Finite Group Databasebeta

← C2×#20252498 #27081324

Subgroups — C2×#20252498 #27081324

Order 12032.

2434 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#10625354C211yesmaximal, characteristic
2#20252498C2116yesmaximal, direct product factor
3C2×Q8×C47⋊C8C211yesmaximal, characteristic, complemented
4C2×C8×D188C213yesmaximal, complemented
5C2×#15689188C213yesmaximal, complemented
6C2×#15689174C213yesmaximal, complemented
7C2×C4⋊C8×D47C213yesmaximal, complemented
8C2×C8×Q8×C47C211yesmaximal, characteristic, complemented
9C22×#10625354C411yescharacteristic
10C22×C8×D47C2216yes—
11C4×#10625354C418yes—
12C4×#10625354C2218yes—
13C2×Q8×Dic47C411yescharacteristic
14C2×C4×D188C2213yes—
15C2×#13093986C413yes—
16C2×#13093982C413yes—
17C2×C42×D47C2213yes—
18Q8×C47⋊C8C2218yescomplemented
19C8×D188C22124yescomplemented
20#15689188C22124yescomplemented
21#15689174C22124yescomplemented
22C2×C4⋊(C47⋊C8)C2213yes—
23C2×Dic47⋊C8C2213yes—
24C2×Dic47⋊C8C2211yescharacteristic
25C4⋊C8×D47C22124yescomplemented
26C2×(C2×D47)⋊C8C2213yes—
27C2×(C2×D47)⋊C8C2213yes—
28C2×C4×C47⋊C8C2213yes—
29C2×C4×Q8×C47C2211yescharacteristic
30C8×Q8×C47C2218yescomplemented
31C2×C4⋊C8×C47C2213yes—
32C2×C4×C8×C47C2213yes—
33C2×#10625354C8116yescomplemented
34C2×#10625354C2×C418yes—
35C22×D188C2×C413yes—
36C22×C4×D47C2×C413yes—
37C22×C4×D47C2×C413yes—
38C22×C4×D47C2313yes—
39C22×C47⋊C8C2311yescharacteristic
40C22×C47⋊C8C2313yes—
41C2×C8×D47—412no—
42C2×C8×D47C23124yes—
43Q8×Dic47C2×C418yes—
44C4×D188C2×C4112yes—
45C4×D188C23112yes—
46#13093986C2×C4124yes—
47#13093982C2×C4124yes—
48C2×C4⋊Dic47C2×C413yes—
49C2×(C2×D47)⋊C4C2×C416yes—
50C2×C4×Dic47C2×C413yes—
51C2×C4×Dic47C2×C413yes—
52C2×C4×Dic47C2311yescharacteristic
53C42×D47C2×C4112yes—
54C42×D47C23112yes—
55C4⋊(C47⋊C8)C23112yes—
56Dic47⋊C8C23112yes—
57Dic47⋊C8C2314yes—
58(C2×D47)⋊C8C23112yes—
59(C2×D47)⋊C8C23112yes—
60C4×C47⋊C8C23112yes—
61C22×Q8×C47C2×C411yescharacteristic
62C22×C8×C47C2311yescharacteristic
63C22×C8×C47C2313yes—
64C4×Q8×C47C2×C414yes—
65C4×Q8×C47C2314yes—
66C2×C4⋊C4×C47C2×C413yes—
67C2×C42×C47C2313yes—
68C4⋊C8×C47C23112yes—
69C4×C8×C47C23112yes—
70C23×D47C22×C413yes—
71C2×Q8×C47C2×C818yescomplemented
72C2×Q8×C47C22×C414yes—
73C22×C4×C47C22×C413yes—
74C22×C4×C47C22×C413yes—
75C22×C4×C47C2411yescharacteristic
76C22×Dic47C22×C411yescharacteristic
77C22×Dic47C22×C413yes—
78C22×Dic47C22×C411yescharacteristic
79#10625354C2×C8164yescomplemented
80C2×D188C2×C8124yescomplemented
81C2×D188C22×C4112yes—
82C2×C4×D47—26no—
83C2×C4×D47—26no—
84C2×C4×D47—46no—
85C2×C4×D47C2×C8124yescomplemented
86C2×C4×D47C22×C416yes—
87C2×C4×D47C22×C4112yes—
88C2×C4×D47C22×C4112yes—
89C2×C4×D47C2416yes—
90C2×C8×C47—26no—
91C2×C8×C47—22no—
92C2×C8×C47C2416yes—
93C2×C8×C47C2412yes—
94C4⋊C4×C47C22×C4112yes—
95C42×C47C22×C416yes—
96C42×C47C2416yes—
97(C2×D47)⋊C4C22×C4124yes—
98C4⋊Dic47C22×C4112yes—
99C4×Dic47C22×C412yes—
100C4×Dic47C22×C4112yes—
101C4×Dic47C22×C4112yes—
102C4×Dic47C2412yes—
103C2×C47⋊C8—26no—
104C2×C47⋊C8—22no—
105C2×C47⋊C8C2412yes—
106C2×C47⋊C8C2416yes—
107C8×D47—424no—
108C23×C47C23×C411yescharacteristic
109C22×D47—26no—
110C22×D47—26no—
111C22×D47C22×C8112yes—
112C22×D47C23×C416yes—
113Q8×C47C22×C8116yescomplemented
114C2×C4×C47—26no—
115C2×C4×C47C22×C8112yes—
116C2×C4×C47C23×C416yes—
117C2×C4×C47C23×C411yescharacteristic
118C2×C4×C47C23×C416yes—
119C2×C4×C47H(5,𝔽2)12yes—
120C2×C4×C472−1+412yes—
121C2×C4×C47C2511yescharacteristic
122C2×Dic47—26no—
123C2×Dic47—22no—
124C2×Dic47C22×C814yes—
125C2×Dic47C23×C412yes—
126C2×Dic47C23×C416yes—
127C2×Dic47C23×C412yes—
128D188C22×C8148yescomplemented
129C4×D47—212no—
130C4×D47—26no—
131C4×D47—26no—
132C4×D47—412no—
133C4×D47C22×C8148yescomplemented
134C8×C47—26no—
135C8×C47—22no—
136C47⋊C8—26no—
137C47⋊C8—22no—
138C2×#2988—471nomaximal, Sylow
139C22×C47#52012yes—
140C22×C47C23×C812yes—
141C22×C47#56812yes—
142C22×C47C24×C411yescharacteristic
143C2×D47—26no—
144C2×D47—224no—
145C2×D47—26no—
146C2×D47C23×C8112yes—
147C4×C47—26no—
148C4×C47#52012yes—
149C4×C47C23×C8112yes—
150C4×C47C2×H(5,𝔽2)11yescharacteristic
151C4×C47C2×2−1+411yescharacteristic
152Dic47—26no—
153Dic47—22no—
154Dic47C23×C814yes—
155C2×C4×#41—471no—
156#2988—4716no—
157C2×C8×Q8—471no—
158C2×C8×Q8D4711yescharacteristic, complemented
159C2×C8×D4—473no—
160C2×C8×D4—473no—
161C2×#431—473no—
162C22×C4⋊C8—473no—
163C2×C47#298814yescomplemented
164C2×C47C2×#52011yescharacteristic
165C2×C47C24×C811yescharacteristic
166C2×C47C2×#56811yescharacteristic
167D47—212no—
168C22×#41—471no—
169C23×C8—476no—
170C4×#41—478no—
171C4×#41—478no—
172C2×C4×Q8—471no—
173C2×C4×Q8C2×D4711yescharacteristic
174C2×C4×D4—473no—
175C2×C4×D4—473no—
176C2×#117—473no—
177C22×C42—473no—
178C8×Q8—478no—
179C8×Q8C2×D4718yescomplemented
180C8×D4—4724no—
181C8×D4—4724no—
182#431—4724no—
183C2×C4⋊C8—473no—
184C2×C4⋊C8—471no—
185C2×C4⋊C8—473no—
186C2×C4⋊C8—4724no—
187C2×C4⋊C8C2×D4713yes—
188C2×C22⋊C8—473no—
189C2×C22⋊C8—473no—
190C2×C4×C8—473no—
191C2×C4×C8C2×D4713yes—
192C47C2×#298811yescharacteristic, complemented, Sylow
193C2×#41—4716no—
194C2×#41—478no—
195C22×Q8C4×D4711yescharacteristic
196C22×D4—473no—
197C23×C4—473no—
198C23×C4—473no—
199C23×C4—473no—
200C22×C8—473no—
201C22×C8—471no—
202C22×C8—4724no—
203C22×C8—18812no—
204C22×C8C22×D4713yes—
205C22×C8C22×D4711yescharacteristic
206C4×Q8—478no—
207C4×Q8C4×D4714yes—
208C4×Q8C22×D4714yes—
209C4×D4—4724no—
210C4×D4—4712no—
211C4×D4—4712no—
212#117—4724no—
213C2×C4⋊C4—473no—
214C2×C4⋊C4C4×D4713yes—
215C2×C22⋊C4—476no—
216C2×C42—473no—
217C2×C42—473no—
218C2×C42—471no—
219C2×C42—4712no—
220C2×C42—4712no—
221C2×C42C22×D4713yes—
222C4⋊C8—4712no—
223C4⋊C8—474no—
224C4⋊C8—4712no—
225C4⋊C8C22×D47112yes—
226C22⋊C8—4712no—
227C22⋊C8—4712no—
228C4×C8—4712no—
229C4×C8C22×D47112yes—
230C24—473no—
231#41—4764no—
232C2×Q8C8×D4718yescomplemented
233C2×Q8C2×C4×D4714yes—
234C2×D4—4724no—
235C2×D4—4712no—
236C22×C4—471no—
237C22×C4—4724no—
238C22×C4—4712no—
239C22×C4—471no—
240C22×C4—473no—
241C22×C4—476no—
242C22×C4—476no—
243C22×C4—4712no—
244C22×C4—946no—
245C22×C4—946no—
246C22×C4—1886no—
247C22×C4C2×C4×D4713yes—
248C22×C4C2×C4×D4713yes—
249C22×C4C23×D4711yescharacteristic
250C2×C8—26no—
251C2×C8—22no—
252C2×C8—472no—
253C2×C8—476no—
254C2×C8—946no—
255C2×C8—942no—
256C2×C8—18824no—
257C2×C8C23×D4712yes—
258C2×C8C23×D4716yes—
259C4⋊C4—4712no—
260C4⋊C4C2×C4×D47112yes—
261C22⋊C4—4724no—
262C42—472no—
263C42—4712no—
264C42—4712no—
265C42—472no—
266C42C2×C4×D4716yes—
267C42C23×D4716yes—
268C23—4712no—
269C23—476no—
270C23—946no—
271C23—946no—
272C23C22×C4×D4711yescharacteristic
273Q8C2×C8×D47116yescomplemented
274D4—4748no—
275C2×C4—26no—
276C2×C4—474no—
277C2×C4—472no—
278C2×C4—4748no—
279C2×C4—472no—
280C2×C4—476no—
281C2×C4—946no—
282C2×C4—942no—
283C2×C4—9412no—
284C2×C4—946no—
285C2×C4—946no—
286C2×C4—18812no—
287C2×C4C2×C8×D47112yes—
288C2×C4C22×C4×D4716yes—
289C2×C4C22×C4×D4711yescharacteristic
290C2×C4C22×C4×D4716yes—
291C2×C4#1309407012yes—
292C2×C4#1309407212yes—
293C2×C4C24×D4711yescharacteristic
294C8—26no—
295C8—22no—
296C8—946no—
297C8—942no—
298C22—4712no—
299C22—946no—
300C22—946no—
301C22—9424no—
302C22#1568960512yes—
303C22C22×C8×D4712yes—
304C22#1568988812yes—
305C22C23×C4×D4711yescharacteristic
306C4—26no—
307C4—474no—
308C4—942no—
309C4—946no—
310C4#1568960512yes—
311C4C22×C8×D47112yes—
312C4C2×#1309407011yescharacteristic
313C4C2×#1309407211yescharacteristic
314C2—9412no—
315C2#2025249814yesdirect product factor
316C2C2×#1568960511yescharacteristic
317C2C23×C8×D4711yescharacteristic
318C2C2×#1568988811yescharacteristic
subgroupG/H
centerC22×C4C23×D47
commutatorC2×C47C24×C8
FrattiniC2×C4C24×D47
FittingC2×C8×Q8×C47C2
socleC23×C47C23×C4
prime radicalC22×#10625354C4
cosocleC2×C4×C47C25
hypercenterC2×C8×Q8D47
hypocenterC47C2×#2988
LCSG ▷ C2×C47 ▷ C47
UCSC1 ◁ C22×C4 ◁ C2×C8×Q8

Complements

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

NH ≅ G/Ntype
2. #20252498C2direct (N × H)
3. C2×Q8×C47⋊C8C2semidirect
4. C2×C8×D188C2semidirect
5. C2×#15689188C2semidirect
6. C2×#15689174C2semidirect
7. C2×C4⋊C8×D47C2semidirect
8. C2×C8×Q8×C47C2semidirect
18. Q8×C47⋊C8C22semidirect
19. C8×D188C22semidirect
20. #15689188C22semidirect
21. #15689174C22semidirect
25. C4⋊C8×D47C22semidirect
30. C8×Q8×C47C22semidirect
33. C2×#10625354C8semidirect
71. C2×Q8×C47C2×C8semidirect
79. #10625354C2×C8semidirect
80. C2×D188C2×C8semidirect
85. C2×C4×D47C2×C8semidirect
113. Q8×C47C22×C8semidirect
128. D188C22×C8semidirect
133. C4×D47C22×C8semidirect
158. C2×C8×Q8D47semidirect
163. C2×C47#2988semidirect
179. C8×Q8C2×D47semidirect
192. C47C2×#2988semidirect
232. C2×Q8C8×D47semidirect
273. Q8C2×C8×D47semidirect
315. C2#20252498direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4×#106253542
#202524982
C2×Q8×C47⋊C82
C2×C8×D1882
C2×#156891882
C2×#156891742
C2×C4⋊C8×D472
C2×C8×Q8×C472
C2×#298847