Finite Group Databasebeta

← #45667559

Subgroups — #45667559

Order 32896.

696 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 696 conjugacy classes of subgroups — too many to draw. Only its 155 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D4×Dic257C211yesmaximal, characteristic, complemented
2C2×#22551552C212yesmaximal, complemented
3C2×#22551550C211yesmaximal, characteristic, complemented
4#30656667C211yesmaximal, characteristic, complemented
5#30656664C211yesmaximal, characteristic, complemented
6#30656660C212yesmaximal, complemented
7#30656652C212yesmaximal, complemented
8#30656645C211yesmaximal, characteristic
9#30656588C211yesmaximal, characteristic, complemented
10C22⋊C4×Dic257C212yesmaximal, complemented
11#394×C257C211yesmaximal, characteristic, complemented
12C22×C22⋊D257C2212yes—
13C23×Dic257C2212yes—
14C22×C4×D257C2211yescharacteristic
15D4×Dic257—24no—
16#22551552—28no—
17#22551550—24no—
18C2×C4⋊Dic257C2211yescharacteristic
19C2×C4⋊Dic257C2211yescharacteristic
20C2×#16861597C2212yes—
21C2×#16861597C2212yes—
22C2×C22⋊Dic257C2212yes—
23C2×C22⋊Dic257C2212yes—
24C2×#16861595C2211yescharacteristic
25C2×#16861595C2211yescharacteristic
26C2×#16861595C2212yes—
27C2×C4×Dic257C2211yescharacteristic
28C2×C4×Dic257C2212yes—
29C2×C4×Dic257C2211yescharacteristic
30#22551499C2212yes—
31#22551499C2212yes—
32#22551499C2211yescharacteristic
33C22×D4×C257C2211yescharacteristic
34C2×C4⋊C4×C257C2211yescharacteristic
35C2×C22⋊C4×C257C2212yes—
36C2×C22⋊C4×C257C2212yes—
37#95×C257C2211yescharacteristic
38C23×D257C2311yescharacteristic
39C2×C22⋊D257—24no—
40C2×C22⋊D257—24no—
41C2×C22⋊D257—24no—
42C22×Dic257—24no—
43C22×Dic257—24no—
44C22×Dic257—24no—
45C22×Dic257C2312yes—
46C22×Dic257C2311yescharacteristic
47C22×Dic257C2311yescharacteristic
48C22×Dic257C2312yes—
49C22×Dic257C2311yescharacteristic
50C2×C4×D257—22no—
51C2×C4×D257—22no—
52C2×C4×D257D414yescomplemented
53C4⋊Dic257—22no—
54C4⋊Dic257—22no—
55#16861597—24no—
56#16861597—24no—
57C22⋊Dic257—24no—
58C22⋊Dic257—24no—
59#16861595—24no—
60#16861595—22no—
61#16861595—22no—
62C4×Dic257—24no—
63C4×Dic257—22no—
64C4×Dic257D414yescomplemented
65C24×C257C2312yes—
66C2×D4×C257—24no—
67C2×D4×C257—22no—
68C22×C4×C257C2312yes—
69C22×C4×C257C2311yescharacteristic
70C22×C4×C257C2311yescharacteristic
71C22×C4×C257C2311yescharacteristic
72C4⋊C4×C257—22no—
73C22⋊C4×C257—24no—
74C22⋊C4×C257—24no—
75C22×D257—21no—
76C22×D257—21no—
77C22×D257—22no—
78C22×D257—22no—
79C22×D257C2×D412yes—
80C22⋊D257—216no—
81C2×Dic257—22no—
82C2×Dic257—22no—
83C2×Dic257—22no—
84C2×Dic257—21no—
85C2×Dic257—22no—
86C2×Dic257—22no—
87C2×Dic257—22no—
88C2×Dic257—22no—
89C2×Dic257—48no—
90C2×Dic257C2×D412yes—
91C2×Dic257C2×D412yes—
92C2×Dic257C2×D412yes—
93C2×Dic257#4114yes—
94C2×Dic257#4112yes—
95C4×D257—28no—
96C23×C257—22no—
97C23×C257—22no—
98C23×C257—22no—
99C23×C257—22no—
100C23×C257—22no—
101C23×C257—22no—
102C23×C257#4114yes—
103C23×C257C2411yescharacteristic
104D4×C257—28no—
105C2×C4×C257—21no—
106C2×C4×C257—22no—
107C2×C4×C257—21no—
108C2×C4×C257—22no—
109C2×C4×C257—21no—
110C2×C4×C257—22no—
111C2×C4×C257—22no—
112C2×C4×C257—22no—
113C2×C4×C257C2×D412yes—
114C2×C4×C257C2×D412yes—
115C22×C257—24no—
116C22×C257—24no—
117C22×C257—24no—
118C22×C257—42no—
119C22×C257—42no—
120C22×C257—42no—
121C22×C257—42no—
122C22×C257C22×D411yescharacteristic
123C22×C257C22×D411yescharacteristic
124C22×C257C2×#4111yescharacteristic
125C22×C257C2×#4112yes—
126C22×C257C2×#4112yes—
127C2×D257—22no—
128C2×D257—24no—
129C2×D257—41no—
130C2×D257—42no—
131C2×D257—41no—
132C4×C257—22no—
133C4×C257—22no—
134C4×C257—41no—
135C4×C257—42no—
136Dic257—24no—
137Dic257—22no—
138Dic257—42no—
139Dic257—41no—
140Dic257#12114yes—
141C2×C257—44no—
142C2×C257C2×#12111yescharacteristic
143C2×C257#52511yescharacteristic
144C2×C257#52512yes—
145C2×C257#52912yes—
146C2×C257#53211yescharacteristic
147D257—42no—
148C257#239211yescharacteristic, complemented, Sylow
149#2392—2571nomaximal, Sylow
150C2×C4×D4—2571no—
151C2×C4×D4—2572no—
152C2×#117—2571no—
153#398—2571no—
154#397—2571no—
155#396—2572no—
156#394—2572no—
157#394D25711yescharacteristic, complemented
158#392—2571no—
159C4×C4⋊C4—2571no—
160C4×C22⋊C4—2572no—
161C22×D4—2572no—
162C22×D4C2×D25711yescharacteristic
163C23×C4—2571no—
164C23×C4—2572no—
165C4×D4—5144no—
166C4×D4—5148no—
167#117—5144no—
168C2×C4⋊C4—2571no—
169C2×C4⋊C4—2572no—
170C2×C4⋊C4—2571no—
171C2×C4⋊C4—2572no—
172C2×C4⋊C4C2×D25711yescharacteristic
173C2×C22⋊C4—2572no—
174C2×C22⋊C4—2572no—
175C2×C22⋊C4—2571no—
176C2×C22⋊C4—2571no—
177C2×C22⋊C4—2572no—
178C2×C22⋊C4C2×D25712yes—
179C2×C22⋊C4C2×D25712yes—
180C2×C42—2571no—
181C2×C42—2571no—
182C2×C42—2572no—
183#95—2572no—
184#95—2572no—
185#95—2571no—
186#95C2×D25711yescharacteristic
187C24—2571no—
188C24C22×D25712yes—
189C2×D4—24no—
190C2×D4—22no—
191C2×D4—5144no—
192C2×D4—5144no—
193C2×D4—5144no—
194C22×C4—2571no—
195C22×C4—2572no—
196C22×C4—2571no—
197C22×C4—2574no—
198C22×C4—2571no—
199C22×C4—2572no—
200C22×C4—5142no—
201C22×C4—5142no—
202C22×C4—5144no—
203C22×C4—5144no—
204C22×C4—5144no—
205C22×C4C22×D25711yescharacteristic
206C22×C4C22×D25712yes—
207C22×C4C22×D25711yescharacteristic
208C22×C4C22×D25711yescharacteristic
209C4⋊C4—22no—
210C4⋊C4—5142no—
211C4⋊C4—5144no—
212C4⋊C4—5144no—
213C4⋊C4—5142no—
214C22⋊C4—24no—
215C22⋊C4—24no—
216C22⋊C4—5142no—
217C22⋊C4—5144no—
218C22⋊C4—5144no—
219C22⋊C4—5144no—
220C22⋊C4—5142no—
221C42—2574no—
222C42—5142no—
223C42—5144no—
224C23—2572no—
225C23—22no—
226C23—22no—
227C23—22no—
228C23—22no—
229C23—22no—
230C23—22no—
231C23—5141no—
232C23—5141no—
233C23—5142no—
234C23—5142no—
235C23#1686161714yes—
236C23C23×D25711yescharacteristic
237D4—28no—
238D4—51416no—
239C2×C4—2572no—
240C2×C4—2572no—
241C2×C4—2572no—
242C2×C4—2572no—
243C2×C4—2574no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—21no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—5148no—
253C2×C4—5142no—
254C2×C4—5142no—
255C2×C4—5142no—
256C2×C4—5141no—
257C2×C4—5142no—
258C2×C4—5142no—
259C2×C4—5142no—
260C2×C4—5142no—
261C2×C4—10288no—
262C2×C4D4×D25712yes—
263C2×C4D4×D25712yes—
264C22—24no—
265C22—24no—
266C22—24no—
267C22—5142no—
268C22—5144no—
269C22—42no—
270C22—42no—
271C22—42no—
272C22—42no—
273C22—10281no—
274C22—10281no—
275C22—10282no—
276C22C2×D4×D25711yescharacteristic
277C22C2×D4×D25711yescharacteristic
278C22#41×D25712yes—
279C22#41×D25711yescharacteristic
280C22C2×#1686161712yes—
281C4—2574no—
282C4—22no—
283C4—22no—
284C4—5142no—
285C4—5144no—
286C4—41no—
287C4—42no—
288C4—10281no—
289C4—10282no—
290C2—44no—
291C2—10282no—
292C2#121×D25711yescharacteristic
293C2#3065724311yescharacteristic
294C2#3065724712yes—
295C2#3065726412yes—
296C2#3065728611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D257
commutator, socle, cosocleC23×C257C24
Fitting#394×C257C2
prime radical#30656645C2
hypercenter#394D257
hypocenterC257#2392
LCSG ▷ C23×C257 ▷ C257
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C2×D4×Dic257C2semidirect
2. C2×#22551552C2semidirect
3. C2×#22551550C2semidirect
4. #30656667C2semidirect
5. #30656664C2semidirect
6. #30656660C2semidirect
7. #30656652C2semidirect
9. #30656588C2semidirect
10. C22⋊C4×Dic257C2semidirect
11. #394×C257C2semidirect
52. C2×C4×D257D4semidirect
64. C4×Dic257D4semidirect
148. C257#2392semidirect
157. #394D257semidirect

Maximal subgroups

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

subgroupindex
C2×D4×Dic2572
C2×#225515522
C2×#225515502
#306566672
#306566642
#306566602
#306566522
#306566452
#306565882
C22⋊C4×Dic2572
#394×C2572
#2392257