Finite Group Databasebeta

← C2×#88965 #11724437

Subgroups — C2×#88965 #11724437

Order 864.

850 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 850 conjugacy classes of subgroups — too many to draw. Only its 156 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×#6295C211yesmaximal, characteristic, complemented
2C2×#6292C212yesmaximal, complemented
3C2×#6289C211yesmaximal, characteristic, complemented
4C2×C3×#627C211yesmaximal, characteristic, complemented
5C2×C3×#624C212yesmaximal, complemented
6#88965C218yesmaximal, direct product factor
7C2×#3799—32nomaximal
8C2×#3798—31nomaximal
9C22×C3⋊(C3⋊D3)C2211yescharacteristic
10C22×C3×C3⋊D3C2212yes—
11C2×C3×C3⋊Dic3C2211yescharacteristic
12C2×C32×Dic3C2211yescharacteristic
13C2×C32×Dic3C2212yes—
14#6295C2214yescomplemented
15#6292C2218yescomplemented
16#6289C2214yescomplemented
17C3×#627C2214yescomplemented
18C3×#624C2218yescomplemented
19C2×C4⋊(C3⋊D3)—32no—
20C2×C4×C3⋊D3—31no—
21C2×C3×C4×D3—32no—
22C2×C3×C4×D3—32no—
23C2×C3×Dic6—32no—
24C2×#627D311yescharacteristic, complemented
25C2×#626—32no—
26C2×#626—32no—
27C2×#626—32no—
28C2×#626—62no—
29C2×#625—31no—
30C2×#624—32no—
31C2×#624—61no—
32C2×#624D312yescomplemented
33#3799—316no—
34#3798—38no—
35C22×C33C2311yescharacteristic
36C2×C3⋊(C3⋊D3)—22no—
37C2×C3⋊(C3⋊D3)C2312yes—
38C2×C3×C3⋊D3—24no—
39C2×C3×C3⋊D3C2314yes—
40C3×C3⋊Dic3C2312yes—
41C32×Dic3C2312yes—
42C32×Dic3C2314yes—
43C2×#239—91no—
44C2×#235—92no—
45C22×C3⋊D3—32no—
46C22×C3⋊D3—31no—
47C22×C3⋊D3—61no—
48C22×C3⋊D3—62no—
49C22×C3⋊D3—121no—
50C22×C3⋊D3C2×D312yes—
51C22×C3×D3—32no—
52C22×C3×D3—32no—
53C22×C3×D3—62no—
54C2×C32×C4—32no—
55C2×C32×C4—31no—
56C2×C3⋊Dic3C2×D311yescharacteristic
57C4⋊(C3⋊D3)—38no—
58C4×C3⋊D3—34no—
59C2×C3×Dic3—21no—
60C2×C3×Dic3—22no—
61C2×C3×Dic3—32no—
62C2×C3×Dic3—61no—
63C2×C3×Dic3C2×D312yes—
64C2×C3×Dic3C2×D312yes—
65C2×C3×Dic3C2×D312yes—
66C3×C4×D3—38no—
67C3×C4×D3—38no—
68C3×Dic6—38no—
69#627C2×D314yescomplemented
70#626—38no—
71#626—38no—
72#626—38no—
73#626—68no—
74#625—34no—
75#624—38no—
76#624—64no—
77#624C2×D318yescomplemented
78C2×C33#4112yescomplemented
79C2×C33C2411yescharacteristic
80C3⋊(C3⋊D3)—22no—
81C3×C3⋊D3—24no—
82C2×C3×Q8—91no—
83C22×C3×C4—92no—
84C2×#67—92no—
85C2×#67—92no—
86#239—98no—
87#235—916no—
88C2×D12—92no—
89C2×D12—92no—
90C2×D12—91no—
91C2×D12—182no—
92C2×C4×D3—32no—
93C2×C4×D3—32no—
94C2×C4×D3—92no—
95C2×C4×D3—91no—
96C2×C4×D3—181no—
97C2×Dic6—32no—
98C22×C32—21no—
99C22×C32—22no—
100C22×C32—41no—
101C22×C32C22×D312yes—
102C22×C32C22×D311yescharacteristic
103C2×C3⋊D3—24no—
104C2×C3⋊D3—34no—
105C2×C3⋊D3—32no—
106C2×C3⋊D3—62no—
107C2×C3⋊D3—64no—
108C2×C3⋊D3—62no—
109C2×C3⋊D3—64no—
110C2×C3⋊D3—64no—
111C2×C3⋊D3—122no—
112C2×C3⋊D3—124no—
113C2×C3⋊D3—124no—
114C2×C3⋊D3C22×D314yes—
115C2×C3×D3—34no—
116C2×C3×D3—34no—
117C2×C3×D3—64no—
118C2×C3×D3—64no—
119C2×C3×D3—64no—
120C2×C3×D3—124no—
121C32×C4—34no—
122C32×C4—32no—
123C3⋊Dic3C22×D312yes—
124C3×Dic3—22no—
125C3×Dic3—24no—
126C3×Dic3—34no—
127C3×Dic3—62no—
128C3×Dic3C22×D314yes—
129C3×Dic3C22×D314yes—
130C3×Dic3C22×D314yes—
131C2×#41—271noSylow
132C33C2×#4111yescharacteristic, complemented, Sylow
133C23×C3—92no—
134C22×D3—32no—
135C22×D3—32no—
136C22×D3—62no—
137C22×D3—91no—
138C22×D3—92no—
139C22×D3—181no—
140C22×D3—182no—
141C22×D3—361no—
142C3×Q8—94no—
143C2×C3×C4—32no—
144C2×C3×C4—32no—
145C2×C3×C4—32no—
146C2×C3×C4—61no—
147C2×C3×C4—62no—
148C2×C3×C4—98no—
149C2×C3×C4—91no—
150#67—98no—
151#67—98no—
152C2×Dic3—32no—
153C2×Dic3—61no—
154C2×Dic3D3212yes—
155C2×Dic3D3211yescharacteristic
156D12—98no—
157D12—98no—
158D12—94no—
159D12—188no—
160C4×D3—38no—
161C4×D3—38no—
162C4×D3—98no—
163C4×D3—94no—
164C4×D3—184no—
165Dic6—38no—
166C2×C32—21no—
167C2×C32—22no—
168C2×C32—22no—
169C2×C32—24no—
170C2×C32—42no—
171C2×C32—41no—
172C2×C32#23514yescomplemented
173C2×C32#23912yescomplemented
174C2×C32C23×D312yes—
175C2×C32C23×D311yescharacteristic
176C3⋊D3—24no—
177C3⋊D3—62no—
178C3⋊D3—64no—
179C3⋊D3—64no—
180C3⋊D3—124no—
181C3⋊D3—124no—
182C3×D3—64no—
183C3×D3—64no—
184C3×D3—124no—
185#41—278no—
186C2×Q8—91no—
187C2×D4—272no—
188C2×D4—271no—
189C22×C4—92no—
190C22×C4—271no—
191C22×C3—21no—
192C22×C3—22no—
193C22×C3—41no—
194C22×C3—94no—
195C22×C3—184no—
196C22×C3C2×D3211yescharacteristic
197C22×C3C2×D3212yes—
198C2×D3—34no—
199C2×D3—34no—
200C2×D3—64no—
201C2×D3—64no—
202C2×D3—64no—
203C2×D3—92no—
204C2×D3—94no—
205C2×D3—124no—
206C2×D3—182no—
207C2×D3—184no—
208C2×D3—184no—
209C2×D3—182no—
210C2×D3—184no—
211C2×D3—362no—
212C2×D3—364no—
213C2×D3—364no—
214C3×C4—34no—
215C3×C4—34no—
216C3×C4—34no—
217C3×C4—62no—
218C3×C4—64no—
219C3×C4—92no—
220Dic3—34no—
221Dic3—62no—
222Dic3C2×D3214yes—
223Dic3C2×D3212yes—
224C32—21no—
225C32—22no—
226C32—41no—
227C32C2×#23512yescomplemented
228C32C2×#23911yescharacteristic, complemented
229C23—92no—
230C23—271no—
231Q8—94no—
232D4—278no—
233D4—274no—
234C2×C4—31no—
235C2×C4—32no—
236C2×C4—98no—
237C2×C4—91no—
238C2×C4—274no—
239C2×C3—21no—
240C2×C3—22no—
241C2×C3—24no—
242C2×C3—22no—
243C2×C3—41no—
244C2×C3—42no—
245C2×C3—184no—
246C2×C3#379812yescomplemented
247C2×C3#379914yescomplemented
248C2×C3C22×D3211yescharacteristic
249C2×C3C22×D3212yes—
250D3—64no—
251D3—64no—
252D3—124no—
253D3—184no—
254D3—182no—
255D3—184no—
256D3—364no—
257D3—364no—
258C22—94no—
259C22—184no—
260C22—272no—
261C22—542no—
262C22D3311yescharacteristic
263C4—32no—
264C4—34no—
265C4—92no—
266C3—22no—
267C3—21no—
268C3—41no—
269C3C2×#379811yescharacteristic, complemented
270C3C2×#379912yescomplemented
271C2—184no—
272C2—542no—
273C2#8896512yesdirect product factor
274C2C2×D3311yescharacteristic
subgroupG/H
center, hypercenterC22D33
commutator, cosocleC2×C33C24
FrattiniC2C2×D33
Fitting, socleC22×C33C23
hypocenterC33C2×#41
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C2×#6295C2semidirect
2. C2×#6292C2semidirect
3. C2×#6289C2semidirect
4. C2×C3×#627C2semidirect
5. C2×C3×#624C2semidirect
6. #88965C2direct (N × H)
14. #6295C22semidirect
15. #6292C22semidirect
16. #6289C22semidirect
17. C3×#627C22semidirect
18. C3×#624C22semidirect
24. C2×#627D3semidirect
32. C2×#624D3semidirect
69. #627C2×D3semidirect
77. #624C2×D3semidirect
78. C2×C33#41semidirect
132. C33C2×#41semidirect
172. C2×C32#235semidirect
173. C2×C32#239semidirect
227. C32C2×#235semidirect
228. C32C2×#239semidirect
246. C2×C3#3798semidirect
247. C2×C3#3799semidirect
269. C3C2×#3798semidirect
270. C3C2×#3799semidirect
273. C2#88965direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#62952
C2×#62922
C2×#62892
C2×C3×#6272
C2×C3×#6242
#889652
C2×#37993
C2×#37983