Finite Group Databasebeta

← #238153

Subgroups — #238153

Order 512.

864 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 864 conjugacy classes of subgroups — too many to draw. Only its 202 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#37642C211yesmaximal, characteristic
2#21894C212yesmaximal
3#21318C211yesmaximal, characteristic
4#10353C214yesmaximal, complemented
5#10350C214yesmaximal, complemented
6#8041C211yesmaximal, characteristic
7#7897C212yesmaximal
8C2×#533C2211yescharacteristic
9C2×#423C2212yes—
10C22×C4×C8C2211yescharacteristic
11#2844—22no—
12#2628—24no—
13#2628—22no—
14#2571—22no—
15#2571C2212yes—
16#2551—22no—
17#2551C2212yes—
18#2285C2212yes—
19#2285C2211yescharacteristic
20#1991—22no—
21#1990C2212yes—
22#1990C2212yes—
23#1988C2214yes—
24#1981C2214yes—
25#1894—24no—
26#1803—22no—
27#1803—22no—
28#1735—22no—
29#1296C2212yes—
30#1296C2212yes—
31#1279C2218yes—
32C22×M16—22no—
33C23×C8—21no—
34#533—22no—
35#533—24no—
36#533Q814yescomplemented
37C2×C4⋊Q8—22no—
38C2×C4⋊Q8—24no—
39C2×C4⋊Q8C2312yes—
40C2×#125—21no—
41C2×C22⋊Q8—22no—
42C2×C4×Q8—22no—
43C2×#117—22no—
44C2×#117—21no—
45C2×#117D412yes—
46C22×C42C2311yescharacteristic
47#423—24no—
48#423—24no—
49C2×C4⋊C8—22no—
50C2×C4⋊C8C2314yes—
51C2×#103—24no—
52C2×#103—24no—
53C2×C4×C8—24no—
54C2×C4×C8—21no—
55C2×C4×C8C2312yes—
56#400—28no—
57#400—24no—
58#399—24no—
59#395—28no—
60#395—24no—
61#393—24no—
62#389—24no—
63#389—24no—
64#389—24no—
65#384D412yes—
66#384D412yes—
67#384C2311yescharacteristic
68#384C2311yescharacteristic
69#382—22no—
70#382—24no—
71#382—22no—
72#382D412yes—
73#380—22no—
74#380—22no—
75#380D414yes—
76C4×C4⋊C4—22no—
77C4×C4⋊C4—22no—
78C4×C4⋊C4C2314yes—
79#340—24no—
80#340—24no—
81#340—24no—
82#340—24no—
83C22×Q8—24no—
84C22×Q8—22no—
85C23×C4C2×D411yescharacteristic
86C23×C4C2×D412yes—
87C2×M16—44no—
88C2×M16—82no—
89C2×M16—82no—
90C22×C8—24no—
91C22×C8—22no—
92C22×C8—24no—
93C22×C8—42no—
94C22×C8—42no—
95C4⋊Q8—22no—
96C4⋊Q8—22no—
97C4⋊Q8—22no—
98C4⋊Q8—84no—
99C4⋊Q8—84no—
100C4⋊Q8—84no—
101C4⋊Q8C2×Q814yes—
102#125—24no—
103#125—42no—
104C22⋊Q8—28no—
105C22⋊Q8—44no—
106C4×Q8—28no—
107C4×Q8—44no—
108#117—24no—
109#117—42no—
110#117—44no—
111#117—82no—
112#117—82no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—24no—
120C2×C4⋊C4—28no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—42no—
124C2×C4⋊C4C2×D414yes—
125C2×C4⋊C4C2×D414yes—
126C2×C4⋊C4C2×D412yes—
127C2×C4⋊C4C2×D412yes—
128C2×C22⋊C4—24no—
129C2×C22⋊C4—22no—
130C2×C22⋊C4—41no—
131C2×C42—22no—
132C2×C42—24no—
133C2×C42—41no—
134C2×C42C2×D412yes—
135C2×C42C2×D411yescharacteristic
136C2×C42C2×Q812yes—
137C2×C42#4114yes—
138C2×C42#4112yes—
139C2×C42C2411yescharacteristic
140C4⋊C8—24no—
141C4⋊C8—24no—
142C4⋊C8—44no—
143#103—84no—
144#103—84no—
145C4×C8—24no—
146C4×C8—42no—
147#95—24no—
148#95—24no—
149#95—28no—
150#95—24no—
151#95—24no—
152#95—24no—
153C24#12011yescharacteristic
154C2×Q8—24no—
155C2×Q8—42no—
156C2×Q8—42no—
157C2×Q8—42no—
158C2×Q8—42no—
159C2×Q8—42no—
160C2×Q8—88no—
161C2×Q8—84no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—24no—
169C22×C4—22no—
170C22×C4—24no—
171C22×C4—22no—
172C22×C4—42no—
173C22×C4#12012yes—
174C22×C4#12011yescharacteristic
175C22×C4C22⋊Q812yes—
176C22×C4C22⋊Q814yes—
177C22×C4#12412yes—
178C22×C4#12414yes—
179C22×C4C22×D411yescharacteristic
180C22×C4C22×D412yes—
181M16—88no—
182C2×C8—24no—
183C2×C8—42no—
184C2×C8—48no—
185C2×C8—42no—
186C2×C8—44no—
187C2×C8—44no—
188C2×C8—44no—
189C4⋊C4—24no—
190C4⋊C4—24no—
191C4⋊C4—24no—
192C4⋊C4—24no—
193C4⋊C4—24no—
194C4⋊C4—24no—
195C4⋊C4—42no—
196C4⋊C4—42no—
197C4⋊C4—42no—
198C4⋊C4—42no—
199C4⋊C4—42no—
200C4⋊C4—42no—
201C4⋊C4—42no—
202C4⋊C4—42no—
203C4⋊C4—48no—
204C4⋊C4—44no—
205C4⋊C4—44no—
206C4⋊C4—82no—
207C4⋊C4—82no—
208C4⋊C4—88no—
209C4⋊C4—84no—
210C22⋊C4—44no—
211C22⋊C4—82no—
212C22⋊C4—84no—
213C42—44no—
214C42—44no—
215C42—44no—
216C42—44no—
217C42C22⋊Q814yes—
218C42C22⋊Q812yes—
219C42#12412yes—
220C42#12414yes—
221C42C22×Q811yescharacteristic
222C42C2×#4112yes—
223C42C2×#4111yescharacteristic
224C23—22no—
225C23—22no—
226C23—21no—
227C23—21no—
228C23#39312yes—
229C23C2×#12011yescharacteristic
230Q8—44no—
231Q8—44no—
232Q8—82no—
233Q8—82no—
234Q8—168no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—24no—
240C2×C4—24no—
241C2×C4—24no—
242C2×C4—24no—
243C2×C4—24no—
244C2×C4—24no—
245C2×C4—44no—
246C2×C4—44no—
247C2×C4—44no—
248C2×C4—44no—
249C2×C4—44no—
250C2×C4—44no—
251C2×C4—44no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—44no—
255C2×C4—44no—
256C2×C4—82no—
257C2×C4—82no—
258C2×C4#39314yes—
259C2×C4#39312yes—
260C2×C4#45512yes—
261C2×C4#45612yes—
262C2×C4C2×C22⋊Q811yescharacteristic
263C2×C4C2×C22⋊Q812yes—
264C2×C4C2×#12411yescharacteristic
265C2×C4C2×#12412yes—
266C2×C4#57612yes—
267C2×C4#57712yes—
268C2×C4#57712yes—
269C2×C4#57812yes—
270C8—44no—
271C8—84no—
272C22—24no—
273C22—22no—
274C22—41no—
275C22—41no—
276C22—42no—
277C22C2×#39311yescharacteristic
278C22#301212yes—
279C22#301312yes—
280C22C2×#45511yescharacteristic
281C22C2×#45611yescharacteristic
282C4—24no—
283C4—21no—
284C4—21no—
285C4—84no—
286C4—82no—
287C4—82no—
288C4—84no—
289C4—84no—
290C4#202814yes—
291C4#308312yes—
292C4#308412yes—
293C4#313612yes—
294C4#313712yes—
295C2—42no—
296C2#1869111yescharacteristic
297C2#1869311yescharacteristic
298C2#1870212yes—
299C2#2345812yes—
300C2C2×#202811yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #10353C2semidirect
5. #10350C2semidirect
36. #533Q8semidirect

Maximal subgroups

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

subgroupindex
#376422
#218942
#213182
#103532
#103502
#80412
#78972