Finite Group Databasebeta

← C2×#20253928 #27085912

Subgroups — C2×#20253928 #27085912

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#20253928C2116yesmaximal, direct product factor
3C2×#15689530C214yesmaximal, complemented
4C2×#15689527C212yesmaximal, complemented
5C2×#15689524C212yesmaximal, complemented
6C2×#15689166C212yesmaximal
7C22×#13093958C211yesmaximal, characteristic
8C2×#15689078C211yesmaximal, characteristic
9C2×C47×#427C211yesmaximal, characteristic
10C22×#10625329C2211yescharacteristic
11#15689751—28no—
12#15689751C22116yes—
13C2×#13094024—22no—
14C2×#13094023C2214yes—
15C2×#13094016—22no—
16C2×#13094015C2214yes—
17C2×#13094001—24no—
18C2×#13093983C2211yescharacteristic
19C22×C4⋊Dic47C2212yes—
20#15689530C22132yescomplemented
21#15689527C22116yescomplemented
22#15689524C22116yescomplemented
23#15689166C22116yes—
24C2×#13093958C2212yes—
25C2×#13093958C2212yes—
26C2×#13093958C2218yes—
27C2×#13093957C2214yes—
28C2×#13093955C2214yes—
29#15689078C2218yes—
30C2×#13093934C2212yes—
31C2×C8×Dic47C2212yes—
32C23×C8×C47C2211yescharacteristic
33C2×#117×C47C2212yes—
34C47×#427C2218yes—
35C2×#107×C47C2211yescharacteristic
36C2×#107×C47C2211yescharacteristic
37C2×#106×C47C2212yes—
38C22×Dic94—24no—
39C23×Dic47—22no—
40C2×#10625329Q818yescomplemented
41C2×#10625329C2314yes—
42C22×C47⋊C8C2312yes—
43#13094024—216no—
44#13094023—28no—
45#13094023C23116yes—
46#13094016—216no—
47#13094015—28no—
48#13094015C23116yes—
49#13094001—232no—
50#13093983D414yes—
51#13093983C2314yes—
52C2×C4⋊Dic47—24no—
53C2×C4⋊Dic47—22no—
54C2×C4⋊Dic47—24no—
55C2×C4⋊Dic47D411yescharacteristic
56C2×C4⋊Dic47D411yescharacteristic
57C2×C4⋊Dic47C2318yes—
58C2×C4⋊Dic47C2314yes—
59C2×Dic47⋊C4—24no—
60C2×Dic47⋊C4—24no—
61C2×C22⋊Dic47D412yes—
62C2×C4×Dic47C2312yes—
63#13093958C2318yes—
64#13093958C2318yes—
65#13093957C23116yes—
66#13093955C23116yes—
67#13093934C2318yes—
68C8×Dic47C2318yes—
69C23×C4×C47C2311yescharacteristic
70C22×C8×C47Q818yescomplemented
71C22×C8×C47C2311yescharacteristic
72C22×C8×C47C2311yescharacteristic
73C22×C8×C47C2314yes—
74#117×C47—24no—
75#117×C47C2318yes—
76C2×C4⋊C4×C47C2312yes—
77C2×C4⋊C4×C47C2312yes—
78C2×C22⋊C4×C47—22no—
79C2×C42×C47—22no—
80#107×C47C2314yes—
81#107×C47C2314yes—
82#106×C47C2318yes—
83C24×C47C2×D411yescharacteristic
84C22×C4×C47—22no—
85C22×C4×C47—22no—
86C22×C4×C47C2×D411yescharacteristic
87C22×C4×C47C2×D412yes—
88C22×C4×C47C2×Q814yes—
89C22×C4×C47C2×Q814yes—
90C22×C4×C47C2411yescharacteristic
91C22×C4×C47C2412yes—
92C22×Dic47—24no—
93C22×Dic47—28no—
94C22×Dic47—84no—
95C22×Dic47C2×D412yes—
96C22×Dic47C2×D412yes—
97C2×Dic94—216no—
98C2×Dic94—88no—
99C2×C8×C47C2×Q814yes—
100C2×C8×C47C2×Q814yes—
101C2×C8×C47C2×Q8116yescomplemented
102C2×C8×C47C2412yes—
103C2×C8×C47C2412yes—
104C4⋊C4×C47—22no—
105C4⋊C4×C47—22no—
106C4⋊C4×C47C2414yes—
107C4⋊C4×C47C2414yes—
108C22⋊C4×C47—28no—
109C42×C47—28no—
110C22⋊Dic47C2×D418yes—
111C4⋊Dic47—24no—
112C4⋊Dic47—28no—
113C4⋊Dic47—216no—
114C4⋊Dic47C2×D414yes—
115C4⋊Dic47C2×D414yes—
116C4⋊Dic47C2418yes—
117Dic47⋊C4—216no—
118Dic47⋊C4—216no—
119C4×Dic47C2×D414yes—
120C4×Dic47C2414yes—
121#10625329C2×Q8116yescomplemented
122C2×C47⋊C8C2×Q818yes—
123C2×C47⋊C8C2414yes—
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×C47—44no—
132C2×C4×C47—44no—
133C2×C4×C47C4⋊Q818yes—
134C2×C4×C47C4⋊Q814yes—
135C2×C4×C47C22×D411yescharacteristic
136C2×C4×C47C22×D412yes—
137C2×C4×C47C22×Q812yes—
138C2×C4×C47C22×Q814yes—
139C2×C4×C47C22×Q812yes—
140C2×C4×C47C22×Q814yes—
141C2×C4×C47C2511yescharacteristic
142C2×Dic47—24no—
143C2×Dic47—24no—
144C2×Dic47—28no—
145C2×Dic47—48no—
146C2×Dic47—88no—
147C2×Dic47C22×D414yes—
148C2×Dic47C22×D414yes—
149Dic94—816no—
150C8×C47C22×Q814yes—
151C8×C47C22×Q814yes—
152C47⋊C8C22×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—44no—
163C4×C47—44no—
164C4×C47C2×C4⋊Q812yes—
165C4×C47C2×C4⋊Q814yes—
166C4×C47C23×Q811yescharacteristic
167C4×C47C23×Q811yescharacteristic
168Dic47—24no—
169Dic47—24no—
170Dic47—48no—
171C2×#544—472no—
172#3164—4716no—
173C2×#501—474no—
174C2×#500—472no—
175C2×#499—472no—
176C2×#428—472no—
177C2×#427D4711yescharacteristic
178C22×#107—471no—
179C2×#405—471no—
180C2×C47—24no—
181C2×C47#316414yescomplemented
182C2×C47C22×C4⋊Q811yescharacteristic
183C2×C47C2×#57611yescharacteristic
184C2×C47C2×#57811yescharacteristic
185C22×M16—471no—
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—471no—
195C2×#117C2×D4712yes—
196C22×C4⋊C4—472no—
197#501—4732no—
198#500—4716no—
199#499—4716no—
200#428—4716no—
201#427C2×D4718yes—
202C2×#107—472no—
203C2×#107—472no—
204C2×#107—478no—
205C2×#107—474no—
206C2×#107C2×D4711yescharacteristic
207C2×#107C2×D4711yescharacteristic
208C2×#106—474no—
209C2×#106C2×D4712yes—
210#405—478no—
211C2×#97—472no—
212C2×C4×C8—472no—
213C47C2×#316411yescharacteristic, complemented, Sylow
214C22×Q8—944no—
215C23×C4—942no—
216C23×C4C22×D4711yescharacteristic
217C2×M16—478no—
218C2×M16—474no—
219C22×C8—472no—
220C22×C8Dic9418yescomplemented
221C22×C8C22×D4711yescharacteristic
222C22×C8C22×D4711yescharacteristic
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—24no—
232#117—474no—
233#117—474no—
234#117C22×D4718yes—
235C2×C4⋊C4—471no—
236C2×C4⋊C4—471no—
237C2×C4⋊C4—474no—
238C2×C4⋊C4—478no—
239C2×C4⋊C4—944no—
240C2×C4⋊C4—944no—
241C2×C4⋊C4—944no—
242C2×C4⋊C4—942no—
243C2×C4⋊C4—944no—
244C2×C4⋊C4C22×D4712yes—
245C2×C4⋊C4C22×D4712yes—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—472no—
248C2×C42—22no—
249C2×C42—472no—
250#107—478no—
251#107—478no—
252#107—4716no—
253#107C22×D4714yes—
254#107C22×D4714yes—
255#106—4716no—
256#106C22×D4718yes—
257#97—478no—
258C4×C8—478no—
259C24D4×D4711yescharacteristic
260C2×Q8—9416no—
261C2×Q8—3768no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—472no—
265C22×C4—472no—
266C22×C4—948no—
267C22×C4—944no—
268C22×C4—3764no—
269C22×C4C2×Dic9414yes—
270C22×C4D4×D4712yes—
271C22×C4D4×D4711yescharacteristic
272C22×C4Q8×D4714yes—
273C22×C4C23×D4712yes—
274C22×C4C23×D4711yescharacteristic
275M16—4716no—
276C2×C8—478no—
277C2×C8—474no—
278C2×C8C2×Dic9414yes—
279C2×C8C2×Dic9414yes—
280C2×C8C2×Dic94116yescomplemented
281C2×C8C23×D4712yes—
282C2×C8C23×D4712yes—
283C4⋊C4—22no—
284C4⋊C4—22no—
285C4⋊C4—474no—
286C4⋊C4—474no—
287C4⋊C4—478no—
288C4⋊C4—9416no—
289C4⋊C4—9416no—
290C4⋊C4—9416no—
291C4⋊C4—948no—
292C4⋊C4—944no—
293C4⋊C4C23×D4714yes—
294C4⋊C4C23×D4714yes—
295C22⋊C4—28no—
296C22⋊C4—478no—
297C42—28no—
298C42—474no—
299C42—474no—
300C23—22no—
301C23—22no—
302C23#1309402314yes—
303C23C2×D4×D4712yes—
304C23C2×D4×D4711yescharacteristic
305Q8—37616no—
306C2×C4—24no—
307C2×C4—24no—
308C2×C4—44no—
309C2×C4—44no—
310C2×C4—474no—
311C2×C4—474no—
312C2×C4—944no—
313C2×C4—944no—
314C2×C4—948no—
315C2×C4—3768no—
316C2×C4—1888no—
317C2×C4#1309402318yes—
318C2×C4#1309402314yes—
319C2×C4C2×D4×D4712yes—
320C2×C4C2×D4×D4711yescharacteristic
321C2×C4C2×Q8×D4712yes—
322C2×C4C2×Q8×D4714yes—
323C2×C4C22×Dic9412yes—
324C2×C4C22×Dic9414yes—
325C2×C4C24×D4711yescharacteristic
326C8—478no—
327C8C22×Dic9414yes—
328C8C22×Dic9414yes—
329C22—22no—
330C22—22no—
331C22—28no—
332C22C2×#1309402314yes—
333C22C2×#1309402312yes—
334C22#1568992512yes—
335C22#1568993712yes—
336C22C22×D4×D4711yescharacteristic
337C4—44no—
338C4—44no—
339C4—944no—
340C4—944no—
341C4—1888no—
342C4C2×#1309402314yes—
343C4C2×#1309402312yes—
344C4C22×Q8×D4711yescharacteristic
345C4C23×Dic9411yescharacteristic
346C2—24no—
347C2#2025392814yesdirect product factor
348C2C22×#1309402311yescharacteristic
349C2C2×#1568992511yescharacteristic
350C2C2×#1568993711yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutator, cosocleC2×C4×C47C25
FrattiniC2×C4C24×D47
FittingC2×C47×#427C2
socleC23×C47C22×D4
prime radicalC24×C47C2×D4
hypercenterC2×#427D47
hypocenterC47C2×#3164
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#427

Complements

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

NH ≅ G/Ntype
2. #20253928C2direct (N × H)
3. C2×#15689530C2semidirect
4. C2×#15689527C2semidirect
5. C2×#15689524C2semidirect
20. #15689530C22semidirect
21. #15689527C22semidirect
22. #15689524C22semidirect
40. C2×#10625329Q8semidirect
70. C22×C8×C47Q8semidirect
101. C2×C8×C47C2×Q8semidirect
121. #10625329C2×Q8semidirect
181. C2×C47#3164semidirect
213. C47C2×#3164semidirect
220. C22×C8Dic94semidirect
280. C2×C8C2×Dic94semidirect
347. C2#20253928direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156897512
#202539282
C2×#156895302
C2×#156895272
C2×#156895242
C2×#156891662
C22×#130939582
C2×#156890782
C2×C47×#4272
C2×#316447