Finite Group Databasebeta

← C2×#20257173 #27124693

Subgroups — C2×#20257173 #27124693

Order 12032.

11126 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13094071C212yesmaximal, complemented
2#20257173C2132yesmaximal, direct product factor
3C2×Q8×D188C214yesmaximal, complemented
4C2×#15689777C2112yesmaximal, complemented
5C2×#15689699C216yesmaximal, complemented
6C2×#15689673C213yesmaximal, complemented
7C22×#13094024C213yesmaximal, complemented
8C2×C4×#41×C47C211yesmaximal, characteristic, complemented
9C2×#13094071—232no—
10C2×#13094071C22132yescomplemented
11C22×#10625350C2216yes—
12C22×#10625352C22112yes—
13C23×Dic94C2216yes—
14C22×Q8×D47C2214yes—
15Q8×D188C22164yescomplemented
16#15689777C221192yescomplemented
17#15689699C22196yescomplemented
18#15689673C22148yescomplemented
19C2×#13094024C22112yes—
20C2×#13094024C2213yes—
21C2×#13094024C2216yes—
22C2×#13094024C22148yescomplemented
23C2×#13094021C2211yescharacteristic
24C2×#13094012C2216yes—
25C2×#13094008C22124yes—
26C2×#13094004C22124yes—
27C2×#13093998C22112yes—
28C2×C4×D188C22112yes—
29C22×C4⋊Dic47C22112yes—
30C22×#41×C47C2211yescharacteristic
31C4×#41×C47C22116yescomplemented
32C2×C4×Q8×C47C2212yes—
33C2×C4×D4×C47C2216yes—
34C2×#117×C47C2213yes—
35C22×C42×C47C2213yes—
36#13094071—2256no—
37C2×#10625350—248no—
38C2×#10625350C23148yes—
39C2×#10625352—296no—
40C2×#10625352C23196yes—
41C22×Dic94—248no—
42C22×Dic94C23112yes—
43C22×Dic94C2316yes—
44C22×Dic94C23148yes—
45C2×Q8×D47—232no—
46C2×Q8×D47C23132yes—
47C22×D188C2314yes—
48C22×D188C2312yes—
49C22×C22⋊D47C23112yes—
50C23×Dic47C23112yes—
51C22×C4×D47C23112yes—
52#13094024C23196yes—
53#13094024C23124yes—
54#13094024C23148yes—
55#13094021C2318yes—
56#13094012C23148yes—
57#13094008C231192yes—
58#13094004C231192yes—
59#13093998C23196yes—
60C4×D188C23196yes—
61C2×C4⋊Dic47C23196yes—
62C2×C4⋊Dic47C23112yes—
63C2×C4⋊Dic47C23124yes—
64C2×(C2×D47)⋊C4C23124yes—
65C2×#41×C47D4116yescomplemented
66C2×#41×C47C2318yes—
67C22×Q8×C47C2311yescharacteristic
68C22×D4×C47C2313yes—
69C23×C4×C47C2316yes—
70C23×C4×C47C2313yes—
71C4×Q8×C47C23116yes—
72C4×D4×C47C23148yes—
73#117×C47C23124yes—
74C2×C4⋊C4×C47C2316yes—
75C2×C22⋊C4×C47C2316yes—
76C2×C42×C47C2311yescharacteristic
77C2×C42×C47C2313yes—
78C2×C42×C47C2316yes—
79C2×C42×C47C23124yes—
80C24×C47C2413yes—
81C23×D47C2414yes—
82#41×C47C2×D4164yescomplemented
83C2×Q8×C47C2×D418yes—
84C2×Q8×C47C2414yes—
85C2×D4×C47C2×D4124yes—
86C2×D4×C47C24112yes—
87C22×C4×C47—224no—
88C22×C4×C47C2×D4124yes—
89C22×C4×C47C2416yes—
90C22×C4×C47C24124yes—
91C22×C4×C47C2411yescharacteristic
92C22×C4×C47C2413yes—
93C22×C4×C47C24112yes—
94C22×C4×C47C2412yes—
95C2×C22⋊D47—248no—
96C2×C22⋊D47C24148yes—
97C22×Dic47—248no—
98C22×Dic47C24112yes—
99C22×Dic47C24148yes—
100Q8×D47—2128no—
101#10625352—2384no—
102#10625350—2192no—
103C2×D188—28no—
104C2×D188—216no—
105C2×D188C24116yes—
106C2×D188C2418yes—
107C2×C4×D47—248no—
108C2×C4×D47C24148yes—
109C2×Dic94—248no—
110C2×Dic94—224no—
111C2×Dic94—2192no—
112C2×Dic94C24148yes—
113C2×Dic94C24124yes—
114C4⋊C4×C47C24124yes—
115C22⋊C4×C47C24124yes—
116C42×C47C2414yes—
117C42×C47C24112yes—
118C42×C47C24124yes—
119(C2×D47)⋊C4C24196yes—
120C4⋊Dic47C24148yes—
121C4⋊Dic47C24196yes—
122C23×C47—212no—
123C23×C47C22×D4112yes—
124C23×C47C2516yes—
125C23×C47C2511yescharacteristic
126C22×D47—28no—
127C22×D47—216no—
128C22×D47C2518yes—
129Q8×C47C22×D4116yes—
130D4×C47C22×D4148yes—
131C2×C4×C47—212no—
132C2×C4×C47—248no—
133C2×C4×C47C22×D414yes—
134C2×C4×C47C22×D4112yes—
135C2×C4×C47C22×D4148yes—
136C2×C4×C472−1+418yescomplemented
137C2×C4×C47C25112yes—
138C2×C4×C47C2512yes—
139C2×C4×C47C2516yes—
140C2×C4×C47C2514yes—
141C22⋊D47—296no—
142C2×Dic47—224no—
143C2×Dic47—296no—
144C2×Dic47C25124yes—
145D188—216no—
146D188—232no—
147C4×D47—296no—
148Dic94—248no—
149Dic94—296no—
150C2×#3499—471nomaximal, Sylow
151C22×C47—212no—
152C22×C47—224no—
153C22×C47C23×D4112yes—
154C22×C47C23×D412yes—
155C22×C47C2×2−1+414yes—
156C22×C47C2611yescharacteristic
157C2×D47—28no—
158C2×D47—216no—
159C2×D47—416no—
160C4×C47—212no—
161C4×C47C23×D414yes—
162C4×C47C23×D4112yes—
163C4×C47C2×2−1+418yescomplemented
164Dic47—224no—
165C22×2−1+4—472no—
166#3499—4732no—
167C2×D4×Q8—474no—
168C2×#547—4712no—
169C2×#536—476no—
170C2×#532—473no—
171C22×C4⋊Q8—473no—
172C2×C4×#41D4711yescharacteristic, complemented
173C2×C47—212no—
174C2×C47#349914yescomplemented
175C2×C47C24×D411yescharacteristic
176C2×C47C22×2−1+412yes—
177D47—48no—
178C2×2−1+4—4732no—
179C2×2−1+4—9432no—
180C22×#41—4712no—
181C22×#41—476no—
182C22×#41C2×D4711yescharacteristic
183C23×Q8—474no—
184C23×Q8—476no—
185D4×Q8—4764no—
186#547—47192no—
187#536—4796no—
188#532—4748no—
189C2×C4⋊Q8—473no—
190C2×C4⋊Q8—476no—
191C2×C4⋊Q8—4712no—
192C2×C4⋊Q8—4748no—
193C2×#127—471no—
194C2×#124—476no—
195C2×#123—4724no—
196C2×C22⋊Q8—4724no—
197C2×#121—4712no—
198C4×#41C2×D47116yescomplemented
199C2×C4×Q8C2×D4712yes—
200C2×C4×D4—4712no—
201C2×C4×D4C2×D4716yes—
202C2×#117C2×D4713yes—
203C22×C4⋊C4—4712no—
204C22×C42C2×D4713yes—
205C47C2×#349911yescharacteristic, complemented, Sylow
2062−1+4—94256no—
207C2×#41—4796no—
208C2×#41—4748no—
209C2×#41—9496no—
210C2×#41—9448no—
211C2×#41D188116yescomplemented
212C2×#41C22×D4718yes—
213C22×Q8—4732no—
214C22×Q8—4712no—
215C22×Q8—476no—
216C22×Q8—4748no—
217C22×Q8—9432no—
218C22×Q8—9448no—
219C22×Q8C22×D4711yescharacteristic
220C22×D4—4712no—
221C22×D4—474no—
222C22×D4—472no—
223C22×D4C22×D4713yes—
224C23×C4—4712no—
225C23×C4—4712no—
226C23×C4C22×D4716yes—
227C23×C4C22×D4713yes—
228C4⋊Q8—4724no—
229C4⋊Q8—4748no—
230C4⋊Q8—4796no—
231#127—478no—
232#124—4748no—
233#123—47192no—
234C22⋊Q8—47192no—
235#121—4796no—
236C4×Q8C22×D47116yes—
237C4×D4—4796no—
238C4×D4C22×D47148yes—
239#117C22×D47124yes—
240C2×C4⋊C4—4712no—
241C2×C4⋊C4—4724no—
242C2×C4⋊C4—4796no—
243C2×C4⋊C4C22×D4716yes—
244C2×C22⋊C4—4724no—
245C2×C22⋊C4C22×D4716yes—
246C2×C42C22×D4713yes—
247C2×C42C22×D4716yes—
248C2×C42C22×D47124yes—
249C2×C42C22×D4711yescharacteristic
250C24—474no—
251C24C23×D4713yes—
252#41—94384no—
253#41—94192no—
254#41C2×D188164yescomplemented
255C2×Q8—4748no—
256C2×Q8—4724no—
257C2×Q8—94128no—
258C2×Q8—9448no—
259C2×Q8—9424no—
260C2×Q8—94192no—
261C2×Q8C2×D18818yes—
262C2×Q8C23×D4714yes—
263C2×D4—4748no—
264C2×D4—4716no—
265C2×D4—478no—
266C2×D4—9448no—
267C2×D4—948no—
268C2×D4—9416no—
269C2×D4C2×D188124yes—
270C2×D4C23×D47112yes—
271C22×C4—224no—
272C22×C4—4748no—
273C22×C4—4712no—
274C22×C4—4748no—
275C22×C4—9448no—
276C22×C4—9448no—
277C22×C4C2×D188124yes—
278C22×C4C23×D47124yes—
279C22×C4C23×D4716yes—
280C22×C4C23×D47112yes—
281C22×C4C23×D4711yescharacteristic
282C22×C4C23×D4713yes—
283C22×C4C23×D4712yes—
284C4⋊C4—4748no—
285C4⋊C4—4796no—
286C4⋊C4C23×D47124yes—
287C22⋊C4—4796no—
288C22⋊C4C23×D47124yes—
289C42C23×D4714yes—
290C42C23×D47112yes—
291C42C23×D47124yes—
292C23—212no—
293C23—478no—
294C23—948no—
295C23—9416no—
296C23C22×D188112yes—
297C23C24×D4716yes—
298C23C24×D4711yescharacteristic
299Q8—9496no—
300Q8—9448no—
301Q8C22×D188116yes—
302D4—9496no—
303D4—9416no—
304D4—9432no—
305D4C22×D188148yes—
306C2×C4—212no—
307C2×C4—248no—
308C2×C4—4724no—
309C2×C4—9424no—
310C2×C4—9496no—
311C2×C4—9496no—
312C2×C4C22×D188112yes—
313C2×C4C22×D18814yes—
314C2×C4C22×D188148yes—
315C2×C4#1309407118yescomplemented
316C2×C4C24×D47112yes—
317C2×C4C24×D4712yes—
318C2×C4C24×D4716yes—
319C2×C4C24×D4714yes—
320C22—212no—
321C22—224no—
322C22—948no—
323C22—9416no—
324C22—18816no—
325C22C23×D188112yes—
326C22C23×D18812yes—
327C22C2×#1309407114yes—
328C22C25×D4711yescharacteristic
329C4—212no—
330C4—9424no—
331C4C23×D188112yes—
332C4C23×D18814yes—
333C4C2×#1309407118yescomplemented
334C2—212no—
335C2—1888no—
336C2#2025717314yesdirect product factor
337C2C24×D18811yescharacteristic
338C2C22×#1309407112yes—
subgroupG/H
centerC23C24×D47
commutator, cosocleC22×C47C26
FrattiniC22C25×D47
FittingC2×C4×#41×C47C2
socleC23×C47C25
hypercenterC2×C4×#41D47
hypocenterC47C2×#3499
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C4×#41

Complements

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

NH ≅ G/Ntype
1. C22×#13094071C2semidirect
2. #20257173C2direct (N × H)
3. C2×Q8×D188C2semidirect
4. C2×#15689777C2semidirect
5. C2×#15689699C2semidirect
6. C2×#15689673C2semidirect
7. C22×#13094024C2semidirect
8. C2×C4×#41×C47C2semidirect
10. C2×#13094071C22semidirect
15. Q8×D188C22semidirect
16. #15689777C22semidirect
17. #15689699C22semidirect
18. #15689673C22semidirect
22. C2×#13094024C22semidirect
31. C4×#41×C47C22semidirect
65. C2×#41×C47D4semidirect
82. #41×C47C2×D4semidirect
136. C2×C4×C472−1+4semidirect
163. C4×C47C2×2−1+4semidirect
172. C2×C4×#41D47semidirect
174. C2×C47#3499semidirect
198. C4×#41C2×D47semidirect
205. C47C2×#3499semidirect
211. C2×#41D188semidirect
254. #41C2×D188semidirect
315. C2×C4#13094071semidirect
333. C4C2×#13094071semidirect
336. C2#20257173direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#130940712
#202571732
C2×Q8×D1882
C2×#156897772
C2×#156896992
C2×#156896732
C22×#130940242
C2×C4×#41×C472
C2×#349947