Finite Group Databasebeta

← #11802848

Subgroups — #11802848

Order 1152.

814 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 814 conjugacy classes of subgroups — too many to draw. Only its 162 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
1C3⋊D3×#117C211yesmaximal, characteristic
2#10593812C212yesmaximal
3#10593758C214yesmaximal, complemented
4#10593747C214yesmaximal
5#10593736C211yesmaximal, characteristic, complemented
6#10593735C211yesmaximal, characteristic, complemented
7C4×C4⋊#157C212yesmaximal, complemented
8C22×C4×C3⋊D3C2211yescharacteristic
9C2×C22⋊#157C2214yes—
10C2×C4⋊#157C2212yes—
11C2×C4⋊#157C2212yes—
12C2×C4⋊#157C2214yes—
13C2×C4×#157C2214yes—
14C32×#117C411yescharacteristic, complemented
15#64135C411yescharacteristic, complemented
16#64101C412yescomplemented
17C4⋊C4×C3⋊D3C2212yes—
18C22⋊C4×C3⋊D3C2212yes—
19#64086C412yescomplemented
20#64081C411yescharacteristic, complemented
21#64081C411yescharacteristic, complemented
22C42×C3⋊D3C2211yescharacteristic
23C42×C3⋊D3C2211yescharacteristic
24#63781C2214yes—
25#63781C2214yes—
26#63781C2214yes—
27D3×#117—62no—
28C23×C3⋊D3C2311yescharacteristic
29C22×#157C2314yes—
30C22×#157C2314yes—
31C22×C32×C4C2×C411yescharacteristic
32C22×C3⋊Dic3C2×C411yescharacteristic
33C2×C4×C3⋊D3—22no—
34C2×C4×C3⋊D3—22no—
35C2×C4×C3⋊D3C2×C411yescharacteristic
36C2×C4×C3⋊D3C2×C411yescharacteristic
37C2×C4×C3⋊D3C2312yes—
38C2×C4×C3⋊D3C2312yes—
39C2×C4×C3⋊D3C2311yescharacteristic
40C2×C4×C3⋊D3C2311yescharacteristic
41C22⋊#157—44no—
42C4⋊#157—24no—
43C4⋊#157—24no—
44C4⋊#157—44no—
45C4×#157—44no—
46C32×C4⋊C4C2×C412yes—
47C32×C22⋊C4C2×C412yes—
48C32×C42C2×C411yescharacteristic
49C32×C42C2×C411yescharacteristic
50#3757C2×C412yes—
51#3752C2×C412yes—
52#3752C2×C412yes—
53C4⋊(C3⋊Dic3)C2×C412yes—
54#3750C2×C412yes—
55#3750C2×C412yes—
56C4×C3⋊Dic3C2×C411yescharacteristic
57C4×C3⋊Dic3C2×C411yescharacteristic
58C4×C3⋊Dic3C2×C412yes—
59C4×C3⋊Dic3C2×C412yes—
60#2314—91nomaximal, Sylow
61C22×C4×D3—62no—
62C3×#117—22no—
63#926—62no—
64#892—64no—
65D3×C4⋊C4—64no—
66D3×C22⋊C4—64no—
67#877—64no—
68#872—62no—
69#872—62no—
70C42×D3—62no—
71C42×D3—62no—
72C23×C32C22×C411yescharacteristic
73C22×C3⋊D3—21no—
74C22×C3⋊D3—21no—
75C22×C3⋊D3—22no—
76C22×C3⋊D3—22no—
77C22×C3⋊D3C22×C411yescharacteristic
78C22×C3⋊D3C2411yescharacteristic
79C2×#157—24no—
80C2×#157—24no—
81C2×#157—24no—
82C2×#157—24no—
83C2×#157—24no—
84C2×#157—24no—
85C2×C32×C4—22no—
86C2×C32×C4C22×C412yes—
87C2×C32×C4C22×C412yes—
88C2×C32×C4C22×C411yescharacteristic
89C2×C32×C4C22×C411yescharacteristic
90C2×C3⋊Dic3—22no—
91C2×C3⋊Dic3C22×C412yes—
92C2×C3⋊Dic3C22×C412yes—
93C2×C3⋊Dic3C22×C411yescharacteristic
94C2×C3⋊Dic3C22×C411yescharacteristic
95C4×C3⋊D3—21no—
96C4×C3⋊D3—21no—
97C4×C3⋊D3—22no—
98C4×C3⋊D3—22no—
99C4×C3⋊D3—22no—
100C4×C3⋊D3—22no—
101C4×C3⋊D3—22no—
102C4×C3⋊D3—22no—
103C4×C3⋊D3#4112yes—
104C4×C3⋊D3#4112yes—
105C2×#117—91no—
106#388—94no—
107#387—94no—
108#384—91no—
109#382—91no—
110#380—92no—
111C4×C4⋊C4—92no—
112C23×D3—62no—
113C22×C3×C4—22no—
114C22×Dic3—62no—
115C2×C4×D3—62no—
116C2×C4×D3—62no—
117C2×C4×D3—62no—
118C2×C4×D3—62no—
119C2×C4×D3—64no—
120C2×C4×D3—64no—
121C2×C4×D3—68no—
122C2×C4×D3—68no—
123C3×C4⋊C4—24no—
124C3×C22⋊C4—24no—
125C3×C42—22no—
126C3×C42—22no—
127#217—64no—
128(C2×D3)⋊C4—64no—
129(C2×D3)⋊C4—64no—
130C4⋊Dic3—64no—
131Dic3⋊C4—64no—
132Dic3⋊C4—64no—
133C4×Dic3—62no—
134C4×Dic3—62no—
135C4×Dic3—64no—
136C4×Dic3—64no—
137C22×C32—21no—
138C22×C32—22no—
139C22×C32C23×C411yescharacteristic
140C2×C3⋊D3—21no—
141C2×C3⋊D3—22no—
142C2×C3⋊D3—41no—
143C2×C3⋊D3—41no—
144C2×C3⋊D3—42no—
145C2×C3⋊D3C2×#4111yescharacteristic
146C2×C3⋊D3C2×#4111yescharacteristic
147C2×C3⋊D3H(5,𝔽2)12yes—
148C2×C3⋊D32−1+412yes—
149#157—44no—
150#157—44no—
151C32×C4—21no—
152C32×C4—22no—
153C32×C4—22no—
154C32×C4#11712yes—
155C3⋊Dic3—21no—
156C3⋊Dic3—22no—
157C3⋊Dic3—22no—
158C3⋊Dic3#11712yes—
159C23×C4—91no—
160#117—91no—
161#117—92no—
162#117—92no—
163#117—91no—
164#117—91no—
165#117#15711yescharacteristic, complemented
166C2×C4⋊C4—92no—
167C2×C4⋊C4—92no—
168C2×C4⋊C4—92no—
169C2×C4⋊C4—94no—
170C2×C22⋊C4—92no—
171C2×C22⋊C4—94no—
172C2×C42—91no—
173C2×C42—91no—
174C2×C42—94no—
175#95—94no—
176#95—94no—
177#95—94no—
178C23×C3—22no—
179C22×D3—62no—
180C22×D3—62no—
181C22×D3—64no—
182C22×D3—64no—
183C22×D3—124no—
184C22×D3—124no—
185C2×C3×C4—28no—
186C2×C3×C4—22no—
187C2×C3×C4—22no—
188C2×C3×C4—24no—
189C2×C3×C4—24no—
190C2×Dic3—62no—
191C2×Dic3—62no—
192C2×Dic3—64no—
193C2×Dic3—64no—
194C2×Dic3—68no—
195C4×D3—64no—
196C4×D3—64no—
197C4×D3—68no—
198C4×D3—64no—
199C4×D3—64no—
200C4×D3—68no—
201C4×D3—124no—
202C4×D3—124no—
203C4×D3—124no—
204C4×D3—124no—
205C2×C32—41no—
206C2×C32C2×#11711yescharacteristic
207C2×C32#52012yes—
208C3⋊D3—41no—
209C3⋊D3#54011yescharacteristic
210C3⋊D3#54111yescharacteristic
211C3⋊D3#54212yes—
212C24—91no—
213C22×C4—91no—
214C22×C4—91no—
215C22×C4—91no—
216C22×C4—92no—
217C22×C4—92no—
218C22×C4—91no—
219C22×C4—91no—
220C22×C4—94no—
221C22×C4—94no—
222C22×C4—182no—
223C22×C4—182no—
224C22×C4C2×#15711yescharacteristic
225C4⋊C4—92no—
226C4⋊C4—92no—
227C4⋊C4—92no—
228C4⋊C4—184no—
229C4⋊C4—184no—
230C4⋊C4—364no—
231C4⋊C4C2×#15712yes—
232C22⋊C4—92no—
233C22⋊C4—92no—
234C22⋊C4—92no—
235C22⋊C4—364no—
236C22⋊C4C2×#15712yes—
237C42—92no—
238C42—92no—
239C42—91no—
240C42—91no—
241C42—364no—
242C42C2×#15711yescharacteristic
243C42C2×#15711yescharacteristic
244C22×C3—24no—
245C22×C3—22no—
246C22×C3—44no—
247C2×D3—62no—
248C2×D3—62no—
249C2×D3—64no—
250C2×D3—64no—
251C2×D3—64no—
252C2×D3—124no—
253C2×D3—124no—
254C2×D3—124no—
255C2×D3—128no—
256C3×C4—24no—
257C3×C4—24no—
258C3×C4—44no—
259C3×C4—44no—
260Dic3—64no—
261Dic3—64no—
262Dic3—124no—
263Dic3—124no—
264C32#231411yescharacteristic, complemented, Sylow
265C23—91no—
266C23—91no—
267C23—182no—
268C23—181no—
269C23—181no—
270C23—182no—
271C23C22×#15711yescharacteristic
272C2×C4—22no—
273C2×C4—91no—
274C2×C4—92no—
275C2×C4—92no—
276C2×C4—92no—
277C2×C4—92no—
278C2×C4—91no—
279C2×C4—182no—
280C2×C4—182no—
281C2×C4—182no—
282C2×C4—181no—
283C2×C4—181no—
284C2×C4—182no—
285C2×C4—182no—
286C2×C4—182no—
287C2×C4—182no—
288C2×C4—184no—
289C2×C4—184no—
290C2×C4—184no—
291C2×C4—184no—
292C2×C4—184no—
293C2×C4—184no—
294C2×C4C22×#15712yes—
295C2×C4C22×#15712yes—
296C2×C4C22×#15711yescharacteristic
297C2×C4C22×#15711yescharacteristic
298C2×C3—22no—
299C2×C3—24no—
300C2×C3—44no—
301D3—62no—
302D3—62no—
303D3—64no—
304D3—124no—
305C22—21no—
306C22—22no—
307C22—91no—
308C22—91no—
309C22—92no—
310C22—92no—
311C22—181no—
312C22—182no—
313C22—361no—
314C22—362no—
315C22—361no—
316C22C23×#15711yescharacteristic
317C4—21no—
318C4—22no—
319C4—22no—
320C4—92no—
321C4—181no—
322C4—182no—
323C4—182no—
324C4—364no—
325C4—364no—
326C4#6428612yes—
327C3—22no—
328C2—41no—
329C2—91no—
330C2—92no—
331C2—91no—
332C2—361no—
333C2C2×#6428611yescharacteristic
334C2#1059665012yes—
subgroupG/H
center, FrattiniC22C23×#157
commutator, socleC22×C32C23×C4
FittingC32×#117C4
prime radicalC23×C3⋊D3C23
cosocleC22×C3⋊D3C24
hypercenter#117#157
hypocenterC32#2314
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ #117

Complements

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

NH ≅ G/Ntype
3. #10593758C2semidirect
5. #10593736C2semidirect
6. #10593735C2semidirect
7. C4×C4⋊#157C2semidirect
14. C32×#117C4semidirect
15. #64135C4semidirect
16. #64101C4semidirect
19. #64086C4semidirect
20. #64081C4semidirect
21. #64081C4semidirect
165. #117#157semidirect
264. C32#2314semidirect

Maximal subgroups

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

subgroupindex
C3⋊D3×#1172
#105938122
#105937582
#105937472
#105937362
#105937352
C4×C4⋊#1572
#23149