Finite Group Databasebeta

← #26375753

Subgroups — #26375753

Order 12032.

2446 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20246353C216yesmaximal, complemented
2#20246309C212yesmaximal, complemented
3#20246257C216yesmaximal
4Q8×C22⋊Dic47C212yesmaximal
5#20245866C216yesmaximal, complemented
6#20245786C213yesmaximal
7C4×Q8×Dic47C212yesmaximal, complemented
8C4×#13093983C213yesmaximal
9C22×C4×Q8×C47C211yesmaximal, characteristic
10C2×Q8×Dic47C2214yes—
11C2×#13093983C2216yes—
12#15689030C22112yes—
13#15689022C22112yes—
14#15689009C22124yes—
15#15688991C2216yes—
16#15688989C2216yes—
17#15688986C2216yes—
18#15688983C2216yes—
19C4⋊C4×Dic47C22112yes—
20C4×C4⋊Dic47C2216yes—
21C4×C4⋊Dic47C2216yes—
22C4×C4⋊Dic47C2216yes—
23C4×C22⋊Dic47C2213yes—
24C4×C22⋊Dic47C2216yes—
25C4×C22⋊Dic47C2211yescharacteristic
26#15688969C2216yes—
27#15688969C2216yes—
28C42×Dic47C2216yes—
29C23×Q8×C47C2211yescharacteristic
30C2×C4×Q8×C47C4116yescomplemented
31C2×C4×Q8×C47C2211yescharacteristic
32C2×C4×Q8×C47C2213yes—
33C2×C4×Q8×C47C2213yes—
34C2×C4×Q8×C47C2211yescharacteristic
35C22×C4⋊C4×C47C2213yes—
36C22×C42×C47C2213yes—
37Q8×Dic47—216no—
38#13093983—224no—
39C2×C4⋊Dic47C23112yes—
40C2×C4⋊Dic47C23112yes—
41C2×C22⋊Dic47C2316yes—
42C2×C22⋊Dic47C2312yes—
43C2×C4×Dic47C2312yes—
44C2×C4×Dic47C2316yes—
45C2×C4×Dic47C23112yes—
46C2×C4×Dic47C23112yes—
47C47⋊#95C2312yes—
48C47⋊#95C2316yes—
49C47⋊#95C23112yes—
50C47⋊#95C23112yes—
51C47⋊#95C23112yes—
52C47⋊#95C23112yes—
53C22×Q8×C47—28no—
54C22×Q8×C47C2×C418yes—
55C22×Q8×C47C2313yes—
56C22×Q8×C47C2311yescharacteristic
57C23×C4×C47C2311yescharacteristic
58C23×C4×C47C2313yes—
59C23×C4×C47C2313yes—
60C4×Q8×C47C2×C4124yes—
61C4×Q8×C47C2×C4124yes—
62C4×Q8×C47C2×C418yes—
63C4×Q8×C47C2×C418yes—
64C2×C4⋊C4×C47C2×C4124yes—
65C2×C4⋊C4×C47C2313yes—
66C2×C4⋊C4×C47C2316yes—
67C2×C4⋊C4×C47C2313yes—
68C2×C42×C47C2×C4124yes—
69C2×C42×C47C2313yes—
70C2×C42×C47C2313yes—
71C2×C42×C47C2313yes—
72C2×C42×C47C2313yes—
73C24×C47C2411yescharacteristic
74C2×Q8×C47—232no—
75C2×Q8×C47—212no—
76C2×Q8×C47C22×C4112yes—
77C2×Q8×C47C22×C414yes—
78C2×Q8×C47#4118yes—
79C22×C4×C47—212no—
80C22×C4×C47—212no—
81C22×C4×C47—24no—
82C22×C4×C47C22×C414yes—
83C22×C4×C47C22×C4112yes—
84C22×C4×C47C22×C4112yes—
85C22×C4×C47C2411yescharacteristic
86C22×C4×C47C2413yes—
87C22×C4×C47C2413yes—
88C22×C4×C47C2413yes—
89C22×C4×C47C2413yes—
90C22×C4×C47C2411yescharacteristic
91C22×Dic47C24112yes—
92C22×Dic47C2414yes—
93C4⋊C4×C47C22×C4124yes—
94C4⋊C4×C47C22×C4112yes—
95C4⋊C4×C47C22×C4112yes—
96C42×C47C22×C4112yes—
97C42×C47C22×C4112yes—
98C42×C47C22×C4112yes—
99C42×C47C22×C4112yes—
100C22⋊Dic47—212no—
101C22⋊Dic47#4118yes—
102C4⋊Dic47—224no—
103C4⋊Dic47—224no—
104C4×Dic47—224no—
105C4×Dic47—24no—
106C4×Dic47—212no—
107C4×Dic47—224no—
108C23×C47—22no—
109C23×C47—22no—
110C23×C47C23×C412yes—
111C23×C47C2×#4114yes—
112C23×C47C2511yescharacteristic
113Q8×C47—224no—
114Q8×C47#117116yescomplemented
115C2×C4×C47—224no—
116C2×C4×C47—26no—
117C2×C4×C47—26no—
118C2×C4×C47—26no—
119C2×C4×C47—22no—
120C2×C4×C47—224no—
121C2×C4×C47—28no—
122C2×C4×C47C23×C412yes—
123C2×C4×C47C23×C416yes—
124C2×C4×C47C23×C416yes—
125C2×C4×C47C23×C416yes—
126C2×C4×C47C23×C416yes—
127C2×C4×C47C23×C412yes—
128C2×C4×C47C2×#41112yes—
129C2×C4×C472−1+414yes—
130C2×Dic47—24no—
131C2×Dic47—24no—
132C2×Dic47—212no—
133C2×Dic47—212no—
134C2×Dic47—212no—
135C2×Dic47C2×#4118yes—
136#34128—471nomaximal, Sylow
137C22×C47—22no—
138C22×C47—22no—
139C22×C47—28no—
140C22×C47C4×#4114yes—
141C22×C47C24×C411yescharacteristic
142C22×C47C22×#4112yes—
143C22×C47C22×#4112yes—
144C22×C47C2×2−1+412yes—
145C4×C47—26no—
146C4×C47—26no—
147C4×C47—26no—
148C4×C47—22no—
149C4×C47C2×#117112yes—
150C4×C47#51914yes—
151Dic47—28no—
152Dic47—412no—
153C22×C4×Q8D4711yescharacteristic
154#2356—476no—
155#2351—472no—
156#2344—476no—
157Q8×C22⋊C4—472no—
158#2295—476no—
159#2283—473no—
160C42×Q8—472no—
161C4×#117—473no—
162C2×C47—24no—
163C2×C47C22×#11711yescharacteristic
164C2×C47C2×C4×#4111yescharacteristic
165C2×C47C2×#51911yescharacteristic
166C2×C47#348214yes—
167C23×Q8C2×D4711yescharacteristic
168C2×C4×Q8—474no—
169C2×C4×Q8Dic47116yescomplemented
170C2×C4×Q8C2×D4711yescharacteristic
171C2×C4×Q8C2×D4713yes—
172C2×C4×Q8C2×D4711yescharacteristic
173C2×C4×Q8C2×D4713yes—
174C2×#117—476no—
175C22×C4⋊C4C2×D4713yes—
176C22×C42C2×D4713yes—
177#391—4712no—
178#389—4712no—
179#387—4724no—
180#383—476no—
181#382—476no—
182#381—476no—
183#380—476no—
184C4×C4⋊C4—4712no—
185C4×C4⋊C4—476no—
186C4×C4⋊C4—476no—
187C4×C4⋊C4—476no—
188C4×C22⋊C4—476no—
189C4×C22⋊C4—471no—
190C4×C22⋊C4—473no—
191#376—476no—
192#376—476no—
193C43—476no—
194C47#3412811yescharacteristic, complemented, Sylow
195C22×Q8—28no—
196C22×Q8C2×Dic4718yes—
197C22×Q8C22×D4713yes—
198C22×Q8C22×D4711yescharacteristic
199C23×C4C22×D4713yes—
200C23×C4C22×D4713yes—
201C23×C4C22×D4711yescharacteristic
202C4×Q8—9416no—
203C4×Q8C2×Dic47124yes—
204C4×Q8C2×Dic47124yes—
205C4×Q8C2×Dic4718yes—
206C4×Q8C2×Dic4718yes—
207#117—9424no—
208C2×C4⋊C4—4712no—
209C2×C4⋊C4—4712no—
210C2×C4⋊C4C2×Dic47124yes—
211C2×C4⋊C4C22×D4713yes—
212C2×C4⋊C4C22×D4716yes—
213C2×C4⋊C4C22×D4713yes—
214C2×C22⋊C4—472no—
215C2×C22⋊C4—476no—
216C2×C42—4712no—
217C2×C42—4712no—
218C2×C42—472no—
219C2×C42—476no—
220C2×C42C2×Dic47124yes—
221C2×C42C22×D4713yes—
222C2×C42C22×D4713yes—
223C2×C42C22×D4713yes—
224C2×C42C22×D4713yes—
225#95—472no—
226#95—476no—
227#95—4712no—
228#95—4712no—
229#95—4712no—
230#95—4712no—
231C24C23×D4711yescharacteristic
232C2×Q8—232no—
233C2×Q8—212no—
234C2×Q8#1062535018yes—
235C2×Q8C22×Dic47112yes—
236C2×Q8C22×Dic4714yes—
237C22×C4—212no—
238C22×C4—212no—
239C22×C4—24no—
240C22×C4—474no—
241C22×C4—4712no—
242C22×C4C22×Dic4714yes—
243C22×C4C22×Dic47112yes—
244C22×C4C22×Dic47112yes—
245C22×C4C23×D4711yescharacteristic
246C22×C4C23×D4713yes—
247C22×C4C23×D4713yes—
248C22×C4C23×D4713yes—
249C22×C4C23×D4713yes—
250C22×C4C23×D4711yescharacteristic
251C4⋊C4—9424no—
252C4⋊C4—9424no—
253C4⋊C4C22×Dic47124yes—
254C4⋊C4C22×Dic47112yes—
255C4⋊C4C22×Dic47112yes—
256C22⋊C4—478no—
257C22⋊C4—9412no—
258C42—9424no—
259C42—9424no—
260C42—944no—
261C42—9412no—
262C42C22×Dic47112yes—
263C42C22×Dic47112yes—
264C42C22×Dic47112yes—
265C42C22×Dic47112yes—
266C23—22no—
267C23—22no—
268C23C23×Dic4712yes—
269C23#41×D4714yes—
270C23C24×D4711yescharacteristic
271Q8—224no—
272Q8#13093983116yescomplemented
273C2×C4—224no—
274C2×C4—26no—
275C2×C4—26no—
276C2×C4—26no—
277C2×C4—22no—
278C2×C4—224no—
279C2×C4—28no—
280C2×C4—478no—
281C2×C4—944no—
282C2×C4—944no—
283C2×C4—9412no—
284C2×C4—9412no—
285C2×C4—9412no—
286C2×C4C23×Dic4712yes—
287C2×C4C23×Dic4716yes—
288C2×C4C23×Dic4716yes—
289C2×C4C23×Dic4716yes—
290C2×C4C23×Dic4716yes—
291C2×C4C23×Dic4712yes—
292C2×C4C2×#10625350112yes—
293C2×C4#1309407214yes—
294C22—22no—
295C22—22no—
296C22—28no—
297C22#41×Dic4714yes—
298C22C24×Dic4711yescharacteristic
299C22C2×#41×D4712yes—
300C22C22×#1062535012yes—
301C22C2×#1309407212yes—
302C4—26no—
303C4—26no—
304C4—26no—
305C4—22no—
306C4—948no—
307C4—18812no—
308C4C2×#13093983112yes—
309C4#1568960114yes—
310C2—24no—
311C2C22×#1309398311yescharacteristic
312C2C2×#41×Dic4711yescharacteristic
313C2C2×#1568960111yescharacteristic
314C2#2025700514yes—
subgroupG/H
centerC22×C4C23×D47
commutatorC22×C47C24×C4
FrattiniC23C24×D47
FittingC22×C4×Q8×C47C2
socle, cosocleC23×C47C25
prime radicalC24×C47C24
hypercenterC22×C4×Q8D47
hypocenterC47#34128
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C22×C4 ◁ C22×C4×Q8

Complements

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

NH ≅ G/Ntype
1. #20246353C2semidirect
2. #20246309C2semidirect
5. #20245866C2semidirect
7. C4×Q8×Dic47C2semidirect
30. C2×C4×Q8×C47C4semidirect
114. Q8×C47#117semidirect
169. C2×C4×Q8Dic47semidirect
194. C47#34128semidirect
272. Q8#13093983semidirect

Maximal subgroups

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

subgroupindex
#202463532
#202463092
#202462572
Q8×C22⋊Dic472
#202458662
#202457862
C4×Q8×Dic472
C4×#130939832
C22×C4×Q8×C472
#3412847