Finite Group Databasebeta

← C2×#20253916 #27085899

Subgroups — C2×#20253916 #27085899

Order 12032.

1854 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689754C212yesmaximal
2#20253916C2116yesmaximal, direct product factor
3C2×#15689530C214yesmaximal, complemented
4C2×#15689523C212yesmaximal, complemented
5C2×#15689520C212yesmaximal, complemented
6C2×#15689163C212yesmaximal
7C2×#15689161C211yesmaximal, characteristic
8C2×#15689078C211yesmaximal, characteristic
9C22×#106×C47C211yesmaximal, characteristic
10C22×#10625329C2211yescharacteristic
11#15689754—28no—
12#15689754C22116yes—
13C2×#13094023—22no—
14C2×#13094023C2214yes—
15C2×#13094015—22no—
16C2×#13094015C2214yes—
17C2×#13094003—24no—
18C2×#13093983C2212yes—
19C2×#13093983C2211yescharacteristic
20#15689530C22132yescomplemented
21#15689523C22116yescomplemented
22#15689520C22116yescomplemented
23#15689163C22116yes—
24#15689161C2218yes—
25C2×#13093958C2212yes—
26C2×#13093957C2214yes—
27C2×#13093956C2211yescharacteristic
28C2×#13093956C2211yescharacteristic
29C2×#13093955C2214yes—
30#15689078C2218yes—
31C2×#13093934C2212yes—
32C2×C8×Dic47C2212yes—
33C23×C8×C47C2211yescharacteristic
34C22×C4⋊C4×C47C2212yes—
35C2×#106×C47C2212yes—
36C2×#106×C47C2212yes—
37C2×#106×C47C2218yes—
38C22×Dic94—24no—
39C2×#10625329Q818yescomplemented
40C2×#10625329C2314yes—
41C22×C47⋊C8C2312yes—
42#13094023—216no—
43#13094023—28no—
44#13094023C23116yes—
45#13094015—216no—
46#13094015—28no—
47#13094015C23116yes—
48#13094003—232no—
49#13093983—24no—
50#13093983D414yes—
51#13093983C2318yes—
52#13093983C2314yes—
53C2×C4⋊Dic47—24no—
54C2×C4⋊Dic47D412yes—
55C2×C4⋊Dic47C2312yes—
56C2×C4⋊Dic47C2312yes—
57C2×Dic47⋊C4—24no—
58C2×Dic47⋊C4—24no—
59C2×C22⋊Dic47—22no—
60C2×C22⋊Dic47D412yes—
61C2×C4×Dic47—22no—
62C2×C4×Dic47C2312yes—
63#13093958C2318yes—
64#13093957C23116yes—
65#13093956C2314yes—
66#13093956C2314yes—
67#13093955C23116yes—
68#13093934C2318yes—
69C8×Dic47C2318yes—
70C23×C4×C47—22no—
71C23×C4×C47C2311yescharacteristic
72C22×C8×C47Q818yescomplemented
73C22×C8×C47C2311yescharacteristic
74C22×C8×C47C2311yescharacteristic
75C22×C8×C47C2314yes—
76C2×C4⋊C4×C47—24no—
77C2×C4⋊C4×C47—22no—
78C2×C4⋊C4×C47C2318yes—
79C2×C4⋊C4×C47C2314yes—
80#106×C47C2318yes—
81#106×C47C2318yes—
82C24×C47C2×D411yescharacteristic
83C22×C4×C47—24no—
84C22×C4×C47—28no—
85C22×C4×C47—84no—
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×Dic47C2×D414yes—
95C2×Dic94—216no—
96C2×Dic94—88no—
97C2×C8×C47C2×Q814yes—
98C2×C8×C47C2×Q814yes—
99C2×C8×C47C2×Q8116yescomplemented
100C2×C8×C47C2412yes—
101C2×C8×C47C2412yes—
102C4⋊C4×C47—28no—
103C4⋊C4×C47—24no—
104C4⋊C4×C47C2418yes—
105C22⋊Dic47—28no—
106C22⋊Dic47C2×D418yes—
107C4⋊Dic47—22no—
108C4⋊Dic47—22no—
109C4⋊Dic47—216no—
110C4⋊Dic47C2×D418yes—
111C4⋊Dic47C2414yes—
112C4⋊Dic47C2414yes—
113Dic47⋊C4—216no—
114Dic47⋊C4—216no—
115C4×Dic47—28no—
116C4×Dic47C2×D414yes—
117C4×Dic47C2414yes—
118#10625329C2×Q8116yescomplemented
119C2×C47⋊C8C2×Q818yes—
120C2×C47⋊C8C2414yes—
121C23×C47—24no—
122C23×C47C4⋊Q814yes—
123C23×C47C22×D412yes—
124C23×C47C22×D411yescharacteristic
125C2×C4×C47—28no—
126C2×C4×C47—48no—
127C2×C4×C47—88no—
128C2×C4×C47C4⋊Q818yes—
129C2×C4×C47C4⋊Q814yes—
130C2×C4×C47C22×D412yes—
131C2×C4×C47C22×D411yescharacteristic
132C2×C4×C47C22×Q812yes—
133C2×C4×C47C22×Q814yes—
134C2×C4×C47C22×Q812yes—
135C2×C4×C47C22×Q814yes—
136C2×C4×C47C2511yescharacteristic
137C2×Dic47—28no—
138C2×Dic47—24no—
139C2×Dic47—24no—
140C2×Dic47—44no—
141C2×Dic47—44no—
142C2×Dic47C22×D418yes—
143Dic94—816no—
144C8×C47C22×Q814yes—
145C8×C47C22×Q814yes—
146C47⋊C8C22×Q818yes—
147C2×#3163—471nomaximal, Sylow
148C22×C47—28no—
149C22×C47—24no—
150C22×C47C2×C4⋊Q814yes—
151C22×C47C2×C4⋊Q812yes—
152C22×C47#57714yes—
153C22×C47C23×D411yescharacteristic
154C4×C47—48no—
155C4×C47C2×C4⋊Q814yes—
156C4×C47C2×C4⋊Q812yes—
157C4×C47C23×Q811yescharacteristic
158C4×C47C23×Q811yescharacteristic
159Dic47—28no—
160Dic47—44no—
161Dic47—44no—
162C2×#544—472no—
163#3163—4716no—
164C2×#501—474no—
165C2×#499—472no—
166C2×#498—472no—
167C2×#428—472no—
168C2×#427—471no—
169C22×#106D4711yescharacteristic
170C2×#405—471no—
171C2×C47—24no—
172C2×C47#316314yescomplemented
173C2×C47C22×C4⋊Q811yescharacteristic
174C2×C47C2×#57712yes—
175C22×M16—471no—
176C23×C8C2×D4711yescharacteristic
177#544—4716no—
178#544—948no—
179C2×C4⋊Q8—474no—
180C2×C4⋊Q8—942no—
181C2×#125—474no—
182C2×#125—942no—
183C2×C22⋊Q8—944no—
184C2×#117—471no—
185C2×#117—472no—
186C22×C4⋊C4C2×D4712yes—
187#501—4732no—
188#499—4716no—
189#498—4716no—
190#428—4716no—
191#427—478no—
192C2×#107—472no—
193C2×#107—474no—
194C2×#106—471no—
195C2×#106—471no—
196C2×#106—474no—
197C2×#106C2×D4712yes—
198C2×#106C2×D4712yes—
199C2×#106C2×D4718yes—
200#405—478no—
201C2×#97—472no—
202C2×C4×C8—472no—
203C47C2×#316311yescharacteristic, complemented, Sylow
204C22×Q8—944no—
205C23×C4—22no—
206C23×C4C22×D4711yescharacteristic
207C2×M16—478no—
208C2×M16—474no—
209C22×C8—472no—
210C22×C8Dic9418yescomplemented
211C22×C8C22×D4711yescharacteristic
212C22×C8C22×D4711yescharacteristic
213C22×C8C22×D4714yes—
214C4⋊Q8—4716no—
215C4⋊Q8—948no—
216C4⋊Q8—9416no—
217#125—4716no—
218#125—948no—
219#125—9416no—
220C22⋊Q8—9432no—
221#117—474no—
222#117—474no—
223#117—478no—
224#117—944no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—472no—
228C2×C4⋊C4—472no—
229C2×C4⋊C4—472no—
230C2×C4⋊C4—944no—
231C2×C4⋊C4—944no—
232C2×C4⋊C4—944no—
233C2×C4⋊C4C22×D4714yes—
234C2×C4⋊C4C22×D4718yes—
235C2×C22⋊C4—472no—
236C2×C22⋊C4—942no—
237C2×C42—472no—
238C2×C42—942no—
239#107—478no—
240#107—4716no—
241#106—474no—
242#106—474no—
243#106—4716no—
244#106C22×D4718yes—
245#106C22×D4718yes—
246#97—478no—
247C4×C8—478no—
248C24D4×D4711yescharacteristic
249C2×Q8—9416no—
250C2×Q8—3768no—
251C22×C4—24no—
252C22×C4—28no—
253C22×C4—84no—
254C22×C4—474no—
255C22×C4—942no—
256C22×C4—942no—
257C22×C4C2×Dic9414yes—
258C22×C4D4×D4712yes—
259C22×C4D4×D4711yescharacteristic
260C22×C4Q8×D4714yes—
261C22×C4C23×D4712yes—
262C22×C4C23×D4711yescharacteristic
263M16—4716no—
264C2×C8—478no—
265C2×C8—474no—
266C2×C8C2×Dic9414yes—
267C2×C8C2×Dic9414yes—
268C2×C8C2×Dic94116yescomplemented
269C2×C8C23×D4712yes—
270C2×C8C23×D4712yes—
271C4⋊C4—24no—
272C4⋊C4—28no—
273C4⋊C4—478no—
274C4⋊C4—474no—
275C4⋊C4—474no—
276C4⋊C4—9416no—
277C4⋊C4—9416no—
278C4⋊C4—9416no—
279C4⋊C4—942no—
280C4⋊C4—942no—
281C4⋊C4C23×D4718yes—
282C22⋊C4—478no—
283C22⋊C4—948no—
284C42—474no—
285C42—474no—
286C42—948no—
287C23—24no—
288C23#1309402314yes—
289C23C2×D4×D4712yes—
290C23C2×D4×D4711yescharacteristic
291Q8—37616no—
292C2×C4—28no—
293C2×C4—48no—
294C2×C4—88no—
295C2×C4—478no—
296C2×C4—948no—
297C2×C4—944no—
298C2×C4—944no—
299C2×C4—1884no—
300C2×C4—1884no—
301C2×C4#1309402318yes—
302C2×C4#1309402314yes—
303C2×C4C2×D4×D4711yescharacteristic
304C2×C4C2×D4×D4712yes—
305C2×C4C2×Q8×D4712yes—
306C2×C4C2×Q8×D4714yes—
307C2×C4C22×Dic9412yes—
308C2×C4C22×Dic9414yes—
309C2×C4C24×D4711yescharacteristic
310C8—478no—
311C8C22×Dic9414yes—
312C8C22×Dic9414yes—
313C22—24no—
314C22—28no—
315C22C2×#1309402314yes—
316C22C2×#1309402312yes—
317C22#1568992914yes—
318C22C22×D4×D4711yescharacteristic
319C4—48no—
320C4—948no—
321C4—1884no—
322C4—1884no—
323C4C2×#1309402312yes—
324C4C2×#1309402314yes—
325C4C22×Q8×D4711yescharacteristic
326C4C23×Dic9411yescharacteristic
327C2—24no—
328C2#2025391614yesdirect product factor
329C2C22×#1309402311yescharacteristic
330C2C2×#1568992912yes—
subgroupG/H
centerC23C2×D4×D47
commutator, cosocleC2×C4×C47C25
FrattiniC2×C4C24×D47
FittingC22×#106×C47C2
socleC23×C47C22×D4
prime radicalC24×C47C2×D4
hypercenterC22×#106D47
hypocenterC47C2×#3163
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#106

Complements

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

NH ≅ G/Ntype
2. #20253916C2direct (N × H)
3. C2×#15689530C2semidirect
4. C2×#15689523C2semidirect
5. C2×#15689520C2semidirect
20. #15689530C22semidirect
21. #15689523C22semidirect
22. #15689520C22semidirect
39. C2×#10625329Q8semidirect
72. C22×C8×C47Q8semidirect
99. C2×C8×C47C2×Q8semidirect
118. #10625329C2×Q8semidirect
172. C2×C47#3163semidirect
203. C47C2×#3163semidirect
210. C22×C8Dic94semidirect
268. C2×C8C2×Dic94semidirect
328. C2#20253916direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156897542
#202539162
C2×#156895302
C2×#156895232
C2×#156895202
C2×#156891632
C2×#156891612
C2×#156890782
C22×#106×C472
C2×#316347