Finite Group Databasebeta

← C2×#89041 #11724613

Subgroups — C2×#89041 #11724613

Order 864.

818 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 818 conjugacy classes of subgroups — too many to draw. Only its 159 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×C3×C4⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
2C2×C3×C4×C3⋊D3C211yesmaximal, characteristic, complemented
3C2×C3×#634C211yesmaximal, characteristic, complemented
4C2×#6295C212yesmaximal, complemented
5C2×#6294C212yesmaximal, complemented
6#89041C218yesmaximal, direct product factor
7C2×#3799—31nomaximal
8C2×#3796—31nomaximal
9C2×#3795—31nomaximal
10C22×C3×C3⋊D3C2212yes—
11C22×C3×C3⋊D3C2211yescharacteristic
12C2×C33×C4C2211yescharacteristic
13C2×C3×C3⋊Dic3C2211yescharacteristic
14C2×C3×C3⋊Dic3C2212yes—
15C3×C4⋊(C3⋊D3)C2214yescomplemented
16C3×C4×C3⋊D3C2214yescomplemented
17C3×#634C2214yescomplemented
18#6295C2218yescomplemented
19#6294C2218yescomplemented
20C2×C4⋊(C3⋊D3)D311yescharacteristic, complemented
21C2×C4×C3⋊D3D311yescharacteristic, complemented
22C2×#634D311yescharacteristic, complemented
23C2×C3×D12—31no—
24C2×C3×D12—31no—
25C2×C3×D12—61no—
26C2×C3×C4×D3—31no—
27C2×C3×C4×D3—31no—
28C2×C3×C4×D3—61no—
29C2×C3×Dic6—31no—
30C2×C3×Dic6—31no—
31C2×C3×Dic6—61no—
32C2×#626—32no—
33C2×#626—32no—
34C2×#625—32no—
35C2×#624—32no—
36C2×D3×Dic3—32no—
37C2×D3×Dic3—32no—
38#3799—38no—
39#3796—38no—
40#3795—38no—
41C22×C33C2311yescharacteristic
42C2×C3×C3⋊D3—22no—
43C2×C3×C3⋊D3—24no—
44C2×C3×C3⋊D3C2314yes—
45C2×C3×C3⋊D3C2312yes—
46C33×C4C2312yes—
47C3×C3⋊Dic3C2312yes—
48C3×C3⋊Dic3C2314yes—
49C2×#239—91no—
50C2×#237—91no—
51C2×#235—91no—
52C22×C3⋊D3C2×D312yes—
53C22×C3⋊D3C2×D311yescharacteristic
54C22×C3×D3—32no—
55C22×C3×D3—32no—
56C22×C3×D3—31no—
57C22×C3×D3—31no—
58C22×C3×D3—61no—
59C22×C3×D3—62no—
60C2×C32×C4—21no—
61C2×C32×C4—21no—
62C2×C32×C4—21no—
63C2×C32×C4—41no—
64C2×C32×C4C2×D311yescharacteristic
65C2×C32×C4C2×D311yescharacteristic
66C2×C32×C4C2×D311yescharacteristic
67C2×C3⋊Dic3C2×D311yescharacteristic
68C2×C3⋊Dic3C2×D312yes—
69C4⋊(C3⋊D3)C2×D314yescomplemented
70C4×C3⋊D3C2×D314yescomplemented
71#634C2×D314yescomplemented
72C2×C3×Dic3—31no—
73C2×C3×Dic3—32no—
74C2×C3×Dic3—31no—
75C2×C3×Dic3—32no—
76C2×C3×Dic3—62no—
77C2×C3×Dic3—61no—
78C3×D12—34no—
79C3×D12—34no—
80C3×D12—64no—
81C3×C4×D3—34no—
82C3×C4×D3—34no—
83C3×C4×D3—64no—
84C3×Dic6—34no—
85C3×Dic6—34no—
86C3×Dic6—64no—
87#626—38no—
88#626—38no—
89#625—38no—
90#624—38no—
91D3×Dic3—38no—
92D3×Dic3—38no—
93C2×C33#4112yescomplemented
94C2×C33C2411yescharacteristic
95C3×C3⋊D3—22no—
96C3×C3⋊D3—24no—
97C2×C3×Q8—91no—
98C2×C3×D4—91no—
99C22×C3×C4—91no—
100C2×#67—92no—
101C2×#67—92no—
102C22×Dic3—92no—
103#239—98no—
104#237—98no—
105#235—98no—
106C2×D12—31no—
107C2×D12—31no—
108C2×D12—61no—
109C2×D12—92no—
110C2×C4×D3—31no—
111C2×C4×D3—31no—
112C2×C4×D3—61no—
113C2×C4×D3—92no—
114C2×C4×D3—92no—
115C2×Dic6—31no—
116C2×Dic6—31no—
117C2×Dic6—61no—
118C22×C32—21no—
119C22×C32—21no—
120C22×C32—21no—
121C22×C32—41no—
122C22×C32C22×D311yescharacteristic
123C22×C32C22×D311yescharacteristic
124C22×C32C22×D311yescharacteristic
125C2×C3⋊D3—22no—
126C2×C3⋊D3—24no—
127C2×C3⋊D3C22×D314yes—
128C2×C3⋊D3C22×D312yes—
129C2×C3×D3—34no—
130C2×C3×D3—34no—
131C2×C3×D3—32no—
132C2×C3×D3—32no—
133C2×C3×D3—62no—
134C2×C3×D3—64no—
135C2×C3×D3—64no—
136C2×C3×D3—64no—
137C2×C3×D3—62no—
138C2×C3×D3—64no—
139C2×C3×D3—62no—
140C2×C3×D3—124no—
141C32×C4—22no—
142C32×C4—22no—
143C32×C4—22no—
144C32×C4—42no—
145C32×C4C22×D312yes—
146C32×C4C22×D312yes—
147C32×C4C22×D312yes—
148C3⋊Dic3C22×D312yes—
149C3⋊Dic3C22×D314yes—
150C3×Dic3—32no—
151C3×Dic3—34no—
152C3×Dic3—32no—
153C3×Dic3—34no—
154C3×Dic3—64no—
155C3×Dic3—62no—
156C2×#41—271noSylow
157C33C2×#4111yescharacteristic, complemented, Sylow
158C23×C3—92no—
159C23×C3—91no—
160C22×D3—31no—
161C22×D3—31no—
162C22×D3—32no—
163C22×D3—32no—
164C22×D3—61no—
165C22×D3—62no—
166C3×Q8—94no—
167C3×D4—94no—
168C2×C3×C4—21no—
169C2×C3×C4—21no—
170C2×C3×C4—21no—
171C2×C3×C4—41no—
172C2×C3×C4—91no—
173C2×C3×C4—92no—
174C2×C3×C4—94no—
175C2×C3×C4D3211yescharacteristic
176C2×C3×C4D3211yescharacteristic
177C2×C3×C4D3211yescharacteristic
178#67—98no—
179#67—98no—
180C2×Dic3—32no—
181C2×Dic3—32no—
182C2×Dic3—31no—
183C2×Dic3—31no—
184C2×Dic3—62no—
185C2×Dic3—61no—
186C2×Dic3—98no—
187D12—34no—
188D12—34no—
189D12—64no—
190D12—98no—
191C4×D3—34no—
192C4×D3—34no—
193C4×D3—64no—
194C4×D3—98no—
195C4×D3—98no—
196Dic6—34no—
197Dic6—34no—
198Dic6—64no—
199C2×C32—21no—
200C2×C32—21no—
201C2×C32—21no—
202C2×C32—22no—
203C2×C32—22no—
204C2×C32—22no—
205C2×C32—42no—
206C2×C32—41no—
207C2×C32#23512yescomplemented
208C2×C32#23712yescomplemented
209C2×C32#23912yescomplemented
210C2×C32C23×D311yescharacteristic
211C2×C32C23×D311yescharacteristic
212C2×C32C23×D311yescharacteristic
213C3⋊D3—22no—
214C3⋊D3—24no—
215C3×D3—64no—
216C3×D3—64no—
217C3×D3—62no—
218C3×D3—64no—
219C3×D3—62no—
220C3×D3—124no—
221#41—278no—
222C2×Q8—91no—
223C2×D4—91no—
224C2×D4—272no—
225C22×C4—91no—
226C22×C4—272no—
227C22×C3—21no—
228C22×C3—21no—
229C22×C3—21no—
230C22×C3—41no—
231C22×C3—94no—
232C22×C3—92no—
233C22×C3—182no—
234C22×C3—184no—
235C22×C3C2×D3211yescharacteristic
236C22×C3C2×D3211yescharacteristic
237C22×C3C2×D3211yescharacteristic
238C2×D3—32no—
239C2×D3—32no—
240C2×D3—34no—
241C2×D3—34no—
242C2×D3—62no—
243C2×D3—64no—
244C2×D3—64no—
245C2×D3—62no—
246C2×D3—64no—
247C2×D3—64no—
248C2×D3—62no—
249C2×D3—124no—
250C3×C4—22no—
251C3×C4—22no—
252C3×C4—22no—
253C3×C4—42no—
254C3×C4—92no—
255C3×C4—94no—
256C3×C4C2×D3212yes—
257C3×C4C2×D3212yes—
258C3×C4C2×D3212yes—
259Dic3—34no—
260Dic3—34no—
261Dic3—32no—
262Dic3—32no—
263Dic3—64no—
264Dic3—62no—
265C32—21no—
266C32—21no—
267C32—21no—
268C32—41no—
269C32C2×#23511yescharacteristic, complemented
270C32C2×#23711yescharacteristic, complemented
271C32C2×#23911yescharacteristic, complemented
272C23—91no—
273C23—92no—
274Q8—94no—
275D4—94no—
276D4—278no—
277C2×C4—91no—
278C2×C4—92no—
279C2×C4—94no—
280C2×C4—278no—
281C2×C4#111311yescharacteristic
282C2×C3—21no—
283C2×C3—21no—
284C2×C3—21no—
285C2×C3—22no—
286C2×C3—22no—
287C2×C3—22no—
288C2×C3—41no—
289C2×C3—42no—
290C2×C3—182no—
291C2×C3—184no—
292C2×C3#379512yescomplemented
293C2×C3#379612yescomplemented
294C2×C3#379912yescomplemented
295C2×C3C22×D3211yescharacteristic
296C2×C3C22×D3211yescharacteristic
297C2×C3C22×D3211yescharacteristic
298D3—64no—
299D3—62no—
300D3—64no—
301D3—64no—
302D3—62no—
303D3—124no—
304C22—92no—
305C22—94no—
306C22—182no—
307C22—184no—
308C22C2×#111311yescharacteristic
309C4—92no—
310C4—94no—
311C4C2×#111312yes—
312C3—21no—
313C3—21no—
314C3—21no—
315C3—41no—
316C3C2×#379511yescharacteristic, complemented
317C3C2×#379611yescharacteristic, complemented
318C3C2×#379911yescharacteristic, complemented
319C2—182no—
320C2—184no—
321C2#8904112yesdirect product factor
322C2C22×#111311yescharacteristic
subgroupG/H
centerC22C2×#1113
commutator, cosocleC2×C33C24
FrattiniC2C22×#1113
FittingC2×C33×C4C22
socleC22×C33C23
hypercenterC2×C4#1113
hypocenterC33C2×#41
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×C3×C4⋊(C3⋊D3)C2semidirect
2. C2×C3×C4×C3⋊D3C2semidirect
3. C2×C3×#634C2semidirect
4. C2×#6295C2semidirect
5. C2×#6294C2semidirect
6. #89041C2direct (N × H)
15. C3×C4⋊(C3⋊D3)C22semidirect
16. C3×C4×C3⋊D3C22semidirect
17. C3×#634C22semidirect
18. #6295C22semidirect
19. #6294C22semidirect
20. C2×C4⋊(C3⋊D3)D3semidirect
21. C2×C4×C3⋊D3D3semidirect
22. C2×#634D3semidirect
69. C4⋊(C3⋊D3)C2×D3semidirect
70. C4×C3⋊D3C2×D3semidirect
71. #634C2×D3semidirect
93. C2×C33#41semidirect
157. C33C2×#41semidirect
207. C2×C32#235semidirect
208. C2×C32#237semidirect
209. C2×C32#239semidirect
269. C32C2×#235semidirect
270. C32C2×#237semidirect
271. C32C2×#239semidirect
292. C2×C3#3795semidirect
293. C2×C3#3796semidirect
294. C2×C3#3799semidirect
316. C3C2×#3795semidirect
317. C3C2×#3796semidirect
318. C3C2×#3799semidirect
321. C2#89041direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C4⋊(C3⋊D3)2
C2×C3×C4×C3⋊D32
C2×C3×#6342
C2×#62952
C2×#62942
#890412
C2×#37993
C2×#37963
C2×#37953