Finite Group Databasebeta

← #120×#157 #11801989

Subgroups — #120×#157 #11801989

Order 1152.

1886 conjugacy classes of subgroups; only the 210 normal subgroups are shown.

normal subgroups only

This group has 1886 conjugacy classes of subgroups — too many to draw. Only its 210 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
1C24×#157C211yesmaximal, characteristic, complemented
2C2×D4×#157C213yesmaximal, complemented
3C3⋊D3×#120C211yesmaximal, characteristic
4#10593847C211yesmaximal, characteristic, complemented
5#10593826C213yesmaximal, complemented
6#10593746C213yesmaximal, complemented
7C22⋊C4×#157C213yesmaximal, complemented
8C24×C3⋊D3C2211yescharacteristic
9C23×#157—212no—
10C23×#157C2213yes—
11C23×#157C2213yes—
12C23×#157C2211yescharacteristic, complemented
13C2×D4×C3⋊D3C2213yes—
14C2×C22⋊#157C2213yes—
15C2×C22⋊#157C2213yes—
16C2×C22⋊#157C2213yescomplemented
17D4×#157—212no—
18C2×C4⋊#157C2213yescomplemented
19C2×C4×#157C2213yescomplemented
20C32×#120C411yescharacteristic, complemented
21#64162C411yescharacteristic, complemented
22#64146C413yescomplemented
23#64091C413yescomplemented
24C22⋊C4×C3⋊D3C2213yes—
25#63781C2216yescomplemented
26D3×#120—62no—
27C24×C32C2×C411yescharacteristic, complemented
28C23×C3⋊D3—26no—
29C23×C3⋊D3—26no—
30C23×C3⋊D3C2×C413yescomplemented
31C23×C3⋊D3C2311yescharacteristic
32C23×C3⋊D3C2313yes—
33C22×#157—232no—
34C22×#157—26no—
35C22×#157—212no—
36C22×#157—212no—
37C22×#157—26no—
38C22×#157D4112yescomplemented
39C22×#157C2313yes—
40C22×#157C2311yescharacteristic
41C22×#157C2313yes—
42C22×#157C2311yescharacteristic
43C2×C32×D4C2×C413yescomplemented
44C2×#638C2×C413yescomplemented
45C2×#638C2×C413yescomplemented
46D4×C3⋊D3—26no—
47D4×C3⋊D3—26no—
48C2×C4⋊(C3⋊D3)C2×C413yescomplemented
49C2×C4×C3⋊D3C2313yes—
50C22⋊#157—26no—
51C22⋊#157—26no—
52C22⋊#157—26no—
53C4⋊#157—26no—
54C4×#157—26no—
55C32×C22⋊C4C2×C413yescomplemented
56#3757C2×C413yescomplemented
57#3752C2×C416yescomplemented
58C4×#120—91nomaximal, Sylow
59C24×D3—62no—
60C2×D3×D4—66no—
61C3×#120—22no—
62#953—62no—
63#937—66no—
64#882—66no—
65D3×C22⋊C4—66no—
66C23×C32—26no—
67C23×C32C22×C413yes—
68C23×C32C22×C411yescharacteristic, complemented
69C22×C3⋊D3—28no—
70C22×C3⋊D3—224no—
71C22×C3⋊D3—26no—
72C22×C3⋊D3—212no—
73C22×C3⋊D3—26no—
74C22×C3⋊D3—26no—
75C22×C3⋊D3—23no—
76C22×C3⋊D3—23no—
77C22×C3⋊D3C22×C413yes—
78C22×C3⋊D3C22×C411yescharacteristic, complemented
79C22×C3⋊D3C2×D416yes—
80C22×C3⋊D3#4116yes—
81C22×C3⋊D3C2411yescharacteristic
82C2×#157—224no—
83C2×#157—26no—
84C2×#157—224no—
85C2×#157—23no—
86C2×#157—23no—
87C2×#157—26no—
88C2×#157—44no—
89C2×#157C2×D416yes—
90C2×#157C2×D416yes—
91C32×D4—26no—
92C2×C32×C4C22×C413yescomplemented
93#638—26no—
94#638—26no—
95C2×C3⋊Dic3C22×C413yescomplemented
96C4⋊(C3⋊D3)—26no—
97C4×C3⋊D3—23no—
98C4×C3⋊D3—23no—
99C24×C4—91no—
100C2×#120—91no—
101C2×C4×D4—93no—
102#386—93no—
103#385—93no—
104#379—91no—
105C4×C22⋊C4—93no—
106C24×C3—22no—
107C23×D3—66no—
108C23×D3—66no—
109C23×D3—62no—
110C23×D3—1212no—
111C23×D3—1212no—
112C2×C3×D4—26no—
113C2×#67—66no—
114C2×#67—66no—
115D3×D4—1212no—
116D3×D4—1212no—
117C2×D12—66no—
118C2×C4×D3—66no—
119C3×C22⋊C4—26no—
120#217—66no—
121(C2×D3)⋊C4—612no—
122C22×C32—28no—
123C22×C32—26no—
124C22×C32—23no—
125C22×C32C4×D416yescomplemented
126C22×C32C23×C411yescharacteristic
127C2×C3⋊D3—224no—
128C2×C3⋊D3—26no—
129C2×C3⋊D3—26no—
130C2×C3⋊D3—26no—
131C2×C3⋊D3—212no—
132C2×C3⋊D3—23no—
133C2×C3⋊D3—41no—
134C2×C3⋊D3—43no—
135C2×C3⋊D3C4×D416yescomplemented
136C2×C3⋊D3C22×D413yes—
137C2×C3⋊D3C2×#4113yes—
138#157—26no—
139#157—41no—
140#157—43no—
141#157#12014yesdirect product factor
142C32×C4—23no—
143C3⋊Dic3—23no—
144C25—91no—
145C22×D4—93no—
146C23×C4—93no—
147C23×C4—91no—
148C23×C4—93no—
149C23×C4—1812no—
150#120—91no—
151#120—93no—
152#120—93no—
153#120#15711yescharacteristic, direct product factor
154C4×D4—1812no—
155C2×C4⋊C4—93no—
156C2×C22⋊C4—93no—
157C2×C22⋊C4—93no—
158C2×C22⋊C4—93no—
159C2×C22⋊C4—93no—
160C2×C42—93no—
161#95—96no—
162C23×C3—26no—
163C23×C3—22no—
164C23×C3—412no—
165C22×D3—62no—
166C22×D3—66no—
167C22×D3—612no—
168C22×D3—612no—
169C22×D3—62no—
170C22×D3—1212no—
171C22×D3—1212no—
172C22×D3—1224no—
173C22×D3—126no—
174C22×D3—126no—
175C22×D3—1212no—
176C22×D3—1216no—
177C22×D3—1248no—
178C3×D4—412no—
179C2×C3×C4—26no—
180#67—1212no—
181#67—1212no—
182C2×Dic3—66no—
183D12—1212no—
184C4×D3—126no—
185C4×D3—126no—
186C2×C32—26no—
187C2×C32—41no—
188C2×C32C2×C4×D413yes—
189C3⋊D3—26no—
190C3⋊D3—41no—
191C3⋊D3C2×#12011yescharacteristic
192C3⋊D3#52513yes—
193C24—93no—
194C24—93no—
195C24—91no—
196C24—186no—
197C24—186no—
198C24C2×#15711yescharacteristic, complemented
199C2×D4—93no—
200C2×D4—93no—
201C2×D4—93no—
202C2×D4—186no—
203C2×D4—186no—
204C2×D4C2×#15713yescomplemented
205C22×C4—91no—
206C22×C4—912no—
207C22×C4—93no—
208C22×C4—93no—
209C22×C4—93no—
210C22×C4—91no—
211C22×C4—1812no—
212C22×C4—1812no—
213C22×C4—186no—
214C22×C4—186no—
215C22×C4—1832no—
216C4⋊C4—186no—
217C22⋊C4—93no—
218C22⋊C4—96no—
219C22⋊C4—186no—
220C22⋊C4—186no—
221C22⋊C4—186no—
222C22⋊C4C2×#15713yescomplemented
223C42—186no—
224C22×C3—212no—
225C22×C3—22no—
226C22×C3—412no—
227C22×C3—46no—
228C22×C3—416no—
229C2×D3—66no—
230C2×D3—66no—
231C2×D3—612no—
232C2×D3—1212no—
233C2×D3—1212no—
234C2×D3—1224no—
235C2×D3—1212no—
236C2×D3—1248no—
237C2×D3—126no—
238C2×D3—242no—
239C2×D3—246no—
240C3×C4—46no—
241Dic3—126no—
242C32C4×#12011yescharacteristic, complemented, Sylow
243C23—26no—
244C23—91no—
245C23—96no—
246C23—96no—
247C23—93no—
248C23—91no—
249C23—186no—
250C23—186no—
251C23—1812no—
252C23—183no—
253C23—183no—
254C23—188no—
255C23—1824no—
256C23—186no—
257C23C22×#15713yes—
258C23C22×#15711yescharacteristic, complemented
259D4—26no—
260D4—186no—
261D4—186no—
262D4—186no—
263C2×C4—96no—
264C2×C4—96no—
265C2×C4—93no—
266C2×C4—183no—
267C2×C4—183no—
268C2×C4—1824no—
269C2×C4—186no—
270C2×C4—1824no—
271C2×C4—183no—
272C2×C4—183no—
273C2×C4—186no—
274C2×C4—364no—
275C2×C4C22×#15713yescomplemented
276C2×C3—26no—
277C2×C3—412no—
278C2×C3—82no—
279D3—62no—
280D3—66no—
281D3—1212no—
282D3—242no—
283C22—28no—
284C22—26no—
285C22—23no—
286C22—93no—
287C22—93no—
288C22—96no—
289C22—186no—
290C22—186no—
291C22—1812no—
292C22—1824no—
293C22—186no—
294C22—183no—
295C22—361no—
296C22—363no—
297C22D4×#15716yescomplemented
298C22C23×#15711yescharacteristic
299C4—23no—
300C4—94no—
301C4—186no—
302C4—183no—
303C4—361no—
304C4—363no—
305C3—22no—
306C2—26no—
307C2—41no—
308C2—91no—
309C2—93no—
310C2—186no—
311C2—361no—
312C2C2×D4×#15713yes—
subgroupG/H
center, FrattiniC22C23×#157
commutator, socleC22×C32C23×C4
FittingC32×#120C4
prime radicalC3⋊D3×#120C2
cosocleC22×C3⋊D3C24
hypercenter#120#157
hypocenterC32C4×#120
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ #120

Complements

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

NH ≅ G/Ntype
1. C24×#157C2semidirect
2. C2×D4×#157C2semidirect
4. #10593847C2semidirect
5. #10593826C2semidirect
6. #10593746C2semidirect
7. C22⋊C4×#157C2semidirect
12. C23×#157C22semidirect
16. C2×C22⋊#157C22semidirect
18. C2×C4⋊#157C22semidirect
19. C2×C4×#157C22semidirect
20. C32×#120C4semidirect
21. #64162C4semidirect
22. #64146C4semidirect
23. #64091C4semidirect
25. #63781C22semidirect
27. C24×C32C2×C4semidirect
30. C23×C3⋊D3C2×C4semidirect
38. C22×#157D4semidirect
43. C2×C32×D4C2×C4semidirect
44. C2×#638C2×C4semidirect
45. C2×#638C2×C4semidirect
48. C2×C4⋊(C3⋊D3)C2×C4semidirect
55. C32×C22⋊C4C2×C4semidirect
56. #3757C2×C4semidirect
57. #3752C2×C4semidirect
68. C23×C32C22×C4semidirect
78. C22×C3⋊D3C22×C4semidirect
92. C2×C32×C4C22×C4semidirect
95. C2×C3⋊Dic3C22×C4semidirect
125. C22×C32C4×D4semidirect
135. C2×C3⋊D3C4×D4semidirect
141. #157#120direct (N × H)
153. #120#157direct (N × H)
198. C24C2×#157semidirect
204. C2×D4C2×#157semidirect
222. C22⋊C4C2×#157semidirect
242. C32C4×#120semidirect
258. C23C22×#157semidirect
275. C2×C4C22×#157semidirect
297. C22D4×#157semidirect

Maximal subgroups

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

subgroupindex
C24×#1572
C2×D4×#1572
C3⋊D3×#1202
#105938472
#105938262
#105937462
C22⋊C4×#1572
C4×#1209