Finite Group Databasebeta

← #124189

Subgroups — #124189

Order 512.

394 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 114 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
1C8×C22⋊Q8C211yesmaximal, characteristic
2#12599C211yesmaximal, characteristic
3#12544C211yesmaximal, characteristic
4#7128C212yesmaximal, complemented
5#7118C212yesmaximal, complemented
6C22×M32—21no—
7C2×C8×Q8C411yescharacteristic
8C22×C4×C8C2211yescharacteristic
9C4×C22⋊Q8C411yescharacteristic
10C2×C22⋊C16—21no—
11C2×#346—21no—
12#1984C2211yescharacteristic
13#1939C2211yescharacteristic
14#1916—21no—
15#1850—21no—
16#1797C411yescharacteristic
17C8×C4⋊C4—21no—
18C8×C4⋊C4C411yescharacteristic
19C8×C22⋊C4—21no—
20#1374—21no—
21#1374C2212yes—
22#1374C2212yes—
23C23×C8D411yescharacteristic
24C23×C8D411yescharacteristic
25C2×C22⋊Q8C811yescharacteristic
26C2×C4×Q8C2×C411yescharacteristic
27C22×C42C2×C411yescharacteristic
28C2×M32—24no—
29C2×M32—42no—
30C2×M32—42no—
31C22×C16—22no—
32C22×C16—22no—
33C8×Q8—42no—
34C2×C4⋊C8—21no—
35C2×C4⋊C8—21no—
36C2×C4⋊C8C2×C411yescharacteristic
37C2×C4⋊C8C2×C411yescharacteristic
38C2×C22⋊C8—21no—
39C2×C4×C8—21no—
40C2×C4×C8—21no—
41C2×C4×C8—42no—
42C2×C4×C8C2×C411yescharacteristic
43C2×C4×C8D411yescharacteristic
44C2×C4×C8D411yescharacteristic
45C2×C4×C8C2311yescharacteristic
46#391C811yescharacteristic
47#389C811yescharacteristic
48#387—21no—
49#385—21no—
50#380C811yescharacteristic
51C4×C4⋊C4—21no—
52C4×C4⋊C4C2×C411yescharacteristic
53C4×C22⋊C4—21no—
54C22⋊C16—24no—
55#346—42no—
56#336—21no—
57#336—21no—
58#336—21no—
59#336—21no—
60#336—21no—
61C22×Q8C2×C811yescharacteristic
62C23×C4C22⋊C411yescharacteristic
63C23×C4C2×C811yescharacteristic
64C23×C4M1611yescharacteristic
65C22×C8—22no—
66C22×C8—22no—
67C22×C8—22no—
68C22×C8—22no—
69C22×C8—21no—
70C22×C8—21no—
71C22×C8—41no—
72C22×C8C22⋊C411yescharacteristic
73C22×C8C22⋊C411yescharacteristic
74C22×C8SD1612yes—
75C22×C8Q1612yes—
76C22×C8C2×D411yescharacteristic
77C22×C8C2×D411yescharacteristic
78C22⋊Q8—22no—
79C22⋊Q8—22no—
80C4×Q8—41no—
81C4×Q8—41no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4C2×C811yescharacteristic
86C2×C4⋊C4C2×C811yescharacteristic
87C2×C4⋊C4C2×C811yescharacteristic
88C2×C22⋊C4—21no—
89C2×C22⋊C4—21no—
90C2×C42—21no—
91C2×C42—21no—
92C2×C42—41no—
93C2×C42—41no—
94C2×C42C22⋊C411yescharacteristic
95C2×C42C2×C811yescharacteristic
96C2×C42M1611yescharacteristic
97C2×C42C22×C411yescharacteristic
98M32—48no—
99C2×C16—24no—
100C2×C16—24no—
101C2×C16—42no—
102C2×C16—42no—
103C2×C16—42no—
104C2×C16—42no—
105C4⋊C8—41no—
106C4⋊C8—41no—
107C4⋊C8—42no—
108C4⋊C8—42no—
109C22⋊C8—42no—
110C4×C8—22no—
111C4×C8—22no—
112C4×C8—41no—
113C4×C8—41no—
114C4×C8—42no—
115C4×C8—42no—
116#95—21no—
117#95—21no—
118#95—21no—
119#95—21no—
120#95—21no—
121C24C22⋊C811yescharacteristic
122C2×Q8—21no—
123C2×Q8—41no—
124C2×Q8—41no—
125C2×Q8M3212yescomplemented
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—41no—
137C22×C4—41no—
138C22×C4—41no—
139C22×C4—41no—
140C22×C4—41no—
141C22×C4C22⋊C811yescharacteristic
142C22×C4C22⋊C811yescharacteristic
143C22×C4C22⋊C811yescharacteristic
144C22×C4#10312yes—
145C22×C4#10412yes—
146C22×C4C2×C22⋊C411yescharacteristic
147C22×C4C22×C811yescharacteristic
148C22×C4C2×M1611yescharacteristic
149C2×C8—21no—
150C2×C8—21no—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—24no—
154C2×C8—24no—
155C2×C8—21no—
156C2×C8—21no—
157C2×C8—21no—
158C2×C8—44no—
159C2×C8—41no—
160C2×C8—41no—
161C2×C8—41no—
162C2×C8—41no—
163C2×C8—41no—
164C2×C8—41no—
165C2×C8#9912yes—
166C2×C8#10312yes—
167C2×C8#10412yes—
168C2×C8C2×SD1611yescharacteristic
169C2×C8C2×Q1611yescharacteristic
170C2×C8#13512yes—
171C4⋊C4—24no—
172C4⋊C4—21no—
173C4⋊C4—41no—
174C4⋊C4—41no—
175C4⋊C4—42no—
176C4⋊C4—42no—
177C4⋊C4M3212yescomplemented
178C22⋊C4—24no—
179C22⋊C4—42no—
180C42—21no—
181C42—21no—
182C42—21no—
183C42—21no—
184C42—41no—
185C42—41no—
186C42—41no—
187C42—41no—
188C42—41no—
189C42—41no—
190C16—44no—
191C16—44no—
192C23—21no—
193C23—21no—
194C23—21no—
195C23—21no—
196C23—21no—
197C23—21no—
198C23Q8⋊C812yes—
199C23C2×C22⋊C811yescharacteristic
200Q8—42no—
201Q8—42no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—44no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4Q8⋊C812yes—
235C2×C4C2×#9911yescharacteristic
236C2×C4#41311yescharacteristic
237C2×C4C2×#10311yescharacteristic
238C2×C4#41611yescharacteristic
239C2×C4C2×#10411yescharacteristic
240C2×C4C2×#10411yescharacteristic
241C2×C4C2×M3211yescharacteristic
242C2×C4C2×M3211yescharacteristic
243C8—21no—
244C8—21no—
245C8—22no—
246C8—41no—
247C8—81no—
248C8—81no—
249C8#32814yes—
250C22—22no—
251C22—22no—
252C22—22no—
253C22—41no—
254C22—41no—
255C22—41no—
256C22—41no—
257C22#145911yescharacteristic
258C22C2×Q8⋊C811yescharacteristic
259C22#147611yescharacteristic
260C22#211211yescharacteristic
261C22#211511yescharacteristic
262C22#211711yescharacteristic
263C22#211711yescharacteristic
264C4—22no—
265C4—21no—
266C4—21no—
267C4—42no—
268C4—41no—
269C4—42no—
270C4—41no—
271C4—41no—
272C4—81no—
273C4—81no—
274C4C2×#32811yescharacteristic
275C4#150111yescharacteristic
276C4#152411yescharacteristic
277C4#154211yescharacteristic
278C2—42no—
279C2#756611yescharacteristic
280C2#1146611yescharacteristic
281C2#1146911yescharacteristic
282C2#1160711yescharacteristic
283C2#1161211yescharacteristic
284C2#1161511yescharacteristic
285C2#1162011yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊C4
commutatorC22×C4C22×C8
Frattini, cosocleC2×C4×C8C23
socleC23C2×C22⋊C8
prime radicalC24C22⋊C8
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C4×C8 ◁ G
LECSG ▷ C2×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7128C2semidirect
5. #7118C2semidirect
125. C2×Q8M32semidirect
177. C4⋊C4M32semidirect

Maximal subgroups

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

subgroupindex
C8×C22⋊Q82
#125992
#125442
#71282
#71182