Finite Group Databasebeta

← #238154

Subgroups — #238154

Order 512.

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

normal subgroups only

This group has 856 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#37643C211yesmaximal, characteristic
2#21894C212yesmaximal
3#21318C211yesmaximal, characteristic
4#10357C214yesmaximal, complemented
5#10356C214yesmaximal, complemented
6#8041C211yesmaximal, characteristic
7#7897C212yesmaximal
8C2×#533C2211yescharacteristic
9C2×#423C2212yes—
10C22×C4×C8C2211yescharacteristic
11#2845—22no—
12#2629—24no—
13#2629—22no—
14#2552—22no—
15#2552—22no—
16#2552C2214yes—
17#2285C2212yes—
18#2285C2211yescharacteristic
19#1991—22no—
20#1990C2212yes—
21#1990C2212yes—
22#1988C2214yes—
23#1981C2214yes—
24#1894—24no—
25#1803—22no—
26#1803—22no—
27#1735—22no—
28#1296C2214yes—
29#1279C2214yes—
30#1279C2214yes—
31C22×M16—22no—
32C23×C8—21no—
33#533—22no—
34#533—24no—
35#533Q814yes—
36C2×C4⋊Q8C2312yes—
37C2×#125—22no—
38C2×#125—24no—
39C2×#125—21no—
40C2×C22⋊Q8—22no—
41C2×C4×Q8—22no—
42C2×#117—22no—
43C2×#117—21no—
44C2×#117D412yes—
45C22×C42C2311yescharacteristic
46#423—24no—
47#423—24no—
48C2×C4⋊C8—22no—
49C2×C4⋊C8C2314yes—
50C2×#103—24no—
51C2×#103—24no—
52C2×C4×C8—24no—
53C2×C4×C8—21no—
54C2×C4×C8C2311yescharacteristic
55C2×C4×C8C2311yescharacteristic
56#398—28no—
57#398—28no—
58#398—28no—
59#396—28no—
60#389—24no—
61#389—24no—
62#389—24no—
63#384—22no—
64#384—24no—
65#384D412yes—
66#384D412yes—
67#384C2312yes—
68#382—22no—
69#382D412yes—
70#380—22no—
71#380—22no—
72#380D414yes—
73C4×C4⋊C4—22no—
74C4×C4⋊C4—22no—
75C4×C4⋊C4C2314yes—
76#340—24no—
77#340—24no—
78#340—24no—
79#340—24no—
80C22×Q8—22no—
81C23×C4C2×D411yescharacteristic
82C23×C4C2×D412yes—
83C2×M16—44no—
84C2×M16—82no—
85C2×M16—82no—
86C22×C8—24no—
87C22×C8—21no—
88C22×C8—21no—
89C22×C8—24no—
90C22×C8—42no—
91C22×C8—42no—
92C4⋊Q8—22no—
93C4⋊Q8—22no—
94C4⋊Q8—22no—
95C4⋊Q8C2×Q814yes—
96#125—24no—
97#125—42no—
98#125—84no—
99#125—84no—
100#125—84no—
101C22⋊Q8—28no—
102C22⋊Q8—44no—
103C4×Q8—28no—
104C4×Q8—44no—
105#117—24no—
106#117—42no—
107#117—44no—
108#117—82no—
109#117—82no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—28no—
117C2×C4⋊C4—28no—
118C2×C4⋊C4—28no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—42no—
122C2×C4⋊C4C2×D414yes—
123C2×C4⋊C4C2×D414yes—
124C2×C4⋊C4C2×D414yes—
125C2×C22⋊C4—24no—
126C2×C22⋊C4—22no—
127C2×C22⋊C4—41no—
128C2×C42—22no—
129C2×C42—24no—
130C2×C42—41no—
131C2×C42C2×D412yes—
132C2×C42C2×D411yescharacteristic
133C2×C42C2×Q812yes—
134C2×C42#4114yes—
135C2×C42#4112yes—
136C2×C42C2411yescharacteristic
137C4⋊C8—24no—
138C4⋊C8—24no—
139C4⋊C8—44no—
140#103—84no—
141#103—84no—
142C4×C8—22no—
143C4×C8—22no—
144C4×C8—42no—
145#95—28no—
146#95—24no—
147#95—24no—
148#95—24no—
149C24#12011yescharacteristic
150C2×Q8—24no—
151C2×Q8—42no—
152C2×Q8—42no—
153C2×Q8—42no—
154C2×Q8—42no—
155C2×Q8—42no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—24no—
163C22×C4—24no—
164C22×C4—24no—
165C22×C4—42no—
166C22×C4#12012yes—
167C22×C4#12011yescharacteristic
168C22×C4C22⋊Q812yes—
169C22×C4C22⋊Q814yes—
170C22×C4#12412yes—
171C22×C4#12414yes—
172C22×C4C22×D411yescharacteristic
173C22×C4C22×D412yes—
174M16—88no—
175C2×C8—22no—
176C2×C8—22no—
177C2×C8—41no—
178C2×C8—41no—
179C2×C8—48no—
180C2×C8—41no—
181C2×C8—41no—
182C2×C8—44no—
183C2×C8—44no—
184C2×C8—44no—
185C4⋊C4—24no—
186C4⋊C4—24no—
187C4⋊C4—24no—
188C4⋊C4—24no—
189C4⋊C4—24no—
190C4⋊C4—24no—
191C4⋊C4—44no—
192C4⋊C4—42no—
193C4⋊C4—42no—
194C4⋊C4—42no—
195C4⋊C4—42no—
196C4⋊C4—42no—
197C4⋊C4—42no—
198C4⋊C4—48no—
199C4⋊C4—44no—
200C4⋊C4—44no—
201C4⋊C4—82no—
202C4⋊C4—82no—
203C4⋊C4—88no—
204C4⋊C4—88no—
205C4⋊C4—88no—
206C22⋊C4—44no—
207C22⋊C4—82no—
208C22⋊C4—84no—
209C42—44no—
210C42—44no—
211C42—44no—
212C42—44no—
213C42C22⋊Q814yes—
214C42C22⋊Q812yes—
215C42#12412yes—
216C42#12414yes—
217C42C22×Q811yescharacteristic
218C42C2×#4112yes—
219C42C2×#4111yescharacteristic
220C23—22no—
221C23—22no—
222C23—21no—
223C23—21no—
224C23#39312yes—
225C23C2×#12011yescharacteristic
226Q8—44no—
227Q8—44no—
228Q8—82no—
229Q8—82no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—24no—
235C2×C4—24no—
236C2×C4—24no—
237C2×C4—24no—
238C2×C4—24no—
239C2×C4—24no—
240C2×C4—44no—
241C2×C4—44no—
242C2×C4—44no—
243C2×C4—44no—
244C2×C4—44no—
245C2×C4—44no—
246C2×C4—44no—
247C2×C4—44no—
248C2×C4—44no—
249C2×C4—44no—
250C2×C4—82no—
251C2×C4—82no—
252C2×C4#39314yes—
253C2×C4#39312yes—
254C2×C4#45512yes—
255C2×C4#45612yes—
256C2×C4C2×C22⋊Q811yescharacteristic
257C2×C4C2×C22⋊Q812yes—
258C2×C4C2×#12411yescharacteristic
259C2×C4C2×#12412yes—
260C2×C4#57612yes—
261C2×C4#57712yes—
262C2×C4#57712yes—
263C2×C4#57812yes—
264C8—42no—
265C8—42no—
266C8—84no—
267C22—24no—
268C22—22no—
269C22—42no—
270C22—41no—
271C22—41no—
272C22C2×#39311yescharacteristic
273C22#301212yes—
274C22#301312yes—
275C22C2×#45511yescharacteristic
276C22C2×#45611yescharacteristic
277C4—24no—
278C4—21no—
279C4—21no—
280C4—84no—
281C4—84no—
282C4—84no—
283C4—84no—
284C4#202814yes—
285C4#308312yes—
286C4#308412yes—
287C4#313612yes—
288C4#313712yes—
289C2—42no—
290C2#1869212yes—
291C2#1870111yescharacteristic
292C2#1870311yescharacteristic
293C2#2345812yes—
294C2C2×#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. #10357C2semidirect
5. #10356C2semidirect

Maximal subgroups

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

subgroupindex
#376432
#218942
#213182
#103572
#103562
#80412
#78972