Finite Group Databasebeta

← #26376339

Subgroups — #26376339

Order 12032.

2806 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20246265C218yesmaximal, complemented
2#20246257C218yesmaximal, complemented
3#20246006C212yesmaximal, complemented
4#20245991C214yesmaximal, complemented
5#20245771C214yesmaximal
6#20245771C212yesmaximal
7#20245745C212yesmaximal, complemented
8C23×C4⋊C4×C47C211yesmaximal, characteristic
9C2×#13093983C2214yes—
10C2×#13093983C2218yes—
11#15689022C22116yescomplemented
12#15689009C22132yescomplemented
13#15688986C2214yes—
14#15688986C2218yes—
15#15688983C2218yes—
16#15688983C2214yes—
17#15688983C2214yes—
18#15688983C2218yes—
19#15688979C2214yes—
20C4⋊C4×Dic47C22116yescomplemented
21C4×C22⋊Dic47C2218yes—
22C4×C22⋊Dic47C2214yes—
23C4×C22⋊Dic47C2214yes—
24C4×C22⋊Dic47C2218yes—
25C24×C4×C47C2211yescharacteristic
26C24×C4×C47C2212yes—
27C22×C4⋊C4×C47C4116yescomplemented
28C22×C4⋊C4×C47C2212yes—
29C22×C4⋊C4×C47C2211yescharacteristic
30C22×C4⋊C4×C47C2214yes—
31C22×C4⋊C4×C47C2211yescharacteristic
32C22×C4⋊C4×C47C2214yes—
33#13093983—216no—
34#13093983—416no—
35C2×C4⋊Dic47C2318yes—
36C2×C4⋊Dic47C23116yes—
37C2×C22⋊Dic47C2318yes—
38C2×C22⋊Dic47C2318yes—
39C2×C22⋊Dic47C2318yes—
40C2×C4×Dic47C2318yes—
41C2×C4×Dic47C23116yes—
42C47⋊#95C23116yes—
43C47⋊#95C2318yes—
44C47⋊#95C2318yes—
45C47⋊#95C23116yes—
46C25×C47C2311yescharacteristic
47C23×C4×C47—28no—
48C23×C4×C47—48no—
49C23×C4×C47C2×C418yes—
50C23×C4×C47C2×C4116yes—
51C23×C4×C47C2312yes—
52C23×C4×C47C2312yes—
53C23×C4×C47C2311yescharacteristic
54C23×C4×C47C2314yes—
55C23×C4×C47C2312yes—
56C23×C4×C47C2314yes—
57C23×C4×C47C2312yes—
58C23×C4×C47C2311yescharacteristic
59C2×C4⋊C4×C47C2×C4132yes—
60C2×C4⋊C4×C47C2×C4116yes—
61C2×C4⋊C4×C47C2×C4132yes—
62C2×C4⋊C4×C47C2×C418yes—
63C2×C4⋊C4×C47C2×C418yes—
64C2×C4⋊C4×C47C2314yes—
65C2×C4⋊C4×C47C2318yes—
66C2×C4⋊C4×C47C2314yes—
67C24×C47—24no—
68C24×C47—24no—
69C24×C47—24no—
70C24×C47C22×C414yes—
71C24×C47C2411yescharacteristic
72C24×C47C2411yescharacteristic
73C24×C47C2411yescharacteristic
74C22×C4×C47—24no—
75C22×C4×C47—216no—
76C22×C4×C47—232no—
77C22×C4×C47—28no—
78C22×C4×C47—28no—
79C22×C4×C47—432no—
80C22×C4×C47—44no—
81C22×C4×C47—48no—
82C22×C4×C47—44no—
83C22×C4×C47C22×C414yes—
84C22×C4×C47C22×C4116yes—
85C22×C4×C47C22×C418yes—
86C22×C4×C47C22×C418yes—
87C22×C4×C47C22×C418yes—
88C22×C4×C47C22×C418yes—
89C22×C4×C47C22×C4116yes—
90C22×C4×C47C22×C414yes—
91C22×C4×C47#4118yes—
92C22×C4×C47C2412yes—
93C22×C4×C47C2414yes—
94C22×C4×C47C2414yes—
95C22×C4×C47C2412yes—
96C22×Dic47C24116yes—
97C4⋊C4×C47C22×C4132yes—
98C4⋊C4×C47C22×C4116yes—
99C4⋊C4×C47C22×C4116yes—
100C22⋊Dic47—216no—
101C22⋊Dic47—216no—
102C22⋊Dic47—216no—
103C4⋊Dic47—216no—
104C4⋊Dic47—416no—
105C4×Dic47—216no—
106C4×Dic47—416no—
107C23×C47—232no—
108C23×C47—28no—
109C23×C47—28no—
110C23×C47—28no—
111C23×C47—22no—
112C23×C47—22no—
113C23×C47—22no—
114C23×C47—22no—
115C23×C47—22no—
116C23×C47—22no—
117C23×C47C23×C412yes—
118C23×C47C23×C412yes—
119C23×C47C23×C412yes—
120C23×C47C2×#4114yes—
121C23×C47H(5,𝔽2)14yes—
122C23×C472−1+414yes—
123C23×C47C2511yescharacteristic
124C2×C4×C47—28no—
125C2×C4×C47—232no—
126C2×C4×C47—24no—
127C2×C4×C47—28no—
128C2×C4×C47—28no—
129C2×C4×C47—216no—
130C2×C4×C47—216no—
131C2×C4×C47—48no—
132C2×C4×C47—416no—
133C2×C4×C47—48no—
134C2×C4×C47#117116yescomplemented
135C2×C4×C47C23×C414yes—
136C2×C4×C47C23×C418yes—
137C2×C4×C47C23×C418yes—
138C2×C4×C47C23×C414yes—
139C2×C4×C47C2×#4118yes—
140C2×Dic47—216no—
141C2×Dic47—216no—
142C2×Dic47—216no—
143#34178—471nomaximal, Sylow
144C22×C47—232no—
145C22×C47—22no—
146C22×C47—22no—
147C22×C47—22no—
148C22×C47—22no—
149C22×C47—22no—
150C22×C47—22no—
151C22×C47—28no—
152C22×C47—28no—
153C22×C47—28no—
154C22×C47C2×#11714yes—
155C22×C47#51814yes—
156C22×C47#51914yes—
157C22×C47C24×C411yescharacteristic
158C22×C47C22×#4112yes—
159C22×C47C2×H(5,𝔽2)12yes—
160C22×C47C2×2−1+412yes—
161C4×C47—24no—
162C4×C47—28no—
163C4×C47—28no—
164C4×C47C2×#11718yes—
165Dic47—416no—
166C23×C4⋊C4D4711yescharacteristic
167#2345—478no—
168#2344—478no—
169#2316—472no—
170#2313—474no—
171#2280—472no—
172#2280—474no—
173#2274—472no—
174C2×C47—24no—
175C2×C47—24no—
176C2×C47—24no—
177C2×C47C22×#11711yescharacteristic
178C2×C47C2×#51811yescharacteristic
179C2×C47C2×#51911yescharacteristic
180C2×C47#349214yes—
181C24×C4C2×D4711yescharacteristic
182C24×C4C2×D4712yes—
183C2×#117—474no—
184C2×#117—478no—
185C22×C4⋊C4Dic47116yescomplemented
186C22×C4⋊C4C2×D4712yes—
187C22×C4⋊C4C2×D4711yescharacteristic
188C22×C4⋊C4C2×D4714yes—
189C22×C4⋊C4C2×D4714yes—
190C22×C4⋊C4C2×D4711yescharacteristic
191#389—4716no—
192#387—4732no—
193#381—474no—
194#381—478no—
195#380—478no—
196#380—474no—
197#380—478no—
198#380—474no—
199#379—474no—
200C4×C4⋊C4—4716no—
201C4×C22⋊C4—478no—
202C4×C22⋊C4—474no—
203C4×C22⋊C4—474no—
204C4×C22⋊C4—478no—
205C47#3417811yescharacteristic, complemented, Sylow
206C25C22×D4711yescharacteristic
207C23×C4—28no—
208C23×C4—48no—
209C23×C4C2×Dic4718yes—
210C23×C4C2×Dic47116yes—
211C23×C4C22×D4714yes—
212C23×C4C22×D4712yes—
213C23×C4C22×D4712yes—
214C23×C4C22×D4711yescharacteristic
215C23×C4C22×D4712yes—
216C23×C4C22×D4714yes—
217C23×C4C22×D4712yes—
218C23×C4C22×D4711yescharacteristic
219#117—9416no—
220#117—18816no—
221C2×C4⋊C4—478no—
222C2×C4⋊C4—4716no—
223C2×C4⋊C4C2×Dic47132yes—
224C2×C4⋊C4C2×Dic47116yes—
225C2×C4⋊C4C2×Dic47132yes—
226C2×C4⋊C4C2×Dic4718yes—
227C2×C4⋊C4C2×Dic4718yes—
228C2×C4⋊C4C22×D4714yes—
229C2×C4⋊C4C22×D4718yes—
230C2×C4⋊C4C22×D4714yes—
231C2×C22⋊C4—478no—
232C2×C22⋊C4—478no—
233C2×C22⋊C4—478no—
234C2×C42—478no—
235C2×C42—4716no—
236#95—4716no—
237#95—478no—
238#95—478no—
239#95—4716no—
240C24—24no—
241C24—24no—
242C24—24no—
243C24C22×Dic4714yes—
244C24C23×D4711yescharacteristic
245C24C23×D4711yescharacteristic
246C24C23×D4711yescharacteristic
247C22×C4—24no—
248C22×C4—216no—
249C22×C4—232no—
250C22×C4—28no—
251C22×C4—28no—
252C22×C4—432no—
253C22×C4—44no—
254C22×C4—48no—
255C22×C4—44no—
256C22×C4—4716no—
257C22×C4#1062535018yes—
258C22×C4C22×Dic4714yes—
259C22×C4C22×Dic47116yes—
260C22×C4C22×Dic4718yes—
261C22×C4C22×Dic4718yes—
262C22×C4C22×Dic4718yes—
263C22×C4C22×Dic4718yes—
264C22×C4C22×Dic47116yes—
265C22×C4C22×Dic4714yes—
266C22×C4C23×D4712yes—
267C22×C4C23×D4714yes—
268C22×C4C23×D4714yes—
269C22×C4C23×D4712yes—
270C4⋊C4—9416no—
271C4⋊C4—18816no—
272C4⋊C4C22×Dic47132yes—
273C4⋊C4C22×Dic47116yes—
274C4⋊C4C22×Dic47116yes—
275C22⋊C4—9416no—
276C22⋊C4—9416no—
277C22⋊C4—9416no—
278C42—9416no—
279C42—18816no—
280C23—232no—
281C23—28no—
282C23—28no—
283C23—28no—
284C23—22no—
285C23—22no—
286C23—22no—
287C23—22no—
288C23—22no—
289C23—22no—
290C23C23×Dic4712yes—
291C23C23×Dic4712yes—
292C23C23×Dic4712yes—
293C23C2×#1062535014yes—
294C23#1309406914yes—
295C23#1309407214yes—
296C23C24×D4711yescharacteristic
297C2×C4—28no—
298C2×C4—232no—
299C2×C4—24no—
300C2×C4—28no—
301C2×C4—28no—
302C2×C4—216no—
303C2×C4—216no—
304C2×C4—48no—
305C2×C4—416no—
306C2×C4—48no—
307C2×C4—9416no—
308C2×C4—9416no—
309C2×C4—9416no—
310C2×C4#13093983116yescomplemented
311C2×C4C23×Dic4714yes—
312C2×C4C23×Dic4718yes—
313C2×C4C23×Dic4718yes—
314C2×C4C23×Dic4714yes—
315C2×C4C2×#1062535018yes—
316C22—232no—
317C22—22no—
318C22—22no—
319C22—22no—
320C22—22no—
321C22—22no—
322C22—22no—
323C22—28no—
324C22—28no—
325C22—28no—
326C22C2×#1309398314yes—
327C22#1568959814yes—
328C22#1568960114yes—
329C22C24×Dic4711yescharacteristic
330C22C22×#1062535012yes—
331C22C2×#1309406912yes—
332C22C2×#1309407212yes—
333C4—24no—
334C4—28no—
335C4—28no—
336C4—18816no—
337C4C2×#1309398318yes—
338C2—24no—
339C2—24no—
340C2—24no—
341C2C22×#1309398311yescharacteristic
342C2C2×#1568959811yescharacteristic
343C2C2×#1568960111yescharacteristic
344C2#2025709714yes—
subgroupG/H
center, FrattiniC23C24×D47
commutatorC22×C47C24×C4
FittingC23×C4⋊C4×C47C2
socle, cosocleC23×C47C25
prime radicalC25×C47C23
hypercenterC23×C4⋊C4D47
hypocenterC47#34178
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #20246265C2semidirect
2. #20246257C2semidirect
3. #20246006C2semidirect
4. #20245991C2semidirect
7. #20245745C2semidirect
11. #15689022C22semidirect
12. #15689009C22semidirect
20. C4⋊C4×Dic47C22semidirect
27. C22×C4⋊C4×C47C4semidirect
134. C2×C4×C47#117semidirect
185. C22×C4⋊C4Dic47semidirect
205. C47#34178semidirect
310. C2×C4#13093983semidirect

Maximal subgroups

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

subgroupindex
#202462652
#202462572
#202460062
#202459912
#202457712
#202457452
C23×C4⋊C4×C472
#3417847