Finite Group Databasebeta

← #45666926

Subgroups — #45666926

Order 32896.

484 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22551550C211yesmaximal, characteristic
2#30656629C211yesmaximal, characteristic
3#30656628C211yesmaximal, characteristic
4#30656627C211yesmaximal, characteristic
5#30656626C211yesmaximal, characteristic
6#30656623C211yesmaximal, characteristic, complemented
7#30656622C211yesmaximal, characteristic, complemented
8#30656620C211yesmaximal, characteristic, complemented
9#30656606C211yesmaximal, characteristic, complemented
10#30656601C211yesmaximal, characteristic, complemented
11#30656596C211yesmaximal, characteristic
12#30656596C211yesmaximal, characteristic
13C4×#16861597C211yesmaximal, characteristic, complemented
14#30656588C211yesmaximal, characteristic, complemented
15#387×C257C211yesmaximal, characteristic, complemented
16C22×C4×D257C2211yescharacteristic
17#22551550C418yescomplemented
18C2×C4⋊Dic257C2211yescharacteristic
19C2×C4⋊Dic257C2211yescharacteristic
20C2×#16861597C2211yescharacteristic
21C2×#16861597C2211yescharacteristic
22C2×#16861597C2211yescharacteristic
23C2×#16861597C2211yescharacteristic
24C2×#16861597C2211yescharacteristic
25C2×#16861597C2211yescharacteristic
26C2×#16861595C2211yescharacteristic
27C2×#16861595C2211yescharacteristic
28C2×#16861595C2211yescharacteristic
29C2×#16861595C2211yescharacteristic
30C2×#16861595C2211yescharacteristic
31C2×#16861595C2211yescharacteristic
32C2×C4×Dic257C2211yescharacteristic
33C2×C4×Dic257C2211yescharacteristic
34C2×C4×Dic257C2211yescharacteristic
35C2×C4×Dic257C2211yescharacteristic
36C2×C4×Dic257C2211yescharacteristic
37#22551499C2211yescharacteristic
38#22551499C2211yescharacteristic
39#22551499C2211yescharacteristic
40#22551499C2211yescharacteristic
41#22551499C2211yescharacteristic
42#22551499C2211yescharacteristic
43#22551499C2211yescharacteristic
44#22551499C2211yescharacteristic
45C2×C4⋊C4×C257C2211yescharacteristic
46C2×C4⋊C4×C257C2211yescharacteristic
47C2×C42×C257C2211yescharacteristic
48#95×C257C2211yescharacteristic
49#95×C257C2211yescharacteristic
50#95×C257C2211yescharacteristic
51#95×C257C2211yescharacteristic
52C23×D257C2311yescharacteristic
53C22×Dic257C2311yescharacteristic
54C22×Dic257C2311yescharacteristic
55C22×Dic257C2311yescharacteristic
56C22×Dic257C2311yescharacteristic
57C22×Dic257C2311yescharacteristic
58C22×Dic257C2311yescharacteristic
59C22×Dic257C2311yescharacteristic
60C2×C4×D257—22no—
61C2×C4×D257—22no—
62C2×C4×D257C2×C414yes—
63C4⋊Dic257—41no—
64C4⋊Dic257C2×C414yes—
65#16861597—22no—
66#16861597—22no—
67#16861597—22no—
68#16861597—22no—
69#16861597—41no—
70#16861597—41no—
71#16861595—41no—
72#16861595—41no—
73#16861595—41no—
74#16861595—41no—
75#16861595C2×C414yes—
76#16861595C2×C414yes—
77C4×Dic257—41no—
78C4×Dic257—41no—
79C4×Dic257—41no—
80C4×Dic257C2×C414yes—
81C4×Dic257C2×C414yes—
82C22×C4×C257C2311yescharacteristic
83C22×C4×C257C2311yescharacteristic
84C22×C4×C257C2311yescharacteristic
85C22×C4×C257C2311yescharacteristic
86C22×C4×C257C2311yescharacteristic
87C22×C4×C257C2311yescharacteristic
88C22×C4×C257C2311yescharacteristic
89C4⋊C4×C257—41no—
90C4⋊C4×C257C2×C414yes—
91C42×C257—41no—
92C22×D257—21no—
93C22×D257—21no—
94C22×D257—21no—
95C22×D257—21no—
96C22×D257—21no—
97C22×D257—21no—
98C22×D257C22×C412yes—
99C2×Dic257—21no—
100C2×Dic257—21no—
101C2×Dic257—21no—
102C2×Dic257—21no—
103C2×Dic257—21no—
104C2×Dic257—21no—
105C2×Dic257—21no—
106C2×Dic257—21no—
107C2×Dic257—21no—
108C2×Dic257—21no—
109C2×Dic257—21no—
110C2×Dic257—21no—
111C2×Dic257—21no—
112C2×Dic257—21no—
113C2×Dic257—21no—
114C2×Dic257—21no—
115C2×Dic257C22×C412yes—
116C2×Dic257C22×C412yes—
117C2×Dic257C22×C412yes—
118C2×Dic257#4112yes—
119C2×Dic257#4112yes—
120C4×D257—28no—
121C23×C257C2411yescharacteristic
122C2×C4×C257—21no—
123C2×C4×C257—21no—
124C2×C4×C257—21no—
125C2×C4×C257—21no—
126C2×C4×C257—21no—
127C2×C4×C257—21no—
128C2×C4×C257—21no—
129C2×C4×C257—21no—
130C2×C4×C257—21no—
131C2×C4×C257—21no—
132C2×C4×C257—21no—
133C2×C4×C257—21no—
134C2×C4×C257—21no—
135C2×C4×C257—21no—
136C2×C4×C257—21no—
137C2×C4×C257—21no—
138C2×C4×C257—21no—
139C2×C4×C257—21no—
140C2×C4×C257C22×C412yes—
141C2×C4×C257C22×C412yes—
142C2×C4×C257C22×C412yes—
143C22×C257C23×C411yescharacteristic
144C22×C257C2×#4111yescharacteristic
145C22×C257C2×#4111yescharacteristic
146C22×C257H(5,𝔽2)11yescharacteristic
147C22×C257H(5,𝔽2)11yescharacteristic
148C22×C2572−1+411yescharacteristic
149C22×C2572−1+411yescharacteristic
150C2×D257—22no—
151C2×D257—22no—
152C2×D257—22no—
153C2×D257—41no—
154C2×D257—41no—
155C2×D257—41no—
156C2×D257—41no—
157C4×C257—22no—
158C4×C257—22no—
159C4×C257—22no—
160C4×C257—41no—
161C4×C257—41no—
162C4×C257—41no—
163C4×C257—41no—
164Dic257—22no—
165Dic257—22no—
166Dic257—41no—
167Dic257—41no—
168Dic257—41no—
169Dic257—41no—
170Dic257#11714yes—
171C2×C257C2×#11711yescharacteristic
172C2×C257#52011yescharacteristic
173C2×C257#52011yescharacteristic
174C2×C257#54011yescharacteristic
175C2×C257#54111yescharacteristic
176C2×C257#54211yescharacteristic
177C2×C257#54211yescharacteristic
178D257—42no—
179C257#231411yescharacteristic, complemented, Sylow
180#2314—2571nomaximal, Sylow
181C2×#117—2571no—
182#388—2571no—
183#388—2571no—
184#388—2571no—
185#388—2571no—
186#387—2571no—
187#387—2571no—
188#387—2571no—
189#387D25711yescharacteristic, complemented
190#384—2571no—
191#382—2571no—
192#380—2571no—
193#380—2571no—
194C4×C4⋊C4—2571no—
195C4×C4⋊C4—2571no—
196C23×C4—2571no—
197#117—2578no—
198C2×C4⋊C4—2571no—
199C2×C4⋊C4—2571no—
200C2×C4⋊C4—2571no—
201C2×C4⋊C4—2571no—
202C2×C4⋊C4—2571no—
203C2×C4⋊C4—2571no—
204C2×C4⋊C4—2571no—
205C2×C4⋊C4—2571no—
206C2×C4⋊C4C2×D25711yescharacteristic
207C2×C4⋊C4C2×D25711yescharacteristic
208C2×C22⋊C4—2571no—
209C2×C22⋊C4—2571no—
210C2×C22⋊C4—2571no—
211C2×C22⋊C4—2571no—
212C2×C22⋊C4—2571no—
213C2×C22⋊C4—2571no—
214C2×C42—2571no—
215C2×C42—2571no—
216C2×C42—2571no—
217C2×C42—2571no—
218C2×C42—2571no—
219C2×C42C2×D25711yescharacteristic
220#95—2571no—
221#95—2571no—
222#95—2571no—
223#95—2571no—
224#95—2571no—
225#95—2571no—
226#95—2571no—
227#95—2571no—
228#95C2×D25711yescharacteristic
229#95C2×D25711yescharacteristic
230#95C2×D25711yescharacteristic
231#95C2×D25711yescharacteristic
232C24—2571no—
233C22×C4—2571no—
234C22×C4—2571no—
235C22×C4—2571no—
236C22×C4—2574no—
237C22×C4—2571no—
238C22×C4—2571no—
239C22×C4—2571no—
240C22×C4—2571no—
241C22×C4—5142no—
242C22×C4—5142no—
243C22×C4C22×D25711yescharacteristic
244C22×C4C22×D25711yescharacteristic
245C22×C4C22×D25711yescharacteristic
246C22×C4C22×D25711yescharacteristic
247C22×C4C22×D25711yescharacteristic
248C22×C4C22×D25711yescharacteristic
249C22×C4C22×D25711yescharacteristic
250C4⋊C4—2574no—
251C4⋊C4—5142no—
252C4⋊C4—5142no—
253C4⋊C4—5142no—
254C4⋊C4—5142no—
255C4⋊C4—41no—
256C4⋊C4—10281no—
257C4⋊C4—10281no—
258C4⋊C4—10281no—
259C4⋊C4C4×D25714yes—
260C22⋊C4—2574no—
261C22⋊C4—2574no—
262C22⋊C4—10281no—
263C22⋊C4—10281no—
264C22⋊C4—10281no—
265C22⋊C4—10281no—
266C42—2574no—
267C42—2574no—
268C42—41no—
269C42—10281no—
270C42—10281no—
271C42—10281no—
272C23—2572no—
273C23—5141no—
274C23—5141no—
275C23—5141no—
276C23—5141no—
277C23—5141no—
278C23—5141no—
279C23C23×D25711yescharacteristic
280C2×C4—2572no—
281C2×C4—2572no—
282C2×C4—2572no—
283C2×C4—2572no—
284C2×C4—2572no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—5148no—
304C2×C4—5141no—
305C2×C4—5141no—
306C2×C4—5141no—
307C2×C4—5141no—
308C2×C4—5141no—
309C2×C4—5141no—
310C2×C4—5141no—
311C2×C4—5141no—
312C2×C4—5141no—
313C2×C4—5141no—
314C2×C4—5141no—
315C2×C4—5141no—
316C2×C4—5141no—
317C2×C4—5141no—
318C2×C4—5141no—
319C2×C4—5141no—
320C2×C4C2×C4×D25712yes—
321C2×C4C2×C4×D25712yes—
322C2×C4C2×C4×D25712yes—
323C22—5142no—
324C22—5142no—
325C22—5142no—
326C22—10281no—
327C22—10281no—
328C22—10281no—
329C22—10281no—
330C22C22×C4×D25711yescharacteristic
331C22#41×D25711yescharacteristic
332C22#41×D25711yescharacteristic
333C22#2255163711yescharacteristic
334C22#2255163811yescharacteristic
335C22#2255163911yescharacteristic
336C22#2255164011yescharacteristic
337C4—2574no—
338C4—22no—
339C4—22no—
340C4—22no—
341C4—5142no—
342C4—5142no—
343C4—41no—
344C4—41no—
345C4—41no—
346C4—41no—
347C4—10281no—
348C4—10281no—
349C4—10281no—
350C4—10281no—
351C2—10282no—
352C2#117×D25711yescharacteristic
353C2#3065721611yescharacteristic
354C2#3065721911yescharacteristic
355C2#3065733911yescharacteristic
356C2#3065734611yescharacteristic
357C2#3065735311yescharacteristic
358C2#3065735711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D257
commutatorC22×C257C23×C4
Fitting#387×C257C2
socle, cosocleC23×C257C24
prime radicalC23×D257C23
hypercenter#387D257
hypocenterC257#2314
LCSG ▷ C22×C257 ▷ C257
UCSC1 ◁ C23 ◁ #387

Complements

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

NH ≅ G/Ntype
6. #30656623C2semidirect
7. #30656622C2semidirect
8. #30656620C2semidirect
9. #30656606C2semidirect
10. #30656601C2semidirect
13. C4×#16861597C2semidirect
14. #30656588C2semidirect
15. #387×C257C2semidirect
17. #22551550C4semidirect
179. C257#2314semidirect
189. #387D257semidirect

Maximal subgroups

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

subgroupindex
C2×#225515502
#306566292
#306566282
#306566272
#306566262
#306566232
#306566222
#306566202
#306566062
#306566012
#306565962
C4×#168615972
#306565882
#387×C2572
#2314257