Finite Group Databasebeta

← #45664519

Subgroups — #45664519

Order 32896.

316 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 109 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×#22551525C211yesmaximal, characteristic, complemented
2C2×#22551523C211yesmaximal, characteristic, complemented
3C2×C22⋊#14523339C211yesmaximal, characteristic
4#30656597C211yesmaximal, characteristic
5#30656499C211yesmaximal, characteristic, complemented
6#30656499C211yesmaximal, characteristic, complemented
7C2×#117×C257C211yesmaximal, characteristic
8C22×#14523339C2211yescharacteristic
9C22×#14523339C2211yescharacteristic
10C2×C4⋊Dic257C2211yescharacteristic
11C2×C4⋊Dic257C2211yescharacteristic
12C2×C22⋊Dic257—21no—
13#22551525—22no—
14#22551523—22no—
15C22⋊#14523339C414yescomplemented
16#22551499—21no—
17C23×C4×C257C2211yescharacteristic
18#117×C257—22no—
19#117×C257—22no—
20C2×C4⋊C4×C257C2211yescharacteristic
21C2×C4⋊C4×C257C2211yescharacteristic
22C2×C22⋊C4×C257—21no—
23C2×C42×C257—21no—
24C22×Dic257—21no—
25C22×Dic257—21no—
26C2×#14523339—21no—
27C2×#14523339—21no—
28C2×#14523339—21no—
29C2×#14523339—21no—
30C2×#14523339C2×C412yes—
31C2×#14523339C2×C412yes—
32C4⋊Dic257—21no—
33C4⋊Dic257—21no—
34C4⋊Dic257—21no—
35C4⋊Dic257—21no—
36C22⋊Dic257—42no—
37C24×C257D411yescharacteristic
38C22×C4×C257—21no—
39C22×C4×C257—21no—
40C22×C4×C257—21no—
41C22×C4×C257—21no—
42C22×C4×C257—21no—
43C22×C4×C257—21no—
44C22×C4×C257C2×C412yes—
45C22×C4×C257D411yescharacteristic
46C22×C4×C257Q812yes—
47C22×C4×C257C2311yescharacteristic
48C4⋊C4×C257—21no—
49C4⋊C4×C257—21no—
50C4⋊C4×C257D412yes—
51C4⋊C4×C257D412yes—
52C22⋊C4×C257—24no—
53C42×C257—24no—
54C2×Dic257—41no—
55C2×Dic257—41no—
56C2×Dic257—41no—
57C2×Dic257—41no—
58C2×Dic257—41no—
59C2×Dic257—41no—
60#14523339—22no—
61#14523339—22no—
62C23×C257—21no—
63C23×C257—21no—
64C23×C257—21no—
65C23×C257—21no—
66C23×C257—21no—
67C23×C257—21no—
68C23×C257C4⋊C412yes—
69C23×C257C2×D411yescharacteristic
70C2×C4×C257—21no—
71C2×C4×C257—21no—
72C2×C4×C257—22no—
73C2×C4×C257—22no—
74C2×C4×C257—22no—
75C2×C4×C257—22no—
76C2×C4×C257—22no—
77C2×C4×C257—22no—
78C2×C4×C257—41no—
79C2×C4×C257—41no—
80C2×C4×C257—41no—
81C2×C4×C257—41no—
82C2×C4×C257C4⋊C412yes—
83C2×C4×C257C22×C411yescharacteristic
84C2×C4×C257C2×D411yescharacteristic
85C2×C4×C257C2×D411yescharacteristic
86C2×C4×C257C2×Q811yescharacteristic
87C2×C4×C257#4111yescharacteristic
88C2×C4×C257#4111yescharacteristic
89C22×C257—22no—
90C22×C257—22no—
91C22×C257—22no—
92C22×C257—41no—
93C22×C257—41no—
94C22×C257—41no—
95C22×C257—41no—
96C22×C257C2×C4⋊C411yescharacteristic
97C22×C257#12011yescharacteristic
98C22×C257#12311yescharacteristic
99C22×C257#13511yescharacteristic
100C22×C257#13511yescharacteristic
101C22×C257AGL(1,ℤ/8)11yescharacteristic
102C22×C257#13711yescharacteristic
103C4×C257—22no—
104C4×C257—42no—
105C4×C257—42no—
106C4×C257C4×D411yescharacteristic
107C4×C257C4×D411yescharacteristic
108C4×C257C22⋊Q811yescharacteristic
109C4×C257C22⋊Q811yescharacteristic
110Dic257—81no—
111Dic257—81no—
112C2×C257—42no—
113C2×C257#38511yescharacteristic
114C2×C257#42711yescharacteristic
115C2×C257#42811yescharacteristic
116C2×C257#44911yescharacteristic
117C2×C257#45211yescharacteristic
118C2×C257#48211yescharacteristic
119C2×C257#48511yescharacteristic
120C257#185411yescharacteristic, complemented, Sylow
121#1854—2571nomaximal, Sylow
122C2×#117D25711yescharacteristic
123C2×#107—2571no—
124C2×#106—2571no—
125C2×C22⋊C8—2571no—
126#380—2571no—
127#340—2571no—
128#340—2571no—
129C23×C4C2×D25711yescharacteristic
130C22×C8—2571no—
131C22×C8—2571no—
132#117—22no—
133#117—22no—
134C2×C4⋊C4—2571no—
135C2×C4⋊C4—2571no—
136C2×C4⋊C4C2×D25711yescharacteristic
137C2×C4⋊C4C2×D25711yescharacteristic
138C2×C22⋊C4—21no—
139C2×C22⋊C4—5141no—
140C2×C42—21no—
141#107—5142no—
142#106—5142no—
143C22⋊C8—2574no—
144#95—5141no—
145C24C22⋊D25711yescharacteristic
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—5141no—
153C22×C4—5141no—
154C22×C4C4×D25712yes—
155C22×C4C22⋊D25711yescharacteristic
156C22×C4Dic51412yes—
157C22×C4C22×D25711yescharacteristic
158C2×C8—2572no—
159C2×C8—2572no—
160C2×C8—5141no—
161C2×C8—5141no—
162C2×C8—5141no—
163C2×C8—5141no—
164C4⋊C4—21no—
165C4⋊C4—21no—
166C4⋊C4—5141no—
167C4⋊C4—5141no—
168C4⋊C4—5141no—
169C4⋊C4—5141no—
170C4⋊C4C22⋊D25712yes—
171C4⋊C4C22⋊D25712yes—
172C22⋊C4—24no—
173C22⋊C4—10282no—
174C42—24no—
175C23—21no—
176C23—21no—
177C23—21no—
178C23—21no—
179C23—21no—
180C23—21no—
181C23#1686159712yes—
182C23C2×C22⋊D25711yescharacteristic
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—10281no—
196C2×C4—10281no—
197C2×C4—10281no—
198C2×C4—10281no—
199C2×C4—10281no—
200C2×C4—10281no—
201C2×C4#1686159712yes—
202C2×C4C2×C4×D25711yescharacteristic
203C2×C4C2×C22⋊D25711yescharacteristic
204C2×C4C2×C22⋊D25711yescharacteristic
205C2×C4C2×Dic51411yescharacteristic
206C2×C4#1686161811yescharacteristic
207C2×C4#1686161811yescharacteristic
208C8—5142no—
209C8—5142no—
210C22—22no—
211C22—22no—
212C22—22no—
213C22—41no—
214C22—41no—
215C22—41no—
216C22—41no—
217C22C2×#1686159711yescharacteristic
218C22#2255156111yescharacteristic
219C22#2255157511yescharacteristic
220C22#2255161211yescharacteristic
221C22#2255161211yescharacteristic
222C22#2255161711yescharacteristic
223C22#2255162211yescharacteristic
224C4—22no—
225C4—42no—
226C4—42no—
227C4—20561no—
228C4—20561no—
229C4C4×C22⋊D25711yescharacteristic
230C4C4×C22⋊D25711yescharacteristic
231C4#2255157111yescharacteristic
232C4#2255157111yescharacteristic
233C2—42no—
234C2#3065661011yescharacteristic
235C2#3065677311yescharacteristic
236C2#3065678011yescharacteristic
237C2#3065688211yescharacteristic
238C2#3065689911yescharacteristic
239C2#3065706111yescharacteristic
240C2#3065707811yescharacteristic
subgroupG/H
centerC23C2×C22⋊D257
commutatorC2×C4×C257C22×C4
FrattiniC22×C4C22×D257
FittingC2×#117×C257C2
socleC23×C257C2×D4
prime radicalC24×C257D4
cosocleC22×C4×C257C23
hypercenterC2×#117D257
hypocenterC257#1854
LCSG ▷ C2×C4×C257 ▷ C2×C257 ▷ C257
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
1. C2×#22551525C2semidirect
2. C2×#22551523C2semidirect
5. #30656499C2semidirect
6. #30656499C2semidirect
15. C22⋊#14523339C4semidirect
120. C257#1854semidirect

Maximal subgroups

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

subgroupindex
C2×#225515252
C2×#225515232
C2×C22⋊#145233392
#306565972
#306564992
C2×#117×C2572
#1854257