Finite Group Databasebeta

← #11807889

Subgroups — #11807889

Order 1152.

850 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 850 conjugacy classes of subgroups — too many to draw. Only its 152 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×#124C211yesmaximal, characteristic
2#10593837C212yesmaximal, complemented
3#10593826C212yesmaximal
4#10593747C214yesmaximal, complemented
5#10593746C214yesmaximal, complemented
6#10593736C212yesmaximal, complemented
7C23×#157C2212yes—
8C2×Q8×C3⋊D3C2211yescharacteristic
9C2×D4×C3⋊D3C2211yescharacteristic
10C2×C22⋊#157C2214yes—
11C2×C22⋊#157C2212yes—
12C2×C4⋊#157C2212yes—
13C2×C4⋊#157C2212yes—
14C2×C4⋊#157C2212yes—
15C2×C4×#157C2212yes—
16C32×#124C411yescharacteristic, complemented
17#64156C411yescharacteristic, complemented
18#64145C411yescharacteristic, complemented
19#64094C414yescomplemented
20C22⋊C4×C3⋊D3C2214yes—
21#64084C411yescharacteristic, complemented
22C42×C3⋊D3C2211yescharacteristic
23#63781C2214yes—
24#63781C2214yes—
25#63781C2214yes—
26D3×#124—62no—
27C23×C3⋊D3C2312yes—
28C22×#157—42no—
29C22×#157—44no—
30C22×#157C2314yes—
31C22×#157C2314yes—
32C2×C32×Q8C2×C411yescharacteristic, complemented
33C2×C32×D4C2×C411yescharacteristic
34C2×#638C2×C412yes—
35Q8×C3⋊D3—41no—
36Q8×C3⋊D3—41no—
37D4×C3⋊D3—41no—
38D4×C3⋊D3—41no—
39C2×C4⋊(C3⋊D3)C2×C411yescharacteristic
40C2×C4×C3⋊D3C2312yes—
41C2×C4×C3⋊D3C2312yes—
42C2×C4×C3⋊D3C2311yescharacteristic
43C2×#634C2×C411yescharacteristic, complemented
44C2×#634C2×C412yescomplemented
45C22⋊#157—44no—
46C22⋊#157—42no—
47C4⋊#157—42no—
48C4⋊#157—42no—
49C4⋊#157—42no—
50C4×#157—42no—
51C32×C22⋊C4C2×C414yescomplemented
52C32×C42C2×C411yescharacteristic, complemented
53#3757C2×C414yescomplemented
54#3752C2×C414yescomplemented
55#3752C2×C414yescomplemented
56C4×C3⋊Dic3C2×C412yescomplemented
57C4×C3⋊Dic3C2×C411yescharacteristic, complemented
58#2379—91nomaximal, Sylow
59C2×D3×Q8—62no—
60C2×D3×D4—62no—
61C3×#124—22no—
62#947—62no—
63#936—62no—
64#885—68no—
65D3×C22⋊C4—68no—
66#875—62no—
67C42×D3—62no—
68C23×C32C22×C412yes—
69C22×C3⋊D3—24no—
70C22×C3⋊D3—22no—
71C22×C3⋊D3—22no—
72C22×C3⋊D3—24no—
73C22×C3⋊D3C22×C412yes—
74C22×C3⋊D3C2411yescharacteristic
75C2×#157—24no—
76C2×#157—24no—
77C2×#157—24no—
78C2×#157—24no—
79C2×#157—24no—
80C2×#157—24no—
81C2×#157—48no—
82C32×Q8—41no—
83C32×D4—41no—
84C2×C32×C4C22×C412yes—
85C2×C32×C4C22×C411yescharacteristic
86C2×C32×C4C22×C412yes—
87#638—42no—
88C2×C3⋊Dic3C22×C412yes—
89C2×C3⋊Dic3C22×C411yescharacteristic
90C2×C3⋊Dic3C22×C412yes—
91C4⋊(C3⋊D3)—41no—
92C4×C3⋊D3—21no—
93C4×C3⋊D3—21no—
94C4×C3⋊D3—22no—
95C4×C3⋊D3—22no—
96C4×C3⋊D3—22no—
97C4×C3⋊D3—22no—
98#634—42no—
99#634—41no—
100C2×#124—91no—
101#391—92no—
102#388—94no—
103#386—92no—
104#385—94no—
105#382—92no—
106C23×D3—64no—
107C2×C3×Q8—22no—
108C2×C3×D4—22no—
109C2×#67—64no—
110D3×Q8—124no—
111D3×Q8—124no—
112D3×D4—124no—
113D3×D4—124no—
114C2×D12—62no—
115C2×C4×D3—62no—
116C2×C4×D3—64no—
117C2×C4×D3—64no—
118C2×Dic6—62no—
119C2×Dic6—64no—
120C3×C22⋊C4—28no—
121C3×C42—22no—
122#217—68no—
123(C2×D3)⋊C4—68no—
124(C2×D3)⋊C4—68no—
125C4×Dic3—64no—
126C4×Dic3—62no—
127C22×C32—22no—
128C22×C32—24no—
129C22×C32C23×C411yescharacteristic
130C2×C3⋊D3—22no—
131C2×C3⋊D3—24no—
132C2×C3⋊D3—42no—
133C2×C3⋊D3—44no—
134C2×C3⋊D3—42no—
135C2×C3⋊D3H(5,𝔽2)12yes—
136C2×C3⋊D3H(5,𝔽2)11yescharacteristic
137C2×C3⋊D3H(5,𝔽2)11yescharacteristic
138C2×C3⋊D32−1+412yes—
139#157—44no—
140#157—44no—
141C32×C4—21no—
142C32×C4—22no—
143C32×C4—22no—
144C3⋊Dic3—21no—
145C3⋊Dic3—22no—
146C3⋊Dic3—22no—
147C22×Q8—91no—
148C22×D4—91no—
149C23×C4—92no—
150#124—91no—
151#124—91no—
152#124—94no—
153#124—91no—
154#124#15711yescharacteristic, complemented
155C2×C4⋊C4—92no—
156C2×C4⋊C4—92no—
157C2×C4⋊C4—92no—
158C2×C22⋊C4—94no—
159C2×C22⋊C4—94no—
160C2×C22⋊C4—92no—
161C2×C42—91no—
162C2×C42—92no—
163#95—94no—
164#95—94no—
165#95—94no—
166C23×C3—24no—
167C22×D3—62no—
168C22×D3—64no—
169C22×D3—124no—
170C22×D3—124no—
171C22×D3—128no—
172C22×D3—128no—
173C3×Q8—44no—
174C3×D4—44no—
175C2×C3×C4—22no—
176C2×C3×C4—24no—
177C2×C3×C4—24no—
178#67—128no—
179C2×Dic3—62no—
180C2×Dic3—64no—
181C2×Dic3—64no—
182D12—124no—
183C4×D3—64no—
184C4×D3—64no—
185C4×D3—124no—
186C4×D3—124no—
187C4×D3—124no—
188C4×D3—124no—
189Dic6—128no—
190Dic6—124no—
191C2×C32—42no—
192C2×C32#51811yescharacteristic
193C2×C32#52012yes—
194C3⋊D3—42no—
195C3⋊D3H(3,𝔽4)11yescharacteristic
196C3⋊D3#56212yes—
197C3⋊D3#56311yescharacteristic
198C24—92no—
199C2×Q8—92no—
200C2×Q8—91no—
201C2×Q8—361no—
202C2×Q8—361no—
203C2×Q8C2×#15711yescharacteristic, complemented
204C2×D4—92no—
205C2×D4—91no—
206C2×D4—361no—
207C2×D4—361no—
208C2×D4C2×#15711yescharacteristic
209C22×C4—91no—
210C22×C4—92no—
211C22×C4—92no—
212C22×C4—94no—
213C22×C4—94no—
214C22×C4—364no—
215C22×C4—362no—
216C4⋊C4—362no—
217C4⋊C4—362no—
218C4⋊C4—362no—
219C22⋊C4—94no—
220C22⋊C4—94no—
221C22⋊C4—94no—
222C22⋊C4—362no—
223C22⋊C4—364no—
224C22⋊C4C2×#15714yescomplemented
225C42—91no—
226C42—92no—
227C42—362no—
228C42C2×#15711yescharacteristic, complemented
229C22×C3—22no—
230C22×C3—44no—
231C22×C3—48no—
232C2×D3—62no—
233C2×D3—62no—
234C2×D3—64no—
235C2×D3—64no—
236C2×D3—124no—
237C2×D3—128no—
238C2×D3—244no—
239C2×D3—248no—
240C2×D3—244no—
241C3×C4—24no—
242C3×C4—44no—
243C3×C4—44no—
244Dic3—64no—
245Dic3—124no—
246Dic3—124no—
247C32#237911yescharacteristic, complemented, Sylow
248C23—91no—
249C23—92no—
250C23—184no—
251C23—182no—
252C23—182no—
253C23—184no—
254C23C22×#15712yes—
255Q8—41no—
256Q8—361no—
257Q8—362no—
258D4—41no—
259D4—361no—
260D4—362no—
261C2×C4—92no—
262C2×C4—91no—
263C2×C4—92no—
264C2×C4—182no—
265C2×C4—182no—
266C2×C4—182no—
267C2×C4—181no—
268C2×C4—181no—
269C2×C4—182no—
270C2×C4—184no—
271C2×C4—184no—
272C2×C4—184no—
273C2×C4—184no—
274C2×C4—184no—
275C2×C4—184no—
276C2×C4—368no—
277C2×C4C22×#15712yes—
278C2×C4C22×#15711yescharacteristic
279C2×C4C22×#15712yes—
280C2×C3—22no—
281C2×C3—24no—
282C2×C3—84no—
283D3—62no—
284D3—64no—
285D3—62no—
286D3—244no—
287C22—22no—
288C22—24no—
289C22—91no—
290C22—91no—
291C22—92no—
292C22—92no—
293C22—182no—
294C22—184no—
295C22—362no—
296C22—364no—
297C22—362no—
298C22C23×#15711yescharacteristic
299C4—21no—
300C4—22no—
301C4—22no—
302C4—181no—
303C4—182no—
304C4—182no—
305C4—364no—
306C4—364no—
307C3—22no—
308C2—42no—
309C2—91no—
310C2—92no—
311C2—91no—
312C2—362no—
313C2#1059664211yescharacteristic
314C2#1059665012yes—
subgroupG/H
center, FrattiniC22C23×#157
commutator, socleC22×C32C23×C4
FittingC32×#124C4
prime radicalC2×D4×C3⋊D3C22
cosocleC22×C3⋊D3C24
hypercenter#124#157
hypocenterC32#2379
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ #124

Complements

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

NH ≅ G/Ntype
2. #10593837C2semidirect
4. #10593747C2semidirect
5. #10593746C2semidirect
6. #10593736C2semidirect
16. C32×#124C4semidirect
17. #64156C4semidirect
18. #64145C4semidirect
19. #64094C4semidirect
21. #64084C4semidirect
32. C2×C32×Q8C2×C4semidirect
43. C2×#634C2×C4semidirect
44. C2×#634C2×C4semidirect
51. C32×C22⋊C4C2×C4semidirect
52. C32×C42C2×C4semidirect
53. #3757C2×C4semidirect
54. #3752C2×C4semidirect
55. #3752C2×C4semidirect
56. C4×C3⋊Dic3C2×C4semidirect
57. C4×C3⋊Dic3C2×C4semidirect
154. #124#157semidirect
203. C2×Q8C2×#157semidirect
224. C22⋊C4C2×#157semidirect
228. C42C2×#157semidirect
247. C32#2379semidirect

Maximal subgroups

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

subgroupindex
C3⋊D3×#1242
#105938372
#105938262
#105937472
#105937462
#105937362
#23799