Finite Group Databasebeta

← #122147

Subgroups — #122147

Order 512.

810 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 810 conjugacy classes of subgroups — too many to draw. Only its 154 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×#2660C211yesmaximal, characteristic
2#8040C212yesmaximal
3#8030C212yesmaximal
4#7910C212yesmaximal
5C22×C4⋊Q8C411yescharacteristic
6C22×#125C411yescharacteristic
7C2×#432C2214yes—
8#2660—24no—
9#2660—24no—
10C2×#398—22no—
11C2×#395—22no—
12C2×#384C411yescharacteristic
13C2×#384C2211yescharacteristic
14C2×#382C411yescharacteristic
15C2×#382C2211yescharacteristic
16C2×#376C2211yescharacteristic
17#1296—24no—
18#1295—24no—
19#1280—22no—
20#1280—22no—
21C23×Q8—21no—
22C2×C4⋊Q8—22no—
23C2×C4⋊Q8—22no—
24C2×C4⋊Q8—42no—
25C2×C4⋊Q8—42no—
26C2×#125—22no—
27C2×#125—22no—
28C2×#125—42no—
29C2×#125—42no—
30C22×C4⋊C4—21no—
31C22×C4⋊C4—21no—
32C22×C4⋊C4—21no—
33C22×C4⋊C4—22no—
34C22×C4⋊C4—22no—
35C22×C4⋊C4C2×C411yescharacteristic
36C22×C4⋊C4C2×C411yescharacteristic
37C22×C4⋊C4C2×C411yescharacteristic
38C22×C4⋊C4C2×C411yescharacteristic
39C22×C42C2×C411yescharacteristic
40C22×C42C2×C411yescharacteristic
41C22×C42C2311yescharacteristic
42#432—24no—
43#432—24no—
44#432—24no—
45#432Q818yescomplemented
46C2×C4⋊C8—24no—
47C2×C22⋊C8—24no—
48C2×#97—24no—
49#398—216no—
50#395—216no—
51#384—22no—
52#384—22no—
53#384—22no—
54#384D414yes—
55#382—22no—
56#382—22no—
57#382—22no—
58#382D414yes—
59#376—22no—
60#376—22no—
61C2×#95—22no—
62C2×#95—22no—
63C2×#95D412yes—
64C2×#95D412yes—
65C22×Q8—24no—
66C22×Q8—42no—
67C22×Q8—42no—
68C22×Q8—84no—
69C23×C4—21no—
70C23×C4—21no—
71C23×C4—21no—
72C23×C4—21no—
73C23×C4C22⋊C411yescharacteristic
74C23×C4C22⋊C411yescharacteristic
75C23×C4C22⋊C412yes—
76C23×C4C22×C411yescharacteristic
77C23×C4C2×D412yes—
78C22×C8—44no—
79C4⋊Q8—48no—
80C4⋊Q8—48no—
81#125—48no—
82#125—48no—
83C2×C4⋊C4—24no—
84C2×C4⋊C4—24no—
85C2×C4⋊C4—24no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—28no—
91C2×C4⋊C4—28no—
92C2×C4⋊C4—41no—
93C2×C4⋊C4—41no—
94C2×C4⋊C4—41no—
95C2×C4⋊C4—41no—
96C2×C4⋊C4—41no—
97C2×C4⋊C4—41no—
98C2×C4⋊C4—41no—
99C2×C4⋊C4—41no—
100C2×C4⋊C4—42no—
101C2×C4⋊C4—42no—
102C2×C4⋊C4—42no—
103C2×C4⋊C4—42no—
104C2×C4⋊C4—42no—
105C2×C4⋊C4—42no—
106C2×C4⋊C4—84no—
107C2×C4⋊C4—84no—
108C2×C4⋊C4C22⋊C412yes—
109C2×C4⋊C4C22⋊C412yes—
110C2×C4⋊C4C22⋊C412yes—
111C2×C4⋊C4C22⋊C412yes—
112C2×C42—22no—
113C2×C42—22no—
114C2×C42—24no—
115C2×C42—21no—
116C2×C42—21no—
117C2×C42—21no—
118C2×C42—21no—
119C2×C42C2×D412yes—
120C2×C42C2×D412yes—
121C2×C42C2×Q814yes—
122C2×C42#4114yes—
123C4⋊C8—28no—
124C4⋊C8—44no—
125C22⋊C8—28no—
126C22⋊C8—44no—
127#97—28no—
128#97—44no—
129#95—28no—
130#95—28no—
131#95—24no—
132#95—24no—
133C24C2×C22⋊C411yescharacteristic
134C2×Q8—48no—
135C2×Q8—88no—
136C2×Q8—88no—
137C2×Q8—88no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—42no—
152C22×C4—42no—
153C22×C4—42no—
154C22×C4—42no—
155C22×C4—42no—
156C22×C4—42no—
157C22×C4—44no—
158C22×C4—44no—
159C22×C4—44no—
160C22×C4—44no—
161C22×C4—41no—
162C22×C4—41no—
163C22×C4—41no—
164C22×C4—41no—
165C22×C4—41no—
166C22×C4—41no—
167C22×C4—41no—
168C22×C4—41no—
169C22×C4#10012yes—
170C22×C4#10012yes—
171C22×C4#10114yes—
172C22×C4C2×C22⋊C411yescharacteristic
173C22×C4C2×C22⋊C411yescharacteristic
174C22×C4C4×Q814yes—
175C22×C4#12014yes—
176C22×C4C22⋊Q814yes—
177C22×C4#12412yes—
178C22×C4C4⋊Q812yes—
179C2×C8—48no—
180C2×C8—84no—
181C2×C8—84no—
182C4⋊C4—42no—
183C4⋊C4—42no—
184C4⋊C4—42no—
185C4⋊C4—42no—
186C4⋊C4—42no—
187C4⋊C4—42no—
188C4⋊C4—48no—
189C4⋊C4—48no—
190C4⋊C4—42no—
191C4⋊C4—42no—
192C4⋊C4—48no—
193C4⋊C4—88no—
194C4⋊C4—88no—
195C42—24no—
196C42—24no—
197C42—24no—
198C42—24no—
199C42C22⋊Q814yes—
200C42C22⋊Q814yes—
201C42#12412yes—
202C42#12412yes—
203C42C4⋊Q812yes—
204C42C4⋊Q812yes—
205C23—22no—
206C23—22no—
207C23#37911yescharacteristic
208C23#39112yes—
209C23C2×#10011yescharacteristic
210C23C2×#10011yescharacteristic
211C23C2×#10112yes—
212Q8—816no—
213C2×C4—24no—
214C2×C4—24no—
215C2×C4—24no—
216C2×C4—24no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—24no—
224C2×C4—44no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—44no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—42no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—44no—
237C2×C4—44no—
238C2×C4—44no—
239C2×C4—44no—
240C2×C4—44no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—82no—
246C2×C4—82no—
247C2×C4—82no—
248C2×C4—82no—
249C2×C4—82no—
250C2×C4—82no—
251C2×C4—82no—
252C2×C4—82no—
253C2×C4#39112yes—
254C2×C4#39112yes—
255C2×C4#39314yes—
256C2×C4#39514yes—
257C8—88no—
258C22—21no—
259C22—21no—
260C22—21no—
261C22—21no—
262C22—22no—
263C22—22no—
264C22—22no—
265C22—22no—
266C22#144611yescharacteristic
267C22#178311yescharacteristic
268C22#178411yescharacteristic
269C22#179512yes—
270C22#179612yes—
271C22#199712yes—
272C22#199712yes—
273C4—22no—
274C4—22no—
275C4—24no—
276C4—24no—
277C4—42no—
278C4—42no—
279C4—44no—
280C4—82no—
281C4—82no—
282C4—82no—
283C4—82no—
284C2—22no—
285C2—22no—
286C2#852211yescharacteristic
287C2#852411yescharacteristic
288C2#852512yes—
289C2#1138211yescharacteristic
290C2#1138812yes—
subgroupG/H
center, socleC23#379
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
45. #432Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#26602
#80402
#80302
#79102