Finite Group Databasebeta

← #124702

Subgroups — #124702

Order 512.

388 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 94 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#21151C211yesmaximal, characteristic
2#12844C211yesmaximal, characteristic
3#7875C211yesmaximal, characteristic
4#7121C212yesmaximal, complemented
5#7120C212yesmaximal, complemented
6C2×#432—21no—
7C2×#427C411yescharacteristic
8C2×#427C411yescharacteristic
9C22×C4×C8C2211yescharacteristic
10#2296—21no—
11C2×C22⋊C16—21no—
12#1942C411yescharacteristic
13#1942C411yescharacteristic
14#1942C2211yescharacteristic
15#1942C2211yescharacteristic
16#1941—21no—
17#1941—21no—
18#1735—21no—
19#1374—21no—
20#1374C2212yes—
21#1277—21no—
22#1277C2212yes—
23C23×C8C2×C411yescharacteristic
24C23×C8D411yescharacteristic
25C2×#117—21no—
26C2×#117—21no—
27C22×C42D411yescharacteristic
28C22×C16—22no—
29#432—24no—
30#432—42no—
31#427—22no—
32#427—22no—
33#427—42no—
34C2×#107C2×C411yescharacteristic
35C2×#107C2×C411yescharacteristic
36C2×#107C2×C411yescharacteristic
37C2×#107C2×C411yescharacteristic
38C2×#106—21no—
39C2×#106—21no—
40C2×C4⋊C8—22no—
41C2×C22⋊C8—41no—
42C2×#97—41no—
43C2×C4×C8—42no—
44C2×C4×C8C2×C411yescharacteristic
45C2×C4×C8D411yescharacteristic
46C2×C4×C8D411yescharacteristic
47C2×C4×C8C2311yescharacteristic
48#389—21no—
49#389—21no—
50#389—21no—
51#389—21no—
52#385—41no—
53#385—41no—
54C4×C4⋊C4—41no—
55C4×C4⋊C4—41no—
56C22⋊C16—42no—
57#340—21no—
58#340—21no—
59#340—21no—
60#340—21no—
61C23×C4C22⋊C411yescharacteristic
62C23×C4D811yescharacteristic
63C23×C4SD1611yescharacteristic
64C22×C8—22no—
65C22×C8—22no—
66C22×C8—21no—
67C22×C8—21no—
68C22×C8—22no—
69C22×C8—22no—
70C22×C8—41no—
71C22×C8—42no—
72C22×C8C22⋊C411yescharacteristic
73C22×C8C22⋊C411yescharacteristic
74C22×C8C22×C411yescharacteristic
75C22×C8C2×D411yescharacteristic
76#117—24no—
77#117—42no—
78#117—42no—
79#117—42no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—41no—
85C2×C4⋊C4—41no—
86C2×C4⋊C4—41no—
87C2×C4⋊C4—41no—
88C2×C22⋊C4—41no—
89C2×C22⋊C4—41no—
90C2×C42—22no—
91C2×C42—22no—
92C2×C42—41no—
93C2×C42—41no—
94C2×C42C22⋊C411yescharacteristic
95C2×C42D811yescharacteristic
96C2×C42SD1611yescharacteristic
97C2×C42C2×D411yescharacteristic
98C2×C16—42no—
99C2×C16—42no—
100C2×C16—42no—
101#107—21no—
102#107—21no—
103#107—41no—
104#107—41no—
105#107M1612yes—
106#107M1612yes—
107#106—24no—
108#106—42no—
109C4⋊C8—24no—
110C4⋊C8—42no—
111C22⋊C8—44no—
112#97—44no—
113C4×C8—41no—
114C4×C8—41no—
115C4×C8—41no—
116C4×C8—41no—
117#95—41no—
118#95—41no—
119#95—41no—
120#95—41no—
121C24#10211yescharacteristic
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—22no—
127C22×C4—22no—
128C22×C4—22no—
129C22×C4—41no—
130C22×C4—41no—
131C22×C4—41no—
132C22×C4—41no—
133C22×C4—41no—
134C22×C4—41no—
135C22×C4—41no—
136C22×C4#9912yes—
137C22×C4#10211yescharacteristic
138C22×C4#10211yescharacteristic
139C22×C4#10211yescharacteristic
140C22×C4#10412yes—
141C22×C4C2×C22⋊C411yescharacteristic
142C22×C4C2×D811yescharacteristic
143C22×C4C2×SD1611yescharacteristic
144C2×C8—24no—
145C2×C8—24no—
146C2×C8—21no—
147C2×C8—21no—
148C2×C8—21no—
149C2×C8—21no—
150C2×C8—21no—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—21no—
154C2×C8—21no—
155C2×C8—44no—
156C2×C8—44no—
157C2×C8—82no—
158C2×C8—82no—
159C2×C8#10112yes—
160C2×C8C2×M1611yescharacteristic
161C2×C8C2×M1611yescharacteristic
162C2×C8#13712yes—
163C4⋊C4—22no—
164C4⋊C4—22no—
165C4⋊C4—41no—
166C4⋊C4—41no—
167C4⋊C4—41no—
168C4⋊C4—41no—
169C4⋊C4—41no—
170C4⋊C4—41no—
171C4⋊C4—82no—
172C4⋊C4—82no—
173C4⋊C4—82no—
174C4⋊C4—82no—
175C22⋊C4—44no—
176C22⋊C4—82no—
177C42—22no—
178C42—22no—
179C42—21no—
180C42—21no—
181C42—21no—
182C42—21no—
183C42—44no—
184C42—82no—
185C16—82no—
186C23—21no—
187C23—21no—
188C23—21no—
189C23—21no—
190C23—21no—
191C23—21no—
192C23#32712yes—
193C23C2×#10211yescharacteristic
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—44no—
212C2×C4—44no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—81no—
216C2×C4—81no—
217C2×C4—81no—
218C2×C4—81no—
219C2×C4—81no—
220C2×C4—81no—
221C2×C4—81no—
222C2×C4—81no—
223C2×C4—81no—
224C2×C4—81no—
225C2×C4#32712yes—
226C2×C4#40711yescharacteristic
227C2×C4C2×#9911yescharacteristic
228C2×C4C2×#10111yescharacteristic
229C2×C4#41911yescharacteristic
230C2×C4C2×#10411yescharacteristic
231C2×C4#42111yescharacteristic
232C2×C4#50812yes—
233C8—22no—
234C8—21no—
235C8—21no—
236C8—22no—
237C8—41no—
238C8—84no—
239C22—22no—
240C22—22no—
241C22—22no—
242C22—41no—
243C22—41no—
244C22—41no—
245C22—41no—
246C22#148611yescharacteristic
247C22C2×#32711yescharacteristic
248C22#154611yescharacteristic
249C22#213811yescharacteristic
250C22#214311yescharacteristic
251C22#214511yescharacteristic
252C22#214811yescharacteristic
253C4—22no—
254C4—22no—
255C4—41no—
256C4—41no—
257C4—41no—
258C4—82no—
259C4—82no—
260C4—161no—
261C4—161no—
262C4#146311yescharacteristic
263C4#149111yescharacteristic
264C4#151111yescharacteristic
265C4#152811yescharacteristic
266C2—42no—
267C2#761011yescharacteristic
268C2#1165911yescharacteristic
269C2#1166511yescharacteristic
270C2#1170311yescharacteristic
271C2#1171011yescharacteristic
272C2#1197011yescharacteristic
273C2#1197311yescharacteristic
subgroupG/H
center, socleC23C2×#102
commutatorC22×C8C22×C4
Frattini, cosocleC2×C4×C8C23
prime radicalC24#102
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G
LECSG ▷ C2×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7121C2semidirect
5. #7120C2semidirect

Maximal subgroups

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

subgroupindex
#211512
#128442
#78752
#71212
#71202