Finite Group Databasebeta

← #141130

Subgroups — #141130

Order 512.

465 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 465 conjugacy classes of subgroups — too many to draw. Only its 96 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#33180C211yesmaximal, characteristic
2#12790C211yesmaximal, characteristic
3C2×#1342C212yesmaximal
4C2×#1340C212yesmaximal
5#10699C211yesmaximal, characteristic
6C2×#494C411yescharacteristic, complemented
7C2×#470C2212yes—
8C2×#467—21no—
9C2×#405—21no—
10C2×#405C2211yescharacteristic
11#2621—21no—
12C2×#364—21no—
13C2×#349C2212yes—
14#1971C412yescomplemented
15#1969—21no—
16#1960—21no—
17#1846C411yescharacteristic, complemented
18C2×#338C2211yescharacteristic
19C2×H(3,ℤ/4)—21no—
20C2×H(3,ℤ/4)C2211yescharacteristic
21#1375—21no—
22#1342—24no—
23#1340—24no—
24C22×Q16—21no—
25C22×Q16C2×C412yes—
26C22×M16—21no—
27C22×M16C2311yescharacteristic
28C23×C8D411yescharacteristic
29C2×C4⋊Q8—21no—
30C2×C22⋊Q8—22no—
31C2×#117—21no—
32C2×#117D411yescharacteristic
33C2×M32—22no—
34#494—42no—
35#470—22no—
36#470—22no—
37#470—22no—
38#470—22no—
39#467—42no—
40#467—42no—
41C2×#107—21no—
42C2×#106C2×C412yes—
43C2×#103—22no—
44C2×#103—22no—
45#405—42no—
46#405—42no—
47#405—42no—
48C2×#97—22no—
49C2×#97C2×C411yescharacteristic
50C2×#97D411yescharacteristic
51C2×C4×C8—22no—
52C2×C4×C8C2×C411yescharacteristic
53C2×C4×C8D411yescharacteristic
54#398—42no—
55#397—42no—
56#391—22no—
57#382—21no—
58#364—42no—
59#349—22no—
60#349—22no—
61#338—21no—
62#338D412yes—
63H(3,ℤ/4)—24no—
64H(3,ℤ/4)—21no—
65H(3,ℤ/4)D412yes—
66C22×Q8—22no—
67C23×C4C2×D411yescharacteristic
68C2×Q16—22no—
69C2×Q16—22no—
70C2×Q16—42no—
71C2×Q16—42no—
72C2×Q16—42no—
73C2×Q16—42no—
74C2×Q16—41no—
75C2×Q16—41no—
76C2×M16—22no—
77C2×M16—22no—
78C2×M16—41no—
79C2×M16—41no—
80C2×M16—41no—
81C2×M16—41no—
82C2×M16—41no—
83C2×M16—41no—
84C2×M16C2×D411yescharacteristic
85C2×M16C2×D411yescharacteristic
86C2×M16#4111yescharacteristic
87C2×M16#4111yescharacteristic
88C22×C8—22no—
89C22×C8—21no—
90C22×C8—21no—
91C22×C8—22no—
92C22×C8C22⋊C411yescharacteristic
93C22×C8C22⋊C411yescharacteristic
94C22×C8D812yes—
95C22×C8SD1612yes—
96C22×C8C22×C411yescharacteristic
97C22×C8C2×D411yescharacteristic
98C4⋊Q8—42no—
99C4⋊Q8—42no—
100C22⋊Q8—42no—
101C22⋊Q8—42no—
102C22⋊Q8—42no—
103C22⋊Q8—42no—
104#117—22no—
105#117—22no—
106#117—22no—
107#117—41no—
108#117—41no—
109#117—41no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—41no—
113C2×C4⋊C4—42no—
114C2×C4⋊C4—42no—
115C2×C22⋊C4—21no—
116C2×C22⋊C4—41no—
117C2×C22⋊C4—42no—
118C2×C42—22no—
119C2×C42C22⋊C411yescharacteristic
120C2×C42C22⋊C411yescharacteristic
121M32—42no—
122M32—42no—
123C2×C16—42no—
124#107—42no—
125#106—22no—
126#106—22no—
127#103—44no—
128#103—42no—
129#103—42no—
130#97—41no—
131#97—41no—
132#97—44no—
133C4×C8—22no—
134C4×C8—22no—
135C4×C8—44no—
136#95—42no—
137#95—44no—
138C24AGL(1,ℤ/8)11yescharacteristic
139C2×Q8—42no—
140C2×Q8—42no—
141C2×Q8—42no—
142C2×Q8—42no—
143C2×Q8—42no—
144C2×Q8—42no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—42no—
152C22×C4—42no—
153C22×C4—42no—
154C22×C4C2×C22⋊C411yescharacteristic
155C22×C4#12111yescharacteristic
156C22×C4#12111yescharacteristic
157C22×C4C2×D811yescharacteristic
158C22×C4C2×SD1611yescharacteristic
159C22×C4AGL(1,ℤ/8)11yescharacteristic
160Q16—82no—
161Q16—82no—
162Q16—82no—
163Q16—82no—
164M16—41no—
165M16—41no—
166M16—41no—
167M16—41no—
168M16—42no—
169M16—42no—
170M16—42no—
171M16—42no—
172C2×C8—21no—
173C2×C8—21no—
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—21no—
178C2×C8—22no—
179C2×C8—22no—
180C2×C8—22no—
181C2×C8—22no—
182C2×C8—24no—
183C2×C8—42no—
184C2×C8—42no—
185C2×C8#9912yes—
186C2×C8#10212yes—
187C2×C8#10212yes—
188C2×C8#10411yescharacteristic
189C2×C8#10411yescharacteristic
190C2×C8C4×D411yescharacteristic
191C2×C8C4×D411yescharacteristic
192C2×C8#12411yescharacteristic
193C2×C8#12711yescharacteristic
194C4⋊C4—42no—
195C4⋊C4—42no—
196C4⋊C4—42no—
197C4⋊C4—42no—
198C4⋊C4—42no—
199C4⋊C4—84no—
200C4⋊C4—82no—
201C4⋊C4—82no—
202C22⋊C4—42no—
203C22⋊C4—42no—
204C22⋊C4—42no—
205C22⋊C4—82no—
206C22⋊C4—82no—
207C42—21no—
208C42—24no—
209C42—21no—
210C42—21no—
211C42—42no—
212C42#10412yes—
213C16—82no—
214C23—21no—
215C23—21no—
216C23—42no—
217C23#41811yescharacteristic
218C23#45911yescharacteristic
219C23#46011yescharacteristic
220Q8—82no—
221Q8—82no—
222Q8—82no—
223Q8—82no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—42no—
236C2×C4—44no—
237C2×C4—42no—
238C2×C4—41no—
239C2×C4—84no—
240C2×C4—84no—
241C2×C4—82no—
242C2×C4—82no—
243C2×C4—82no—
244C2×C4—82no—
245C2×C4#39011yescharacteristic
246C2×C4C2×#9911yescharacteristic
247C2×C4#41011yescharacteristic
248C2×C4C2×#10211yescharacteristic
249C2×C4C2×#10411yescharacteristic
250C2×C4C2×#10411yescharacteristic
251C2×C4#45911yescharacteristic
252C2×C4#46011yescharacteristic
253C8—21no—
254C8—21no—
255C8—21no—
256C8—21no—
257C8—22no—
258C8—44no—
259C8#32712yes—
260C8#32712yes—
261C22—22no—
262C22—21no—
263C22—21no—
264C22—41no—
265C22—41no—
266C22—42no—
267C22—81no—
268C22#150511yescharacteristic
269C22#150711yescharacteristic
270C22#198211yescharacteristic
271C4—21no—
272C4—21no—
273C4—42no—
274C4—81no—
275C4—84no—
276C4—162no—
277C4—162no—
278C4C2×#32711yescharacteristic
279C4#150211yescharacteristic
280C4#196211yescharacteristic
281C4#196411yescharacteristic
282C2—21no—
283C2—21no—
284C2—41no—
285C2—41no—
286C2#971211yescharacteristic
287C2#1177112yes—
subgroupG/H
center, socleC22#1505
commutatorC22×C8C22×C4
Frattini, cosocleC22×M16C23
prime radicalC24AGL(1,ℤ/8)
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#405 ◁ G
LECSG ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
6. C2×#494C4semidirect
14. #1971C4semidirect
17. #1846C4semidirect

Maximal subgroups

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

subgroupindex
#331802
#127902
C2×#13422
C2×#13402
#106992