Finite Group Databasebeta

← C2×#20253927 #27085911

Subgroups — C2×#20253927 #27085911

Order 12032.

1854 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689751C212yesmaximal
2#20253927C2116yesmaximal, direct product factor
3C2×#15689532C214yesmaximal, complemented
4C2×#15689528C212yesmaximal, complemented
5C2×#15689525C212yesmaximal, complemented
6C2×#15689165C212yesmaximal
7C2×#15689161C211yesmaximal, characteristic
8C22×#13093958C211yesmaximal, characteristic
9C2×C47×#405C211yesmaximal, characteristic
10#15689751—28no—
11#15689751C22116yes—
12C2×#13094024C2214yes—
13C2×#13094023—22no—
14C2×#13094016C2214yes—
15C2×#13094015—22no—
16C2×#13094001—24no—
17C2×#13093983C2212yes—
18C22×C4⋊Dic47C2212yes—
19#15689532C22132yescomplemented
20#15689528C22116yescomplemented
21#15689525C22116yescomplemented
22#15689165C22116yes—
23#15689161C2218yes—
24C2×#13093958C2212yes—
25C2×#13093958C2212yes—
26C2×#13093958C2218yes—
27C2×#13093958C2211yescharacteristic
28C2×#13093958C2211yescharacteristic
29C2×#13093958C2214yes—
30C2×#13093956C2212yes—
31C2×#13093956C2214yes—
32C22×M16×C47C2211yescharacteristic
33C23×C8×C47C2211yescharacteristic
34C2×#117×C47C2211yescharacteristic
35C47×#405C2218yes—
36C2×#97×C47C2212yes—
37C2×C4×C8×C47C2212yes—
38C22×Dic94—24no—
39C23×Dic47—22no—
40#13094024—28no—
41#13094024C23116yes—
42#13094023—216no—
43#13094016—28no—
44#13094016C23116yes—
45#13094015—216no—
46#13094001—232no—
47#13093983—24no—
48#13093983C2318yes—
49C2×C4⋊Dic47—24no—
50C2×C4⋊Dic47—22no—
51C2×C4⋊Dic47—24no—
52C2×C4⋊Dic47C2318yes—
53C2×C4⋊Dic47C2314yes—
54C2×C4⋊Dic47C2312yes—
55C2×C4⋊Dic47C2312yes—
56C2×Dic47⋊C4—24no—
57C2×Dic47⋊C4—24no—
58C2×C22⋊Dic47—22no—
59C2×C4×Dic47—22no—
60#13093958C2314yes—
61#13093958C2318yes—
62#13093958C2314yes—
63#13093958C23116yes—
64#13093958C2318yes—
65#13093956C23116yes—
66#13093956C2318yes—
67C23×C4×C47C2311yescharacteristic
68C2×M16×C47Q818yescomplemented
69C2×M16×C47C2314yes—
70C22×C8×C47Q818yescomplemented
71C22×C8×C47C2311yescharacteristic
72C22×C8×C47C2311yescharacteristic
73C22×C8×C47C2312yes—
74C22×C8×C47C2314yes—
75#117×C47D414yes—
76#117×C47C2314yes—
77C2×C4⋊C4×C47D411yescharacteristic
78C2×C4⋊C4×C47D411yescharacteristic
79C2×C22⋊C4×C47D412yes—
80C2×C42×C47C2312yes—
81#97×C47C2318yes—
82C4×C8×C47C2318yes—
83C24×C47C2×D411yescharacteristic
84C22×C4×C47C2×D412yes—
85C22×C4×C47C2×D412yes—
86C22×C4×C47C2×D411yescharacteristic
87C22×C4×C47C2×D412yes—
88C22×C4×C47C2×Q814yes—
89C22×C4×C47C2×Q814yes—
90C22×C4×C47C2411yescharacteristic
91C22×C4×C47C2412yes—
92C22×Dic47—22no—
93C22×Dic47—22no—
94C22×Dic47—24no—
95C22×Dic47—28no—
96C22×Dic47—84no—
97C2×Dic94—216no—
98C2×Dic94—88no—
99M16×C47C2×Q8116yescomplemented
100C2×C8×C47C2×Q814yes—
101C2×C8×C47C2×Q814yes—
102C2×C8×C47C2×Q818yes—
103C2×C8×C47C2×Q8116yescomplemented
104C2×C8×C47C2412yes—
105C2×C8×C47C2412yes—
106C2×C8×C47C2414yes—
107C4⋊C4×C47C2×D414yes—
108C4⋊C4×C47C2×D414yes—
109C22⋊C4×C47C2×D418yes—
110C42×C47C2×D414yes—
111C42×C47C2414yes—
112C22⋊Dic47—28no—
113C4⋊Dic47—28no—
114C4⋊Dic47—24no—
115C4⋊Dic47—216no—
116C4⋊Dic47—22no—
117C4⋊Dic47—22no—
118C4⋊Dic47C2414yes—
119C4⋊Dic47C2418yes—
120C4⋊Dic47C2414yes—
121Dic47⋊C4—216no—
122Dic47⋊C4—216no—
123C4×Dic47—28no—
124C23×C47—22no—
125C23×C47—22no—
126C23×C47C4⋊Q814yes—
127C23×C47C22×D412yes—
128C23×C47C22×D411yescharacteristic
129C2×C4×C47—24no—
130C2×C4×C47—24no—
131C2×C4×C47C4⋊Q818yes—
132C2×C4×C47C4⋊Q814yes—
133C2×C4×C47C22×D414yes—
134C2×C4×C47C22×D414yes—
135C2×C4×C47C22×D412yes—
136C2×C4×C47C22×D411yescharacteristic
137C2×C4×C47C22×Q812yes—
138C2×C4×C47C22×Q812yes—
139C2×C4×C47C22×Q814yes—
140C2×C4×C47C22×Q814yes—
141C2×C4×C47C2511yescharacteristic
142C2×Dic47—24no—
143C2×Dic47—24no—
144C2×Dic47—28no—
145C2×Dic47—44no—
146C2×Dic47—48no—
147C2×Dic47—44no—
148C2×Dic47—88no—
149Dic94—816no—
150C8×C47C22×Q814yes—
151C8×C47C22×Q814yes—
152C8×C47C22×Q818yes—
153C2×#3164—471nomaximal, Sylow
154C22×C47—28no—
155C22×C47—22no—
156C22×C47—22no—
157C22×C47C2×C4⋊Q814yes—
158C22×C47C2×C4⋊Q812yes—
159C22×C47#57612yes—
160C22×C47#57812yes—
161C22×C47C23×D411yescharacteristic
162C4×C47—24no—
163C4×C47—24no—
164C4×C47C2×C4⋊Q814yes—
165C4×C47C2×C4⋊Q812yes—
166C4×C47C23×Q811yescharacteristic
167C4×C47C23×Q811yescharacteristic
168Dic47—44no—
169Dic47—48no—
170Dic47—44no—
171C2×#544—472no—
172#3164—4716no—
173C2×#501—474no—
174C2×#500—472no—
175C2×#499—472no—
176C2×#428—472no—
177C2×#427—471no—
178C22×#107—471no—
179C2×#405D4711yescharacteristic
180C2×C47—24no—
181C2×C47#316414yescomplemented
182C2×C47C22×C4⋊Q811yescharacteristic
183C2×C47C2×#57611yescharacteristic
184C2×C47C2×#57811yescharacteristic
185C22×M16C2×D4711yescharacteristic
186C23×C8C2×D4711yescharacteristic
187#544—4716no—
188#544—948no—
189C2×C4⋊Q8—474no—
190C2×C4⋊Q8—942no—
191C2×#125—474no—
192C2×#125—942no—
193C2×C22⋊Q8—944no—
194C2×#117—472no—
195C2×#117C2×D4711yescharacteristic
196C22×C4⋊C4—472no—
197#501—4732no—
198#500—4716no—
199#499—4716no—
200#428—4716no—
201#427—478no—
202C2×#107—471no—
203C2×#107—472no—
204C2×#107—471no—
205C2×#107—474no—
206C2×#107—472no—
207C2×#107—478no—
208C2×#106—474no—
209C2×#106—472no—
210#405C2×D4718yes—
211C2×#97C2×D4712yes—
212C2×C4×C8C2×D4712yes—
213C47C2×#316411yescharacteristic, complemented, Sylow
214C22×Q8—944no—
215C23×C4—942no—
216C23×C4C22×D4711yescharacteristic
217C2×M16Dic9418yescomplemented
218C2×M16C22×D4714yes—
219C22×C8Dic9418yescomplemented
220C22×C8C22×D4711yescharacteristic
221C22×C8C22×D4711yescharacteristic
222C22×C8C22×D4712yes—
223C22×C8C22×D4714yes—
224C4⋊Q8—4716no—
225C4⋊Q8—9416no—
226C4⋊Q8—948no—
227#125—4716no—
228#125—9416no—
229#125—948no—
230C22⋊Q8—9432no—
231#117—478no—
232#117—944no—
233#117D18814yes—
234#117C22×D4714yes—
235C2×C4⋊C4—474no—
236C2×C4⋊C4—478no—
237C2×C4⋊C4—472no—
238C2×C4⋊C4—472no—
239C2×C4⋊C4—942no—
240C2×C4⋊C4—944no—
241C2×C4⋊C4—944no—
242C2×C4⋊C4—944no—
243C2×C4⋊C4—944no—
244C2×C4⋊C4D18811yescharacteristic
245C2×C4⋊C4D18811yescharacteristic
246C2×C22⋊C4—942no—
247C2×C22⋊C4D18812yes—
248C2×C42—942no—
249C2×C42C22×D4712yes—
250#107—478no—
251#107—478no—
252#107—474no—
253#107—474no—
254#107—4716no—
255#106—478no—
256#106—4716no—
257#97C22×D4718yes—
258C4×C8C22×D4718yes—
259C24C2×D18811yescharacteristic
260C2×Q8—9416no—
261C2×Q8—3768no—
262C22×C4—948no—
263C22×C4—942no—
264C22×C4—942no—
265C22×C4—944no—
266C22×C4—3764no—
267C22×C4C2×Dic9414yes—
268C22×C4C2×Dic9414yes—
269C22×C4C2×D18812yes—
270C22×C4C2×D18812yes—
271C22×C4C2×D18812yes—
272C22×C4C2×D18811yescharacteristic
273C22×C4C23×D4712yes—
274C22×C4C23×D4711yescharacteristic
275M16C2×Dic94116yescomplemented
276C2×C8C2×Dic9414yes—
277C2×C8C2×Dic9414yes—
278C2×C8C2×Dic9418yes—
279C2×C8C2×Dic94116yescomplemented
280C2×C8C23×D4712yes—
281C2×C8C23×D4714yes—
282C2×C8C23×D4712yes—
283C4⋊C4—474no—
284C4⋊C4—478no—
285C4⋊C4—474no—
286C4⋊C4—9416no—
287C4⋊C4—9416no—
288C4⋊C4—948no—
289C4⋊C4—944no—
290C4⋊C4—942no—
291C4⋊C4—942no—
292C4⋊C4—9416no—
293C4⋊C4C2×D18814yes—
294C4⋊C4C2×D18814yes—
295C22⋊C4—948no—
296C22⋊C4C2×D18818yes—
297C42—948no—
298C42C2×D18814yes—
299C42C23×D4714yes—
300C23—22no—
301C23—22no—
302C23#1309402414yes—
303C23C22×D18812yes—
304C23C22×D18811yescharacteristic
305Q8—37616no—
306C2×C4—24no—
307C2×C4—24no—
308C2×C4—944no—
309C2×C4—944no—
310C2×C4—948no—
311C2×C4—3768no—
312C2×C4—1888no—
313C2×C4—1884no—
314C2×C4—1884no—
315C2×C4#1309402418yes—
316C2×C4#1309402414yes—
317C2×C4C22×D18814yes—
318C2×C4C22×D18814yes—
319C2×C4C22×D18812yes—
320C2×C4C22×D18811yescharacteristic
321C2×C4C22×Dic9412yes—
322C2×C4C22×Dic9412yes—
323C2×C4C22×Dic9414yes—
324C2×C4C22×Dic9414yes—
325C2×C4C24×D4711yescharacteristic
326C8C22×Dic9414yes—
327C8C22×Dic9414yes—
328C8C22×Dic9418yes—
329C22—22no—
330C22—22no—
331C22—28no—
332C22C2×#1309402414yes—
333C22C2×#1309402412yes—
334C22#1568992812yes—
335C22#1568994012yes—
336C22C23×D18811yescharacteristic
337C4—24no—
338C4—24no—
339C4—1888no—
340C4—1884no—
341C4—1884no—
342C4C2×#1309402414yes—
343C4C2×#1309402412yes—
344C4C23×Dic9411yescharacteristic
345C4C23×Dic9411yescharacteristic
346C2—24no—
347C2#2025392714yesdirect product factor
348C2C22×#1309402411yescharacteristic
349C2C2×#1568992811yescharacteristic
350C2C2×#1568994011yescharacteristic
subgroupG/H
centerC23C22×D188
commutator, cosocleC2×C4×C47C25
FrattiniC2×C4C24×D47
FittingC2×C47×#405C2
socleC23×C47C22×D4
prime radicalC24×C47C2×D4
hypercenterC2×#405D47
hypocenterC47C2×#3164
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#405

Complements

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

NH ≅ G/Ntype
2. #20253927C2direct (N × H)
3. C2×#15689532C2semidirect
4. C2×#15689528C2semidirect
5. C2×#15689525C2semidirect
19. #15689532C22semidirect
20. #15689528C22semidirect
21. #15689525C22semidirect
68. C2×M16×C47Q8semidirect
70. C22×C8×C47Q8semidirect
99. M16×C47C2×Q8semidirect
103. C2×C8×C47C2×Q8semidirect
181. C2×C47#3164semidirect
213. C47C2×#3164semidirect
217. C2×M16Dic94semidirect
219. C22×C8Dic94semidirect
275. M16C2×Dic94semidirect
279. C2×C8C2×Dic94semidirect
347. C2#20253927direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156897512
#202539272
C2×#156895322
C2×#156895282
C2×#156895252
C2×#156891652
C2×#156891612
C22×#130939582
C2×C47×#4052
C2×#316447