Finite Group Databasebeta

← #98916

Subgroups — #98916

Order 512.

357 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 357 conjugacy classes of subgroups — too many to draw. Only its 102 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#21266C211yesmaximal, characteristic
2#8785C211yesmaximal, characteristic, complemented
3C4×#334C211yesmaximal, characteristic, complemented
4#7574C212yesmaximal
5#7022C212yesmaximal, complemented
6C2×#433—21no—
7#2293C2211yescharacteristic
8#1954C411yescharacteristic
9#1954C411yescharacteristic
10#1939C411yescharacteristic
11#1939C411yescharacteristic
12#1923—21no—
13#1923—21no—
14#1831C2212yes—
15#1767C2211yescharacteristic
16C4×C4⋊C8C2212yes—
17#1750—21no—
18C42×C8C2211yescharacteristic
19C2×#334—21no—
20#1277—21no—
21#1277—21no—
22#1277—21no—
23#1271—22no—
24#1270—22no—
25C2×C4×D4C2×C411yescharacteristic
26C2×#117—21no—
27#433—44no—
28C2×C4⋊C8—22no—
29C2×C4⋊C8—21no—
30C2×C4⋊C8—21no—
31C2×C4⋊C8—22no—
32C2×C4⋊C8C2×C411yescharacteristic
33C2×C4⋊C8C2×C411yescharacteristic
34C2×C22⋊C8—21no—
35C2×C22⋊C8—21no—
36C2×C22⋊C8—21no—
37C2×C22⋊C8—21no—
38C2×#97—21no—
39C2×C4×C8—22no—
40C2×C4×C8C2×C411yescharacteristic
41C2×C4×C8C2×C411yescharacteristic
42C2×C4×C8D411yescharacteristic
43C2×C4×C8D411yescharacteristic
44C2×C4×C8D411yescharacteristic
45C2×C4×C8D411yescharacteristic
46#390D411yescharacteristic
47#390Q811yescharacteristic
48#389D411yescharacteristic
49#389Q811yescharacteristic
50#388—21no—
51#388—21no—
52C4×C4⋊C4C2×C411yescharacteristic
53C4×C22⋊C4—21no—
54C43C2311yescharacteristic
55#336—21no—
56#336—21no—
57#336—21no—
58#336—22no—
59#336—22no—
60#336—22no—
61#334—24no—
62C22×D4C4⋊C411yescharacteristic
63C23×C4—21no—
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×D4—22no—
73C4×D4—22no—
74#117—42no—
75#117—42no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4C4⋊C411yescharacteristic
79C2×C4⋊C4C4⋊C411yescharacteristic
80C2×C4⋊C4C4⋊C411yescharacteristic
81C2×C22⋊C4—21no—
82C2×C22⋊C4—21no—
83C2×C22⋊C4—21no—
84C2×C22⋊C4—21no—
85C2×C22⋊C4—21no—
86C2×C42—21no—
87C2×C42C22×C411yescharacteristic
88C2×C42C2×D411yescharacteristic
89C2×C42C2×D411yescharacteristic
90C2×C42C2×D411yescharacteristic
91C2×C42C2×Q811yescharacteristic
92C2×C42#4111yescharacteristic
93C2×C42#4111yescharacteristic
94C4⋊C8—24no—
95C4⋊C8—22no—
96C4⋊C8—22no—
97C4⋊C8—42no—
98C4⋊C8—42no—
99C4⋊C8—42no—
100C4⋊C8—42no—
101C4⋊C8—42no—
102C22⋊C8—42no—
103C22⋊C8—42no—
104C22⋊C8—42no—
105C22⋊C8—42no—
106#97—42no—
107C4×C8—22no—
108C4×C8—22no—
109C4×C8—22no—
110C4×C8—22no—
111C4×C8—22no—
112C4×C8—41no—
113C4×C8—82no—
114#95—21no—
115#95—21no—
116#95—21no—
117C24—21no—
118C2×D4—22no—
119C2×D4—41no—
120C2×D4—41no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—41no—
125C22×C4—41no—
126C22×C4—41no—
127C22×C4—41no—
128C22×C4—41no—
129C22×C4—41no—
130C22×C4C2×C22⋊C411yescharacteristic
131C22×C4C2×C4⋊C411yescharacteristic
132C22×C4#11711yescharacteristic
133C22×C4#12111yescharacteristic
134C22×C4#12111yescharacteristic
135C22×C4C22⋊Q811yescharacteristic
136C22×C4C22⋊Q811yescharacteristic
137C2×C8—21no—
138C2×C8—21no—
139C2×C8—21no—
140C2×C8—21no—
141C2×C8—21no—
142C2×C8—21no—
143C2×C8—21no—
144C2×C8—21no—
145C2×C8—21no—
146C2×C8—21no—
147C2×C8—21no—
148C2×C8—44no—
149C2×C8—41no—
150C2×C8—41no—
151C2×C8—41no—
152C2×C8—41no—
153C2×C8—41no—
154C2×C8—41no—
155C2×C8—82no—
156C2×C8—82no—
157C2×C8—82no—
158C2×C8—82no—
159C2×C8—82no—
160C2×C8#10412yes—
161C4⋊C4—22no—
162C4⋊C4—41no—
163C4⋊C4—41no—
164C4⋊C4—42no—
165C4⋊C4—42no—
166C22⋊C4—42no—
167C22⋊C4—42no—
168C22⋊C4—42no—
169C22⋊C4—42no—
170C22⋊C4—42no—
171C42—21no—
172C42—21no—
173C42—22no—
174C42—41no—
175C42—41no—
176C42—41no—
177C42—41no—
178C42—41no—
179C42#13112yes—
180C42#13112yes—
181C42#13512yes—
182C42AGL(1,ℤ/8)12yes—
183C23—41no—
184C23—41no—
185C23—41no—
186C23—41no—
187C23—41no—
188C23—41no—
189C23—41no—
190C23#38011yescharacteristic
191D4—82no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—81no—
215C2×C4—81no—
216C2×C4—81no—
217C2×C4—81no—
218C2×C4#40811yescharacteristic
219C2×C4#40811yescharacteristic
220C2×C4#41611yescharacteristic
221C2×C4#41811yescharacteristic
222C2×C4C2×#10411yescharacteristic
223C2×C4#42111yescharacteristic
224C2×C4#42412yes—
225C2×C4#43312yes—
226C2×C4#46312yes—
227C2×C4#47112yes—
228C2×C4#47812yes—
229C8—22no—
230C8—41no—
231C8—41no—
232C8—41no—
233C8—84no—
234C8—81no—
235C8—81no—
236C8—162no—
237C22—81no—
238C22—81no—
239C22—81no—
240C22—81no—
241C22—81no—
242C22—81no—
243C22—81no—
244C22#179811yescharacteristic
245C22#180411yescharacteristic
246C22#180611yescharacteristic
247C22#181211yescharacteristic
248C22#181311yescharacteristic
249C22#181411yescharacteristic
250C22#181511yescharacteristic
251C4—22no—
252C4—22no—
253C4—22no—
254C4—22no—
255C4—22no—
256C4—41no—
257C4—81no—
258C4—81no—
259C4—81no—
260C4#148211yescharacteristic
261C4#148411yescharacteristic
262C4#149411yescharacteristic
263C4#149611yescharacteristic
264C2—161no—
265C2#856911yescharacteristic
266C2#860311yescharacteristic
267C2#860711yescharacteristic
268C2#867811yescharacteristic
269C2#868611yescharacteristic
270C2#869311yescharacteristic
271C2#869411yescharacteristic
subgroupG/H
center, socleC23#380
commutatorC2×C42C22×C4
Frattini, cosocleC43C23
prime radicalC22×D4C4⋊C4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C43 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #8785C2semidirect
3. C4×#334C2semidirect
5. #7022C2semidirect

Maximal subgroups

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

subgroupindex
#212662
#87852
C4×#3342
#75742
#70222