Finite Group Databasebeta

← #45667591

Subgroups — #45667591

Order 32896.

528 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 159 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×Q8×Dic257C211yesmaximal, characteristic, complemented
2C2×#22551550C211yesmaximal, characteristic
3#30656676C211yesmaximal, characteristic, complemented
4#30656674C211yesmaximal, characteristic
5#30656669C212yesmaximal, complemented
6#30656659C212yesmaximal
7#30656648C211yesmaximal, characteristic
8C4⋊C4×Dic257C212yesmaximal, complemented
9#30656588C211yesmaximal, characteristic, complemented
10#30656584C212yesmaximal
11#395×C257C211yesmaximal, characteristic, complemented
12C22×C4×D257C2211yescharacteristic
13Q8×Dic257—24no—
14#22551550—24no—
15C2×C4⋊Dic257C2212yes—
16C2×C4⋊Dic257C2212yes—
17C2×C4⋊Dic257C2211yescharacteristic
18C2×C4⋊Dic257C2211yescharacteristic
19C2×#16861597C2212yes—
20C2×#16861595C2212yes—
21C2×#16861595C2212yes—
22C2×#16861595C2211yescharacteristic
23C2×#16861595C2211yescharacteristic
24C2×C4×Dic257C2212yes—
25C2×C4×Dic257C2212yes—
26C2×C4×Dic257C2211yescharacteristic
27C2×C4×Dic257C2211yescharacteristic
28#22551499C2212yes—
29#22551499C2211yescharacteristic
30#22551499C2212yes—
31#22551499C2212yes—
32C22×Q8×C257C2211yescharacteristic
33C2×C4⋊C4×C257C2212yes—
34C2×C4⋊C4×C257C2211yescharacteristic
35#95×C257C2212yes—
36#95×C257C2211yescharacteristic
37C23×D257C2311yescharacteristic
38C22×Dic257C2312yes—
39C22×Dic257C2311yescharacteristic
40C22×Dic257C2312yes—
41C22×Dic257C2311yescharacteristic
42C22×Dic257C2311yescharacteristic
43C2×C4×D257—22no—
44C2×C4×D257—22no—
45C2×C4×D257Q814yescomplemented
46C4⋊Dic257—24no—
47C4⋊Dic257—22no—
48C4⋊Dic257—22no—
49C4⋊Dic257—24no—
50#16861597—24no—
51#16861595—22no—
52#16861595—22no—
53#16861595—24no—
54#16861595—24no—
55C4×Dic257—24no—
56C4×Dic257—22no—
57C4×Dic257—24no—
58C4×Dic257D414yes—
59C2×Q8×C257—22no—
60C2×Q8×C257—24no—
61C22×C4×C257C2311yescharacteristic
62C22×C4×C257C2311yescharacteristic
63C22×C4×C257C2311yescharacteristic
64C22×C4×C257C2312yes—
65C22×C4×C257C2312yes—
66C4⋊C4×C257—22no—
67C4⋊C4×C257—24no—
68C22×D257—21no—
69C22×D257—22no—
70C22×D257—22no—
71C22×D257—21no—
72C22×D257C2×Q812yes—
73C2×Dic257—22no—
74C2×Dic257—21no—
75C2×Dic257—22no—
76C2×Dic257—22no—
77C2×Dic257—22no—
78C2×Dic257—22no—
79C2×Dic257—22no—
80C2×Dic257—22no—
81C2×Dic257—22no—
82C2×Dic257C2×D412yes—
83C2×Dic257C2×D412yes—
84C2×Dic257C2×Q812yes—
85C2×Dic257#4112yes—
86C4×D257—28no—
87C23×C257C2411yescharacteristic
88Q8×C257—28no—
89C2×C4×C257—21no—
90C2×C4×C257—22no—
91C2×C4×C257—22no—
92C2×C4×C257—22no—
93C2×C4×C257—22no—
94C2×C4×C257—22no—
95C2×C4×C257—21no—
96C2×C4×C257—21no—
97C2×C4×C257—22no—
98C2×C4×C257C2×D412yes—
99C2×C4×C257C2×Q812yes—
100C2×C4×C257#4114yes—
101C2×C4×C257#4114yes—
102C22×C257C22×D411yescharacteristic
103C22×C257C22×Q811yescharacteristic
104C22×C257C2×#4111yescharacteristic
105C22×C257C2×#4112yes—
106C22×C257C2×#4112yes—
107C2×D257—22no—
108C2×D257—24no—
109C2×D257—41no—
110C2×D257—42no—
111C2×D257—41no—
112C4×C257—22no—
113C4×C257—22no—
114C4×C257—24no—
115C4×C257—24no—
116C4×C257—41no—
117Dic257—22no—
118Dic257—42no—
119Dic257—41no—
120Dic257—42no—
121Dic257C22⋊Q814yes—
122C2×C257C2×C22⋊Q811yescharacteristic
123C2×C257#52511yescharacteristic
124C2×C257#52912yes—
125C2×C257#53211yescharacteristic
126C2×C257#53312yes—
127D257—42no—
128C257#239611yescharacteristic, complemented, Sylow
129#2396—2571nomaximal, Sylow
130C2×C4×Q8—2571no—
131C2×#117—2571no—
132#400—2571no—
133#399—2571no—
134#398—2572no—
135#396—2572no—
136#395D25711yescharacteristic, complemented
137#393—2571no—
138C4×C4⋊C4—2572no—
139C4×C4⋊C4—2571no—
140C4×C22⋊C4—2572no—
141C22×Q8C2×D25711yescharacteristic
142C23×C4—2571no—
143C4×Q8—5144no—
144#117—5144no—
145C2×C4⋊C4—2571no—
146C2×C4⋊C4—2572no—
147C2×C4⋊C4—2571no—
148C2×C4⋊C4—2572no—
149C2×C4⋊C4—2572no—
150C2×C4⋊C4C2×D25711yescharacteristic
151C2×C4⋊C4C2×D25712yes—
152C2×C22⋊C4—2571no—
153C2×C22⋊C4—2571no—
154C2×C22⋊C4—2572no—
155C2×C22⋊C4—2572no—
156C2×C42—2571no—
157C2×C42—2571no—
158C2×C42—2572no—
159C2×C42—2572no—
160#95—2572no—
161#95—2572no—
162#95—2571no—
163#95—2572no—
164#95C2×D25712yes—
165#95C2×D25711yescharacteristic
166C24—2571no—
167C2×Q8—22no—
168C2×Q8—24no—
169C22×C4—2571no—
170C22×C4—2571no—
171C22×C4—2574no—
172C22×C4—2572no—
173C22×C4—2571no—
174C22×C4—2572no—
175C22×C4—5142no—
176C22×C4—5142no—
177C22×C4C22×D25711yescharacteristic
178C22×C4C22×D25711yescharacteristic
179C22×C4C22×D25711yescharacteristic
180C22×C4C22×D25712yes—
181C22×C4C22×D25712yes—
182C4⋊C4—22no—
183C4⋊C4—24no—
184C4⋊C4—5142no—
185C4⋊C4—5144no—
186C4⋊C4—5142no—
187C4⋊C4—5144no—
188C4⋊C4—5144no—
189C22⋊C4—5142no—
190C22⋊C4—5142no—
191C22⋊C4—5144no—
192C22⋊C4—5144no—
193C42—2574no—
194C42—5142no—
195C42—5144no—
196C42—5144no—
197C23—2572no—
198C23—5141no—
199C23—5141no—
200C23—5142no—
201C23—5142no—
202C23C23×D25711yescharacteristic
203Q8—28no—
204C2×C4—2572no—
205C2×C4—2572no—
206C2×C4—2572no—
207C2×C4—2572no—
208C2×C4—22no—
209C2×C4—21no—
210C2×C4—22no—
211C2×C4—21no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—21no—
216C2×C4—22no—
217C2×C4—5148no—
218C2×C4—5142no—
219C2×C4—5141no—
220C2×C4—5142no—
221C2×C4—5142no—
222C2×C4—5142no—
223C2×C4—5142no—
224C2×C4—5142no—
225C2×C4—5142no—
226C2×C4—5142no—
227C2×C4D4×D25712yes—
228C2×C4Q8×D25712yes—
229C2×C4#1686161714yes—
230C2×C4#1686161914yes—
231C22—5144no—
232C22—5142no—
233C22—10281no—
234C22—10281no—
235C22—10282no—
236C22C2×D4×D25711yescharacteristic
237C22C2×Q8×D25711yescharacteristic
238C22#41×D25711yescharacteristic
239C22C2×#1686161712yes—
240C22C2×#1686161912yes—
241C4—2574no—
242C4—22no—
243C4—22no—
244C4—24no—
245C4—24no—
246C4—5142no—
247C4—41no—
248C4—10281no—
249C4—10282no—
250C4—10282no—
251C2—10282no—
252C2C22⋊Q8×D25711yescharacteristic
253C2#3065724311yescharacteristic
254C2#3065727312yes—
255C2#3065728811yescharacteristic
256C2#3065729212yes—
subgroupG/H
center, FrattiniC23C23×D257
commutator, socle, cosocleC23×C257C24
Fitting#395×C257C2
prime radicalC23×D257C23
hypercenter#395D257
hypocenterC257#2396
LCSG ▷ C23×C257 ▷ C257
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
1. C2×Q8×Dic257C2semidirect
3. #30656676C2semidirect
5. #30656669C2semidirect
8. C4⋊C4×Dic257C2semidirect
9. #30656588C2semidirect
11. #395×C257C2semidirect
45. C2×C4×D257Q8semidirect
128. C257#2396semidirect
136. #395D257semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×Dic2572
C2×#225515502
#306566762
#306566742
#306566692
#306566592
#306566482
C4⋊C4×Dic2572
#306565882
#306565842
#395×C2572
#2396257