Finite Group Databasebeta

← #26374965

Subgroups — #26374965

Order 12032.

2858 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20246031C218yesmaximal, complemented
2#20246006C212yesmaximal, complemented
3#20245986C214yesmaximal, complemented
4#20245970C212yesmaximal
5#20245853C218yesmaximal, complemented
6#20245771C214yesmaximal, complemented
7C4×#13093983C212yesmaximal, complemented
8C22×C4×D4×C47C211yesmaximal, characteristic
9C2×#13093983C2214yes—
10C2×#13093983C2218yescomplemented
11#15689026C22116yes—
12#15689022C22116yescomplemented
13#15689016C22132yescomplemented
14#15688986C2218yescomplemented
15#15688983C2218yes—
16#15688983C2218yescomplemented
17#15688979C2218yes—
18C4×C4⋊Dic47C2214yescomplemented
19C4×C4⋊Dic47C2214yescomplemented
20C4×C22⋊Dic47C2214yes—
21C4×C22⋊Dic47C2214yes—
22C4×C22⋊Dic47C2218yescomplemented
23#15688969C2214yescomplemented
24C42×Dic47C2214yescomplemented
25C23×D4×C47C2211yescharacteristic
26C24×C4×C47C2212yes—
27C2×C4×D4×C47C4116yescomplemented
28C2×C4×D4×C47C2214yes—
29C2×C4×D4×C47C2214yes—
30C22×C4⋊C4×C47C2211yescharacteristic
31C22×C22⋊C4×C47C2212yes—
32C22×C42×C47C2211yescharacteristic
33#13093983—216no—
34#13093983—416no—
35C2×C4⋊Dic47C2318yes—
36C2×C4⋊Dic47C23116yes—
37C2×C22⋊Dic47C2318yes—
38C2×C22⋊Dic47C23116yes—
39C2×C22⋊Dic47C23116yes—
40C2×C4×Dic47C2318yes—
41C2×C4×Dic47C2318yes—
42C2×C4×Dic47C2318yes—
43C47⋊#95C2318yes—
44C47⋊#95C2318yes—
45C47⋊#95C23116yes—
46C25×C47C2312yes—
47C22×D4×C47—28no—
48C22×D4×C47C2×C418yes—
49C22×D4×C47C2314yes—
50C23×C4×C47—48no—
51C23×C4×C47C2×C4116yes—
52C23×C4×C47C2311yescharacteristic
53C23×C4×C47C2314yes—
54C23×C4×C47C2311yescharacteristic
55C23×C4×C47C2312yes—
56C23×C4×C47C2314yes—
57C23×C4×C47C2311yescharacteristic
58C4×D4×C47C2×C4132yes—
59C4×D4×C47C2×C4132yes—
60C2×C4⋊C4×C47C2×C418yes—
61C2×C4⋊C4×C47C2314yes—
62C2×C22⋊C4×C47C2×C4116yes—
63C2×C22⋊C4×C47C2318yes—
64C2×C42×C47C2×C418yes—
65C2×C42×C47C2312yes—
66C2×C42×C47C2312yes—
67C24×C47—28no—
68C24×C47—44no—
69C24×C47—44no—
70C24×C47C22×C418yes—
71C24×C47C2414yes—
72C24×C47C2411yescharacteristic
73C2×D4×C47—232no—
74C2×D4×C47—216no—
75C2×D4×C47C22×C4116yes—
76C22×C4×C47—24no—
77C22×C4×C47—216no—
78C22×C4×C47—432no—
79C22×C4×C47—44no—
80C22×C4×C47—48no—
81C22×C4×C47—42no—
82C22×C4×C47C22×C414yes—
83C22×C4×C47C22×C4116yes—
84C22×C4×C47C22×C414yes—
85C22×C4×C47C22×C418yes—
86C22×C4×C47C22×C4116yes—
87C22×C4×C47C22×C414yes—
88C22×C4×C47#4118yes—
89C22×C4×C47C2412yes—
90C22×C4×C47C2411yescharacteristic
91C22×C4×C47C2414yes—
92C22×C4×C47C2412yes—
93C22×C4×C47C2411yescharacteristic
94C22×Dic47C24116yes—
95C4⋊C4×C47C22×C4116yes—
96C22⋊C4×C47C22×C4132yes—
97C42×C47C22×C418yes—
98C42×C47C22×C418yes—
99C22⋊Dic47—216no—
100C22⋊Dic47—416no—
101C22⋊Dic47—416no—
102C4⋊Dic47—216no—
103C4⋊Dic47—416no—
104C4×Dic47—216no—
105C4×Dic47—216no—
106C4×Dic47—48no—
107C23×C47—22no—
108C23×C47—22no—
109C23×C47—216no—
110C23×C47—28no—
111C23×C47—28no—
112C23×C47—28no—
113C23×C47—48no—
114C23×C47—48no—
115C23×C47—432no—
116C23×C47C23×C418yes—
117C23×C47C23×C412yes—
118C23×C47C2×#4114yes—
119C23×C47C2511yescharacteristic
120D4×C47—232no—
121C2×C4×C47—28no—
122C2×C4×C47—22no—
123C2×C4×C47—24no—
124C2×C4×C47—28no—
125C2×C4×C47—232no—
126C2×C4×C47—416no—
127C2×C4×C47—48no—
128C2×C4×C47—44no—
129C2×C4×C47#117116yescomplemented
130C2×C4×C47C23×C414yes—
131C2×C4×C47C23×C412yes—
132C2×C4×C47C23×C418yes—
133C2×C4×C47C23×C414yes—
134C2×C4×C47C23×C412yes—
135C2×C4×C47C2×#4118yes—
136C2×C4×C47H(5,𝔽2)14yes—
137C2×Dic47—216no—
138C2×Dic47—216no—
139C2×Dic47—216no—
140#34058—471nomaximal, Sylow
141C22×C47—22no—
142C22×C47—22no—
143C22×C47—28no—
144C22×C47—28no—
145C22×C47—28no—
146C22×C47—28no—
147C22×C47—432no—
148C22×C47—416no—
149C22×C47C2×#11714yes—
150C22×C47C24×C411yescharacteristic
151C22×C47C22×#4112yes—
152C22×C47C2×H(5,𝔽2)12yes—
153C22×C47#58512yes—
154C4×C47—22no—
155C4×C47—28no—
156C4×C47—24no—
157C4×C47C2×#11718yes—
158C4×C47#51814yes—
159Dic47—416no—
160C22×C4×D4D4711yescharacteristic
161#2319—478no—
162#2316—472no—
163#2312—474no—
164#2309—472no—
165#2293—478no—
166#2280—474no—
167C4×#117—472no—
168C2×C47—24no—
169C2×C47—48no—
170C2×C47C22×#11711yescharacteristic
171C2×C47C2×#51811yescharacteristic
172C2×C47#342811yescharacteristic
173C2×C47#346014yes—
174C23×D4C2×D4711yescharacteristic
175C24×C4C2×D4712yes—
176C2×C4×D4Dic47116yescomplemented
177C2×C4×D4C2×D4714yes—
178C2×C4×D4C2×D4714yes—
179C2×#117—474no—
180C2×#117—478no—
181C22×C4⋊C4C2×D4711yescharacteristic
182C22×C22⋊C4C2×D4712yes—
183C22×C42C2×D4711yescharacteristic
184#390—4716no—
185#389—4716no—
186#388—4732no—
187#381—478no—
188#380—478no—
189#380—478no—
190#379—478no—
191C4×C4⋊C4—474no—
192C4×C4⋊C4—474no—
193C4×C22⋊C4—474no—
194C4×C22⋊C4—474no—
195C4×C22⋊C4—478no—
196#376—474no—
197C43—474no—
198C47#3405811yescharacteristic, complemented, Sylow
199C25C22×D4712yes—
200C22×D4—28no—
201C22×D4C2×Dic4718yes—
202C22×D4C22×D4714yes—
203C23×C4—48no—
204C23×C4C2×Dic47116yes—
205C23×C4C22×D4714yes—
206C23×C4C22×D4711yescharacteristic
207C23×C4C22×D4711yescharacteristic
208C23×C4C22×D4712yes—
209C23×C4C22×D4714yes—
210C23×C4C22×D4711yescharacteristic
211C4×D4C2×Dic47132yes—
212C4×D4C2×Dic47132yes—
213#117—9416no—
214#117—18816no—
215C2×C4⋊C4—478no—
216C2×C4⋊C4—4716no—
217C2×C4⋊C4C2×Dic4718yes—
218C2×C4⋊C4C22×D4714yes—
219C2×C22⋊C4—478no—
220C2×C22⋊C4—4716no—
221C2×C22⋊C4—4716no—
222C2×C22⋊C4C2×Dic47116yes—
223C2×C22⋊C4C22×D4718yes—
224C2×C42—478no—
225C2×C42—478no—
226C2×C42—478no—
227C2×C42C2×Dic4718yes—
228C2×C42C22×D4712yes—
229C2×C42C22×D4712yes—
230#95—478no—
231#95—478no—
232#95—4716no—
233C24—28no—
234C24—44no—
235C24—44no—
236C24C22×Dic4718yes—
237C24C23×D4714yes—
238C24C23×D4711yescharacteristic
239C2×D4—232no—
240C2×D4—216no—
241C2×D4C22×Dic47116yes—
242C22×C4—216no—
243C22×C4—24no—
244C22×C4—432no—
245C22×C4—44no—
246C22×C4—48no—
247C22×C4—42no—
248C22×C4—4716no—
249C22×C4#1062535018yes—
250C22×C4C22×Dic4714yes—
251C22×C4C22×Dic47116yes—
252C22×C4C22×Dic4714yes—
253C22×C4C22×Dic4718yes—
254C22×C4C22×Dic47116yes—
255C22×C4C22×Dic4714yes—
256C22×C4C23×D4711yescharacteristic
257C22×C4C23×D4712yes—
258C22×C4C23×D4714yes—
259C22×C4C23×D4712yes—
260C22×C4C23×D4711yescharacteristic
261C4⋊C4—9416no—
262C4⋊C4—18816no—
263C4⋊C4C22×Dic47116yes—
264C22⋊C4—9416no—
265C22⋊C4—18816no—
266C22⋊C4—18816no—
267C22⋊C4C22×Dic47132yes—
268C42—9416no—
269C42—9416no—
270C42—1888no—
271C42C22×Dic4718yes—
272C42C22×Dic4718yes—
273C23—22no—
274C23—22no—
275C23—216no—
276C23—28no—
277C23—28no—
278C23—28no—
279C23—48no—
280C23—48no—
281C23—432no—
282C23C23×Dic4718yes—
283C23C23×Dic4712yes—
284C23C2×#1062535014yes—
285C23C24×D4711yescharacteristic
286D4—232no—
287C2×C4—28no—
288C2×C4—22no—
289C2×C4—24no—
290C2×C4—28no—
291C2×C4—232no—
292C2×C4—416no—
293C2×C4—48no—
294C2×C4—44no—
295C2×C4—9416no—
296C2×C4—9416no—
297C2×C4—9416no—
298C2×C4#13093983116yescomplemented
299C2×C4C23×Dic4714yes—
300C2×C4C23×Dic4712yes—
301C2×C4C23×Dic4718yes—
302C2×C4C23×Dic4714yes—
303C2×C4C23×Dic4712yes—
304C2×C4C2×#1062535018yes—
305C2×C4#1309406914yes—
306C22—28no—
307C22—22no—
308C22—22no—
309C22—28no—
310C22—28no—
311C22—28no—
312C22—432no—
313C22—416no—
314C22C2×#1309398314yes—
315C22C24×Dic4711yescharacteristic
316C22C22×#1062535012yes—
317C22C2×#1309406912yes—
318C22#1568995912yes—
319C4—22no—
320C4—28no—
321C4—24no—
322C4—18816no—
323C4C2×#1309398318yes—
324C4#1568959814yes—
325C2—24no—
326C2—48no—
327C2C22×#1309398311yescharacteristic
328C2C2×#1568959811yescharacteristic
329C2#2025659111yescharacteristic
330C2#2025681014yes—
subgroupG/H
centerC22×C4C23×D47
commutatorC22×C47C24×C4
FrattiniC23C24×D47
FittingC22×C4×D4×C47C2
socle, cosocleC23×C47C25
prime radicalC23×D4×C47C22
hypercenterC22×C4×D4D47
hypocenterC47#34058
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C22×C4 ◁ C22×C4×D4

Complements

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

NH ≅ G/Ntype
1. #20246031C2semidirect
2. #20246006C2semidirect
3. #20245986C2semidirect
5. #20245853C2semidirect
6. #20245771C2semidirect
7. C4×#13093983C2semidirect
10. C2×#13093983C22semidirect
12. #15689022C22semidirect
13. #15689016C22semidirect
14. #15688986C22semidirect
16. #15688983C22semidirect
18. C4×C4⋊Dic47C22semidirect
19. C4×C4⋊Dic47C22semidirect
22. C4×C22⋊Dic47C22semidirect
23. #15688969C22semidirect
24. C42×Dic47C22semidirect
27. C2×C4×D4×C47C4semidirect
129. C2×C4×C47#117semidirect
176. C2×C4×D4Dic47semidirect
198. C47#34058semidirect
298. C2×C4#13093983semidirect

Maximal subgroups

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

subgroupindex
#202460312
#202460062
#202459862
#202459702
#202458532
#202457712
C4×#130939832
C22×C4×D4×C472
#3405847