Finite Group Databasebeta

← #97500

Subgroups — #97500

Order 512.

716 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 716 conjugacy classes of subgroups — too many to draw. Only its 152 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#59832C211yesmaximal, characteristic
2#9870C212yesmaximal
3#9637C211yesmaximal, characteristic
4#8587C211yesmaximal, characteristic
5#8586C212yesmaximal
6C2×Q82—21no—
7#2734C414yes—
8#2582—21no—
9#2551—22no—
10#2546—22no—
11#2451—24no—
12#2365—22no—
13C4×C4⋊Q8C2212yes—
14C4×C4⋊Q8C2211yescharacteristic
15#1842C2211yescharacteristic
16#1842C2212yes—
17C42×C8C2211yescharacteristic
18C2×#333—21no—
19C2×#332—22no—
20C2×Q8⋊C8—22no—
21#1296—22no—
22#1296—21no—
23#1279—24no—
24Q82—44no—
25Q82—42no—
26Q82—42no—
27C2×C4⋊Q8—24no—
28C2×C4⋊Q8—24no—
29C2×C4⋊Q8—22no—
30C2×C4⋊Q8D411yescharacteristic
31C2×C4⋊Q8D412yes—
32C2×C4×Q8—24no—
33C2×C4×Q8—22no—
34C2×C4×Q8D412yes—
35C2×C4⋊C8—22no—
36C2×C4⋊C8—22no—
37C2×C4⋊C8—22no—
38C2×C4⋊C8—21no—
39C2×C4⋊C8—22no—
40C2×C4×C8—21no—
41C2×C4×C8—22no—
42#398—24no—
43#398—24no—
44#395—28no—
45#391—24no—
46#391—24no—
47#391—22no—
48#391C2×C412yes—
49#389—24no—
50#389—24no—
51#389—22no—
52#389—22no—
53#389C2×C414yes—
54#387—28no—
55#384D411yescharacteristic
56#384D412yes—
57C4×C4⋊C4—22no—
58C4×C4⋊C4—22no—
59C4×C4⋊C4—22no—
60C4×C4⋊C4D414yes—
61C43C2311yescharacteristic
62#333—42no—
63#332—44no—
64Q8⋊C8—44no—
65C22×Q8—24no—
66C22×Q8—22no—
67C22×Q8C22⋊C412yes—
68C22×C8—41no—
69C22×C8—42no—
70C22×C8—41no—
71C4⋊Q8—22no—
72C4⋊Q8—22no—
73C4⋊Q8—22no—
74C4⋊Q8—48no—
75C4⋊Q8—48no—
76C4⋊Q8—48no—
77C4⋊Q8—48no—
78C4⋊Q8—82no—
79C4⋊Q8—82no—
80C4⋊Q8D812yes—
81C4⋊Q8SD1612yes—
82C4⋊Q8SD1614yes—
83C4⋊Q8Q1614yes—
84C4×Q8—42no—
85C4×Q8—42no—
86C4×Q8—48no—
87C4×Q8—48no—
88C4×Q8—42no—
89C4×Q8—42no—
90C4×Q8—42no—
91C4×Q8—42no—
92C2×C4⋊C4—22no—
93C2×C4⋊C4—22no—
94C2×C4⋊C4—28no—
95C2×C4⋊C4—24no—
96C2×C4⋊C4—22no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4—24no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4C22⋊C414yes—
102C2×C4⋊C4C22⋊C412yes—
103C2×C4⋊C4C22⋊C414yes—
104C2×C42—24no—
105C2×C42—24no—
106C2×C42—22no—
107C2×C42C22×C411yescharacteristic
108C2×C42C2×D411yescharacteristic
109C2×C42C2×D412yes—
110C2×C42C2×D411yescharacteristic
111C2×C42C2×D412yes—
112C4⋊C8—42no—
113C4⋊C8—44no—
114C4⋊C8—42no—
115C4⋊C8—42no—
116C4⋊C8—42no—
117C4⋊C8—42no—
118C4⋊C8—42no—
119C4⋊C8—42no—
120C4×C8—22no—
121C4×C8—24no—
122C4×C8—42no—
123C4×C8—41no—
124#95—24no—
125#95—28no—
126#95—24no—
127#95—24no—
128#95—24no—
129C2×Q8—22no—
130C2×Q8—24no—
131C2×Q8—24no—
132C2×Q8—48no—
133C2×Q8—42no—
134C2×Q8—42no—
135C2×Q8—42no—
136C2×Q8—42no—
137C2×Q8—42no—
138C2×Q8—42no—
139C2×Q8—84no—
140C2×Q8—84no—
141C2×Q8#10214yes—
142C22×C4—24no—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4—24no—
146C22×C4C2×C22⋊C411yescharacteristic
147C22×C4C2×C22⋊C412yes—
148C22×C4#12011yescharacteristic
149C22×C4#12012yes—
150C22×C4#12011yescharacteristic
151C2×C8—42no—
152C2×C8—44no—
153C2×C8—82no—
154C2×C8—82no—
155C2×C8—82no—
156C2×C8—81no—
157C2×C8—82no—
158C4⋊C4—24no—
159C4⋊C4—24no—
160C4⋊C4—24no—
161C4⋊C4—24no—
162C4⋊C4—24no—
163C4⋊C4—44no—
164C4⋊C4—42no—
165C4⋊C4—416no—
166C4⋊C4—44no—
167C4⋊C4—44no—
168C4⋊C4—42no—
169C4⋊C4—42no—
170C4⋊C4—42no—
171C4⋊C4—42no—
172C4⋊C4—42no—
173C4⋊C4—42no—
174C4⋊C4—82no—
175C4⋊C4—84no—
176C4⋊C4#10318yes—
177C42—21no—
178C42—22no—
179C42—21no—
180C42—22no—
181C42—41no—
182C42—48no—
183C42—44no—
184C42—44no—
185C42—42no—
186C42—42no—
187C42C2×D811yescharacteristic
188C42C2×SD1611yescharacteristic
189C42C2×SD1612yes—
190C42C2×Q1612yes—
191C42AGL(1,ℤ/8)12yes—
192C42#13712yes—
193C42#13712yes—
194C23#37911yescharacteristic
195Q8—44no—
196Q8—44no—
197Q8—44no—
198Q8—44no—
199Q8—44no—
200Q8—82no—
201Q8—168no—
202C2×C4—21no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—22no—
208C2×C4—24no—
209C2×C4—28no—
210C2×C4—42no—
211C2×C4—44no—
212C2×C4—44no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—44no—
216C2×C4—44no—
217C2×C4—44no—
218C2×C4—44no—
219C2×C4C2×#10211yescharacteristic
220C2×C4C2×#10312yes—
221C2×C4#41811yescharacteristic
222C2×C4#41912yes—
223C2×C4#44712yes—
224C2×C4#44812yes—
225C2×C4#44812yes—
226C2×C4#45012yes—
227C2×C4#45112yes—
228C2×C4#45112yes—
229C2×C4#45311yescharacteristic
230C2×C4#45412yes—
231C2×C4#45511yescharacteristic
232C2×C4#45612yes—
233C8—44no—
234C8—161no—
235C8—162no—
236C22#178611yescharacteristic
237C22#178712yes—
238C22#179111yescharacteristic
239C22#179211yescharacteristic
240C22#179312yes—
241C4—21no—
242C4—21no—
243C4—22no—
244C4—22no—
245C4—41no—
246C4—44no—
247C4—48no—
248C4—82no—
249C4—84no—
250C4#162311yescharacteristic
251C4#162712yes—
252C4#163311yescharacteristic
253C4#163712yes—
254C4#164512yes—
255C4#164912yes—
256C4#165112yes—
257C2#835311yescharacteristic
258C2#835612yes—
259C2#844712yes—
260C2#845111yescharacteristic
261C2#853111yescharacteristic
subgroupG/H
center, socle, prime radicalC23#379
commutatorC2×C42C22×C4
Frattini, cosocleC43C23
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C43 ▷ C23 ▷ C1

Maximal subgroups

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

subgroupindex
#598322
#98702
#96372
#85872
#85862