Finite Group Databasebeta

← #11784041

Subgroups — #11784041

Order 1152.

486 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 486 conjugacy classes of subgroups — too many to draw. Only its 98 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×Q8×#157C211yesmaximal, characteristic
2C2×#63784C211yesmaximal, characteristic
3C2×#63767C211yesmaximal, characteristic
4C4⋊C4×#157C211yesmaximal, characteristic
5C3⋊D3×#103C211yesmaximal, characteristic
6#10590182C211yesmaximal, characteristic
7#10590163C211yesmaximal, characteristic
8C2×Q8×C3⋊D3C2211yescharacteristic
9Q8×#157—22no—
10Q8×#157C414yescomplemented
11C2×C4⋊#157—21no—
12C2×C4⋊#157C2211yescharacteristic
13C2×C4⋊#157C2211yescharacteristic
14C2×C4×#157—21no—
15C2×C4×#157—21no—
16C2×C4×#157C2211yescharacteristic
17C2×#3787C2211yescharacteristic
18C2×C8×C3⋊D3C2211yescharacteristic
19C4⋊C4×C3⋊D3C2211yescharacteristic
20#63784C414yescomplemented
21#63781—21no—
22#63767—22no—
23C32×#103C411yescharacteristic, complemented
24#63662C411yescharacteristic, complemented
25#63643C411yescharacteristic, complemented
26#63637C411yescharacteristic, complemented
27D3×#103—62no—
28C22×#157—21no—
29C22×#157—21no—
30C22×#157D411yescharacteristic
31C22×#157D411yescharacteristic
32C2×C32×Q8C2×C411yescharacteristic
33Q8×C3⋊D3—21no—
34Q8×C3⋊D3—21no—
35Q8×C3⋊D3C2×C412yes—
36Q8×C3⋊D3C2×C412yes—
37C2×C4×C3⋊D3—21no—
38C2×C4×C3⋊D3—21no—
39C2×C4×C3⋊D3C2311yescharacteristic
40C2×#634—21no—
41C2×#634C2×C411yescharacteristic
42C2×#622—21no—
43C4⋊#157—24no—
44C4⋊#157—21no—
45C4⋊#157—21no—
46C4⋊#157C2×C412yes—
47C4⋊#157C2×C412yes—
48C4×#157—24no—
49C4×#157—42no—
50C4×#157C2×C412yes—
51C4×#157D412yes—
52#3787—21no—
53#3787C2×C412yes—
54C2×C32×C8C2×C411yescharacteristic
55C32×C4⋊C4C2×C411yescharacteristic
56C4⋊(C3⋊Dic3)C2×C411yescharacteristic
57#3750—21no—
58C2×C3⋊(C3⋊C8)C2×C411yescharacteristic
59C8×C3⋊D3—21no—
60C8×C3⋊D3—21no—
61#1763—91nomaximal, Sylow
62C2×D3×Q8—62no—
63C2×D3×C8—62no—
64D3×C4⋊C4—62no—
65C3×#103—22no—
66#835—62no—
67#816—62no—
68#810—62no—
69C22×C3⋊D3C2×D411yescharacteristic
70C2×#157—21no—
71C2×#157—21no—
72C2×#157—21no—
73C2×#157—21no—
74C2×#157—22no—
75C2×#157—22no—
76C2×#157—41no—
77C2×#157—41no—
78C2×#157—41no—
79C2×#157—41no—
80C2×#157C22⋊C412yes—
81C2×#157C22⋊C412yes—
82C32×Q8—21no—
83C32×Q8C4212yescomplemented
84C2×C32×C4—21no—
85C2×C32×C4—21no—
86C2×C32×C4C22×C411yescharacteristic
87C2×C3⋊Dic3—21no—
88C2×C3⋊Dic3—21no—
89C2×C3⋊Dic3#4111yescharacteristic
90C4×C3⋊D3—22no—
91C4×C3⋊D3—22no—
92C4×C3⋊D3—41no—
93C4×C3⋊D3—41no—
94C4×C3⋊D3C22×C411yescharacteristic
95C4×C3⋊D3C22×C411yescharacteristic
96C4×C3⋊D3C2×D411yescharacteristic
97C4×C3⋊D3#4111yescharacteristic
98#634—21no—
99#634—22no—
100#634—22no—
101#634C4212yescomplemented
102#622—22no—
103C32×C8—21no—
104C3⋊(C3⋊C8)—21no—
105C2×C4×Q8—91no—
106C2×#103—91no—
107C2×#103—91no—
108C2×#97—91no—
109C4×C4⋊C4—91no—
110#340—91no—
111#340—91no—
112C2×C3×Q8—22no—
113D3×Q8—64no—
114D3×Q8—64no—
115D3×Q8—122no—
116D3×Q8—122no—
117C2×C4×D3—62no—
118C2×C4×D3—122no—
119C2×C4×D3—122no—
120C2×Dic6—62no—
121C2×Dic6—122no—
122C2×C3×C8—22no—
123C3×C4⋊C4—22no—
124C4⋊Dic3—62no—
125Dic3⋊C4—122no—
126C2×C3⋊C8—62no—
127D3×C8—122no—
128D3×C8—122no—
129C22×C32C4×D411yescharacteristic
130C2×C3⋊D3C2×C22⋊C411yescharacteristic
131C2×C3⋊D3C4×D411yescharacteristic
132C2×C3⋊D3AGL(1,ℤ/8)11yescharacteristic
133C2×C3⋊D3#13711yescharacteristic
134C2×C3⋊D3#13711yescharacteristic
135C2×C3⋊D3#13711yescharacteristic
136#157—22no—
137#157—22no—
138#157—42no—
139#157—42no—
140C32×C4—22no—
141C32×C4—41no—
142C32×C4C2×C4211yescharacteristic
143C32×C4C4×D411yescharacteristic
144C3⋊Dic3—22no—
145C3⋊Dic3—41no—
146C3⋊Dic3#11711yescharacteristic
147C3⋊Dic3C4×D411yescharacteristic
148C22×Q8—91no—
149C22×C8—91no—
150C22×C8—91no—
151C4×Q8—94no—
152C4×Q8—182no—
153C2×C4⋊C4—91no—
154C2×C4⋊C4—91no—
155C2×C4⋊C4—91no—
156C2×C4⋊C4—181no—
157C2×C42—91no—
158C2×C42—181no—
159C2×C42—181no—
160#103—94no—
161#103—91no—
162#103—91no—
163#103—91no—
164#103#15711yescharacteristic, complemented
165#97—182no—
166#95—181no—
167C22×D3—62no—
168C3×Q8—24no—
169C3×Q8—42no—
170C2×C3×C4—22no—
171C2×C3×C4—42no—
172C2×C3×C4—42no—
173C2×Dic3—62no—
174C2×Dic3—122no—
175C2×Dic3—122no—
176C4×D3—62no—
177C4×D3—62no—
178C4×D3—62no—
179C4×D3—62no—
180C4×D3—124no—
181C4×D3—124no—
182C4×D3—242no—
183C4×D3—242no—
184Dic6—64no—
185Dic6—124no—
186Dic6—124no—
187Dic6—122no—
188C3×C8—42no—
189C3⋊C8—122no—
190C2×C32C4×C22⋊C411yescharacteristic
191C2×C32#44011yescharacteristic
192C2×C32#44111yescharacteristic
193C3⋊D3#41711yescharacteristic
194C3⋊D3#41911yescharacteristic
195C3⋊D3#44011yescharacteristic
196C3⋊D3#44111yescharacteristic
197C2×Q8—92no—
198C2×Q8—92no—
199C2×Q8—91no—
200C2×Q8—181no—
201C2×Q8—181no—
202C2×Q8—181no—
203C2×Q8C2×#15711yescharacteristic
204C22×C4—91no—
205C22×C4—91no—
206C22×C4—91no—
207C22×C4—181no—
208C22×C4—181no—
209C22×C4—181no—
210C22×C4—181no—
211C2×C8—92no—
212C2×C8—91no—
213C2×C8—181no—
214C2×C8—181no—
215C2×C8—181no—
216C2×C8—181no—
217C2×C8C2×#15711yescharacteristic
218C4⋊C4—92no—
219C4⋊C4—92no—
220C4⋊C4—91no—
221C4⋊C4—181no—
222C4⋊C4—181no—
223C4⋊C4—181no—
224C4⋊C4—184no—
225C4⋊C4C2×#15711yescharacteristic
226C42—92no—
227C42—92no—
228C42—184no—
229C42—362no—
230C22×C3—22no—
231C2×D3—62no—
232C2×D3—62no—
233C2×D3—62no—
234C2×D3—62no—
235C2×D3—62no—
236C2×D3—62no—
237C3×C4—22no—
238C3×C4—22no—
239C3×C4—44no—
240C3×C4—82no—
241Dic3—62no—
242Dic3—62no—
243Dic3—124no—
244Dic3—242no—
245C32#176311yescharacteristic, complemented, Sylow
246C23—91no—
247Q8—21no—
248Q8—92no—
249Q8—182no—
250Q8—182no—
251Q8—181no—
252Q8C4×#15712yescomplemented
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—91no—
256C2×C4—91no—
257C2×C4—91no—
258C2×C4—91no—
259C2×C4—91no—
260C2×C4—92no—
261C2×C4—92no—
262C2×C4—182no—
263C2×C4—182no—
264C2×C4—181no—
265C2×C4—181no—
266C2×C4—181no—
267C2×C4—181no—
268C2×C4—181no—
269C2×C4—182no—
270C2×C4—182no—
271C2×C4—181no—
272C2×C4—361no—
273C2×C4—361no—
274C2×C4—361no—
275C2×C4—361no—
276C2×C4—361no—
277C2×C4—361no—
278C2×C4C22×#15711yescharacteristic
279C8—21no—
280C8—182no—
281C8—181no—
282C2×C3—22no—
283C2×C3—22no—
284C2×C3—22no—
285D3—62no—
286D3—62no—
287D3—62no—
288D3—62no—
289C22—91no—
290C22—91no—
291C22—91no—
292C22—91no—
293C22—91no—
294C22—91no—
295C22D4×#15711yescharacteristic
296C4—22no—
297C4—41no—
298C4—91no—
299C4—91no—
300C4—182no—
301C4—182no—
302C4—182no—
303C4—362no—
304C4—362no—
305C4—361no—
306C4C2×C4×#15711yescharacteristic
307C4D4×#15711yescharacteristic
308C3—22no—
309C2—91no—
310C2—91no—
311C2—91no—
312C2—91no—
313C2C22⋊C4×#15711yescharacteristic
314C2#1059378211yescharacteristic
315C2#1059378611yescharacteristic
subgroupG/H
centerC22D4×#157
commutatorC32×C4C2×C42
FrattiniC2×C4C22×#157
FittingC32×#103C4
socleC22×C32C4×D4
prime radicalC22×C3⋊D3C2×D4
cosocleC2×C4×C3⋊D3C23
hypercenter#103#157
hypocenterC32#1763
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4 ◁ #103

Complements

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

NH ≅ G/Ntype
10. Q8×#157C4semidirect
20. #63784C4semidirect
23. C32×#103C4semidirect
24. #63662C4semidirect
25. #63643C4semidirect
26. #63637C4semidirect
83. C32×Q8C42semidirect
101. #634C42semidirect
164. #103#157semidirect
245. C32#1763semidirect
252. Q8C4×#157semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×#1572
C2×#637842
C2×#637672
C4⋊C4×#1572
C3⋊D3×#1032
#105901822
#105901632
#17639