Finite Group Databasebeta

← #238062

Subgroups — #238062

Order 512.

908 conjugacy classes of subgroups; only the 178 normal subgroups are shown.

normal subgroups only

This group has 908 conjugacy classes of subgroups — too many to draw. Only its 178 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#37480C211yesmaximal, characteristic
2#20705C212yesmaximal
3#20688C211yesmaximal, characteristic
4#10387C218yesmaximal, complemented
5#8112C211yesmaximal, characteristic
6#7946C212yesmaximal
7C22×#117C2211yescharacteristic
8C2×#407C2212yes—
9#2842—21no—
10#2840—21no—
11#2665—24no—
12#2628—22no—
13#2607C2214yes—
14#2439—24no—
15#2280C2211yescharacteristic
16#2280C2211yescharacteristic
17#2280C2212yes—
18#1826—24no—
19#1826C2218yes—
20#1820C2214yes—
21#1819—21no—
22#1819—21no—
23#1740—22no—
24#1733—24no—
25#1306C2214yes—
26#1284C2218yes—
27C24×C4C2311yescharacteristic
28C22×M16—22no—
29C2×C4⋊Q8—22no—
30C2×#117—21no—
31C2×#117—21no—
32C2×#117—21no—
33C2×#117—22no—
34C2×#117—22no—
35C2×#117—22no—
36C2×#117—22no—
37C2×#117D412yes—
38C2×#117D414yes—
39C2×#117Q812yescomplemented
40C2×#117C2312yes—
41C22×C4⋊C4Q812yescomplemented
42C22×C22⋊C4—21no—
43C22×C42—21no—
44#407—22no—
45#407—22no—
46#407—22no—
47#407C2314yes—
48C2×C22⋊C8—22no—
49C2×C22⋊C8—22no—
50#400—24no—
51#399—24no—
52#396—28no—
53#395—24no—
54#393—28no—
55#393—24no—
56#391—28no—
57#389—24no—
58#387—28no—
59#382—22no—
60#381—22no—
61#381C2312yes—
62#381C2312yes—
63#380—22no—
64#380—22no—
65#380—22no—
66#380—22no—
67#379—22no—
68#379—21no—
69#379D411yescharacteristic
70#379D411yescharacteristic
71C4×C22⋊C4—22no—
72C4×C22⋊C4—22no—
73C4×C22⋊C4D414yes—
74C4×C22⋊C4C2314yes—
75#342—24no—
76#342—24no—
77#339—216no—
78C25C2×D411yescharacteristic
79C22×Q8—24no—
80C23×C4—22no—
81C23×C4—44no—
82C23×C4C2×D411yescharacteristic
83C23×C4C2×D412yes—
84C23×C4C2×D412yes—
85C23×C4C2×Q811yescharacteristic
86C23×C4C2×Q811yescharacteristic
87C23×C4#4112yes—
88C23×C4#4112yes—
89C23×C4#4111yescharacteristic
90C23×C4#4111yescharacteristic
91C23×C4C2411yescharacteristic
92C2×M16—24no—
93C2×M16—42no—
94C2×M16—48no—
95C22×C8—42no—
96C4⋊Q8—84no—
97#117—24no—
98#117—24no—
99#117—24no—
100#117—24no—
101#117—41no—
102#117—41no—
103#117—42no—
104#117—42no—
105#117—41no—
106#117—41no—
107#117—48no—
108#117—44no—
109#117—44no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—24no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—24no—
117C2×C4⋊C4—28no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—41no—
120C2×C4⋊C4C2×Q814yes—
121C2×C22⋊C4—22no—
122C2×C22⋊C4—22no—
123C2×C22⋊C4—22no—
124C2×C22⋊C4—24no—
125C2×C22⋊C4—22no—
126C2×C22⋊C4—22no—
127C2×C22⋊C4—24no—
128C2×C22⋊C4—24no—
129C2×C22⋊C4—42no—
130C2×C22⋊C4—42no—
131C2×C22⋊C4—42no—
132C2×C22⋊C4C2×D412yes—
133C2×C22⋊C4C2×D412yes—
134C2×C42—22no—
135C2×C42—22no—
136C2×C42—22no—
137C2×C42—42no—
138C2×C42—42no—
139C2×C42—41no—
140C2×C42C2×D418yes—
141C22⋊C8—24no—
142C22⋊C8—24no—
143C22⋊C8—42no—
144C22⋊C8—42no—
145#95—24no—
146#95—24no—
147#95—28no—
148#95—24no—
149#95—24no—
150#95—24no—
151#95—24no—
152C24—42no—
153C24—82no—
154C24#12012yes—
155C24C22⋊Q811yescharacteristic
156C24C22⋊Q811yescharacteristic
157C24#12412yes—
158C24C22×D411yescharacteristic
159C2×Q8—84no—
160C2×Q8—88no—
161C22×C4—24no—
162C22×C4—28no—
163C22×C4—22no—
164C22×C4—28no—
165C22×C4—22no—
166C22×C4—24no—
167C22×C4—42no—
168C22×C4—42no—
169C22×C4—44no—
170C22×C4—44no—
171C22×C4—42no—
172C22×C4—42no—
173C22×C4—44no—
174C22×C4—42no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4—42no—
178C22×C4—84no—
179C22×C4#12011yescharacteristic
180C22×C4#12011yescharacteristic
181C22×C4C22⋊Q812yes—
182C22×C4C22⋊Q812yes—
183C22×C4C22⋊Q811yescharacteristic
184C22×C4C22⋊Q811yescharacteristic
185C22×C4C22⋊Q812yes—
186C22×C4C22⋊Q812yes—
187C22×C4#12412yes—
188C22×C4#12412yes—
189C22×C4#12411yescharacteristic
190C22×C4#12411yescharacteristic
191C22×C4#12412yes—
192C22×C4#12412yes—
193C22×C4C22×D412yes—
194C22×C4C22×Q811yescharacteristic
195C22×C4C2×#4112yes—
196C22×C4C2×#4111yescharacteristic
197M16—48no—
198M16—84no—
199C2×C8—44no—
200C2×C8—84no—
201C4⋊C4—24no—
202C4⋊C4—24no—
203C4⋊C4—42no—
204C4⋊C4—42no—
205C4⋊C4—48no—
206C4⋊C4—82no—
207C4⋊C4—82no—
208C4⋊C4—88no—
209C4⋊C4—84no—
210C22⋊C4—42no—
211C22⋊C4—42no—
212C22⋊C4—44no—
213C22⋊C4—42no—
214C22⋊C4—42no—
215C22⋊C4—44no—
216C22⋊C4—84no—
217C22⋊C4—82no—
218C22⋊C4—84no—
219C22⋊C4—84no—
220C22⋊C4—82no—
221C22⋊C4—84no—
222C42—48no—
223C42—44no—
224C42—42no—
225C42—42no—
226C42—44no—
227C42—82no—
228C23—22no—
229C23—24no—
230C23—41no—
231C23—42no—
232C23—42no—
233C23—42no—
234C23—42no—
235C23—42no—
236C23—41no—
237C23—42no—
238C23—42no—
239C23—42no—
240C23—88no—
241C23#39312yes—
242C23#39312yes—
243C23C2×#12011yescharacteristic
244C23C2×C22⋊Q811yescharacteristic
245C23C2×#12411yescharacteristic
246Q8—168no—
247C2×C4—24no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—24no—
252C2×C4—24no—
253C2×C4—24no—
254C2×C4—24no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—42no—
264C2×C4—44no—
265C2×C4—48no—
266C2×C4—48no—
267C2×C4—44no—
268C2×C4—44no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—48no—
272C2×C4—84no—
273C2×C4—84no—
274C2×C4—84no—
275C2×C4—82no—
276C2×C4—84no—
277C2×C4#39311yescharacteristic
278C2×C4#39311yescharacteristic
279C2×C4#39311yescharacteristic
280C2×C4#39311yescharacteristic
281C2×C4#45814yes—
282C2×C4C2×C22⋊Q812yes—
283C2×C4C2×#12412yes—
284C8—84no—
285C22—22no—
286C22—21no—
287C22—21no—
288C22—22no—
289C22—22no—
290C22—22no—
291C22—22no—
292C22—22no—
293C22—22no—
294C22—22no—
295C22—22no—
296C22—44no—
297C22—82no—
298C22—84no—
299C22—84no—
300C22—82no—
301C22C2×#39311yescharacteristic
302C22C2×#45812yes—
303C4—22no—
304C4—22no—
305C4—21no—
306C4—21no—
307C4—82no—
308C4—88no—
309C4—82no—
310C4—84no—
311C4—162no—
312C4#203012yes—
313C4#203012yes—
314C2—22no—
315C2—41no—
316C2—41no—
317C2—42no—
318C2—42no—
319C2C2×#203011yescharacteristic
320C2#2347212yes—
subgroupG/H
center, socleC22C2×#393
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #10387C2semidirect
39. C2×#117Q8semidirect
41. C22×C4⋊C4Q8semidirect

Maximal subgroups

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

subgroupindex
#374802
#207052
#206882
#103872
#81122
#79462