Finite Group Databasebeta

← #101287

Subgroups — #101287

Order 512.

579 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 579 conjugacy classes of subgroups — too many to draw. Only its 86 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#23455C211yesmaximal, characteristic
2#14489C211yesmaximal, characteristic
3#8333C212yesmaximal
4#7595C211yesmaximal, characteristic
5#7037C212yesmaximal
6C2×#533C2211yescharacteristic
7C2×#419—21no—
8C2×#391C2211yescharacteristic
9#2026—22no—
10#2024C414yescomplemented
11#1817C2212yes—
12#1736—21no—
13#1728C2211yescharacteristic
14C2×#328—22no—
15C2×#328—22no—
16#1446—21no—
17#1443—22no—
18#1436—21no—
19#1434—22no—
20#1434C2212yes—
21#1272—22no—
22C23×Q8D411yescharacteristic
23C22×M16—21no—
24#533—44no—
25C2×C4⋊Q8C2×C412yes—
26C2×#125—21no—
27C2×C22⋊Q8—22no—
28C2×C4×Q8—22no—
29C2×#117—21no—
30C22×C4⋊C4D411yescharacteristic
31C22×C42—21no—
32C22×C42C2311yescharacteristic
33#419—44no—
34C2×#103—24no—
35C2×C22⋊C8—22no—
36#391—22no—
37#391C2×C412yes—
38#391D412yes—
39C2×#95—22no—
40C2×#95—22no—
41C2×#95—22no—
42C2×#95—22no—
43C2×#95—21no—
44C2×#95—22no—
45C2×#95—21no—
46C2×#95—21no—
47C2×#95D412yes—
48C2×#95D412yes—
49#340—24no—
50#336—21no—
51#336—22no—
52#336—22no—
53#336C2×C412yes—
54#328—44no—
55#328—44no—
56C22×Q8—21no—
57C22×Q8—22no—
58C22×Q8—22no—
59C22×Q8—22no—
60C22×Q8—41no—
61C22×Q8—41no—
62C22×Q8—42no—
63C22×Q8C22⋊C412yes—
64C23×C4—21no—
65C23×C4—21no—
66C23×C4—22no—
67C23×C4—22no—
68C23×C4C2×D411yescharacteristic
69C23×C4C2×D412yes—
70C2×M16—41no—
71C2×M16—41no—
72C2×M16—42no—
73C2×M16—42no—
74C22×C8—22no—
75C22×C8—42no—
76C4⋊Q8—44no—
77#125—44no—
78C22⋊Q8—24no—
79C22⋊Q8—24no—
80C22⋊Q8—42no—
81C22⋊Q8—42no—
82C4×Q8—44no—
83C4×Q8—44no—
84#117—24no—
85#117—42no—
86C2×C4⋊C4—22no—
87C2×C4⋊C4—22no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—42no—
90C2×C4⋊C4—41no—
91C2×C4⋊C4—41no—
92C2×C4⋊C4C22⋊C412yes—
93C2×C22⋊C4—41no—
94C2×C42—42no—
95C2×C42—41no—
96C2×C42—44no—
97C2×C42—81no—
98C2×C42C22×C411yescharacteristic
99C2×C42C2×D411yescharacteristic
100C2×C42#4112yes—
101#103—48no—
102C22⋊C8—42no—
103C22⋊C8—42no—
104#95—24no—
105#95—24no—
106#95—42no—
107#95—42no—
108#95—44no—
109#95—44no—
110#95—42no—
111#95—42no—
112#95—81no—
113#95—81no—
114#95—81no—
115#95—82no—
116C24#12011yescharacteristic
117C2×Q8—24no—
118C2×Q8—22no—
119C2×Q8—22no—
120C2×Q8—42no—
121C2×Q8—42no—
122C2×Q8—42no—
123C2×Q8—44no—
124C2×Q8—44no—
125C2×Q8—42no—
126C2×Q8—81no—
127C2×Q8—81no—
128C2×Q8—82no—
129C2×Q8—82no—
130C2×Q8—82no—
131C2×Q8—82no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—22no—
135C22×C4—22no—
136C22×C4—22no—
137C22×C4—24no—
138C22×C4—22no—
139C22×C4—22no—
140C22×C4—42no—
141C22×C4—42no—
142C22×C4—42no—
143C22×C4—42no—
144C22×C4—42no—
145C22×C4—42no—
146C22×C4—42no—
147C22×C4—42no—
148C22×C4—42no—
149C22×C4—42no—
150C22×C4—42no—
151C22×C4—41no—
152C22×C4—41no—
153C22×C4—41no—
154C22×C4—41no—
155C22×C4—42no—
156C22×C4—42no—
157C22×C4—42no—
158C22×C4—42no—
159C22×C4—42no—
160C22×C4—42no—
161C22×C4—42no—
162C22×C4—42no—
163C22×C4—42no—
164C22×C4C2×C22⋊C411yescharacteristic
165C22×C4C4×D412yes—
166C22×C4#12112yes—
167C22×C4#12311yescharacteristic
168C22×C4#13512yes—
169C22×C4#13712yes—
170M16—82no—
171M16—82no—
172C2×C8—42no—
173C2×C8—42no—
174C2×C8—42no—
175C2×C8—82no—
176C2×C8—82no—
177C2×C8—82no—
178C4⋊C4—24no—
179C4⋊C4—24no—
180C4⋊C4—42no—
181C4⋊C4—42no—
182C4⋊C4—42no—
183C4⋊C4—44no—
184C4⋊C4—44no—
185C4⋊C4—81no—
186C22⋊C4—44no—
187C42—41no—
188C42—41no—
189C42—42no—
190C42—44no—
191C42—84no—
192C23—22no—
193C23—22no—
194C23#38611yescharacteristic
195C23#45212yes—
196C23#45511yescharacteristic
197C23#45511yescharacteristic
198C23B(4,𝔽2)12yes—
199Q8—44no—
200Q8—82no—
201Q8—84no—
202Q8—84no—
203Q8—84no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—44no—
219C2×C4—44no—
220C2×C4—44no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—44no—
225C2×C4—42no—
226C2×C4—44no—
227C2×C4—82no—
228C2×C4—82no—
229C2×C4—81no—
230C2×C4—82no—
231C2×C4—82no—
232C2×C4—82no—
233C2×C4—81no—
234C2×C4—82no—
235C2×C4—82no—
236C2×C4—82no—
237C2×C4—82no—
238C2×C4—82no—
239C2×C4—82no—
240C2×C4—82no—
241C2×C4—82no—
242C2×C4—82no—
243C2×C4—82no—
244C2×C4—82no—
245C2×C4—82no—
246C2×C4—82no—
247C2×C4#41611yescharacteristic
248C2×C4#41911yescharacteristic
249C2×C4#48512yes—
250C8—82no—
251C8—162no—
252C22—22no—
253C22—22no—
254C22—21no—
255C22—21no—
256C22—22no—
257C22—22no—
258C22—21no—
259C22—21no—
260C22#160612yes—
261C22#160612yes—
262C22#187311yescharacteristic
263C22#188511yescharacteristic
264C22#188911yescharacteristic
265C22#190512yes—
266C22#190612yes—
267C4—21no—
268C4—21no—
269C4—22no—
270C4—42no—
271C4—42no—
272C4—84no—
273C4—84no—
274C4—82no—
275C4—82no—
276C4—162no—
277C2—22no—
278C2—22no—
279C2#901611yescharacteristic
280C2#917812yes—
281C2#942611yescharacteristic
282C2#943911yescharacteristic
283C2#952412yes—
subgroupG/H
center, socleC23#386
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#120
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
10. #2024C4semidirect

Maximal subgroups

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

subgroupindex
#234552
#144892
#83332
#75952
#70372