Finite Group Databasebeta

← C22×#15689148 #27079892

Subgroups — C22×#15689148 #27079892

Order 12032.

3006 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#10625324C211yesmaximal, characteristic
2C23×C8×D47C211yesmaximal, characteristic
3C22×#13093981C211yesmaximal, characteristic
4C2×#15689148C2124yesmaximal, direct product factor
5C22×C4⋊(C47⋊C8)C211yesmaximal, characteristic, complemented
6C22×Dic47⋊C8C211yesmaximal, characteristic, complemented
7C22×Dic47⋊C8C211yesmaximal, characteristic, complemented
8C22×C4⋊C8×C47C211yesmaximal, characteristic, complemented
9C23×C4×D47C2211yescharacteristic
10C22×#10625324—28no—
11C22×#10625324C22112yes—
12C23×C47⋊C8C2211yescharacteristic
13C23×C47⋊C8C2211yescharacteristic
14C22×C8×D47—28no—
15C22×C8×D47C22112yes—
16C2×#13093981C4112yes—
17C2×#13093981C22112yes—
18C22×Dic47⋊C4C411yescharacteristic
19C22×Dic47⋊C4C411yescharacteristic
20C22×(C2×D47)⋊C4C411yescharacteristic
21C22×(C2×D47)⋊C4C411yescharacteristic
22C22×C4×Dic47C2211yescharacteristic
23#15689148C22164yesdirect product factor
24C2×C4⋊(C47⋊C8)C22112yescomplemented
25C2×Dic47⋊C8C22112yescomplemented
26C2×Dic47⋊C8C22112yescomplemented
27C23×C8×C47C2211yescharacteristic
28C23×C8×C47C2211yescharacteristic
29C22×C42×C47C2211yescharacteristic
30C2×C4⋊C8×C47C22112yescomplemented
31C24×D47C2×C411yescharacteristic
32C23×Dic47C2×C411yescharacteristic
33C23×Dic47C2×C411yescharacteristic
34C23×Dic47C2×C411yescharacteristic
35C22×C4×D47C2×C416yes—
36C22×C4×D47D418yes—
37C22×C4×D47Q818yes—
38C22×C4×D47C2316yes—
39C2×#10625324—248no—
40C2×#10625324C23116yes—
41C22×C47⋊C8—24no—
42C22×C47⋊C8—24no—
43C22×C47⋊C8C2316yes—
44C22×C47⋊C8C2316yes—
45C2×C8×D47—248no—
46C2×C8×D47C23116yes—
47#13093981C2×C4148yes—
48#13093981C23116yes—
49C2×Dic47⋊C4C2×C4112yes—
50C2×Dic47⋊C4C2×C4112yes—
51C2×(C2×D47)⋊C4C2×C4112yes—
52C2×(C2×D47)⋊C4C2×C4112yes—
53C2×C4×Dic47C2×C416yes—
54C2×C4×Dic47C2316yes—
55C4⋊(C47⋊C8)C23116yescomplemented
56Dic47⋊C8C23116yescomplemented
57Dic47⋊C8C23116yescomplemented
58C23×C4×C47C2×C411yescharacteristic
59C23×C4×C47C2×C411yescharacteristic
60C23×C4×C47C2311yescharacteristic
61C22×C8×C47—24no—
62C22×C8×C47—24no—
63C22×C8×C47C2316yes—
64C22×C8×C47C2316yes—
65C2×C42×C47C2×C416yes—
66C2×C42×C47C2316yes—
67C4⋊C8×C47C23116yescomplemented
68C24×C47C22×C411yescharacteristic
69C23×D47—24no—
70C23×D47—24no—
71C23×D47C4⋊C418yes—
72C23×D47C22×C416yes—
73C22×C4×C47—24no—
74C22×C4×C47—24no—
75C22×C4×C47C22×C416yes—
76C22×C4×C47C22×C416yes—
77C22×C4×C47C22×C413yes—
78C22×C4×C47C2×D414yes—
79C22×C4×C47C2×Q814yes—
80C22×C4×C47C2413yes—
81C22×Dic47—24no—
82C22×Dic47—24no—
83C22×Dic47C4⋊C418yes—
84C22×Dic47C22×C416yes—
85C22×Dic47C22×C416yes—
86C22×Dic47C22×C416yes—
87C2×C4×D47C4⋊C4148yes—
88C2×C4×D47C22×C4112yes—
89C2×C4×D47C2×D4124yes—
90C2×C4×D47C2×Q8124yes—
91C2×C4×D47C2414yes—
92C2×C8×C47—212no—
93C2×C8×C47—212no—
94C2×C8×C47C2414yes—
95C2×C8×C47C2414yes—
96C42×C47C22×C4112yes—
97C42×C47C2414yes—
98(C2×D47)⋊C4C22×C4116yes—
99(C2×D47)⋊C4C22×C4116yes—
100Dic47⋊C4C22×C4116yes—
101Dic47⋊C4C22×C4116yes—
102C4×Dic47C22×C4112yes—
103C4×Dic47C2414yes—
104C2×C47⋊C8—212no—
105C2×C47⋊C8—212no—
106C2×C47⋊C8C2414yes—
107C2×C47⋊C8C2414yes—
108#10625324—232no—
109C8×D47—232no—
110C23×C47C2×C4⋊C414yes—
111C23×C47#13114yes—
112C23×C47#13114yes—
113C23×C47C23×C413yes—
114C22×D47—212no—
115C22×D47—212no—
116C22×D47—232no—
117C22×D47C2×C4⋊C4124yes—
118C22×D47C23×C414yes—
119C2×C4×C47—212no—
120C2×C4×C47—212no—
121C2×C4×C47C2×C4⋊C4112yes—
122C2×C4×C47C23×C414yes—
123C2×C4×C47C23×C414yes—
124C2×C4×C47C23×C413yes—
125C2×C4×C47C22×D416yes—
126C2×C4×C47C22×Q816yes—
127C2×C4×C47C2511yescharacteristic
128C2×Dic47—212no—
129C2×Dic47—212no—
130C2×Dic47C2×C4⋊C4124yes—
131C2×Dic47C23×C414yes—
132C2×Dic47C23×C414yes—
133C2×Dic47C23×C414yes—
134C4×D47C2×C4⋊C4148yes—
135C4×D47C22×D418yes—
136C4×D47C22×Q818yes—
137C8×C47—24no—
138C8×C47—24no—
139C47⋊C8—24no—
140C47⋊C8—24no—
141C22×#424—471nomaximal, Sylow
142C22×C47#424116yescomplemented
143C22×C47C22×C4⋊C416yes—
144C22×C47C2×#13116yes—
145C22×C47C2×#13116yes—
146C22×C47C24×C411yescharacteristic
147C2×D47—248no—
148C2×D47—24no—
149C2×D47—24no—
150C2×D47C22×C4⋊C418yes—
151C4×C47—24no—
152C4×C47—24no—
153C4×C47C22×C4⋊C416yes—
154C4×C47C23×D411yescharacteristic
155C4×C47C23×Q811yescharacteristic
156Dic47—24no—
157Dic47—24no—
158Dic47C22×C4⋊C418yes—
159C23×M16—471no—
160C24×C8—471no—
161C22×#117—471no—
162C2×#424—4724no—
163C22×C4⋊C8—471no—
164C22×C4⋊C8—471no—
165C22×C4⋊C8—471no—
166C22×C4⋊C8D4711yescharacteristic, complemented
167C2×C47C2×#424112yescomplemented
168C2×C47C23×C4⋊C411yescharacteristic
169C2×C47C22×#13111yescharacteristic
170C2×C47C22×#13111yescharacteristic
171D47—28no—
172C24×C4—471no—
173C22×M16—4712no—
174C22×M16—948no—
175C23×C8—471no—
176C23×C8—471no—
177C23×C8—4712no—
178C23×C8—948no—
179C23×C8C2×D4711yescharacteristic
180C23×C8C2×D4711yescharacteristic
181C2×#117—4712no—
182C2×#117—4712no—
183C22×C4⋊C4—471no—
184C22×C4⋊C4—471no—
185C22×C22⋊C4—471no—
186C22×C22⋊C4—471no—
187C22×C42—471no—
188C22×C42C2×D4711yescharacteristic
189#424—4764no—
190C2×C4⋊C8—4712no—
191C2×C4⋊C8—4712no—
192C2×C4⋊C8—4712no—
193C2×C4⋊C8C2×D47112yescomplemented
194C47C22×#42411yescharacteristic, complemented, Sylow
195C25—471no—
196C23×C4—471no—
197C23×C4—476no—
198C23×C4—478no—
199C23×C4—478no—
200C23×C4—476no—
201C23×C4—471no—
202C23×C4—471no—
203C23×C4C4×D4711yescharacteristic
204C23×C4C4×D4711yescharacteristic
205C23×C4C22×D4711yescharacteristic
206C2×M16—4716no—
207C2×M16—9448no—
208C22×C8—24no—
209C22×C8—24no—
210C22×C8—476no—
211C22×C8—476no—
212C22×C8—4716no—
213C22×C8—9448no—
214C22×C8—944no—
215C22×C8—944no—
216C22×C8C22×D4716yes—
217C22×C8C22×D4716yes—
218#117—4748no—
219#117—4716no—
220C2×C4⋊C4—4712no—
221C2×C4⋊C4—4712no—
222C2×C22⋊C4—4712no—
223C2×C22⋊C4—4712no—
224C2×C42—476no—
225C2×C42—476no—
226C2×C42C4×D4716yes—
227C2×C42C22×D4716yes—
228C4⋊C8—4716no—
229C4⋊C8—4716no—
230C4⋊C8—4716no—
231C4⋊C8C22×D47116yescomplemented
232C24—478no—
233C24—476no—
234C24—944no—
235C24—944no—
236C24C2×C4×D4711yescharacteristic
237C22×C4—24no—
238C22×C4—24no—
239C22×C4—478no—
240C22×C4—476no—
241C22×C4—4748no—
242C22×C4—4712no—
243C22×C4—4724no—
244C22×C4—4724no—
245C22×C4—474no—
246C22×C4—476no—
247C22×C4—476no—
248C22×C4—944no—
249C22×C4—944no—
250C22×C4C2×C4×D4716yes—
251C22×C4C2×C4×D4716yes—
252C22×C4C2×C4×D4713yes—
253C22×C4D4×D4714yes—
254C22×C4Q8×D4714yes—
255C22×C4C23×D4713yes—
256M16—9432no—
257C2×C8—212no—
258C2×C8—212no—
259C2×C8—474no—
260C2×C8—474no—
261C2×C8—9432no—
262C2×C8—9412no—
263C2×C8—9412no—
264C2×C8C23×D4714yes—
265C2×C8C23×D4714yes—
266C4⋊C4—4716no—
267C4⋊C4—4716no—
268C22⋊C4—4716no—
269C22⋊C4—4716no—
270C42—4712no—
271C42—474no—
272C42C2×C4×D47112yes—
273C42C23×D4714yes—
274C23—4724no—
275C23—474no—
276C23—9412no—
277C23—9412no—
278C23—9432no—
279C23C4⋊C4×D4714yes—
280C23#1309403014yes—
281C23#1309403214yes—
282C23C22×C4×D4713yes—
283C2×C4—212no—
284C2×C4—212no—
285C2×C4—4724no—
286C2×C4—474no—
287C2×C4—4748no—
288C2×C4—478no—
289C2×C4—478no—
290C2×C4—474no—
291C2×C4—474no—
292C2×C4—9412no—
293C2×C4—9412no—
294C2×C4C4⋊C4×D47112yes—
295C2×C4C22×C4×D4714yes—
296C2×C4C22×C4×D4714yes—
297C2×C4C22×C4×D4713yes—
298C2×C4C2×D4×D4716yes—
299C2×C4C2×Q8×D4716yes—
300C2×C4C24×D4711yescharacteristic
301C8—24no—
302C8—24no—
303C8—944no—
304C8—944no—
305C22—478no—
306C22—9448no—
307C22—944no—
308C22—944no—
309C22#15689148116yesdirect product factor
310C22C2×C4⋊C4×D4716yes—
311C22C2×#1309403016yes—
312C22C2×#1309403216yes—
313C22C23×C4×D4711yescharacteristic
314C4—24no—
315C4—24no—
316C4—478no—
317C4—944no—
318C4—944no—
319C4C2×C4⋊C4×D4716yes—
320C4C22×D4×D4711yescharacteristic
321C4C22×Q8×D4711yescharacteristic
322C2—948no—
323C2C2×#15689148112yesdirect product factor
324C2C22×C4⋊C4×D4711yescharacteristic
325C2C22×#1309403011yescharacteristic
326C2C22×#1309403211yescharacteristic
subgroupG/H
centerC23×C4C22×D47
commutatorC22×C47C24×C4
FrattiniC2×C4C24×D47
FittingC22×C4⋊C8×C47C2
socleC24×C47C22×C4
prime radicalC24×D47C2×C4
cosocleC2×C4×C47C25
hypercenterC22×C4⋊C8D47
hypocenterC47C22×#424
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C23×C4 ◁ C22×C4⋊C8

Complements

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

NH ≅ G/Ntype
4. C2×#15689148C2direct (N × H)
5. C22×C4⋊(C47⋊C8)C2semidirect
6. C22×Dic47⋊C8C2semidirect
7. C22×Dic47⋊C8C2semidirect
8. C22×C4⋊C8×C47C2semidirect
23. #15689148C22direct (N × H)
24. C2×C4⋊(C47⋊C8)C22semidirect
25. C2×Dic47⋊C8C22semidirect
26. C2×Dic47⋊C8C22semidirect
30. C2×C4⋊C8×C47C22semidirect
55. C4⋊(C47⋊C8)C23semidirect
56. Dic47⋊C8C23semidirect
57. Dic47⋊C8C23semidirect
67. C4⋊C8×C47C23semidirect
142. C22×C47#424semidirect
166. C22×C4⋊C8D47semidirect
167. C2×C47C2×#424semidirect
193. C2×C4⋊C8C2×D47semidirect
194. C47C22×#424semidirect
231. C4⋊C8C22×D47semidirect
309. C22#15689148direct (N × H)
323. C2C2×#15689148direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#106253242
C23×C8×D472
C22×#130939812
C2×#156891482
C22×C4⋊(C47⋊C8)2
C22×Dic47⋊C82
C22×C4⋊C8×C472
C22×#42447