Finite Group Databasebeta

← #26376518

Subgroups — #26376518

Order 12032.

2350 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20246345C216yesmaximal, complemented
2Q8×C4⋊Dic47C212yesmaximal, complemented
3#20246257C216yesmaximal
4Q8×C22⋊Dic47C212yesmaximal
5#20246039C216yesmaximal, complemented
6#20245795C213yesmaximal
7#20245786C213yesmaximal
8#20245753C212yesmaximal, complemented
9C22×C4×Q8×C47C211yesmaximal, characteristic
10C2×Q8×Dic47C2214yes—
11C2×#13093983C2216yes—
12#15689030C22112yes—
13#15689022C22112yes—
14#15689009C22124yes—
15#15688993C2216yes—
16#15688991C2216yes—
17#15688989C2216yes—
18#15688989C2216yes—
19#15688986C2216yes—
20#15688983C2211yescharacteristic
21#15688983C2213yes—
22#15688983C2216yes—
23C4⋊C4×Dic47C22112yes—
24C4×C4⋊Dic47C2216yes—
25C4×C4⋊Dic47C2216yes—
26C4×C4⋊Dic47C2216yes—
27C4×C22⋊Dic47C2216yes—
28#15688969C2216yes—
29C23×Q8×C47C2211yescharacteristic
30C2×C4×Q8×C47C4116yescomplemented
31C2×C4×Q8×C47C2213yes—
32C2×C4×Q8×C47C2211yescharacteristic
33C2×C4×Q8×C47C2213yes—
34C2×C4×Q8×C47C2211yescharacteristic
35C22×C4⋊C4×C47C2213yes—
36C22×C42×C47C2213yes—
37Q8×Dic47—216no—
38#13093983—412no—
39C2×C4⋊Dic47C2312yes—
40C2×C4⋊Dic47C2316yes—
41C2×C4⋊Dic47C23112yes—
42C2×C4⋊Dic47C23112yes—
43C2×C22⋊Dic47C2312yes—
44C2×C22⋊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×C47C2313yes—
58C23×C4×C47C2311yescharacteristic
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—26no—
80C22×C4×C47—26no—
81C22×C4×C47—212no—
82C22×C4×C47—24no—
83C22×C4×C47C22×C414yes—
84C22×C4×C47C22×C4112yes—
85C22×C4×C47C22×C4112yes—
86C22×C4×C47C2411yescharacteristic
87C22×C4×C47C2413yes—
88C22×C4×C47C2413yes—
89C22×C4×C47C2413yes—
90C22×C4×C47C2413yes—
91C22×C4×C47C2411yescharacteristic
92C22×Dic47C2414yes—
93C22×Dic47C24112yes—
94C4⋊C4×C47C22×C4124yes—
95C4⋊C4×C47C22×C4112yes—
96C4⋊C4×C47C22×C4112yes—
97C42×C47C22×C4112yes—
98C42×C47C22×C4112yes—
99C42×C47C22×C4112yes—
100C42×C47C22×C4112yes—
101C22⋊Dic47—212no—
102C22⋊Dic47—24no—
103C4⋊Dic47—224no—
104C4⋊Dic47—24no—
105C4⋊Dic47—212no—
106C4⋊Dic47—412no—
107C4×Dic47—224no—
108C4×Dic47—412no—
109C23×C47—22no—
110C23×C47—22no—
111C23×C47C23×C412yes—
112C23×C47H(5,𝔽2)12yes—
113C23×C472−1+412yes—
114C23×C47C2511yescharacteristic
115Q8×C47—224no—
116Q8×C47#117116yescomplemented
117C2×C4×C47—224no—
118C2×C4×C47—26no—
119C2×C4×C47—23no—
120C2×C4×C47—23no—
121C2×C4×C47—23no—
122C2×C4×C47—23no—
123C2×C4×C47—22no—
124C2×C4×C47—224no—
125C2×C4×C47—28no—
126C2×C4×C47C23×C412yes—
127C2×C4×C47C23×C416yes—
128C2×C4×C47C23×C416yes—
129C2×C4×C47C23×C416yes—
130C2×C4×C47C23×C416yes—
131C2×C4×C47C23×C412yes—
132C2×C4×C47C2×#41112yes—
133C2×C4×C472−1+414yes—
134C2×Dic47—24no—
135C2×Dic47—212no—
136C2×Dic47—212no—
137C2×Dic47—212no—
138C2×Dic47—24no—
139C2×Dic47—24no—
140#34196—471nomaximal, Sylow
141C22×C47—22no—
142C22×C47—22no—
143C22×C47—24no—
144C22×C47—24no—
145C22×C47#52014yes—
146C22×C47C24×C411yescharacteristic
147C22×C47C22×#4112yes—
148C22×C47C2×H(5,𝔽2)11yescharacteristic
149C22×C47C2×2−1+411yescharacteristic
150C22×C47C2×2−1+412yes—
151C4×C47—26no—
152C4×C47—22no—
153C4×C47—26no—
154C4×C47—26no—
155C4×C47C2×#117112yes—
156C4×C47#51914yes—
157Dic47—412no—
158Dic47—44no—
159C22×C4×Q8D4711yescharacteristic
160#2355—476no—
161Q8×C4⋊C4—472no—
162#2344—476no—
163Q8×C22⋊C4—472no—
164#2320—476no—
165#2285—473no—
166#2283—473no—
167#2276—472no—
168C2×C47—24no—
169C2×C47C22×#11711yescharacteristic
170C2×C47C2×#51911yescharacteristic
171C2×C47C2×#52011yescharacteristic
172C2×C47#352212yes—
173C2×C47#352312yes—
174C23×Q8C2×D4711yescharacteristic
175C2×C4×Q8—474no—
176C2×C4×Q8Dic47116yescomplemented
177C2×C4×Q8C2×D4713yes—
178C2×C4×Q8C2×D4711yescharacteristic
179C2×C4×Q8C2×D4713yes—
180C2×C4×Q8C2×D4711yescharacteristic
181C2×#117—476no—
182C22×C4⋊C4C2×D4713yes—
183C22×C42C2×D4713yes—
184#391—4712no—
185#389—4712no—
186#387—4724no—
187#384—476no—
188#383—476no—
189#382—476no—
190#382—476no—
191#381—476no—
192#380—476no—
193#380—471no—
194#380—473no—
195C4×C4⋊C4—4712no—
196C4×C4⋊C4—476no—
197C4×C4⋊C4—476no—
198C4×C4⋊C4—476no—
199C4×C22⋊C4—476no—
200#376—476no—
201C47#3419611yescharacteristic, complemented, Sylow
202C22×Q8—28no—
203C22×Q8C2×Dic4718yes—
204C22×Q8C22×D4713yes—
205C22×Q8C22×D4711yescharacteristic
206C23×C4C22×D4713yes—
207C23×C4C22×D4713yes—
208C23×C4C22×D4711yescharacteristic
209C4×Q8—9416no—
210C4×Q8C2×Dic47124yes—
211C4×Q8C2×Dic47124yes—
212C4×Q8C2×Dic4718yes—
213C4×Q8C2×Dic4718yes—
214#117—18812no—
215C2×C4⋊C4—4712no—
216C2×C4⋊C4—472no—
217C2×C4⋊C4—476no—
218C2×C4⋊C4—4712no—
219C2×C4⋊C4C2×Dic47124yes—
220C2×C4⋊C4C22×D4713yes—
221C2×C4⋊C4C22×D4716yes—
222C2×C4⋊C4C22×D4713yes—
223C2×C22⋊C4—476no—
224C2×C22⋊C4—472no—
225C2×C42—4712no—
226C2×C42—4712no—
227C2×C42C2×Dic47124yes—
228C2×C42C22×D4713yes—
229C2×C42C22×D4713yes—
230C2×C42C22×D4713yes—
231C2×C42C22×D4713yes—
232#95—472no—
233#95—476no—
234#95—4712no—
235#95—4712no—
236#95—4712no—
237#95—4712no—
238C24C23×D4711yescharacteristic
239C2×Q8—232no—
240C2×Q8—212no—
241C2×Q8#1062535018yes—
242C2×Q8C22×Dic47112yes—
243C2×Q8C22×Dic4714yes—
244C22×C4—26no—
245C22×C4—26no—
246C22×C4—212no—
247C22×C4—24no—
248C22×C4—474no—
249C22×C4—4712no—
250C22×C4C22×Dic4714yes—
251C22×C4C22×Dic47112yes—
252C22×C4C22×Dic47112yes—
253C22×C4C23×D4711yescharacteristic
254C22×C4C23×D4713yes—
255C22×C4C23×D4713yes—
256C22×C4C23×D4713yes—
257C22×C4C23×D4713yes—
258C22×C4C23×D4711yescharacteristic
259C4⋊C4—9424no—
260C4⋊C4—944no—
261C4⋊C4—9412no—
262C4⋊C4—18812no—
263C4⋊C4C22×Dic47124yes—
264C4⋊C4C22×Dic47112yes—
265C4⋊C4C22×Dic47112yes—
266C22⋊C4—9412no—
267C22⋊C4—944no—
268C42—9424no—
269C42—18812no—
270C42C22×Dic47112yes—
271C42C22×Dic47112yes—
272C42C22×Dic47112yes—
273C42C22×Dic47112yes—
274C23—22no—
275C23—22no—
276C23C23×Dic4712yes—
277C23#1309407012yes—
278C23#1309407112yes—
279C23C24×D4711yescharacteristic
280Q8—224no—
281Q8#13093983116yescomplemented
282C2×C4—224no—
283C2×C4—26no—
284C2×C4—23no—
285C2×C4—23no—
286C2×C4—23no—
287C2×C4—23no—
288C2×C4—22no—
289C2×C4—224no—
290C2×C4—28no—
291C2×C4—944no—
292C2×C4—9412no—
293C2×C4—9412no—
294C2×C4—944no—
295C2×C4—9412no—
296C2×C4—944no—
297C2×C4C23×Dic4712yes—
298C2×C4C23×Dic4716yes—
299C2×C4C23×Dic4716yes—
300C2×C4C23×Dic4716yes—
301C2×C4C23×Dic4716yes—
302C2×C4C23×Dic4712yes—
303C2×C4C2×#10625350112yes—
304C2×C4#1309407214yes—
305C22—22no—
306C22—22no—
307C22—24no—
308C22—24no—
309C22#1568960614yes—
310C22C24×Dic4711yescharacteristic
311C22C22×#1062535012yes—
312C22C2×#1309407011yescharacteristic
313C22C2×#1309407111yescharacteristic
314C22C2×#1309407212yes—
315C4—26no—
316C4—22no—
317C4—26no—
318C4—26no—
319C4—18812no—
320C4—1884no—
321C4C2×#13093983112yes—
322C4#1568960114yes—
323C2—24no—
324C2C22×#1309398311yescharacteristic
325C2C2×#1568960111yescharacteristic
326C2C2×#1568960611yescharacteristic
327C2#2025747312yes—
328C2#2025748412yes—
subgroupG/H
center, FrattiniC23C24×D47
commutatorC22×C47C24×C4
FittingC22×C4×Q8×C47C2
socle, cosocleC23×C47C25
prime radicalC24×C47C24
hypercenterC22×C4×Q8D47
hypocenterC47#34196
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×C4×Q8

Complements

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

NH ≅ G/Ntype
1. #20246345C2semidirect
2. Q8×C4⋊Dic47C2semidirect
5. #20246039C2semidirect
8. #20245753C2semidirect
30. C2×C4×Q8×C47C4semidirect
116. Q8×C47#117semidirect
176. C2×C4×Q8Dic47semidirect
201. C47#34196semidirect
281. Q8#13093983semidirect

Maximal subgroups

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

subgroupindex
#202463452
Q8×C4⋊Dic472
#202462572
Q8×C22⋊Dic472
#202460392
#202457952
#202457862
#202457532
C22×C4×Q8×C472
#3419647