Finite Group Databasebeta

← #108481

Subgroups — #108481

Order 512.

708 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 708 conjugacy classes of subgroups — too many to draw. Only its 150 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#20377C211yesmaximal, characteristic
2#10716C212yesmaximal
3#8104C212yesmaximal
4#8055C211yesmaximal, characteristic
5#8049C211yesmaximal, characteristic
6C22×#117C2211yescharacteristic
7C2×#407C2212yes—
8C2×#407C2211yescharacteristic
9C2×#407C2211yescharacteristic
10C2×C4×M16C412yescomplemented
11#2280C2212yes—
12#1750—24no—
13C2×H(3,ℤ/4)—24no—
14#1728C412yescomplemented
15#1305—24no—
16#1298C414yescomplemented
17#1297C414yescomplemented
18C24×C4C2311yescharacteristic
19C22×M16—22no—
20C22×M16C2×C412yes—
21C22×M16C2×C412yes—
22C2×#117—22no—
23C2×#117—22no—
24C2×#117—44no—
25C2×#117D414yes—
26C22×C4⋊C4—21no—
27C22×C22⋊C4—21no—
28C22×C42C2×C411yescharacteristic
29C22×C42C2×C411yescharacteristic
30#407—22no—
31#407—22no—
32#407—22no—
33#407—22no—
34#407—22no—
35#407—22no—
36#407C2×C412yes—
37#407C2×C412yes—
38#407D412yes—
39#407Q812yes—
40C2×C22⋊C8—22no—
41C2×C22⋊C8—22no—
42C2×C22⋊C8—22no—
43C2×C22⋊C8C2×C414yes—
44C4×M16—44no—
45C2×#97—22no—
46C2×C4×C8—22no—
47#381—22no—
48#380—22no—
49#380—22no—
50#379—22no—
51C4×C22⋊C4—22no—
52C4×C22⋊C4C2×C414yes—
53H(3,ℤ/4)—48no—
54#336—24no—
55C25#4111yescharacteristic
56C23×C4—21no—
57C23×C4—21no—
58C23×C4—42no—
59C23×C4—42no—
60C23×C4C22×C411yescharacteristic
61C23×C4C22×C411yescharacteristic
62C23×C4C22×C411yescharacteristic
63C23×C4C2×D412yes—
64C23×C4C2×D411yescharacteristic
65C23×C4C2×Q811yescharacteristic
66C23×C4#4112yes—
67C23×C4#4112yes—
68C23×C4#4112yes—
69C23×C4#4111yescharacteristic
70C2×M16—22no—
71C2×M16—22no—
72C2×M16—44no—
73C2×M16—44no—
74C2×M16—42no—
75C2×M16—42no—
76C2×M16—42no—
77C2×M16—42no—
78C2×M16—42no—
79C2×M16—42no—
80C2×M16C4214yes—
81C2×M16C4⋊C414yes—
82C22×C8—22no—
83C22×C8—22no—
84C22×C8—24no—
85#117—28no—
86#117—44no—
87#117—48no—
88#117—44no—
89#117—44no—
90C2×C4⋊C4—24no—
91C2×C4⋊C4—44no—
92C2×C4⋊C4—42no—
93C2×C22⋊C4—24no—
94C2×C22⋊C4—24no—
95C2×C22⋊C4—44no—
96C2×C22⋊C4—42no—
97C2×C22⋊C4—42no—
98C2×C42—21no—
99C2×C42—24no—
100C2×C42—21no—
101C2×C42—24no—
102C2×C42—42no—
103C2×C42C22⋊C412yes—
104C2×C42C22⋊C412yes—
105C22⋊C8—24no—
106C22⋊C8—24no—
107C22⋊C8—24no—
108C22⋊C8—24no—
109C22⋊C8—24no—
110C22⋊C8—24no—
111C22⋊C8—42no—
112C22⋊C8—42no—
113#97—44no—
114C4×C8—44no—
115#95—24no—
116#95—42no—
117C24—41no—
118C24—41no—
119C24—82no—
120C24#11711yescharacteristic
121C24C4×D411yescharacteristic
122C24C4×Q811yescharacteristic
123C24#12312yes—
124C24#12612yes—
125C22×C4—22no—
126C22×C4—24no—
127C22×C4—24no—
128C22×C4—22no—
129C22×C4—22no—
130C22×C4—24no—
131C22×C4—22no—
132C22×C4—42no—
133C22×C4—42no—
134C22×C4—42no—
135C22×C4—42no—
136C22×C4—44no—
137C22×C4—44no—
138C22×C4—41no—
139C22×C4—42no—
140C22×C4—41no—
141C22×C4—42no—
142C22×C4—42no—
143C22×C4—44no—
144C22×C4—82no—
145C22×C4—82no—
146C22×C4C2×C4211yescharacteristic
147C22×C4C2×C22⋊C411yescharacteristic
148C22×C4C2×C4⋊C411yescharacteristic
149C22×C4#11711yescharacteristic
150C22×C4#11711yescharacteristic
151C22×C4#11711yescharacteristic
152C22×C4#11712yes—
153C22×C4#11712yes—
154C22×C4C4×D412yes—
155C22×C4C4×D412yes—
156C22×C4C4×D411yescharacteristic
157C22×C4C4×Q811yescharacteristic
158C22×C4#12112yes—
159C22×C4C22⋊Q812yes—
160C22×C4#12312yes—
161C22×C4#12412yes—
162C22×C4#12512yes—
163C22×C4#12612yes—
164M16—44no—
165M16—44no—
166M16—44no—
167M16—44no—
168M16—82no—
169M16—82no—
170M16—84no—
171C2×C8—24no—
172C2×C8—24no—
173C2×C8—42no—
174C2×C8—42no—
175C2×C8—42no—
176C2×C8—42no—
177C2×C8—48no—
178C2×C8—44no—
179C4⋊C4—48no—
180C4⋊C4—84no—
181C22⋊C4—48no—
182C22⋊C4—82no—
183C22⋊C4—82no—
184C22⋊C4—84no—
185C22⋊C4—84no—
186C42—24no—
187C42—22no—
188C42—24no—
189C42—42no—
190C42—48no—
191C42#9914yes—
192C23—21no—
193C23—21no—
194C23—22no—
195C23—22no—
196C23—42no—
197C23—42no—
198C23—42no—
199C23—41no—
200C23—41no—
201C23—42no—
202C23—42no—
203C23—42no—
204C23—42no—
205C23—42no—
206C23—84no—
207C23—84no—
208C23C4×C4⋊C411yescharacteristic
209C23#38111yescharacteristic
210C23#38311yescharacteristic
211C23#38712yes—
212C23#38812yes—
213C2×C4—21no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—21no—
217C2×C4—24no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—24no—
221C2×C4—24no—
222C2×C4—24no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—24no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—42no—
231C2×C4—42no—
232C2×C4—42no—
233C2×C4—44no—
234C2×C4—44no—
235C2×C4—42no—
236C2×C4—84no—
237C2×C4—84no—
238C2×C4—84no—
239C2×C4—84no—
240C2×C4—84no—
241C2×C4#37611yescharacteristic
242C2×C4C4×C22⋊C411yescharacteristic
243C2×C4#38011yescharacteristic
244C2×C4#38211yescharacteristic
245C2×C4#38712yes—
246C2×C4#38812yes—
247C2×C4C2×#9912yes—
248C2×C4#41012yes—
249C8—44no—
250C8—44no—
251C8—84no—
252C22—22no—
253C22—22no—
254C22—21no—
255C22—21no—
256C22—22no—
257C22—22no—
258C22—22no—
259C22—22no—
260C22—22no—
261C22—22no—
262C22—22no—
263C22—42no—
264C22—42no—
265C22—82no—
266C22—82no—
267C22—82no—
268C22—84no—
269C22—82no—
270C22#143411yescharacteristic
271C22#181911yescharacteristic
272C22#182011yescharacteristic
273C4—22no—
274C4—21no—
275C4—21no—
276C4—22no—
277C4—44no—
278C4—82no—
279C4—164no—
280C4C4×#9912yes—
281C4#179912yes—
282C2—21no—
283C2—21no—
284C2—42no—
285C2—41no—
286C2—41no—
287C2—42no—
288C2#1080111yescharacteristic
289C2#1081012yes—
subgroupG/H
centerC2×C4C4×C22⋊C4
commutatorC22×C4C2×C42
Frattini, cosocleC24×C4C23
socleC22#1434
prime radicalC25#41
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×#117 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
10. C2×C4×M16C4semidirect
14. #1728C4semidirect
16. #1298C4semidirect
17. #1297C4semidirect

Maximal subgroups

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

subgroupindex
#203772
#107162
#81042
#80552
#80492