Finite Group Databasebeta

← C2×#20245997 #26374873

Subgroups — C2×#20245997 #26374873

Order 12032.

2678 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20245997C2116yesmaximal, direct product factor
2C2×#15689016C214yesmaximal
3C2×#15689009C214yesmaximal, complemented
4C2×#15688993C211yesmaximal, characteristic, complemented
5C2×#15688989C211yesmaximal, characteristic, complemented
6C2×#15688983C212yesmaximal
7C2×C4×C4⋊Dic47C212yesmaximal, complemented
8C22×#117×C47C211yesmaximal, characteristic
9C22×C4⋊Dic47C2212yes—
10C22×C4⋊Dic47C2212yes—
11C22×C4⋊Dic47C2214yes—
12C22×C22⋊Dic47C2214yes—
13C22×C4×Dic47C2214yes—
14#15689016C22132yes—
15#15689009C22132yescomplemented
16#15688993C2218yescomplemented
17#15688989C2218yescomplemented
18#15688983C22116yes—
19C4×C4⋊Dic47C22116yescomplemented
20C2×C47⋊#95C2214yes—
21C2×C47⋊#95C2214yes—
22C2×C47⋊#95C2214yes—
23C24×C4×C47C2211yescharacteristic
24C2×#117×C47C4116yescomplemented
25C2×#117×C47C2218yes—
26C22×C4⋊C4×C47C2212yes—
27C22×C22⋊C4×C47C2212yes—
28C22×C42×C47C2211yescharacteristic
29C22×C42×C47C2211yescharacteristic
30C23×Dic47C2314yes—
31C23×Dic47C2314yes—
32C2×C4⋊Dic47—28no—
33C2×C4⋊Dic47—28no—
34C2×C4⋊Dic47—48no—
35C2×C4⋊Dic47C2318yes—
36C2×C4⋊Dic47C2318yes—
37C2×C4⋊Dic47C23116yes—
38C2×C22⋊Dic47—48no—
39C2×C22⋊Dic47C23116yes—
40C2×C4×Dic47—48no—
41C2×C4×Dic47C23116yes—
42C47⋊#95C23116yes—
43C47⋊#95C23116yes—
44C47⋊#95C23116yes—
45C25×C47C2311yescharacteristic
46C23×C4×C47—24no—
47C23×C4×C47—24no—
48C23×C4×C47C2×C418yes—
49C23×C4×C47C2311yescharacteristic
50C23×C4×C47C2312yes—
51C23×C4×C47C2312yes—
52C23×C4×C47C2314yes—
53C23×C4×C47C2311yescharacteristic
54#117×C47C2×C4164yescomplemented
55C2×C4⋊C4×C47C2×C4116yes—
56C2×C4⋊C4×C47C2318yes—
57C2×C22⋊C4×C47C2×C4116yes—
58C2×C22⋊C4×C47C2318yes—
59C2×C42×C47C2×C418yes—
60C2×C42×C47C2×C418yes—
61C2×C42×C47C2314yes—
62C2×C42×C47C2314yes—
63C24×C47—22no—
64C24×C47—22no—
65C24×C47—24no—
66C24×C47—24no—
67C24×C47C22×C414yes—
68C24×C47C2412yes—
69C24×C47C2411yescharacteristic
70C22×C4×C47—22no—
71C22×C4×C47—22no—
72C22×C4×C47—24no—
73C22×C4×C47—24no—
74C22×C4×C47—24no—
75C22×C4×C47—24no—
76C22×C4×C47—232no—
77C22×C4×C47—28no—
78C22×C4×C47—28no—
79C22×C4×C47C22×C418yes—
80C22×C4×C47C22×C418yes—
81C22×C4×C47C22×C414yes—
82C22×C4×C47C22×C4116yes—
83C22×C4×C47C22×C414yes—
84C22×C4×C47#4114yes—
85C22×C4×C47#4114yes—
86C22×C4×C47C2414yes—
87C22×C4×C47C2414yes—
88C22×C4×C47C2412yes—
89C22×C4×C47C2412yes—
90C22×Dic47—28no—
91C22×Dic47—28no—
92C22×Dic47—28no—
93C22×Dic47—28no—
94C22×Dic47—28no—
95C22×Dic47—28no—
96C22×Dic47C2418yes—
97C22×Dic47C2418yes—
98C4⋊C4×C47C22×C4132yes—
99C22⋊C4×C47C22×C4132yes—
100C42×C47C22×C4116yes—
101C42×C47C22×C4116yes—
102C22⋊Dic47—416no—
103C4⋊Dic47—216no—
104C4⋊Dic47—216no—
105C4⋊Dic47—416no—
106C4×Dic47—416no—
107C23×C47—28no—
108C23×C47—216no—
109C23×C47—22no—
110C23×C47—22no—
111C23×C47—24no—
112C23×C47—24no—
113C23×C47—44no—
114C23×C47—44no—
115C23×C47—48no—
116C23×C47C23×C414yes—
117C23×C47C23×C412yes—
118C23×C47C2×#4112yes—
119C23×C47C2×#4112yes—
120C23×C47H(5,𝔽2)14yes—
121C23×C472−1+414yes—
122C23×C47C2511yescharacteristic
123C2×C4×C47—28no—
124C2×C4×C47—216no—
125C2×C4×C47—216no—
126C2×C4×C47—22no—
127C2×C4×C47—22no—
128C2×C4×C47—24no—
129C2×C4×C47—24no—
130C2×C4×C47—24no—
131C2×C4×C47—24no—
132C2×C4×C47—232no—
133C2×C4×C47#117116yes—
134C2×C4×C47C23×C418yes—
135C2×C4×C47C23×C418yes—
136C2×C4×C47C23×C414yes—
137C2×C4×C47C23×C414yes—
138C2×C4×C47C2×#4114yes—
139C2×C4×C47C2×#4114yes—
140C2×Dic47—28no—
141C2×Dic47—28no—
142C2×Dic47—28no—
143C2×Dic47—28no—
144C2×Dic47—28no—
145C2×Dic47—28no—
146C2×Dic47—416no—
147C2×Dic47—416no—
148C2×#2314—471nomaximal, Sylow
149C22×C47—24no—
150C22×C47—28no—
151C22×C47—416no—
152C22×C47—42no—
153C22×C47—42no—
154C22×C47—44no—
155C22×C47C2×#11714yes—
156C22×C47#52018yes—
157C22×C47#54014yes—
158C22×C47#54114yes—
159C22×C47#54218yes—
160C22×C47C24×C411yescharacteristic
161C22×C47C22×#4111yescharacteristic
162C22×C47C22×#4111yescharacteristic
163C22×C47C2×H(5,𝔽2)12yes—
164C22×C47C2×2−1+412yes—
165C4×C47—24no—
166C4×C47—28no—
167C4×C47—28no—
168C4×C47C2×#11718yes—
169Dic47—48no—
170Dic47—48no—
171C22×#117D4711yescharacteristic
172#2314—4716no—
173C2×#388—474no—
174C2×#387—474no—
175C2×#384—471no—
176C2×#382—471no—
177C2×#380—472no—
178C2×C4×C4⋊C4—472no—
179C2×C47—44no—
180C2×C47#231418yescomplemented
181C2×C47C22×#11711yescharacteristic
182C2×C47C2×#52012yes—
183C2×C47C2×#54011yescharacteristic
184C2×C47C2×#54111yescharacteristic
185C2×C47C2×#54212yes—
186C24×C4C2×D4711yescharacteristic
187C2×#117Dic47116yescomplemented
188C2×#117C2×D4718yes—
189C22×C4⋊C4—472no—
190C22×C4⋊C4—472no—
191C22×C4⋊C4—474no—
192C22×C4⋊C4C2×D4712yes—
193C22×C22⋊C4—474no—
194C22×C22⋊C4C2×D4712yes—
195C22×C42—474no—
196C22×C42C2×D4711yescharacteristic
197C22×C42C2×D4711yescharacteristic
198#388—4732no—
199#387—4732no—
200#384—478no—
201#382—478no—
202#380—4716no—
203C4×C4⋊C4—4716no—
204C2×#95—474no—
205C2×#95—474no—
206C2×#95—474no—
207C47C2×#231411yescharacteristic, complemented, Sylow
208C25C22×D4711yescharacteristic
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—474no—
212C23×C4—474no—
213C23×C4C2×Dic4718yes—
214C23×C4C22×D4714yes—
215C23×C4C22×D4712yes—
216C23×C4C22×D4712yes—
217C23×C4C22×D4711yescharacteristic
218C23×C4C22×D4711yescharacteristic
219#117C2×Dic47164yescomplemented
220C2×C4⋊C4—478no—
221C2×C4⋊C4—478no—
222C2×C4⋊C4—4716no—
223C2×C4⋊C4—948no—
224C2×C4⋊C4—948no—
225C2×C4⋊C4—1888no—
226C2×C4⋊C4C2×Dic47116yes—
227C2×C4⋊C4C22×D4718yes—
228C2×C22⋊C4—4716no—
229C2×C22⋊C4—1888no—
230C2×C22⋊C4C2×Dic47116yes—
231C2×C22⋊C4C22×D4718yes—
232C2×C42—4716no—
233C2×C42—1888no—
234C2×C42C2×Dic4718yes—
235C2×C42C2×Dic4718yes—
236C2×C42C22×D4714yes—
237C2×C42C22×D4714yes—
238#95—4716no—
239#95—4716no—
240#95—4716no—
241C24—22no—
242C24—22no—
243C24—24no—
244C24—24no—
245C24C22×Dic4714yes—
246C24C23×D4712yes—
247C24C23×D4711yescharacteristic
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—24no—
253C22×C4—24no—
254C22×C4—232no—
255C22×C4—28no—
256C22×C4—28no—
257C22×C4—478no—
258C22×C4—478no—
259C22×C4—948no—
260C22×C4—948no—
261C22×C4—948no—
262C22×C4—948no—
263C22×C4—948no—
264C22×C4—948no—
265C22×C4#1062535014yes—
266C22×C4#1062535014yes—
267C22×C4C22×Dic4718yes—
268C22×C4C22×Dic4718yes—
269C22×C4C22×Dic4714yes—
270C22×C4C22×Dic47116yes—
271C22×C4C22×Dic4714yes—
272C22×C4C23×D4712yes—
273C22×C4C23×D4714yes—
274C22×C4C23×D4714yes—
275C22×C4C23×D4712yes—
276C4⋊C4—9416no—
277C4⋊C4—9416no—
278C4⋊C4—18816no—
279C4⋊C4C22×Dic47132yes—
280C22⋊C4—18816no—
281C22⋊C4C22×Dic47132yes—
282C42—18816no—
283C42C22×Dic47116yes—
284C42C22×Dic47116yes—
285C23—28no—
286C23—216no—
287C23—22no—
288C23—22no—
289C23—24no—
290C23—24no—
291C23—44no—
292C23—44no—
293C23—48no—
294C23C23×Dic4714yes—
295C23C23×Dic4712yes—
296C23C2×#1062535012yes—
297C23C2×#1062535012yes—
298C23#1309407014yes—
299C23#1309407114yes—
300C23C24×D4711yescharacteristic
301C2×C4—28no—
302C2×C4—216no—
303C2×C4—216no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—24no—
307C2×C4—24no—
308C2×C4—24no—
309C2×C4—24no—
310C2×C4—232no—
311C2×C4—948no—
312C2×C4—948no—
313C2×C4—948no—
314C2×C4—948no—
315C2×C4—948no—
316C2×C4—948no—
317C2×C4—18816no—
318C2×C4—18816no—
319C2×C4#13093983116yes—
320C2×C4C23×Dic4718yes—
321C2×C4C23×Dic4718yes—
322C2×C4C23×Dic4714yes—
323C2×C4C23×Dic4714yes—
324C2×C4C2×#1062535014yes—
325C2×C4C2×#1062535014yes—
326C22—24no—
327C22—28no—
328C22—42no—
329C22—416no—
330C22—42no—
331C22—44no—
332C22C2×#1309398314yes—
333C22#1568960618yes—
334C22#1568972814yes—
335C22#1568973514yes—
336C22#1568974418yes—
337C22C24×Dic4711yescharacteristic
338C22C22×#1062535011yescharacteristic
339C22C22×#1062535011yescharacteristic
340C22C2×#1309407012yes—
341C22C2×#1309407112yes—
342C4—24no—
343C4—28no—
344C4—28no—
345C4—1888no—
346C4—1888no—
347C4C2×#1309398318yes—
348C2—44no—
349C2#2024599718yesdirect product factor
350C2C22×#1309398311yescharacteristic
351C2C2×#1568960612yes—
352C2C2×#1568972811yescharacteristic
353C2C2×#1568973511yescharacteristic
354C2C2×#1568974412yes—
subgroupG/H
centerC24C23×D47
commutatorC22×C47C24×C4
FrattiniC23C24×D47
FittingC22×#117×C47C2
socleC24×C47C24
prime radicalC25×C47C23
cosocleC23×C47C25
hypercenterC22×#117D47
hypocenterC47C2×#2314
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C24 ◁ C22×#117

Complements

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

NH ≅ G/Ntype
1. #20245997C2direct (N × H)
3. C2×#15689009C2semidirect
4. C2×#15688993C2semidirect
5. C2×#15688989C2semidirect
7. C2×C4×C4⋊Dic47C2semidirect
15. #15689009C22semidirect
16. #15688993C22semidirect
17. #15688989C22semidirect
19. C4×C4⋊Dic47C22semidirect
24. C2×#117×C47C4semidirect
54. #117×C47C2×C4semidirect
180. C2×C47#2314semidirect
187. C2×#117Dic47semidirect
207. C47C2×#2314semidirect
219. #117C2×Dic47semidirect
349. C2#20245997direct (N × H)

Maximal subgroups

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

subgroupindex
#202459972
C2×#156890162
C2×#156890092
C2×#156889932
C2×#156889892
C2×#156889832
C2×C4×C4⋊Dic472
C22×#117×C472
C2×#231447