Finite Group Databasebeta

← C2×#26012440 #36370015

Subgroups — C2×#26012440 #36370015

Order 22144.

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⋊#10621924C211yesmaximal, characteristic
2C22×C4×#10621924C211yesmaximal, characteristic
3#26012440C218yesmaximal, direct product factor
4C2×(C2×#10621924)⋊C4C211yesmaximal, characteristic
5C2×(C2×#10621924)⋊C4C211yesmaximal, characteristic
6C2×(C2×#10621924)⋊C4C211yesmaximal, characteristic
7C2×(C2×#10621924)⋊C4C211yesmaximal, characteristic
8C2×C4⋊C4×D173C211yesmaximal, characteristic
9C23×#10621924C2211yescharacteristic
10C23×#10621924C2211yescharacteristic
11C23×#10621924C2211yescharacteristic
12C23×#10621924C2211yescharacteristic
13C2×C4⋊#10621924—24no—
14C2×C4⋊#10621924C2214yes—
15C2×C4×#10621924—24no—
16C2×C4×#10621924C2214yes—
17(C2×#10621924)⋊C4C2214yes—
18(C2×#10621924)⋊C4C2214yes—
19(C2×#10621924)⋊C4C2214yes—
20(C2×#10621924)⋊C4C2214yes—
21C22×C4×D173C2211yescharacteristic
22C22×C4×D173C2211yescharacteristic
23C22×C4×D173C2211yescharacteristic
24C2×C4⋊Dic173C411yescharacteristic, complemented
25C2×#14583805C411yescharacteristic, complemented
26C2×#14583805C411yescharacteristic, complemented
27C4⋊C4×D173C414yescomplemented
28C4⋊C4×D173C2214yes—
29C2×C4⋊C4×C173C411yescharacteristic, complemented
30C22×#10621924—22no—
31C22×#10621924—22no—
32C22×#10621924—22no—
33C22×#10621924—22no—
34C22×#10621924—22no—
35C22×#10621924—22no—
36C22×#10621924—22no—
37C22×#10621924—22no—
38C22×#10621924—22no—
39C22×#10621924—22no—
40C22×#10621924D414yes—
41C22×#10621924Q814yes—
42C22×#10621924C2312yes—
43C22×#10621924C2312yes—
44C22×#10621924C2312yes—
45C22×#10621924C2312yes—
46C4⋊#10621924—28no—
47C4×#10621924—28no—
48C23×D173C2311yescharacteristic
49C22×Dic173C2×C411yescharacteristic
50C22×Dic173C2×C411yescharacteristic
51C22×Dic173C2×C411yescharacteristic
52C2×C4×D173—22no—
53C2×C4×D173—22no—
54C2×C4×D173—22no—
55C2×C4×D173—22no—
56C2×C4×D173—22no—
57C2×C4×D173—22no—
58C2×C4×D173C2×C412yes—
59C2×C4×D173C2×C412yes—
60C2×C4×D173C2×C412yes—
61C2×C4×D173C2312yes—
62C2×C4×D173C2312yes—
63C2×C4×D173C2312yes—
64C4⋊Dic173C2×C414yescomplemented
65#14583805C2×C414yescomplemented
66#14583805C2×C414yescomplemented
67C22×C4×C173C2×C411yescharacteristic
68C22×C4×C173C2×C411yescharacteristic
69C22×C4×C173C2×C411yescharacteristic
70C4⋊C4×C173C2×C414yescomplemented
71C2×#10621924—28no—
72C2×#10621924—28no—
73C2×#10621924—22no—
74C2×#10621924—22no—
75C2×#10621924—22no—
76C2×#10621924—22no—
77C2×#10621924—22no—
78C2×#10621924—22no—
79C2×#10621924—22no—
80C2×#10621924—22no—
81C2×#10621924—22no—
82C2×#10621924—22no—
83C2×#10621924—44no—
84C2×#10621924—44no—
85C2×#10621924C2×D414yes—
86C2×#10621924C2×Q814yes—
87C22×D173C22×C411yescharacteristic
88C22×D173C2×D412yes—
89C22×D173C2×Q812yes—
90C22×D173#4112yes—
91C22×D173#4112yes—
92C22×D173#4112yes—
93C22×D173#4112yes—
94C22×D173C2411yescharacteristic
95C2×Dic173—22no—
96C2×Dic173—22no—
97C2×Dic173—22no—
98C2×Dic173C22×C412yes—
99C2×Dic173C22×C412yes—
100C2×Dic173C22×C412yes—
101C4×D173—24no—
102C4×D173—24no—
103C4×D173—24no—
104C4×D173—22no—
105C4×D173—22no—
106C4×D173—22no—
107C4×D173—22no—
108C4×D173—22no—
109C4×D173—22no—
110C23×C173C22×C411yescharacteristic
111C2×C4×C173—22no—
112C2×C4×C173—22no—
113C2×C4×C173—22no—
114C2×C4×C173C22×C412yes—
115C2×C4×C173C22×C412yes—
116C2×C4×C173C22×C412yes—
117C22×C173#11712yes—
118C22×C173C4×D412yes—
119C22×C173C4×Q812yes—
120C22×C173C23×C411yescharacteristic
121C2×D173#11712yes—
122C2×D173C4×D412yes—
123C2×D173C4×Q812yes—
124C2×D173C22⋊Q814yes—
125C2×D173#12314yes—
126C2×D173#12514yes—
127C2×D173#12614yes—
128C2×D173C22×D411yescharacteristic
129C2×D173C22×Q811yescharacteristic
130C2×D173C2×#4111yescharacteristic
131C2×D173C2×#4111yescharacteristic
132C2×D173C2×#4111yescharacteristic
133C2×D173C2×#4111yescharacteristic
134#10621924—24no—
135#10621924—24no—
136#10621924—42no—
137#10621924—42no—
138C4×C173—22no—
139C4×C173—22no—
140C4×C173—22no—
141Dic173—22no—
142Dic173—22no—
143Dic173—22no—
144C2×C173#38714yescomplemented
145C2×C173C2×#11711yescharacteristic
146C2×C173C2×C4×D411yescharacteristic
147C2×C173C2×C4×Q811yescharacteristic
148D173#38714yescomplemented
149D173C2×C22⋊Q811yescharacteristic
150D173C2×#12311yescharacteristic
151D173C2×#12511yescharacteristic
152D173C2×#12611yescharacteristic
153C173C2×#38711yescharacteristic, complemented, Sylow
154C2×#387—1731nomaximal, Sylow
155C22×C4⋊C4—1731no—
156C22×C4⋊C4—1731no—
157C22×C42—1731no—
158#387—1738no—
159C2×#95—1731no—
160C2×#95—1731no—
161C2×#95—1731no—
162C2×#95—1731no—
163C23×C4—1731no—
164C23×C4—1731no—
165C23×C4—1731no—
166C23×C4—1731no—
167C23×C4—1731no—
168C23×C4—1731no—
169C23×C4—1731no—
170C2×C4⋊C4—3464no—
171C2×C4⋊C4—1734no—
172C2×C4⋊C4—1731no—
173C2×C4⋊C4—1731no—
174C2×C4⋊C4—1731no—
175C2×C4⋊C4—1734no—
176C2×C4⋊C4—1734no—
177C2×C4⋊C4#1062192411yescharacteristic, complemented
178C2×C42—3464no—
179C2×C42—1734no—
180#95—1734no—
181#95—1734no—
182#95—1734no—
183#95—1734no—
184C24—1731no—
185C22×C4—3462no—
186C22×C4—3462no—
187C22×C4—3462no—
188C22×C4—3462no—
189C22×C4—3462no—
190C22×C4—3462no—
191C22×C4—3462no—
192C22×C4—3462no—
193C22×C4—3462no—
194C22×C4—3462no—
195C22×C4—3462no—
196C22×C4—3462no—
197C22×C4—3462no—
198C22×C4—3462no—
199C22×C4—3462no—
200C22×C4—3462no—
201C22×C4—1732no—
202C22×C4—1732no—
203C22×C4—1734no—
204C22×C4—1734no—
205C22×C4—1732no—
206C22×C4—1732no—
207C22×C4—1731no—
208C22×C4—1731no—
209C22×C4—1731no—
210C22×C4—1732no—
211C22×C4—1732no—
212C22×C4—1732no—
213C22×C4—1732no—
214C22×C4—1732no—
215C22×C4—1732no—
216C22×C4C2×#1062192411yescharacteristic
217C22×C4C2×#1062192411yescharacteristic
218C22×C4C2×#1062192411yescharacteristic
219C4⋊C4—3468no—
220C4⋊C4—1734no—
221C4⋊C4—1734no—
222C4⋊C4—1734no—
223C4⋊C4C2×#1062192414yescomplemented
224C42—3468no—
225C23—1732no—
226C23—1732no—
227C23—1732no—
228C23—1732no—
229C23—1732no—
230C23—1732no—
231C23—1731no—
232C23—1731no—
233C23C22×#1062192411yescharacteristic
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—3462no—
238C2×C4—3462no—
239C2×C4—3462no—
240C2×C4—3462no—
241C2×C4—3464no—
242C2×C4—3462no—
243C2×C4—3462no—
244C2×C4—3464no—
245C2×C4—3464no—
246C2×C4—3462no—
247C2×C4—3468no—
248C2×C4—3468no—
249C2×C4—3462no—
250C2×C4—3462no—
251C2×C4—3462no—
252C2×C4—3462no—
253C2×C4—3462no—
254C2×C4—3462no—
255C2×C4—3462no—
256C2×C4—3462no—
257C2×C4—3462no—
258C2×C4—3462no—
259C2×C4—3462no—
260C2×C4—3462no—
261C2×C4—1734no—
262C2×C4—1734no—
263C2×C4—1732no—
264C2×C4—1732no—
265C2×C4—1732no—
266C2×C4—6924no—
267C2×C4—6924no—
268C2×C4C22×#1062192412yes—
269C2×C4C22×#1062192412yes—
270C2×C4C22×#1062192412yes—
271C22—1734no—
272C22—1734no—
273C22—1734no—
274C22—1734no—
275C22—1731no—
276C22—1731no—
277C22—1732no—
278C22—1731no—
279C22—1731no—
280C22—1732no—
281C22—1731no—
282C22—1731no—
283C22—1732no—
284C22#1914975812yes—
285C22D4×#1062192412yes—
286C22Q8×#1062192412yes—
287C22C23×#1062192411yescharacteristic
288C4—22no—
289C4—22no—
290C4—22no—
291C4—3464no—
292C4—3464no—
293C4—3462no—
294C4—3462no—
295C4—3462no—
296C4—6922no—
297C4—6922no—
298C2—1731no—
299C2—1731no—
300C2—1731no—
301C2—1731no—
302C2—1734no—
303C2#2601244014yesdirect product factor
304C2C2×#1914975811yescharacteristic
305C2C2×D4×#1062192411yescharacteristic
306C2C2×Q8×#1062192411yescharacteristic
subgroupG/H
centerC23C22×#10621924
commutatorC22×C173C23×C4
FrattiniC22C23×#10621924
FittingC2×C4⋊C4×C173C4
socleC23×C173C22×C4
prime radicalC23×D173C23
cosocleC22×D173C24
hypercenterC2×C4⋊C4#10621924
hypocenterC173C2×#387
LCSG ▷ C22×C173 ▷ C173
UCSC1 ◁ C23 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. #26012440C2direct (N × H)
24. C2×C4⋊Dic173C4semidirect
25. C2×#14583805C4semidirect
26. C2×#14583805C4semidirect
27. C4⋊C4×D173C4semidirect
29. C2×C4⋊C4×C173C4semidirect
64. C4⋊Dic173C2×C4semidirect
65. #14583805C2×C4semidirect
66. #14583805C2×C4semidirect
70. C4⋊C4×C173C2×C4semidirect
144. C2×C173#387semidirect
148. D173#387semidirect
153. C173C2×#387semidirect
177. C2×C4⋊C4#10621924semidirect
223. C4⋊C4C2×#10621924semidirect
303. C2#26012440direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4⋊#106219242
C22×C4×#106219242
#260124402
C2×(C2×#10621924)⋊C42
C2×C4⋊C4×D1732
C2×#387173