Finite Group Databasebeta

← C2×#20257194 #27124459

Subgroups — C2×#20257194 #27124459

Order 12032.

6826 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20257194C2132yesmaximal, direct product factor
2C2×Q8×Dic94C214yesmaximal, complemented
3C2×#15689852C214yesmaximal, complemented
4C2×#15689849C214yesmaximal, complemented
5C2×#15689849C214yesmaximal, complemented
6C2×#15689751C216yesmaximal
7C2×#15689676C216yesmaximal
8C2×#15689601C211yesmaximal, characteristic
9C22×Q8×Dic47C211yesmaximal, characteristic
10C2×C47×#519C211yesmaximal, characteristic
11Q8×Dic94C22164yescomplemented
12#15689852C22164yescomplemented
13#15689849C22164yescomplemented
14#15689849C22164yescomplemented
15#15689751C22196yes—
16#15689676C22196yes—
17C2×#13094024C22112yes—
18C2×#13094023C22112yes—
19C2×#13094023C2216yes—
20C2×#13094023C2216yes—
21C2×#13094016C22112yes—
22C2×#13094015C22112yes—
23C2×#13094015C2216yes—
24C2×#13094015C2216yes—
25C2×#13094001C22112yes—
26C2×#13094001C2214yes—
27#15689601C22116yes—
28C2×Q8×Dic47C22116yes—
29C2×Q8×Dic47C2214yes—
30C2×Q8×Dic47C2214yes—
31C2×Q8×Dic47C2214yes—
32C2×Q8×Dic47C2214yes—
33C2×C4×Dic94C22112yes—
34C2×C4×Dic94C22112yes—
35C2×#13093983C2216yes—
36C22×C4⋊Dic47C2213yes—
37C22×C4×Dic47C2213yes—
38C23×Q8×C47C2211yescharacteristic
39C47×#519C22116yes—
40C2×C4×Q8×C47C2214yes—
41C2×C4×Q8×C47C2214yes—
42C2×#117×C47C2216yes—
43C22×Dic94C2314yes—
44C22×Dic94C23112yes—
45C23×Dic47C2311yescharacteristic
46C23×Dic47C2313yes—
47#13094024C23196yes—
48#13094023C23196yes—
49#13094023C23148yes—
50#13094023C23148yes—
51#13094016C23196yes—
52#13094015C23196yes—
53#13094015C23148yes—
54#13094015C23148yes—
55#13094001C23196yes—
56#13094001C23132yes—
57Q8×Dic47C23132yes—
58Q8×Dic47C23132yes—
59Q8×Dic47C23132yes—
60Q8×Dic47C23132yes—
61C4×Dic94C23196yes—
62C4×Dic94C23196yes—
63#13093983C23148yes—
64C2×C4⋊Dic47C23124yes—
65C2×C4⋊Dic47C23112yes—
66C2×C4⋊Dic47C2316yes—
67C2×C4⋊Dic47C2316yes—
68C2×C4⋊Dic47C2314yes—
69C2×C4⋊Dic47C23112yes—
70C2×Dic47⋊C4C2314yes—
71C2×Dic47⋊C4C2314yes—
72C2×Dic47⋊C4C23112yes—
73C2×Dic47⋊C4C23112yes—
74C2×C22⋊Dic47C2314yes—
75C2×C4×Dic47C2316yes—
76C2×C4×Dic47C2316yes—
77C2×C4×Dic47C23124yes—
78C2×C4×Dic47C23112yes—
79C22×Q8×C47Q8116yescomplemented
80C22×Q8×C47C2314yes—
81C22×Q8×C47C2318yes—
82C23×C4×C47C2313yes—
83C4×Q8×C47C23132yes—
84C4×Q8×C47C23132yes—
85#117×C47C23148yes—
86C2×C4⋊C4×C47C2316yes—
87C2×C4⋊C4×C47C2316yes—
88C2×C22⋊C4×C47C2314yes—
89C2×C42×C47C23112yes—
90C24×C47C2411yescharacteristic
91C2×Q8×C47C2×Q8132yes—
92C2×Q8×C47C2×Q8164yescomplemented
93C2×Q8×C47C24116yes—
94C22×C4×C47C2×Q8124yes—
95C22×C4×C47C2414yes—
96C22×C4×C47C2414yes—
97C22×C4×C47C2416yes—
98C22×C4×C47C24112yes—
99C22×Dic47—24no—
100C22×Dic47—46no—
101C22×Dic47C2411yescharacteristic
102C22×Dic47C2414yes—
103C22×Dic47C2414yes—
104C22×Dic47C2414yes—
105C22×Dic47C2416yes—
106C22×Dic47C24112yes—
107C22×Dic47C2411yescharacteristic
108C2×Dic94—216no—
109C2×Dic94—424no—
110C2×Dic94C24116yes—
111C2×Dic94C24148yes—
112C4⋊C4×C47C24124yes—
113C4⋊C4×C47C24124yes—
114C22⋊C4×C47C24116yes—
115C42×C47C24148yes—
116C22⋊Dic47C24116yes—
117C4⋊Dic47C24148yes—
118C4⋊Dic47C24124yes—
119C4⋊Dic47C24124yes—
120C4⋊Dic47C24116yes—
121C4⋊Dic47C24148yes—
122Dic47⋊C4C24116yes—
123Dic47⋊C4C24116yes—
124Dic47⋊C4C24148yes—
125Dic47⋊C4C24148yes—
126C4×Dic47C24124yes—
127C4×Dic47C24124yes—
128C4×Dic47C24148yes—
129C23×C47—22no—
130C23×C47—22no—
131C23×C47C22×Q814yes—
132C23×C47C2512yes—
133C23×C47C2511yescharacteristic
134Q8×C47C22×Q8164yes—
135C2×C4×C47—28no—
136C2×C4×C47—28no—
137C2×C4×C47C22×Q8124yes—
138C2×C4×C47C22×Q8148yes—
139C2×C4×C47C2518yes—
140C2×C4×C47C2518yes—
141C2×C4×C47C25112yes—
142C2×Dic47—28no—
143C2×Dic47—22no—
144C2×Dic47—28no—
145C2×Dic47—212no—
146C2×Dic47—28no—
147C2×Dic47—412no—
148C2×Dic472−1+414yes—
149C2×Dic47C2512yes—
150C2×Dic47C2518yes—
151C2×Dic47C2518yes—
152C2×Dic47C25112yes—
153C2×Dic47C2512yes—
154Dic94—232no—
155Dic94—448no—
156C2×#3502—471nomaximal, Sylow
157C22×C47—22no—
158C22×C47—22no—
159C22×C47—28no—
160C22×C47C23×Q814yes—
161C22×C47C23×Q812yes—
162C22×C47C2×2−1+412yes—
163C22×C47#58512yes—
164C22×C47C2611yescharacteristic
165C4×C47—28no—
166C4×C47—28no—
167C4×C47C23×Q8124yes—
168Dic47—28no—
169Dic47—22no—
170Dic47—28no—
171Dic47—212no—
172Dic47C2×2−1+414yes—
173#3502—4732no—
174C2×Q82—474no—
175C2×#557—474no—
176C2×#557—474no—
177C2×#557—474no—
178C2×#544—476no—
179C2×#533—476no—
180C2×#519—471no—
181C2×#519D4711yescharacteristic
182C22×C4×Q8—471no—
183C2×C47—24no—
184C2×C47#350214yescomplemented
185C2×C47C24×Q811yescharacteristic
186C2×C47C22×2−1+411yescharacteristic
187C2×C47C2×#58511yescharacteristic
188C23×Q8C2×D4711yescharacteristic
189Q82—4764no—
190#557—4764no—
191#557—4764no—
192#557—4764no—
193#544—4796no—
194#533—4796no—
195C2×C4⋊Q8—476no—
196C2×C4⋊Q8—476no—
197C2×C4⋊Q8—4712no—
198C2×C4⋊Q8—4712no—
199C2×#125—476no—
200C2×#125—476no—
201C2×#125—4712no—
202C2×#125—4712no—
203C2×C22⋊Q8—474no—
204C2×C22⋊Q8—4712no—
205#519—4716no—
206#519C2×D47116yes—
207C2×C4×Q8—474no—
208C2×C4×Q8—474no—
209C2×C4×Q8—4716no—
210C2×C4×Q8—474no—
211C2×C4×Q8—474no—
212C2×C4×Q8—4712no—
213C2×C4×Q8—4712no—
214C2×C4×Q8C2×D4714yes—
215C2×C4×Q8C2×D4714yes—
216C2×#117—476no—
217C2×#117C2×D4716yes—
218C22×C4⋊C4—473no—
219C22×C42—473no—
220C47C2×#350211yescharacteristic, complemented, Sylow
221C22×Q8—474no—
222C22×Q8—4712no—
223C22×Q8Dic94116yescomplemented
224C22×Q8C22×D4714yes—
225C22×Q8C22×D4718yes—
226C23×C4—471no—
227C23×C4—473no—
228C23×C4C22×D4713yes—
229C4⋊Q8—4748no—
230C4⋊Q8—4748no—
231C4⋊Q8—4796no—
232C4⋊Q8—4796no—
233#125—4748no—
234#125—4748no—
235#125—4796no—
236#125—4796no—
237C22⋊Q8—4732no—
238C22⋊Q8—4796no—
239C4×Q8—4732no—
240C4×Q8—4732no—
241C4×Q8—4732no—
242C4×Q8—4732no—
243C4×Q8—4796no—
244C4×Q8—4796no—
245C4×Q8C22×D47132yes—
246C4×Q8C22×D47132yes—
247#117—4748no—
248#117C22×D47148yes—
249C2×C4⋊C4—4712no—
250C2×C4⋊C4—4724no—
251C2×C4⋊C4—476no—
252C2×C4⋊C4—476no—
253C2×C4⋊C4—474no—
254C2×C4⋊C4—474no—
255C2×C4⋊C4—474no—
256C2×C4⋊C4—4712no—
257C2×C4⋊C4—4712no—
258C2×C4⋊C4—4712no—
259C2×C4⋊C4C22×D4716yes—
260C2×C4⋊C4C22×D4716yes—
261C2×C22⋊C4—474no—
262C2×C22⋊C4C22×D4714yes—
263C2×C42—4712no—
264C2×C42—476no—
265C2×C42—476no—
266C2×C42—4724no—
267C2×C42C22×D47112yes—
268C24C23×D4711yescharacteristic
269C2×Q8—4716no—
270C2×Q8—4748no—
271C2×Q8—9416no—
272C2×Q8—18824no—
273C2×Q8C2×Dic94132yes—
274C2×Q8C2×Dic94164yescomplemented
275C2×Q8C23×D47116yes—
276C22×C4—471no—
277C22×C4—474no—
278C22×C4—471no—
279C22×C4—474no—
280C22×C4—474no—
281C22×C4—476no—
282C22×C4—4712no—
283C22×C4—944no—
284C22×C4—1886no—
285C22×C4C2×Dic94124yes—
286C22×C4C23×D4714yes—
287C22×C4C23×D4714yes—
288C22×C4C23×D47112yes—
289C22×C4C23×D4716yes—
290C4⋊C4—4716no—
291C4⋊C4—4716no—
292C4⋊C4—4748no—
293C4⋊C4—4748no—
294C4⋊C4—4748no—
295C4⋊C4—4724no—
296C4⋊C4—4724no—
297C4⋊C4—4716no—
298C4⋊C4—4748no—
299C4⋊C4C23×D47124yes—
300C4⋊C4C23×D47124yes—
301C22⋊C4—4716no—
302C22⋊C4C23×D47116yes—
303C42—4724no—
304C42—4724no—
305C42—4748no—
306C42C23×D47148yes—
307C23—22no—
308C23—22no—
309C23C22×Dic9414yes—
310C23C24×D4712yes—
311C23C24×D4711yescharacteristic
312Q8—9432no—
313Q8—18848no—
314Q8C22×Dic94164yes—
315C2×C4—28no—
316C2×C4—28no—
317C2×C4—474no—
318C2×C4—472no—
319C2×C4—472no—
320C2×C4—478no—
321C2×C4—478no—
322C2×C4—4712no—
323C2×C4—948no—
324C2×C4—942no—
325C2×C4—948no—
326C2×C4—9412no—
327C2×C4—948no—
328C2×C4—18812no—
329C2×C4C22×Dic94124yes—
330C2×C4C22×Dic94148yes—
331C2×C4C24×D4718yes—
332C2×C4C24×D4718yes—
333C2×C4C24×D47112yes—
334C22—22no—
335C22—22no—
336C22—28no—
337C22C23×Dic9414yes—
338C22C23×Dic9412yes—
339C222−1+4×D4712yes—
340C22#1568996012yes—
341C22C25×D4711yescharacteristic
342C4—28no—
343C4—28no—
344C4—474no—
345C4—942no—
346C4—948no—
347C4—9412no—
348C4—948no—
349C4C23×Dic94124yes—
350C2—24no—
351C2#2025719414yesdirect product factor
352C2C24×Dic9411yescharacteristic
353C2C2×2−1+4×D4711yescharacteristic
354C2C2×#1568996011yescharacteristic
subgroupG/H
centerC23C24×D47
commutator, cosocleC22×C47C26
FrattiniC22C25×D47
FittingC2×C47×#519C2
socleC23×C47C25
prime radicalC24×C47C24
hypercenterC2×#519D47
hypocenterC47C2×#3502
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×#519

Complements

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

NH ≅ G/Ntype
1. #20257194C2direct (N × H)
2. C2×Q8×Dic94C2semidirect
3. C2×#15689852C2semidirect
4. C2×#15689849C2semidirect
5. C2×#15689849C2semidirect
11. Q8×Dic94C22semidirect
12. #15689852C22semidirect
13. #15689849C22semidirect
14. #15689849C22semidirect
79. C22×Q8×C47Q8semidirect
92. C2×Q8×C47C2×Q8semidirect
184. C2×C47#3502semidirect
220. C47C2×#3502semidirect
223. C22×Q8Dic94semidirect
274. C2×Q8C2×Dic94semidirect
351. C2#20257194direct (N × H)

Maximal subgroups

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

subgroupindex
#202571942
C2×Q8×Dic942
C2×#156898522
C2×#156898492
C2×#156897512
C2×#156896762
C2×#156896012
C22×Q8×Dic472
C2×C47×#5192
C2×#350247