Finite Group Databasebeta

← #45667557

Subgroups — #45667557

Order 32896.

696 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 696 conjugacy classes of subgroups — too many to draw. Only its 151 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×#22551554C211yesmaximal, characteristic, complemented
2C2×#22551552C212yesmaximal, complemented
3C2×#22551548C211yesmaximal, characteristic, complemented
4#30656668C211yesmaximal, characteristic, complemented
5#30656661C212yesmaximal, complemented
6#30656655C211yesmaximal, characteristic, complemented
7#30656652C212yesmaximal, complemented
8#30656646C211yesmaximal, characteristic
9#30656588C211yesmaximal, characteristic, complemented
10#30656584C212yesmaximal, complemented
11#397×C257C211yesmaximal, characteristic, complemented
12C22×D1028C2211yescharacteristic
13C22×C22⋊D257C2212yes—
14C23×Dic257C2211yescharacteristic
15C22×C4×D257C2212yes—
16#22551554—24no—
17#22551552—28no—
18#22551548—24no—
19C2×#16861597C2212yes—
20C2×#16861597C2211yescharacteristic
21C2×#16861597C2211yescharacteristic
22C2×#16861597C2212yes—
23C2×C22⋊Dic257C2211yescharacteristic
24C2×#16861595C2212yes—
25C2×#16861595C2212yes—
26C2×#16861595C2212yes—
27C2×#16861595C2212yes—
28C2×C4×Dic257C2211yescharacteristic
29C2×C4×Dic257C2212yes—
30C2×C4×Dic257C2211yescharacteristic
31#22551499C2212yes—
32#22551499C2211yescharacteristic
33C2×C4⋊C4×C257C2211yescharacteristic
34C2×C22⋊C4×C257C2211yescharacteristic
35C2×C22⋊C4×C257C2212yes—
36#95×C257C2212yes—
37#95×C257C2211yescharacteristic
38C23×D257C2312yes—
39C2×D1028—24no—
40C2×D1028—22no—
41C2×C22⋊D257—24no—
42C2×C22⋊D257—24no—
43C2×C22⋊D257—24no—
44C22×Dic257—22no—
45C22×Dic257—22no—
46C22×Dic257D414yescomplemented
47C22×Dic257C2311yescharacteristic
48C22×Dic257C2311yescharacteristic
49C22×Dic257C2311yescharacteristic
50C22×Dic257C2312yes—
51C22×Dic257C2311yescharacteristic
52C2×C4×D257—24no—
53C2×C4×D257—24no—
54C2×C4×D257—24no—
55#16861597—24no—
56#16861597—22no—
57#16861597—24no—
58#16861597—22no—
59C22⋊Dic257—22no—
60#16861595—24no—
61#16861595—24no—
62#16861595—24no—
63#16861595—24no—
64C4×Dic257—24no—
65C4×Dic257—22no—
66C4×Dic257D414yescomplemented
67C24×C257C2311yescharacteristic
68C22×C4×C257C2312yes—
69C22×C4×C257C2312yes—
70C22×C4×C257C2311yescharacteristic
71C22×C4×C257C2311yescharacteristic
72C4⋊C4×C257—22no—
73C22⋊C4×C257—24no—
74C22⋊C4×C257—22no—
75C22×D257—22no—
76C22×D257—22no—
77C22×D257—22no—
78C22×D257—22no—
79C22×D257—22no—
80C22×D257—22no—
81C22×D257#4114yes—
82D1028—28no—
83C22⋊D257—216no—
84C2×Dic257—22no—
85C2×Dic257—21no—
86C2×Dic257—22no—
87C2×Dic257—21no—
88C2×Dic257—22no—
89C2×Dic257—28no—
90C2×Dic257—21no—
91C2×Dic257—22no—
92C2×Dic257C2×D412yes—
93C2×Dic257C2×D412yes—
94C2×Dic257C2×D412yes—
95C2×Dic257C2×D412yes—
96C2×Dic257#4114yes—
97C2×Dic257#4112yes—
98C4×D257—48no—
99C23×C257—21no—
100C23×C257—21no—
101C23×C257—22no—
102C23×C257—22no—
103C23×C257C2×D412yes—
104C23×C257C2411yescharacteristic
105C2×C4×C257—22no—
106C2×C4×C257—22no—
107C2×C4×C257—21no—
108C2×C4×C257—22no—
109C2×C4×C257—22no—
110C2×C4×C257—22no—
111C2×C4×C257—22no—
112C2×C4×C257—22no—
113C2×C4×C257—22no—
114C2×C4×C257C2×D412yes—
115C22×C257—22no—
116C22×C257—24no—
117C22×C257—41no—
118C22×C257—42no—
119C22×C257—41no—
120C22×C257C22×D411yescharacteristic
121C22×C257C22×D411yescharacteristic
122C22×C257C2×#4111yescharacteristic
123C22×C257C2×#4112yes—
124C22×C257C2×#4112yes—
125C2×D257—24no—
126C2×D257—24no—
127C2×D257—24no—
128C2×D257—42no—
129C2×D257—42no—
130C2×D257—42no—
131C2×D257—42no—
132C4×C257—22no—
133C4×C257—42no—
134C4×C257—42no—
135C4×C257—41no—
136Dic257—22no—
137Dic257—24no—
138Dic257—22no—
139Dic257—41no—
140Dic257#12114yes—
141C2×C257—42no—
142C2×C257C2×#12111yescharacteristic
143C2×C257#52512yes—
144C2×C257#52511yescharacteristic
145C2×C257#52912yes—
146C2×C257#53211yescharacteristic
147D257—44no—
148C257#239211yescharacteristic, complemented, Sylow
149#2392—2571nomaximal, Sylow
150C2×C4×D4—2571no—
151C2×C4×D4—2572no—
152C2×#117—2571no—
153#398—2571no—
154#397D25711yescharacteristic, complemented
155#396—2572no—
156#394—2572no—
157#394—2571no—
158#392—2571no—
159C4×C4⋊C4—2571no—
160C4×C22⋊C4—2572no—
161C22×D4—2572no—
162C22×D4—2571no—
163C23×C4—2571no—
164C23×C4—2572no—
165C4×D4—5144no—
166C4×D4—5148no—
167#117—5144no—
168C2×C4⋊C4—2571no—
169C2×C4⋊C4—2571no—
170C2×C4⋊C4—2572no—
171C2×C4⋊C4—2572no—
172C2×C4⋊C4C2×D25711yescharacteristic
173C2×C22⋊C4—2571no—
174C2×C22⋊C4—2572no—
175C2×C22⋊C4—2572no—
176C2×C22⋊C4—2572no—
177C2×C22⋊C4—2572no—
178C2×C22⋊C4C2×D25711yescharacteristic
179C2×C22⋊C4C2×D25712yes—
180C2×C42—2571no—
181C2×C42—2571no—
182C2×C42—2572no—
183#95—2572no—
184#95—2571no—
185#95C2×D25712yes—
186#95C2×D25711yescharacteristic
187C24—2572no—
188C24C22×D25711yescharacteristic
189C2×D4—5144no—
190C2×D4—5144no—
191C2×D4—5144no—
192C2×D4—5144no—
193C2×D4—5142no—
194C22×C4—2571no—
195C22×C4—2574no—
196C22×C4—2572no—
197C22×C4—2571no—
198C22×C4—2571no—
199C22×C4—2571no—
200C22×C4—5144no—
201C22×C4—5144no—
202C22×C4—5144no—
203C22×C4—5142no—
204C22×C4—5142no—
205C22×C4C22×D25712yes—
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—22no—
216C22⋊C4—5144no—
217C22⋊C4—5144no—
218C22⋊C4—5144no—
219C22⋊C4—5142no—
220C22⋊C4—5144no—
221C42—2574no—
222C42—5142no—
223C42—5144no—
224C23—2574no—
225C23—21no—
226C23—21no—
227C23—22no—
228C23—22no—
229C23—5142no—
230C23—5142no—
231C23—5142no—
232C23—5142no—
233C23—5142no—
234C23—5142no—
235C23D4×D25712yes—
236C23C23×D25711yescharacteristic
237D4—5148no—
238D4—51416no—
239C2×C4—2572no—
240C2×C4—2572no—
241C2×C4—2572no—
242C2×C4—2572no—
243C2×C4—2572no—
244C2×C4—2574no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—21no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—5148no—
255C2×C4—5142no—
256C2×C4—5141no—
257C2×C4—5142no—
258C2×C4—5141no—
259C2×C4—5142no—
260C2×C4—5141no—
261C2×C4—5142no—
262C2×C4—10288no—
263C2×C4D4×D25712yes—
264C22—22no—
265C22—24no—
266C22—5144no—
267C22—5144no—
268C22—5144no—
269C22—41no—
270C22—41no—
271C22—42no—
272C22—10282no—
273C22—10282no—
274C22—10282no—
275C22—10282no—
276C22C2×D4×D25711yescharacteristic
277C22C2×D4×D25711yescharacteristic
278C22#41×D25712yes—
279C22#41×D25711yescharacteristic
280C22#41×D25712yes—
281C4—2574no—
282C4—22no—
283C4—5142no—
284C4—5142no—
285C4—5144no—
286C4—41no—
287C4—42no—
288C4—42no—
289C4—10281no—
290C2—42no—
291C2—10284no—
292C2#121×D25711yescharacteristic
293C2#3065724511yescharacteristic
294C2#3065724512yes—
295C2#3065727012yes—
296C2#3065728611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D257
commutator, socle, cosocleC23×C257C24
Fitting#397×C257C2
prime radical#30656646C2
hypercenter#397D257
hypocenterC257#2392
LCSG ▷ C23×C257 ▷ C257
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
1. C2×#22551554C2semidirect
2. C2×#22551552C2semidirect
3. C2×#22551548C2semidirect
4. #30656668C2semidirect
5. #30656661C2semidirect
6. #30656655C2semidirect
7. #30656652C2semidirect
9. #30656588C2semidirect
10. #30656584C2semidirect
11. #397×C257C2semidirect
46. C22×Dic257D4semidirect
66. C4×Dic257D4semidirect
148. C257#2392semidirect
154. #397D257semidirect

Maximal subgroups

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

subgroupindex
C2×#225515542
C2×#225515522
C2×#225515482
#306566682
#306566612
#306566552
#306566522
#306566462
#306565882
#306565842
#397×C2572
#2392257