Finite Group Databasebeta

← #98915

Subgroups — #98915

Order 512.

324 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 110 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#21341C211yesmaximal, characteristic
2#8807C211yesmaximal, characteristic
3C4×#335C211yesmaximal, characteristic
4#7577C212yesmaximal
5#7022C212yesmaximal
6#2295C2211yescharacteristic
7#1984C411yescharacteristic
8#1984C411yescharacteristic
9#1939C411yescharacteristic
10#1939C411yescharacteristic
11#1916—21no—
12#1916—21no—
13#1831C2212yes—
14#1768—21no—
15#1767C2211yescharacteristic
16C4×C4⋊C8C2212yes—
17#1746—21no—
18C42×C8C2211yescharacteristic
19C2×#335—21no—
20C2×#335—21no—
21C2×#334—21no—
22#1277—21no—
23#1272—22no—
24#1271—22no—
25C2×C4×Q8C2×C411yescharacteristic
26C2×C4⋊C8—22no—
27C2×C4⋊C8—21no—
28C2×C4⋊C8—21no—
29C2×C4⋊C8—22no—
30C2×C4⋊C8C2×C411yescharacteristic
31C2×C4⋊C8C2×C411yescharacteristic
32C2×#97—21no—
33C2×C4×C8—22no—
34C2×C4×C8C2×C411yescharacteristic
35C2×C4×C8C2×C411yescharacteristic
36C2×C4×C8D411yescharacteristic
37C2×C4×C8D411yescharacteristic
38C2×C4×C8D411yescharacteristic
39C2×C4×C8D411yescharacteristic
40#391D411yescharacteristic
41#391Q811yescharacteristic
42#389D411yescharacteristic
43#389Q811yescharacteristic
44#387—21no—
45#387—21no—
46C4×C4⋊C4—21no—
47C4×C4⋊C4C2×C411yescharacteristic
48#376—21no—
49C43C2311yescharacteristic
50#336—21no—
51#336—21no—
52#336—21no—
53#336—21no—
54#336—21no—
55#336—21no—
56#336—21no—
57#336—22no—
58#336—22no—
59#336—22no—
60#335—24no—
61#335—42no—
62#334—42no—
63C22×Q8C4⋊C411yescharacteristic
64C22×C8—21no—
65C22×C8—21no—
66C22×C8—42no—
67C22×C8—42no—
68C22×C8C22⋊C411yescharacteristic
69C22×C8C22⋊C411yescharacteristic
70C22×C8C22⋊C411yescharacteristic
71C22×C8C22⋊C411yescharacteristic
72C4×Q8—22no—
73C4×Q8—22no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4C4⋊C411yescharacteristic
78C2×C4⋊C4C4⋊C411yescharacteristic
79C2×C4⋊C4C4⋊C411yescharacteristic
80C2×C42—21no—
81C2×C42—21no—
82C2×C42C22×C411yescharacteristic
83C2×C42C2×D411yescharacteristic
84C2×C42C2×D411yescharacteristic
85C2×C42C2×D411yescharacteristic
86C2×C42C2×Q811yescharacteristic
87C2×C42#4111yescharacteristic
88C2×C42#4111yescharacteristic
89C4⋊C8—24no—
90C4⋊C8—22no—
91C4⋊C8—22no—
92C4⋊C8—42no—
93C4⋊C8—42no—
94C4⋊C8—42no—
95C4⋊C8—42no—
96C4⋊C8—42no—
97#97—41no—
98#97—41no—
99C4×C8—22no—
100C4×C8—22no—
101C4×C8—22no—
102C4×C8—21no—
103C4×C8—21no—
104C4×C8—41no—
105C4×C8—82no—
106C4×C8SD1612yes—
107C4×C8Q1612yes—
108#95—21no—
109#95—21no—
110#95—21no—
111#95—21no—
112#95—21no—
113#95—21no—
114#95—21no—
115C2×Q8—22no—
116C2×Q8—41no—
117C2×Q8—41no—
118C22×C4—21no—
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4C2×C22⋊C411yescharacteristic
123C22×C4C2×C4⋊C411yescharacteristic
124C22×C4#11711yescharacteristic
125C22×C4#12111yescharacteristic
126C22×C4#12111yescharacteristic
127C22×C4C22⋊Q811yescharacteristic
128C22×C4C22⋊Q811yescharacteristic
129C2×C8—21no—
130C2×C8—21no—
131C2×C8—21no—
132C2×C8—21no—
133C2×C8—21no—
134C2×C8—21no—
135C2×C8—21no—
136C2×C8—21no—
137C2×C8—21no—
138C2×C8—44no—
139C2×C8—41no—
140C2×C8—41no—
141C2×C8—41no—
142C2×C8—41no—
143C2×C8—41no—
144C2×C8—41no—
145C2×C8—82no—
146C2×C8—82no—
147C2×C8—82no—
148C2×C8—82no—
149C2×C8—82no—
150C2×C8#10312yes—
151C2×C8#10312yes—
152C2×C8#10412yes—
153C4⋊C4—22no—
154C4⋊C4—41no—
155C4⋊C4—41no—
156C4⋊C4—42no—
157C4⋊C4—42no—
158C4⋊C4—41no—
159C4⋊C4—41no—
160C42—21no—
161C42—21no—
162C42—21no—
163C42—21no—
164C42—41no—
165C42—41no—
166C42—41no—
167C42—41no—
168C42—41no—
169C42—41no—
170C42—41no—
171C42#13112yes—
172C42#13112yes—
173C42C2×SD1611yescharacteristic
174C42C2×Q1611yescharacteristic
175C42AGL(1,ℤ/8)11yescharacteristic
176C42#13711yescharacteristic
177C23#38011yescharacteristic
178Q8—82no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4#40811yescharacteristic
205C2×C4#40811yescharacteristic
206C2×C4C2×#10311yescharacteristic
207C2×C4#41711yescharacteristic
208C2×C4C2×#10411yescharacteristic
209C2×C4#42111yescharacteristic
210C2×C4#42412yes—
211C2×C4#43312yes—
212C2×C4#46011yescharacteristic
213C2×C4#46211yescharacteristic
214C2×C4#47211yescharacteristic
215C2×C4#47311yescharacteristic
216C2×C4#47511yescharacteristic
217C2×C4#47711yescharacteristic
218C8—22no—
219C8—22no—
220C8—22no—
221C8—41no—
222C8—84no—
223C8—81no—
224C8—81no—
225C8—162no—
226C22#179811yescharacteristic
227C22#180311yescharacteristic
228C22#180611yescharacteristic
229C22#181211yescharacteristic
230C22#181311yescharacteristic
231C22#181411yescharacteristic
232C22#181611yescharacteristic
233C4—22no—
234C4—22no—
235C4—21no—
236C4—21no—
237C4—21no—
238C4—21no—
239C4—21no—
240C4—21no—
241C4—41no—
242C4—81no—
243C4—81no—
244C4—81no—
245C4—81no—
246C4#148111yescharacteristic
247C4#148311yescharacteristic
248C4#149311yescharacteristic
249C4#149511yescharacteristic
250C2#856811yescharacteristic
251C2#860311yescharacteristic
252C2#860611yescharacteristic
253C2#867711yescharacteristic
254C2#868511yescharacteristic
255C2#869011yescharacteristic
256C2#869711yescharacteristic
subgroupG/H
center, socle, prime radicalC23#380
commutatorC2×C42C22×C4
Frattini, cosocleC43C23
LCSG ▷ C2×C42 ▷ C22 ▷ 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
#213412
#88072
C4×#3352
#75772
#70222