Finite Group Databasebeta

← #45664516

Subgroups — #45664516

Order 32896.

316 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 99 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×#22551550C211yesmaximal, characteristic
2C2×#22551523C211yesmaximal, characteristic, complemented
3C2×#22551504C211yesmaximal, characteristic
4#30656594C211yesmaximal, characteristic
5#30656500C211yesmaximal, characteristic, complemented
6#30656499C211yesmaximal, characteristic, complemented
7C2×#107×C257C211yesmaximal, characteristic, complemented
8C22×C4×D257C2211yescharacteristic
9C22×#14523339C2211yescharacteristic
10#22551550—22no—
11#22551550—22no—
12C2×C4⋊Dic257C2211yescharacteristic
13C2×C4⋊Dic257C2211yescharacteristic
14C2×#16861595—21no—
15C2×#16861595—21no—
16C2×C4×Dic257—21no—
17#22551523—22no—
18#22551504C414yescomplemented
19#22551499—21no—
20C22×C8×C257C2211yescharacteristic
21C2×C4⋊C4×C257C2211yescharacteristic
22C2×C4⋊C4×C257C2211yescharacteristic
23#107×C257—22no—
24C23×D257D411yescharacteristic
25C22×Dic257—21no—
26C22×Dic257—21no—
27C22×Dic257D411yescharacteristic
28C2×C4×D257—21no—
29C2×C4×D257—21no—
30C2×C4×D257—21no—
31C2×C4×D257—21no—
32C2×C4×D257C2×C412yes—
33C2×C4×D257Q812yes—
34C2×#14523339—21no—
35C2×#14523339—21no—
36C2×#14523339C2×C412yes—
37C4⋊Dic257—21no—
38C4⋊Dic257—21no—
39C4⋊Dic257—21no—
40C4⋊Dic257D412yes—
41#16861595—24no—
42#16861595—42no—
43C4×Dic257—24no—
44C22×C4×C257—21no—
45C22×C4×C257—21no—
46C22×C4×C257C2311yescharacteristic
47C2×C8×C257—21no—
48C2×C8×C257—21no—
49C2×C8×C257C2×C412yes—
50C4⋊C4×C257—21no—
51C4⋊C4×C257—21no—
52C4⋊C4×C257—21no—
53C4⋊C4×C257D412yes—
54C22×D257—21no—
55C22×D257—21no—
56C22×D257—21no—
57C22×D257—21no—
58C22×D257—21no—
59C22×D257—21no—
60C22×D257C4⋊C412yes—
61C2×Dic257—21no—
62C2×Dic257—21no—
63C2×Dic257—22no—
64C2×Dic257—41no—
65C2×Dic257—41no—
66C2×Dic257—41no—
67C2×Dic257—41no—
68C2×Dic257—41no—
69C2×Dic257C4⋊C412yes—
70C4×D257—22no—
71C4×D257—22no—
72C4×D257—22no—
73C4×D257—22no—
74#14523339—22no—
75C23×C257C2×D411yescharacteristic
76C2×C4×C257—22no—
77C2×C4×C257—41no—
78C2×C4×C257—41no—
79C2×C4×C257—41no—
80C2×C4×C257—41no—
81C2×C4×C257—41no—
82C2×C4×C257C22×C411yescharacteristic
83C2×C4×C257C2×D411yescharacteristic
84C2×C4×C257C2×D411yescharacteristic
85C2×C4×C257C2×Q811yescharacteristic
86C2×C4×C257#4111yescharacteristic
87C2×C4×C257#4111yescharacteristic
88C8×C257—22no—
89C22×C257C2×C4⋊C411yescharacteristic
90C22×C257#12011yescharacteristic
91C22×C257#12311yescharacteristic
92C22×C257#13511yescharacteristic
93C22×C257#13511yescharacteristic
94C22×C257AGL(1,ℤ/8)11yescharacteristic
95C22×C257#13711yescharacteristic
96C2×D257—22no—
97C2×D257—22no—
98C2×D257—22no—
99C2×D257—41no—
100C2×D257—41no—
101C2×D257—41no—
102C2×D257—41no—
103C4×C257—42no—
104C4×C257—81no—
105C4×C257C4×D411yescharacteristic
106C4×C257C4×D411yescharacteristic
107C4×C257C22⋊Q811yescharacteristic
108C4×C257C22⋊Q811yescharacteristic
109Dic257—22no—
110Dic257—42no—
111Dic257—81no—
112C2×C257#38511yescharacteristic
113C2×C257#42711yescharacteristic
114C2×C257#42811yescharacteristic
115C2×C257#44911yescharacteristic
116C2×C257#45211yescharacteristic
117C2×C257#48211yescharacteristic
118C2×C257#48511yescharacteristic
119D257—42no—
120C257#185411yescharacteristic, complemented, Sylow
121#1854—2571nomaximal, Sylow
122C2×#117—2571no—
123C2×#107D25711yescharacteristic, complemented
124C2×#106—2571no—
125C2×C22⋊C8—2571no—
126#380—2571no—
127#340—2571no—
128#340—2571no—
129C23×C4—2571no—
130C22×C8—2571no—
131C22×C8C2×D25711yescharacteristic
132#117—5142no—
133#117—5142no—
134C2×C4⋊C4—2571no—
135C2×C4⋊C4—2571no—
136C2×C4⋊C4C2×D25711yescharacteristic
137C2×C4⋊C4C2×D25711yescharacteristic
138C2×C22⋊C4—5141no—
139C2×C22⋊C4—5141no—
140C2×C42—5141no—
141#107—22no—
142#106—5142no—
143C22⋊C8—2574no—
144#95—5141no—
145C24—2571no—
146C22×C4—2571no—
147C22×C4—2572no—
148C22×C4—2572no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—5141no—
152C22×C4—5141no—
153C22×C4—5141no—
154C22×C4—5141no—
155C22×C4—5141no—
156C22×C4—5141no—
157C22×C4C22×D25711yescharacteristic
158C2×C8—2572no—
159C2×C8—21no—
160C2×C8—21no—
161C2×C8—5141no—
162C2×C8—5141no—
163C2×C8C4×D25712yes—
164C4⋊C4—2572no—
165C4⋊C4—21no—
166C4⋊C4—21no—
167C4⋊C4—21no—
168C4⋊C4—5141no—
169C4⋊C4—5141no—
170C4⋊C4—5141no—
171C4⋊C4C22⋊D25712yes—
172C22⋊C4—5144no—
173C22⋊C4—10282no—
174C42—5144no—
175C23—2572no—
176C23—5141no—
177C23—5141no—
178C23—5141no—
179C23—5141no—
180C23—5141no—
181C23—5141no—
182C23D4×D25711yescharacteristic
183C2×C4—2572no—
184C2×C4—22no—
185C2×C4—5141no—
186C2×C4—5141no—
187C2×C4—5142no—
188C2×C4—5142no—
189C2×C4—5142no—
190C2×C4—5142no—
191C2×C4—5142no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—10281no—
198C2×C4—10281no—
199C2×C4—10281no—
200C2×C4—10281no—
201C2×C4—10281no—
202C2×C4C2×C4×D25711yescharacteristic
203C2×C4D4×D25711yescharacteristic
204C2×C4C2×C22⋊D25711yescharacteristic
205C2×C4Q8×D25711yescharacteristic
206C2×C4#1686161811yescharacteristic
207C2×C4#1686161911yescharacteristic
208C8—22no—
209C8—5142no—
210C22—5142no—
211C22—5142no—
212C22—5142no—
213C22—10281no—
214C22—10281no—
215C22—10281no—
216C22—10281no—
217C22C4⋊C4×D25711yescharacteristic
218C22#2255156011yescharacteristic
219C22#2255157711yescharacteristic
220C22#2255161111yescharacteristic
221C22#2255161511yescharacteristic
222C22#2255161611yescharacteristic
223C22#2255162411yescharacteristic
224C4—5142no—
225C4—42no—
226C4—10282no—
227C4—81no—
228C4—20561no—
229C4C4×C22⋊D25711yescharacteristic
230C4#2255155411yescharacteristic
231C4#2255156811yescharacteristic
232C4#2255157011yescharacteristic
233C2—10282no—
234C2#3065660811yescharacteristic
235C2#3065677411yescharacteristic
236C2#3065677811yescharacteristic
237C2#3065687911yescharacteristic
238C2#3065689811yescharacteristic
239C2#3065706411yescharacteristic
240C2#3065708011yescharacteristic
subgroupG/H
centerC23D4×D257
commutatorC2×C4×C257C22×C4
FrattiniC22×C4C22×D257
FittingC2×#107×C257C2
socleC23×C257C2×D4
prime radicalC23×D257D4
cosocleC22×C4×C257C23
hypercenterC2×#107D257
hypocenterC257#1854
LCSG ▷ C2×C4×C257 ▷ C2×C257 ▷ C257
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#107

Complements

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

NH ≅ G/Ntype
2. C2×#22551523C2semidirect
5. #30656500C2semidirect
6. #30656499C2semidirect
7. C2×#107×C257C2semidirect
18. #22551504C4semidirect
120. C257#1854semidirect
123. C2×#107D257semidirect

Maximal subgroups

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

subgroupindex
C2×#225515502
C2×#225515232
C2×#225515042
#306565942
#306565002
#306564992
C2×#107×C2572
#1854257