—
233C2×#41—93no—
234C2×#41C2×C3⋊D311yescharacteristic
235C22×D4—96no—
236C22×D4—93no—
237C23×C4—96no—
238C4×Q8—91no—
239C4×Q8—91no—
240C4×Q8—91no—
241C4×Q8—93no—
242C4×Q8C2×C3⋊D312yescomplemented
243C4×D4—93no—
244C4×D4—93no—
245C4×D4—912no—
246C4×D4—96no—
247C4×D4—93no—
248C4×D4—96no—
249C4×D4—96no—
250C4×D4—93no—
251C4×D4—96no—
252C4×D4—96no—
253C4×D4—96no—
254C4×D4—96no—
255C4×D4C2×C3⋊D316yescomplemented
256#117—93no—
257#117—96no—
258#117—96no—
259#117C2×C3⋊D313yescomplemented
260C2×C4⋊C4—96no—
261C2×C4⋊C4C2×C3⋊D313yescomplemented
262C2×C22⋊C4—96no—
263C2×C22⋊C4—912no—
264C2×C42—93no—
265C2×C42—96no—
266C23×C3C23×D3112yes—
267C22×D3—324no—
268C22×D3—396no—
269C22×D3—1224no—
270C22×D3—1224no—
271C3×Q8C2×C4×D3116yescomplemented
272C3×D4C2×C4×D3148yescomplemented
273C2×C3×C4—424no—
274C2×C3×C4C2×C4×D3148yescomplemented
275C2×C3×C4C23×D318yes—
276C2×C3×C4C23×D3124yes—
277C2×C3×C4C23×D314yes—
278C2×C3×C4C23×D3112yes—
279#67—396no—
280C2×Dic3—34no—
281C2×Dic3—38no—
282C2×Dic3—34no—
283C2×Dic3—312no—
284C2×Dic3—312no—
285C2×Dic3—624no—
286C2×Dic3—1212no—
287D12—348no—
288D12—396no—
289C4×D3—396no—
290C4×D3—648no—
291C4×D3—1224no—
292C4×D3—1248no—
293Dic6—316no—
294C2×C32—26no—
295C2×C32C24×C411yescharacteristic
296C2×C32C2×H(5,𝔽2)11yescharacteristic
297C2×C32#58511yescharacteristic
298C3⋊D3—212no—
299C24—96no—
300#41—916no—
301#41—924no—
302#41C4×C3⋊D318yescomplemented
303C2×Q8—91no—
304C2×Q8C22×C3⋊D311yescharacteristic
305C2×D4—96no—
306C2×D4—948no—
307C2×D4—93no—
308C2×D4—96no—
309C2×D4—924no—
310C2×D4C22×C3⋊D313yes—
311C22×C4—96no—
312C22×C4—93no—
313C22×C4—93no—
314C22×C4—96no—
315C22×C4—96no—
316C22×C4—912no—
317C22×C4—3612no—
318C22×C4C22×C3⋊D316yes—
319C22×C4C22×C3⋊D313yes—
320C4⋊C4—96no—
321C4⋊C4—93no—
322C4⋊C4—92no—
323C4⋊C4—93no—
324C4⋊C4C22×C3⋊D316yes—
325C4⋊C4C22×C3⋊D313yes—
326C4⋊C4C22×C3⋊D311yescharacteristic
327C22⋊C4—96no—
328C22⋊C4—96no—
329C22⋊C4—912no—
330C22⋊C4—96no—
331C22⋊C4C22×C3⋊D316yes—
332C42—92no—
333C42—91no—
334C42—93no—
335C42—96no—
336C42—93no—
337C42—93no—
338C42C22×C3⋊D316yes—
339C22×C3—212no—
340C22×C3—212no—
341C22×C3C22×C4×D3124yes—
342C22×C3C24×D314yes—
343C2×D3—348no—
344C2×D3—624no—
345C2×D3—624no—
346C2×D3—1296no—
347C3×C4—224no—
348C3×C4—28no—
349C3×C4C22×C4×D318yes—
350C3×C4C22×C4×D3124yes—
351Dic3—316no—
352Dic3—38no—
353Dic3—612no—
354Dic3—612no—
355Dic3—64no—
356C32C4×H(5,𝔽2)11yescharacteristic, complemented, Sylow
357C23—96no—
358C23—924no—
359C23—366no—
360C23—366no—
361C23C23×C3⋊D313yes—
362Q8—94no—
363Q8C2×C4×C3⋊D314yescomplemented
364D4—924no—
365D4—924no—
366D4—912no—
367D4C2×C4×C3⋊D3112yescomplemented
368C2×C4—46no—
369C2×C4—91no—
370C2×C4—92no—
371C2×C4—924no—
372C2×C4—93no—
373C2×C4—91no—
374C2×C4—93no—
375C2×C4—1812no—
376C2×C4—186no—
377C2×C4—3612no—
378C2×C4—366no—
379C2×C4—363no—
380C2×C4C2×C4×C3⋊D3112yescomplemented
381C2×C4C23×C3⋊D312yes—
382C2×C4C23×C3⋊D316yes—
383C2×C4C23×C3⋊D311yescharacteristic
384C2×C4C23×C3⋊D313yes—
385C2×C3—224no—
386C2×C3C23×C4×D314yes—
387C2×C3D3×H(5,𝔽2)14yes—
388C2×C3#596814yes—
389D3—648no—
390C22—23no—
391C22—23no—
392C22—912no—
393C22—186no—
394C22—186no—
395C22—3624no—
396C22C22×C4×C3⋊D316yes—
397C22C24×C3⋊D311yescharacteristic
398C4—26no—
399C4—22no—
400C4—94no—
401C4—92no—
402C4—183no—
403C4—183no—
404C4—181no—
405C4C22×C4×C3⋊D316yes—
406C4C22×C4×C3⋊D312yes—
407C3#8588614yescomplemented
408C2—26no—
409C2—1812no—
410C2C23×C4×C3⋊D311yescharacteristic
411C2C3⋊D3×H(5,𝔽2)11yescharacteristic
412C2#1059676011yescharacteristic
subgroupG/H
center, FrattiniC22C24×C3⋊D3
commutatorC2×C32C24×C4
FittingC32×#520C2
socle, cosocleC22×C32C25
prime radicalC2×#64366C2
hypercenter#520C3⋊D3
hypocenterC32C4×H(5,𝔽2)
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ #520

Complements

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

NH ≅ G/Ntype
2. C32×#520C2semidirect
3. #41×C3⋊Dic3C2semidirect
4. #10596045C2semidirect
5. C4×#3832C2semidirect
6. #10596016C2semidirect
7. C4×D4×C3⋊D3C2semidirect
8. #10595993C2semidirect
9. #10595992C2semidirect
10. #10595975C2semidirect
11. C2×#64102C2semidirect
14. #64366C4semidirect
19. C22×C4×C3⋊D3C22semidirect
20. C32×C4×Q8C22semidirect
21. C32×C4×D4C22semidirect
22. C32×#117C22semidirect
23. C2×C32×C4⋊C4C22semidirect
24. Q8×C3⋊Dic3C22semidirect
25. Q8×C3⋊Dic3C22semidirect
26. D4×C3⋊Dic3C22semidirect
27. D4×C3⋊Dic3C22semidirect
28. C4×#638C22semidirect
29. C2×#3752C22semidirect
30. #64135C22semidirect
31. C2×C4×C3⋊Dic3C22semidirect
32. #64102C22semidirect
33. #64102C22semidirect
34. #64102C22semidirect
35. #64102C22semidirect
36. #64102C22semidirect
37. #64101C22semidirect
38. C4⋊C4×C3⋊D3C22semidirect
39. #64096C22semidirect
40. #64096C22semidirect
41. #64090C22semidirect
42. #64090C22semidirect
43. C22⋊C4×C3⋊D3C22semidirect
44. C4×C4⋊(C3⋊D3)C22semidirect
45. C4×C4⋊(C3⋊D3)C22semidirect
46. #64081C22semidirect
47. C42×C3⋊D3C22semidirect
49. C3×#520D3semidirect
60. C32×#41C2×C4semidirect
68. #3832C2×C4semidirect
69. D4×C3⋊D3C2×C4semidirect
70. #3828C2×C4semidirect
71. C2×C4⋊(C3⋊D3)C2×C4semidirect
107. C3×C4×Q8C2×D3semidirect
108. C3×C4×D4C2×D3semidirect
109. C3×#117C2×D3semidirect
110. C2×C3×C4⋊C4C2×D3semidirect
138. C22×C3⋊D3C22×C4semidirect
140. C32×Q8C22×C4semidirect
141. C32×D4C22×C4semidirect
143. C2×C32×C4C22×C4semidirect
148. #638C22×C4semidirect
156. C4⋊(C3⋊D3)C22×C4semidirect
157. C4⋊(C3⋊D3)C22×C4semidirect
161. C4×C3⋊D3C22×C4semidirect
162. #634C22×C4semidirect
166. #520C3⋊D3semidirect
175. C3×#41C4×D3semidirect
230. C3⋊Dic3H(5,𝔽2)semidirect
242. C4×Q8C2×C3⋊D3semidirect
255. C4×D4C2×C3⋊D3semidirect
259. #117C2×C3⋊D3semidirect
261. C2×C4⋊C4C2×C3⋊D3semidirect
271. C3×Q8C2×C4×D3semidirect
272. C3×D4C2×C4×D3semidirect
274. C2×C3×C4C2×C4×D3semidirect
302. #41C4×C3⋊D3semidirect
356. C32C4×H(5,𝔽2)semidirect
363. Q8C2×C4×C3⋊D3semidirect
367. D4C2×C4×C3⋊D3semidirect
380. C2×C4C2×C4×C3⋊D3semidirect
407. C3#85886semidirect

Maximal subgroups

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

subgroupindex
C2×#643662
C32×#5202
#41×C3⋊Dic32
#105960452
C4×#38322
#105960162
C4×D4×C3⋊D32
#105959932
#105959922
#105959752
C2×#641022
#858863