Finite Group Databasebeta

← C2×#10593744 #11858781

Subgroups — C2×#10593744 #11858781

Order 1152.

742 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 742 conjugacy classes of subgroups — too many to draw. Only its 170 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×#64086C211yesmaximal, characteristic
2C2×C22⋊#622C211yesmaximal, characteristic
3C2×C22⋊#622C211yesmaximal, characteristic
4C2×#63779C211yesmaximal, characteristic, complemented
5C2×#63777C211yesmaximal, characteristic, complemented
6C2×C4⋊#622C211yesmaximal, characteristic, complemented
7C2×C4×#622C211yesmaximal, characteristic, complemented
8#10593744C218yesmaximal, direct product factor
9C23×C3⋊Dic3C2211yescharacteristic
10C22×#622C2211yescharacteristic
11C22×#622C2211yescharacteristic
12C22×#622C2211yescharacteristic
13C22×#622C2211yescharacteristic
14C2×C32×C22⋊C4C411yescharacteristic
15C2×#3757C411yescharacteristic
16C2×#3750C411yescharacteristic
17C2×#3750C411yescharacteristic
18C2×C4×C3⋊Dic3C2211yescharacteristic
19C2×C4×C3⋊Dic3C2211yescharacteristic
20#64086C414yes—
21#64086C2214yes—
22C22⋊#622C2214yes—
23C22⋊#622C2214yes—
24#63779C2214yescomplemented
25#63777C2214yescomplemented
26C4⋊#622C2214yescomplemented
27C4×#622C2214yescomplemented
28C2×#877—62no—
29C24×C32C2×C411yescharacteristic
30C22×C32×C4C2×C411yescharacteristic
31C22×C32×C4C2×C411yescharacteristic
32C22×C3⋊Dic3—22no—
33C22×C3⋊Dic3—22no—
34C22×C3⋊Dic3C2×C412yes—
35C22×C3⋊Dic3C2×C411yescharacteristic
36C22×C3⋊Dic3C2×C411yescharacteristic
37C22×C3⋊Dic3C2×C411yescharacteristic
38C22×C3⋊Dic3C2312yes—
39C22×C3⋊Dic3C2311yescharacteristic
40C2×#622—22no—
41C2×#622—22no—
42C2×#622—22no—
43C2×#622—22no—
44C2×#622C2312yes—
45C2×#622C2312yes—
46C2×#622C2312yes—
47C2×#622C2312yes—
48C32×C22⋊C4C2×C414yes—
49#3757C2×C414yes—
50#3750C2×C414yes—
51#3750C2×C414yes—
52C4×C3⋊Dic3C2×C412yes—
53C4×C3⋊Dic3C2×C412yes—
54C4×C3⋊Dic3C2312yes—
55C4×C3⋊Dic3C2312yes—
56C2×#433—91nomaximal, Sylow
57C23×Dic3—62no—
58C2×C3×C22⋊C4—22no—
59C2×#217—62no—
60C2×Dic3⋊C4—62no—
61C2×Dic3⋊C4—62no—
62C2×C4×Dic3—62no—
63C2×C4×Dic3—62no—
64#877—68no—
65#877—68no—
66C23×C32—22no—
67C23×C32—22no—
68C23×C32—22no—
69C23×C32C22×C412yes—
70C23×C32C22×C411yescharacteristic
71C2×C32×C4—22no—
72C2×C32×C4—22no—
73C2×C32×C4C22×C412yes—
74C2×C32×C4C22×C412yes—
75C2×C3⋊Dic3—22no—
76C2×C3⋊Dic3—24no—
77C2×C3⋊Dic3—22no—
78C2×C3⋊Dic3—22no—
79C2×C3⋊Dic3—22no—
80C2×C3⋊Dic3—22no—
81C2×C3⋊Dic3C22×C412yes—
82C2×C3⋊Dic3C22×C412yes—
83C2×C3⋊Dic3C22×C412yes—
84C2×C3⋊Dic3C22×C411yescharacteristic
85C2×C3⋊Dic3#4112yes—
86C2×C3⋊Dic3#4112yes—
87C2×C3⋊Dic3C2411yescharacteristic
88#622—22no—
89#622—22no—
90#622—22no—
91#622—22no—
92C2×#117—91no—
93#433—98no—
94C2×C4⋊C8—91no—
95C2×C4⋊C8—91no—
96C2×C22⋊C8—91no—
97C2×C22⋊C8—91no—
98C2×#97—91no—
99C2×C4×C8—91no—
100C24×C3—22no—
101C22×C3×C4—22no—
102C22×C3×C4—22no—
103C22×Dic3—64no—
104C22×Dic3—64no—
105C22×Dic3—62no—
106C22×Dic3—62no—
107C22×Dic3—62no—
108C22×Dic3—62no—
109C22×Dic3—68no—
110C22×Dic3—68no—
111C3×C22⋊C4—28no—
112#217—68no—
113Dic3⋊C4—68no—
114Dic3⋊C4—68no—
115C4×Dic3—64no—
116C4×Dic3—64no—
117C4×Dic3—64no—
118C4×Dic3—64no—
119C22×C32—22no—
120C22×C32—22no—
121C22×C32—22no—
122C22×C32—44no—
123C22×C32#11712yes—
124C22×C32#13112yes—
125C22×C32#13112yes—
126C22×C32C23×C411yescharacteristic
127C32×C4—22no—
128C32×C4—22no—
129C3⋊Dic3—22no—
130C3⋊Dic3—22no—
131C3⋊Dic3—22no—
132C3⋊Dic3#11712yes—
133C3⋊Dic3C2×#4111yescharacteristic
134C3⋊Dic3C2×#4111yescharacteristic
135C23×C4—91no—
136C22×C8—91no—
137C22×C8—91no—
138C22×C8—91no—
139C22×C8—91no—
140#117—94no—
141#117—94no—
142C2×C4⋊C4—91no—
143C2×C4⋊C4—91no—
144C2×C22⋊C4—91no—
145C2×C22⋊C4#15711yescharacteristic
146C2×C42—91no—
147C2×C42—91no—
148C4⋊C8—94no—
149C4⋊C8—94no—
150C22⋊C8—94no—
151C22⋊C8—94no—
152#97—94no—
153C4×C8—94no—
154C23×C3—28no—
155C23×C3—24no—
156C23×C3—22no—
157C23×C3—44no—
158C23×C3—44no—
159C2×C3×C4—24no—
160C2×C3×C4—24no—
161C2×C3×C4—44no—
162C2×C3×C4—44no—
163C2×Dic3—62no—
164C2×Dic3—64no—
165C2×Dic3—64no—
166C2×Dic3—64no—
167C2×Dic3—62no—
168C2×Dic3—64no—
169C2×Dic3—64no—
170C2×Dic3—616no—
171C2×Dic3—68no—
172C2×Dic3—68no—
173C2×Dic3—68no—
174C2×Dic3—124no—
175C2×Dic3—124no—
176C2×C32—42no—
177C2×C32#43314yescomplemented
178C2×C32C2×#11711yescharacteristic
179C2×C32C2×#13111yescharacteristic
180C2×C32C2×#13111yescharacteristic
181C24C2×#15711yescharacteristic
182C22×C4—91no—
183C22×C4—92no—
184C22×C4—92no—
185C22×C4—91no—
186C22×C4—91no—
187C22×C4—91no—
188C22×C4—182no—
189C22×C4—182no—
190C22×C4C2×#15711yescharacteristic
191C22×C4C2×#15711yescharacteristic
192C2×C8—92no—
193C2×C8—92no—
194C2×C8—92no—
195C2×C8—92no—
196C2×C8—182no—
197C2×C8—182no—
198C2×C8—182no—
199C2×C8—182no—
200C4⋊C4—94no—
201C4⋊C4—94no—
202C22⋊C4—94no—
203C22⋊C4C2×#15714yes—
204C42—92no—
205C42—92no—
206C42—92no—
207C42—92no—
208C22×C3—28no—
209C22×C3—24no—
210C22×C3—24no—
211C22×C3—24no—
212C22×C3—22no—
213C22×C3—44no—
214C22×C3—44no—
215C22×C3—416no—
216C3×C4—44no—
217C3×C4—44no—
218Dic3—64no—
219Dic3—62no—
220Dic3—62no—
221Dic3—68no—
222Dic3—124no—
223Dic3—124no—
224C32C2×#43311yescharacteristic, complemented, Sylow
225C23—22no—
226C23—22no—
227C23—22no—
228C23C22×#15712yes—
229C23C22×#15711yescharacteristic
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—92no—
233C2×C4—92no—
234C2×C4—91no—
235C2×C4—91no—
236C2×C4—92no—
237C2×C4—92no—
238C2×C4—92no—
239C2×C4—182no—
240C2×C4—182no—
241C2×C4—184no—
242C2×C4—182no—
243C2×C4—182no—
244C2×C4—182no—
245C2×C4C22×#15712yes—
246C2×C4C22×#15712yes—
247C8—182no—
248C8—182no—
249C8—182no—
250C8—182no—
251C2×C3—22no—
252C2×C3—22no—
253C2×C3—22no—
254C2×C3—28no—
255C2×C3—48no—
256C22—22no—
257C22—22no—
258C22—22no—
259C22—44no—
260C22#6428612yes—
261C22#6428712yes—
262C22#6428712yes—
263C22C23×#15711yescharacteristic
264C4—22no—
265C4—22no—
266C4—91no—
267C4—91no—
268C4—92no—
269C4—182no—
270C4—182no—
271C4—182no—
272C3—22no—
273C2—42no—
274C2#1059374414yesdirect product factor
275C2C2×#6428611yescharacteristic
276C2C2×#6428711yescharacteristic
277C2C2×#6428711yescharacteristic
subgroupG/H
centerC23C22×#157
commutatorC22×C32C23×C4
FrattiniC22C23×#157
FittingC2×C32×C22⋊C4C4
socleC23×C32C22×C4
prime radicalC24×C32C2×C4
cosocleC2×C3⋊Dic3C24
hypercenterC2×C22⋊C4#157
hypocenterC32C2×#433
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
4. C2×#63779C2semidirect
5. C2×#63777C2semidirect
6. C2×C4⋊#622C2semidirect
7. C2×C4×#622C2semidirect
8. #10593744C2direct (N × H)
24. #63779C22semidirect
25. #63777C22semidirect
26. C4⋊#622C22semidirect
27. C4×#622C22semidirect
177. C2×C32#433semidirect
224. C32C2×#433semidirect
274. C2#10593744direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#640862
C2×C22⋊#6222
C2×#637792
C2×#637772
C2×C4⋊#6222
C2×C4×#6222
#105937442
C2×#4339