Finite Group Databasebeta

← #11874455

Subgroups — #11874455

Order 1152.

1968 conjugacy classes of subgroups; only the 277 normal subgroups are shown.

normal subgroups only

This group has 1968 conjugacy classes of subgroups — too many to draw. Only its 277 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#64123C214yesmaximal, complemented
2#10596196C211yesmaximal, characteristic, complemented
3C3⋊D3×#127C211yesmaximal, characteristic, complemented
4C32×#492C211yesmaximal, characteristic, complemented
5#10592875C214yesmaximal, complemented
6#10592784C211yesmaximal, characteristic, complemented
7#10592772C211yesmaximal, characteristic, complemented
8#10592394C211yesmaximal, characteristic, complemented
9C4×#3743C211yesmaximal, characteristic, complemented
10#81761—34nomaximal
11C2×#3832C2212yes—
12C2×D4×C3⋊D3—22no—
13C2×D4×C3⋊D3C2212yes—
14C2×C32×SD16C2214yescomplemented
15C32×C4⋊Q8C2211yescharacteristic, complemented
16C32×#127C2211yescharacteristic
17#64156C2212yescomplemented
18C2×#3755C2214yescomplemented
19#64149—22no—
20#64149C2212yes—
21C2×#3753C2214yescomplemented
22#64123—216no—
23C2×#3745C2214yescomplemented
24C2×#3743C2212yes—
25#64104—22no—
26#64102—22no—
27#64083C2211yescharacteristic
28#64083C2211yescharacteristic, complemented
29C42×C3⋊D3C2211yescharacteristic
30C32×C4×C8C2211yescharacteristic, complemented
31#63642C2212yescomplemented
32C4×C3⋊(C3⋊C8)C2211yescharacteristic, complemented
33C2×#914—316no—
34#5725—34no—
35D3×#127—34no—
36C3×#492D314yescomplemented
37#5176—316no—
38#5085—34no—
39#5073—34no—
40#4695—34no—
41C4×ΓL(1,𝔽25)—34no—
42C23×C3⋊D3—22no—
43C2×C32×Q8C2312yes—
44C2×C32×D4—22no—
45C2×C32×D4C2312yes—
46C2×#638—24no—
47C2×#638—22no—
48#3832—28no—
49D4×C3⋊D3—28no—
50D4×C3⋊D3—48no—
51C2×C4⋊(C3⋊D3)—22no—
52C2×C4⋊(C3⋊D3)—22no—
53C2×C4⋊(C3⋊D3)—22no—
54C2×C4⋊(C3⋊D3)C2312yes—
55C2×C4⋊(C3⋊D3)C2312yes—
56C2×C4×C3⋊D3—22no—
57C2×C4×C3⋊D3D412yes—
58C2×C4×C3⋊D3C2311yescharacteristic
59C32×SD16—28no—
60C2×C32×C8C2312yes—
61C32×C4⋊C4—22no—
62C32×C42C2311yescharacteristic
63#3755—28no—
64#3753—28no—
65#3752—24no—
66C4×C3⋊Dic3D412yes—
67C4×C3⋊Dic3C2311yescharacteristic
68C2×C3⋊(C3⋊C8)C2312yes—
69#3745—28no—
70#3743D418yescomplemented
71#3151—91noSylow
72C2×#239—38no—
73C2×D3×D4—38no—
74C2×D3×D4—68no—
75C2×C3×SD16C2×D3116yescomplemented
76C3×C4⋊Q8C2×D314yescomplemented
77C3×#127C2×D314yes—
78#947—38no—
79C2×#215—316no—
80#940—38no—
81#940—68no—
82C2×#213—316no—
83#914—664no—
84C2×D24—316no—
85C2×ΓL(1,𝔽25)—38no—
86#895—68no—
87#893—68no—
88#874—34no—
89#874—34no—
90C42×D3—34no—
91C3×C4×C8C2×D314yescomplemented
92#815—38no—
93C4×C3⋊C8—34no—
94C23×C32—22no—
95C22×C3⋊D3—22no—
96C22×C3⋊D3—22no—
97C22×C3⋊D3—48no—
98C22×C3⋊D3—44no—
99C22×C3⋊D3C2×D411yescharacteristic
100C32×Q8—24no—
101C32×D4—24no—
102C32×D4—44no—
103C2×C32×C4—22no—
104C2×C32×C4C2×D412yes—
105C2×C32×C4C2411yescharacteristic
106#638—48no—
107#638—44no—
108C2×C3⋊Dic3C2×D412yes—
109C2×C3⋊Dic3C2×D411yescharacteristic
110C4⋊(C3⋊D3)—24no—
111C4⋊(C3⋊D3)—24no—
112C4⋊(C3⋊D3)—44no—
113C4⋊(C3⋊D3)—44no—
114C4⋊(C3⋊D3)—44no—
115C4×C3⋊D3—24no—
116C4×C3⋊D3—44no—
117C4×C3⋊D3C2×D414yes—
118C32×C8C2×D414yescomplemented
119C3⋊(C3⋊C8)C2×D414yescomplemented
120C2×AGL(1,ℤ/8)—94no—
121#532—91no—
122C2×#127—91no—
123#496—94no—
124#493—91no—
125#493—91no—
126#492—91no—
127#492C3⋊D311yescharacteristic, complemented
128C4×M16—91no—
129C23×D3—68no—
130C2×C3×Q8C22×D318yes—
131C2×C3×D4—28no—
132C2×C3×D4C22×D318yes—
133C2×#67—616no—
134C2×#67—68no—
135#239—632no—
136D3×D4—632no—
137D3×D4—1232no—
138C2×D12—38no—
139C2×D12—38no—
140C2×D12—68no—
141C2×D12—68no—
142C2×D12—68no—
143C2×C4×D3—34no—
144C2×C4×D3—38no—
145C2×C4×D3—68no—
146C3×SD16—232no—
147C2×C3×C8C22×D318yes—
148C3×C4⋊C4—28no—
149C3×C42C22×D314yes—
150#215—632no—
151#213—632no—
152(C2×D3)⋊C4—616no—
153C4×Dic3—38no—
154C4×Dic3—34no—
155C2×C3⋊C8—38no—
156D24—632no—
157ΓL(1,𝔽25)—332no—
158C22×C32—42no—
159C22×C32—44no—
160C22×C32C22×D411yescharacteristic
161C2×C3⋊D3—22no—
162C2×C3⋊D3—42no—
163C2×C3⋊D3—42no—
164C2×C3⋊D3—44no—
165C2×C3⋊D3—44no—
166C2×C3⋊D3—88no—
167C2×C3⋊D3#12712yes—
168C32×C4—42no—
169C32×C4AGL(1,ℤ/8)14yescomplemented
170C32×C4C22×D412yes—
171C3⋊Dic3—22no—
172C3⋊Dic3#12712yes—
173C2×#41—92no—
174C22×D4—92no—
175C22×D4—182no—
176AGL(1,ℤ/8)—1816no—
177C2×SD16—94no—
178C2×SD16C2×C3⋊D314yescomplemented
179C2×D8—94no—
180C2×D8—94no—
181C2×M16—92no—
182C4⋊Q8C2×C3⋊D311yescharacteristic, complemented
183#127—92no—
184#127—91no—
185#127—91no—
186#127—182no—
187#127C2×C3⋊D311yescharacteristic
188#124—92no—
189#121—182no—
190C4×D4—182no—
191C2×C42—91no—
192#97—92no—
193C4×C8—91no—
194C4×C8C2×C3⋊D311yescharacteristic, complemented
195C23×C3—28no—
196C22×D3—34no—
197C22×D3—68no—
198C22×D3—68no—
199C22×D3—1216no—
200C22×D3—1232no—
201C3×Q8—216no—
202C3×D4—216no—
203C3×D4—416no—
204C2×C3×C4—28no—
205C2×C3×C4D3×D418yes—
206C2×C3×C4C23×D314yes—
207#67—1216no—
208#67—1232no—
209C2×Dic3—34no—
210C2×Dic3—38no—
211D12—616no—
212D12—616no—
213D12—1216no—
214D12—1216no—
215D12—1216no—
216C4×D3—316no—
217C4×D3—616no—
218C4×D3—1216no—
219C3×C8D3×D4116yescomplemented
220C3⋊C8—316no—
221C2×C32—82no—
222C2×C32C2×#12711yescharacteristic
223C2×C32C2×AGL(1,ℤ/8)12yes—
224C3⋊D3—22no—
225C3⋊D3—82no—
226C3⋊D3—82no—
227C24—182no—
228#41—188no—
229C2×Q8C22×C3⋊D312yes—
230C2×D4—22no—
231C2×D4—92no—
232C2×D4—92no—
233C2×D4—182no—
234C2×D4—188no—
235C2×D4—184no—
236C2×D4—182no—
237C2×D4—182no—
238C2×D4—182no—
239C2×D4—368no—
240C2×D4C22×C3⋊D312yes—
241C22×C4—91no—
242C22×C4—92no—
243C22×C4—182no—
244SD16—28no—
245SD16—188no—
246D8—188no—
247D8—188no—
248M16—98no—
249C2×C8—92no—
250C2×C8C22×C3⋊D312yes—
251C4⋊C4—22no—
252C22⋊C4—184no—
253C42—91no—
254C42—92no—
255C42C22×C3⋊D311yescharacteristic
256C22×C3—48no—
257C22×C3—416no—
258C22×C3C2×D3×D414yes—
259C2×D3—38no—
260C2×D3—68no—
261C2×D3—128no—
262C2×D3—128no—
263C2×D3—1216no—
264C2×D3—1216no—
265C2×D3—2432no—
266C3×C4—48no—
267C3×C4#914116yescomplemented
268C3×C4C2×D3×D418yes—
269Dic3—38no—
270Dic3—68no—
271C32#315111yescharacteristic, complemented, Sylow
272C23—22no—
273C23—91no—
274C23—182no—
275C23—182no—
276C23—364no—
277C23—368no—
278Q8—24no—
279D4—24no—
280D4—44no—
281D4—184no—
282D4—184no—
283D4—364no—
284D4—364no—
285D4—368no—
286D4—364no—
287D4—364no—
288C2×C4—22no—
289C2×C4—91no—
290C2×C4—94no—
291C2×C4—92no—
292C2×C4—184no—
293C2×C4—364no—
294C2×C4D4×C3⋊D312yes—
295C2×C4C23×C3⋊D311yescharacteristic
296C8—94no—
297C8D4×C3⋊D314yescomplemented
298C2×C3—88no—
299C2×C3D3×#12714yes—
300C2×C3C2×#91418yes—
301D3—68no—
302D3—248no—
303D3—248no—
304C22—42no—
305C22—44no—
306C22—92no—
307C22—182no—
308C22—362no—
309C22—362no—
310C22—364no—
311C22—364no—
312C22—728no—
313C22C2×D4×C3⋊D311yescharacteristic
314C4—42no—
315C4—92no—
316C4—182no—
317C4#6412314yescomplemented
318C4C2×D4×C3⋊D312yes—
319C3#8176114yescomplemented
320C2—82no—
321C2—182no—
322C2—722no—
323C2—722no—
324C2C3⋊D3×#12711yescharacteristic
325C2C2×#6412312yes—
subgroupG/H
centerC22C2×D4×C3⋊D3
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C23×C3⋊D3
FittingC32×#492C2
socleC22×C32C22×D4
hypercenter#492C3⋊D3
hypocenterC32#3151
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ #492

Complements

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

NH ≅ G/Ntype
1. C2×#64123C2semidirect
2. #10596196C2semidirect
3. C3⋊D3×#127C2semidirect
4. C32×#492C2semidirect
5. #10592875C2semidirect
6. #10592784C2semidirect
7. #10592772C2semidirect
8. #10592394C2semidirect
9. C4×#3743C2semidirect
14. C2×C32×SD16C22semidirect
15. C32×C4⋊Q8C22semidirect
17. #64156C22semidirect
18. C2×#3755C22semidirect
21. C2×#3753C22semidirect
23. C2×#3745C22semidirect
28. #64083C22semidirect
30. C32×C4×C8C22semidirect
31. #63642C22semidirect
32. C4×C3⋊(C3⋊C8)C22semidirect
36. C3×#492D3semidirect
70. #3743D4semidirect
75. C2×C3×SD16C2×D3semidirect
76. C3×C4⋊Q8C2×D3semidirect
91. C3×C4×C8C2×D3semidirect
118. C32×C8C2×D4semidirect
119. C3⋊(C3⋊C8)C2×D4semidirect
127. #492C3⋊D3semidirect
169. C32×C4AGL(1,ℤ/8)semidirect
178. C2×SD16C2×C3⋊D3semidirect
182. C4⋊Q8C2×C3⋊D3semidirect
194. C4×C8C2×C3⋊D3semidirect
219. C3×C8D3×D4semidirect
267. C3×C4#914semidirect
271. C32#3151semidirect
297. C8D4×C3⋊D3semidirect
317. C4#64123semidirect
319. C3#81761semidirect

Maximal subgroups

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

subgroupindex
C2×#641232
#105961962
C3⋊D3×#1272
C32×#4922
#105928752
#105927842
#105927722
#105923942
C4×#37432
#817613