Finite Group Databasebeta

← C2×#20266126 #27187584

Subgroups — C2×#20266126 #27187584

Order 12928.

712 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4⋊#87938C211yesmaximal, characteristic
2C22×C4×#87938C211yesmaximal, characteristic
3#20266126C218yesmaximal, direct product factor
4C2×(C2×#87938)⋊C4C211yesmaximal, characteristic
5C2×(C2×#87938)⋊C4C211yesmaximal, characteristic
6C2×(C2×#87938)⋊C4C211yesmaximal, characteristic
7C2×(C2×#87938)⋊C4C211yesmaximal, characteristic
8C2×C4⋊C4×D101C211yesmaximal, characteristic
9C23×#87938C2211yescharacteristic
10C23×#87938C2211yescharacteristic
11C23×#87938C2211yescharacteristic
12C23×#87938C2211yescharacteristic
13C2×C4⋊#87938—24no—
14C2×C4⋊#87938C2214yes—
15C2×C4×#87938—24no—
16C2×C4×#87938C2214yes—
17(C2×#87938)⋊C4C2214yes—
18(C2×#87938)⋊C4C2214yes—
19(C2×#87938)⋊C4C2214yes—
20(C2×#87938)⋊C4C2214yes—
21C22×C4×D101C2211yescharacteristic
22C22×C4×D101C2211yescharacteristic
23C22×C4×D101C2211yescharacteristic
24C2×C4⋊Dic101C411yescharacteristic, complemented
25C2×#13108053C411yescharacteristic, complemented
26C2×#13108053C411yescharacteristic, complemented
27C4⋊C4×D101C414yescomplemented
28C4⋊C4×D101C2214yes—
29C2×C4⋊C4×C101C411yescharacteristic, complemented
30C22×#87938—22no—
31C22×#87938—22no—
32C22×#87938—22no—
33C22×#87938—22no—
34C22×#87938—22no—
35C22×#87938—22no—
36C22×#87938—22no—
37C22×#87938—22no—
38C22×#87938—22no—
39C22×#87938—22no—
40C22×#87938D414yes—
41C22×#87938Q814yes—
42C22×#87938C2312yes—
43C22×#87938C2312yes—
44C22×#87938C2312yes—
45C22×#87938C2312yes—
46C4⋊#87938—28no—
47C4×#87938—28no—
48C23×D101C2311yescharacteristic
49C22×Dic101C2×C411yescharacteristic
50C22×Dic101C2×C411yescharacteristic
51C22×Dic101C2×C411yescharacteristic
52C2×C4×D101—22no—
53C2×C4×D101—22no—
54C2×C4×D101—22no—
55C2×C4×D101—22no—
56C2×C4×D101—22no—
57C2×C4×D101—22no—
58C2×C4×D101C2×C412yes—
59C2×C4×D101C2×C412yes—
60C2×C4×D101C2×C412yes—
61C2×C4×D101C2312yes—
62C2×C4×D101C2312yes—
63C2×C4×D101C2312yes—
64C4⋊Dic101C2×C414yescomplemented
65#13108053C2×C414yescomplemented
66#13108053C2×C414yescomplemented
67C22×C4×C101C2×C411yescharacteristic
68C22×C4×C101C2×C411yescharacteristic
69C22×C4×C101C2×C411yescharacteristic
70C4⋊C4×C101C2×C414yescomplemented
71C23×C101C22×C411yescharacteristic
72C22×D101C22×C411yescharacteristic
73C22×D101C2×D412yes—
74C22×D101C2×Q812yes—
75C22×D101#4112yes—
76C22×D101#4112yes—
77C22×D101#4112yes—
78C22×D101#4112yes—
79C22×D101C2411yescharacteristic
80C2×#87938—28no—
81C2×#87938—28no—
82C2×#87938—22no—
83C2×#87938—22no—
84C2×#87938—22no—
85C2×#87938—22no—
86C2×#87938—22no—
87C2×#87938—22no—
88C2×#87938—22no—
89C2×#87938—22no—
90C2×#87938—22no—
91C2×#87938—22no—
92C2×#87938—44no—
93C2×#87938—44no—
94C2×#87938C2×D414yes—
95C2×#87938C2×Q814yes—
96C2×C4×C101—22no—
97C2×C4×C101—22no—
98C2×C4×C101—22no—
99C2×C4×C101C22×C412yes—
100C2×C4×C101C22×C412yes—
101C2×C4×C101C22×C412yes—
102C2×Dic101—22no—
103C2×Dic101—22no—
104C2×Dic101—22no—
105C2×Dic101C22×C412yes—
106C2×Dic101C22×C412yes—
107C2×Dic101C22×C412yes—
108C4×D101—24no—
109C4×D101—24no—
110C4×D101—24no—
111C4×D101—22no—
112C4×D101—22no—
113C4×D101—22no—
114C4×D101—22no—
115C4×D101—22no—
116C4×D101—22no—
117C22×C101#11712yes—
118C22×C101C4×D412yes—
119C22×C101C4×Q812yes—
120C22×C101C23×C411yescharacteristic
121C2×D101#11712yes—
122C2×D101C4×D412yes—
123C2×D101C4×Q812yes—
124C2×D101C22⋊Q814yes—
125C2×D101#12314yes—
126C2×D101#12514yes—
127C2×D101#12614yes—
128C2×D101C22×D411yescharacteristic
129C2×D101C22×Q811yescharacteristic
130C2×D101C2×#4111yescharacteristic
131C2×D101C2×#4111yescharacteristic
132C2×D101C2×#4111yescharacteristic
133C2×D101C2×#4111yescharacteristic
134#87938—24no—
135#87938—24no—
136#87938—42no—
137#87938—42no—
138C4×C101—22no—
139C4×C101—22no—
140C4×C101—22no—
141Dic101—22no—
142Dic101—22no—
143Dic101—22no—
144C2×C101#38714yescomplemented
145C2×C101C2×#11711yescharacteristic
146C2×C101C2×C4×D411yescharacteristic
147C2×C101C2×C4×Q811yescharacteristic
148D101#38714yescomplemented
149D101C2×C22⋊Q811yescharacteristic
150D101C2×#12311yescharacteristic
151D101C2×#12511yescharacteristic
152D101C2×#12611yescharacteristic
153C2×#387—1011nomaximal, Sylow
154C101C2×#38711yescharacteristic, complemented, Sylow
155C22×C4⋊C4—1011no—
156C22×C4⋊C4—1011no—
157C22×C42—1011no—
158#387—1018no—
159C2×#95—1011no—
160C2×#95—1011no—
161C2×#95—1011no—
162C2×#95—1011no—
163C23×C4—1011no—
164C23×C4—1011no—
165C23×C4—1011no—
166C23×C4—1011no—
167C23×C4—1011no—
168C23×C4—1011no—
169C23×C4—1011no—
170C2×C4⋊C4—1011no—
171C2×C4⋊C4—1011no—
172C2×C4⋊C4—1011no—
173C2×C4⋊C4—1014no—
174C2×C4⋊C4—1014no—
175C2×C4⋊C4—1014no—
176C2×C4⋊C4—2024no—
177C2×C4⋊C4#8793811yescharacteristic, complemented
178C2×C42—1014no—
179C2×C42—2024no—
180#95—1014no—
181#95—1014no—
182#95—1014no—
183#95—1014no—
184C24—1011no—
185C22×C4—1012no—
186C22×C4—1012no—
187C22×C4—1014no—
188C22×C4—1014no—
189C22×C4—1011no—
190C22×C4—1011no—
191C22×C4—1011no—
192C22×C4—1012no—
193C22×C4—1012no—
194C22×C4—1012no—
195C22×C4—1012no—
196C22×C4—1012no—
197C22×C4—1012no—
198C22×C4—1012no—
199C22×C4—1012no—
200C22×C4—2022no—
201C22×C4—2022no—
202C22×C4—2022no—
203C22×C4—2022no—
204C22×C4—2022no—
205C22×C4—2022no—
206C22×C4—2022no—
207C22×C4—2022no—
208C22×C4—2022no—
209C22×C4—2022no—
210C22×C4—2022no—
211C22×C4—2022no—
212C22×C4—2022no—
213C22×C4—2022no—
214C22×C4—2022no—
215C22×C4—2022no—
216C22×C4C2×#8793811yescharacteristic
217C22×C4C2×#8793811yescharacteristic
218C22×C4C2×#8793811yescharacteristic
219C4⋊C4—1014no—
220C4⋊C4—1014no—
221C4⋊C4—1014no—
222C4⋊C4—2028no—
223C4⋊C4C2×#8793814yescomplemented
224C42—2028no—
225C23—1012no—
226C23—1012no—
227C23—1012no—
228C23—1012no—
229C23—1012no—
230C23—1012no—
231C23—1011no—
232C23—1011no—
233C23C22×#8793811yescharacteristic
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—1014no—
238C2×C4—1014no—
239C2×C4—1012no—
240C2×C4—1012no—
241C2×C4—1012no—
242C2×C4—4044no—
243C2×C4—4044no—
244C2×C4—2022no—
245C2×C4—2022no—
246C2×C4—2022no—
247C2×C4—2022no—
248C2×C4—2024no—
249C2×C4—2022no—
250C2×C4—2022no—
251C2×C4—2024no—
252C2×C4—2024no—
253C2×C4—2022no—
254C2×C4—2022no—
255C2×C4—2022no—
256C2×C4—2022no—
257C2×C4—2022no—
258C2×C4—2022no—
259C2×C4—2022no—
260C2×C4—2022no—
261C2×C4—2022no—
262C2×C4—2022no—
263C2×C4—2028no—
264C2×C4—2028no—
265C2×C4—2022no—
266C2×C4—2022no—
267C2×C4—2022no—
268C2×C4C22×#8793812yes—
269C2×C4C22×#8793812yes—
270C2×C4C22×#8793812yes—
271C22—1011no—
272C22—1011no—
273C22—1012no—
274C22—1011no—
275C22—1011no—
276C22—1012no—
277C22—1011no—
278C22—1011no—
279C22—1012no—
280C22—1014no—
281C22—1014no—
282C22—1014no—
283C22—1014no—
284C22#1569233412yes—
285C22D4×#8793812yes—
286C22Q8×#8793812yes—
287C22C23×#8793811yescharacteristic
288C4—22no—
289C4—22no—
290C4—22no—
291C4—4042no—
292C4—4042no—
293C4—2024no—
294C4—2024no—
295C4—2022no—
296C4—2022no—
297C4—2022no—
298C2—1011no—
299C2—1011no—
300C2—1011no—
301C2—1011no—
302C2—1014no—
303C2#2026612614yesdirect product factor
304C2C2×#1569233411yescharacteristic
305C2C2×D4×#8793811yescharacteristic
306C2C2×Q8×#8793811yescharacteristic
subgroupG/H
centerC23C22×#87938
commutatorC22×C101C23×C4
FrattiniC22C23×#87938
FittingC2×C4⋊C4×C101C4
socleC23×C101C22×C4
prime radicalC23×D101C23
cosocleC22×D101C24
hypercenterC2×C4⋊C4#87938
hypocenterC101C2×#387
LCSG ▷ C22×C101 ▷ C101
UCSC1 ◁ C23 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. #20266126C2direct (N × H)
24. C2×C4⋊Dic101C4semidirect
25. C2×#13108053C4semidirect
26. C2×#13108053C4semidirect
27. C4⋊C4×D101C4semidirect
29. C2×C4⋊C4×C101C4semidirect
64. C4⋊Dic101C2×C4semidirect
65. #13108053C2×C4semidirect
66. #13108053C2×C4semidirect
70. C4⋊C4×C101C2×C4semidirect
144. C2×C101#387semidirect
148. D101#387semidirect
154. C101C2×#387semidirect
177. C2×C4⋊C4#87938semidirect
223. C4⋊C4C2×#87938semidirect
303. C2#20266126direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4⋊#879382
C22×C4×#879382
#202661262
C2×(C2×#87938)⋊C42
C2×C4⋊C4×D1012
C2×#387101