Finite Group Databasebeta

← #45664762

Subgroups — #45664762

Order 32896.

442 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 87 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
1#30657249C211yesmaximal, characteristic
2C2×C42×D257C211yesmaximal, characteristic, complemented
3C2×#22551520C211yesmaximal, characteristic, complemented
4C2×#22551518C211yesmaximal, characteristic, complemented
5#30656700C211yesmaximal, characteristic, complemented
6#30656496C211yesmaximal, characteristic, complemented
7#339×C257C211yesmaximal, characteristic, complemented
8C2×#16861618C2211yescharacteristic
9C22×C4×D257—21no—
10C22×C4×D257C2211yescharacteristic
11C2×#16861602C2211yescharacteristic
12#22551576—21no—
13#22551572C412yescomplemented
14#22551567C412yescomplemented
15#22551562—21no—
16C4×D1028—22no—
17C2×C4×Dic257—21no—
18C2×C4×Dic257C2211yescharacteristic
19C42×D257—22no—
20C42×D257—22no—
21#22551520—22no—
22#22551518—22no—
23#22551504—22no—
24C2×M16×C257C2211yescharacteristic
25#117×C257C2211yescharacteristic
26C2×C42×C257C2211yescharacteristic
27C23×D257D411yescharacteristic
28#16861618—21no—
29#16861618—21no—
30#16861618—21no—
31#16861618—21no—
32C2×Dic514C2×C411yescharacteristic, complemented
33C2×D1028—22no—
34C2×D1028C2×C411yescharacteristic, complemented
35C2×C22⋊D257—21no—
36C22×Dic257—21no—
37C22×Dic257D411yescharacteristic
38C2×C4×D257—28no—
39C2×C4×D257—24no—
40C2×C4×D257—21no—
41C2×C4×D257—21no—
42C2×C4×D257—21no—
43C2×C4×D257—21no—
44C2×C4×D257—21no—
45C2×C4×D257C2×C412yes—
46C2×C4×D257D412yes—
47#16861602—21no—
48#16861602—21no—
49C2×#14523339—21no—
50C4⋊Dic257—22no—
51#16861595—22no—
52#16861595—22no—
53C4×Dic257—24no—
54C4×Dic257—21no—
55C4×Dic257D412yescomplemented
56C22×C4×C257—21no—
57C22×C4×C257C2311yescharacteristic
58M16×C257—21no—
59M16×C257—21no—
60C2×C8×C257—21no—
61C4⋊C4×C257C2×C412yescomplemented
62C22⋊C4×C257—21no—
63C42×C257—21no—
64C42×C257—21no—
65C42×C257D412yescomplemented
66C22×D257—21no—
67C22×D257—21no—
68C22×D257—24no—
69C22×D257—21no—
70C22×D257C22⋊C412yes—
71Dic514—21no—
72Dic514—21no—
73D1028—21no—
74D1028—21no—
75D1028—44no—
76C22⋊D257—41no—
77C22⋊D257—41no—
78C2×Dic257—24no—
79C2×Dic257—21no—
80C2×Dic257—22no—
81C2×Dic257—21no—
82C2×Dic257—21no—
83C2×Dic257C22⋊C412yes—
84C4×D257—216no—
85C4×D257—22no—
86C4×D257—22no—
87C4×D257—22no—
88C4×D257—22no—
89C4×D257—41no—
90C4×D257—41no—
91#14523339—41no—
92C23×C257C2×D411yescharacteristic
93C2×C4×C257—24no—
94C2×C4×C257—22no—
95C2×C4×C257—22no—
96C2×C4×C257C22×C411yescharacteristic
97C2×C4×C257C2×D411yescharacteristic
98C2×C4×C257C2×D411yescharacteristic
99C2×C4×C257C2×D411yescharacteristic
100C2×C4×C257#4111yescharacteristic
101C2×C4×C257#4111yescharacteristic
102C8×C257—41no—
103C22×C257—22no—
104C22×C257C2×C22⋊C411yescharacteristic
105C22×C257#12011yescharacteristic
106C22×C257#12311yescharacteristic
107C2×D257—24no—
108C2×D257—24no—
109C2×D257—22no—
110C2×D257—22no—
111C2×D257—41no—
112C2×D257—42no—
113C2×D257#10414yescomplemented
114C4×C257—24no—
115C4×C257—42no—
116C4×C257C4×D411yescharacteristic
117C4×C257C4×D411yescharacteristic
118C4×C257#12111yescharacteristic
119C4×C257#12111yescharacteristic
120Dic257—24no—
121Dic257—22no—
122Dic257—41no—
123C2×C257—21no—
124C2×C257—21no—
125C2×C257#38611yescharacteristic
126C2×C257C2×#10412yes—
127D257—24no—
128D257—81no—
129C257#189611yescharacteristic, complemented, Sylow
130#1896—2571nomaximal, Sylow
131#525—2571no—
132C22×C42—2571no—
133C2×#104—2571no—
134C2×#104—2571no—
135#407—2571no—
136#339—2571no—
137#339D25711yescharacteristic, complemented
138C2×#41—2571no—
139C23×C4—2571no—
140C23×C4—5141no—
141C2×M16—2571no—
142C2×M16C2×D25711yescharacteristic
143#123—5141no—
144C22⋊Q8—2572no—
145#121—2572no—
146#120—5141no—
147C4×D4—5142no—
148#117C2×D25711yescharacteristic
149C2×C42—2571no—
150C2×C42—5142no—
151C2×C42—5141no—
152C2×C42—5142no—
153C2×C42C2×D25711yescharacteristic
154#104—5142no—
155#104—5142no—
156C22⋊C8—5142no—
157C24—2571no—
158#41—5141no—
159#41—5141no—
160#41—5141no—
161#41—5141no—
162C2×Q8—2571no—
163C2×D4—2571no—
164C2×D4—5142no—
165C2×D4—5141no—
166C22×C4—2571no—
167C22×C4—2572no—
168C22×C4—2572no—
169C22×C4—21no—
170C22×C4—5144no—
171C22×C4—5148no—
172C22×C4—5141no—
173C22×C4—5141no—
174C22×C4—5141no—
175C22×C4—5141no—
176C22×C4—5141no—
177C22×C4—5141no—
178C22×C4C22×D25711yescharacteristic
179M16—21no—
180M16—21no—
181M16—5141no—
182M16—5141no—
183C2×C8—21no—
184C2×C8—5141no—
185C4⋊C4—5142no—
186C4⋊C4C4×D25712yescomplemented
187C22⋊C4—21no—
188C22⋊C4—5142no—
189C22⋊C4—5142no—
190C42—2572no—
191C42—21no—
192C42—21no—
193C42—5144no—
194C42—5141no—
195C42D102812yescomplemented
196C23—2572no—
197C23—5144no—
198C23—5141no—
199C23—5141no—
200C23—5141no—
201C23D4×D25711yescharacteristic
202Q8—5141no—
203Q8—5141no—
204D4—5141no—
205D4—5141no—
206D4—10284no—
207D4—10281no—
208D4—10281no—
209C2×C4—2572no—
210C2×C4—24no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—51416no—
214C2×C4—5144no—
215C2×C4—5142no—
216C2×C4—5141no—
217C2×C4—5141no—
218C2×C4—5142no—
219C2×C4—5142no—
220C2×C4—5142no—
221C2×C4—5142no—
222C2×C4—5141no—
223C2×C4—10281no—
224C2×C4—10281no—
225C2×C4C2×C4×D25711yescharacteristic
226C2×C4D4×D25711yescharacteristic
227C2×C4D4×D25711yescharacteristic
228C2×C4C2×D102811yescharacteristic
229C2×C4#1686161811yescharacteristic
230C2×C4#1686161911yescharacteristic
231C8—41no—
232C8—10281no—
233C22—2574no—
234C22—22no—
235C22—5144no—
236C22—5144no—
237C22—5142no—
238C22—5142no—
239C22—10281no—
240C22—10282no—
241C22C22⋊C4×D25711yescharacteristic
242C22#2255156211yescharacteristic
243C22#2255157711yescharacteristic
244C4—24no—
245C4—5144no—
246C4—5142no—
247C4—42no—
248C4—10281no—
249C4#2255155411yescharacteristic
250C4C4×D102811yescharacteristic
251C4#2255156511yescharacteristic
252C4#2255156711yescharacteristic
253C2—21no—
254C2—21no—
255C2—5144no—
256C2—20561no—
257C2#3065661511yescharacteristic
258C2#104×D25712yes—
subgroupG/H
centerC2×C4D4×D257
commutatorC2×C4×C257C22×C4
FrattiniC22×C4C22×D257
Fitting#339×C257C2
socleC22×C257C2×C22⋊C4
prime radical#30657249C2
cosocleC22×C4×C257C23
hypercenter#339D257
hypocenterC257#1896
LCSG ▷ C2×C4×C257 ▷ C2×C257 ▷ C257
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ #339

Complements

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

NH ≅ G/Ntype
2. C2×C42×D257C2semidirect
3. C2×#22551520C2semidirect
4. C2×#22551518C2semidirect
5. #30656700C2semidirect
6. #30656496C2semidirect
7. #339×C257C2semidirect
13. #22551572C4semidirect
14. #22551567C4semidirect
32. C2×Dic514C2×C4semidirect
34. C2×D1028C2×C4semidirect
55. C4×Dic257D4semidirect
61. C4⋊C4×C257C2×C4semidirect
65. C42×C257D4semidirect
113. C2×D257#104semidirect
129. C257#1896semidirect
137. #339D257semidirect
186. C4⋊C4C4×D257semidirect
195. C42D1028semidirect

Maximal subgroups

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

subgroupindex
#306572492
C2×C42×D2572
C2×#225515202
C2×#225515182
#306567002
#306564962
#339×C2572
#1896257