Finite Group Databasebeta

← #241269

Subgroups — #241269

Order 512.

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

normal subgroups only

This group has 778 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#37609C211yesmaximal, characteristic
2#20823C212yesmaximal
3#20765C211yesmaximal, characteristic
4#10586C212yesmaximal, complemented
5#10585C212yesmaximal, complemented
6#10584C214yesmaximal, complemented
7#8029C212yesmaximal
8C2×#1295C211yesmaximal, characteristic
9C2×C4×M16C2211yescharacteristic
10C22×#97C2212yes—
11#2845C2212yes—
12#2844C2212yes—
13#2666—24no—
14#2627—22no—
15#2612—22no—
16#2569—22no—
17#2549—22no—
18#2285C2211yescharacteristic
19#2283C2212yes—
20C2×#382C2211yescharacteristic
21#1847C2214yes—
22#1836C2212yes—
23#1836C2212yes—
24#1836C2214yes—
25#1835—22no—
26#1808—22no—
27#1807—22no—
28#1776—24no—
29#1736—22no—
30#1295C2212yes—
31#1295C2214yes—
32#1295C2214yes—
33#1295C2211yescharacteristic
34#1295C2211yescharacteristic
35C22×M16—21no—
36C23×C8—22no—
37C2×C22⋊Q8—22no—
38C2×#117—21no—
39C22×C4⋊C4D412yes—
40C22×C42C2311yescharacteristic
41C2×#107—24no—
42C2×#106—24no—
43C4×M16—22no—
44C4×M16Q814yescomplemented
45C2×#97—22no—
46C2×#97—24no—
47C2×#97Q818yescomplemented
48C2×#97C2312yes—
49C2×#97C2312yes—
50C2×#97C2312yes—
51C2×C4×C8—21no—
52#400—22no—
53#400—24no—
54#400—22no—
55#398—24no—
56#398—24no—
57#398—24no—
58#398—24no—
59#395—22no—
60#395—22no—
61#395—24no—
62#391—24no—
63#389—24no—
64#387—24no—
65#387—24no—
66#385—24no—
67#384C2312yes—
68#383—22no—
69#382—22no—
70#382—21no—
71#382D412yes—
72#382C2311yescharacteristic
73#382C2314yes—
74#382C2311yescharacteristic
75#381—22no—
76#380—22no—
77#380—22no—
78#380D412yes—
79#380D412yes—
80C4×C4⋊C4—22no—
81C4×C4⋊C4D414yes—
82C4×C22⋊C4—22no—
83#376—22no—
84C2×#95—22no—
85#340—24no—
86#340—24no—
87#340—24no—
88#340—24no—
89C22×Q8—22no—
90C23×C4—22no—
91C23×C4C2×D411yescharacteristic
92C23×C4C2×D412yes—
93C2×M16—24no—
94C2×M16—42no—
95C2×M16—42no—
96C22×C8—22no—
97C22×C8—28no—
98C22×C8—22no—
99C22×C8—22no—
100C22×C8—48no—
101C22⋊Q8—84no—
102#117—44no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—24no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—24no—
107C2×C4⋊C4—24no—
108C2×C4⋊C4—24no—
109C2×C4⋊C4—24no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—44no—
112C2×C4⋊C4C2×D412yes—
113C2×C4⋊C4C2×D412yes—
114C2×C4⋊C4C2×D414yes—
115C2×C4⋊C4C2×D414yes—
116C2×C22⋊C4—22no—
117C2×C22⋊C4—22no—
118C2×C22⋊C4—41no—
119C2×C42—24no—
120C2×C42—41no—
121C2×C42C2×D411yescharacteristic
122C2×C42C2×D412yes—
123C2×C42C2×Q814yes—
124C2×C42C2×Q812yes—
125C2×C42#4112yes—
126C2×C42C2411yescharacteristic
127#107—48no—
128#106—48no—
129#97—28no—
130#97—22no—
131#97—22no—
132#97—22no—
133#97C2×Q814yes—
134#97C2×Q814yes—
135#97C2×Q814yes—
136C4×C8—24no—
137#95—22no—
138#95—22no—
139#95—24no—
140#95—24no—
141#95—24no—
142#95—22no—
143#95—22no—
144#95—24no—
145#95—22no—
146#95—22no—
147#95—24no—
148#95—24no—
149#95—42no—
150C24#12011yescharacteristic
151C2×Q8—82no—
152C2×Q8—84no—
153C22×C4—22no—
154C22×C4—22no—
155C22×C4—24no—
156C22×C4—22no—
157C22×C4—24no—
158C22×C4—22no—
159C22×C4—44no—
160C22×C4—44no—
161C22×C4—42no—
162C22×C4—84no—
163C22×C4#12012yes—
164C22×C4#12011yescharacteristic
165C22×C4C22⋊Q812yes—
166C22×C4C22⋊Q814yes—
167C22×C4C4⋊Q812yes—
168C22×C4C4⋊Q814yes—
169C22×C4C22×D411yescharacteristic
170C22×C4C22×D412yes—
171M16—48no—
172C2×C8—28no—
173C2×C8—24no—
174C2×C8—24no—
175C2×C8—24no—
176C2×C8—416no—
177C2×C8—44no—
178C2×C8—42no—
179C2×C8—42no—
180C4⋊C4—44no—
181C4⋊C4—44no—
182C4⋊C4—42no—
183C4⋊C4—42no—
184C4⋊C4—44no—
185C4⋊C4—44no—
186C4⋊C4—44no—
187C4⋊C4—82no—
188C4⋊C4—84no—
189C4⋊C4—84no—
190C4⋊C4—84no—
191C4⋊C4—84no—
192C4⋊C4—82no—
193C22⋊C4—44no—
194C22⋊C4—44no—
195C22⋊C4—82no—
196C42—84no—
197C42—82no—
198C42C22⋊Q812yes—
199C42C22⋊Q814yes—
200C42C4⋊Q812yes—
201C42C4⋊Q814yes—
202C42C22×Q812yes—
203C42C22×Q811yescharacteristic
204C42C2×#4111yescharacteristic
205C23—24no—
206C23#39512yes—
207C23#45412yes—
208C23#45512yes—
209C23C2×#12011yescharacteristic
210Q8—164no—
211C2×C4—24no—
212C2×C4—24no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—24no—
218C2×C4—24no—
219C2×C4—44no—
220C2×C4—44no—
221C2×C4—44no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—44no—
225C2×C4—44no—
226C2×C4—44no—
227C2×C4—44no—
228C2×C4—44no—
229C2×C4—44no—
230C2×C4—44no—
231C2×C4—82no—
232C2×C4—84no—
233C2×C4—164no—
234C2×C4#39512yes—
235C2×C4#39514yes—
236C2×C4C2×C22⋊Q811yescharacteristic
237C2×C4C2×C22⋊Q812yes—
238C2×C4C2×C4⋊Q811yescharacteristic
239C2×C4C2×C4⋊Q812yes—
240C8—28no—
241C8—44no—
242C8—44no—
243C22—24no—
244C22—24no—
245C22—24no—
246C22#206114yes—
247C22C2×#39511yescharacteristic
248C22C2×#45511yescharacteristic
249C22C2×#45411yescharacteristic
250C22#301814yes—
251C4—21no—
252C4—21no—
253C4—22no—
254C4—22no—
255C4—22no—
256C4—24no—
257C4—82no—
258C4—82no—
259C4—84no—
260C4—84no—
261C4—162no—
262C2—24no—
263C2#1907912yes—
264C2#1908212yes—
265C2C2×#206111yescharacteristic
266C2#2356012yes—
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. #10586C2semidirect
5. #10585C2semidirect
6. #10584C2semidirect
44. C4×M16Q8semidirect
47. C2×#97Q8semidirect

Maximal subgroups

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

subgroupindex
#376092
#208232
#207652
#105862
#105852
#105842
#80292
C2×#12952