Finite Group Databasebeta

← #125083

Subgroups — #125083

Order 512.

368 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 90 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#33130C211yesmaximal, characteristic
2#12844C211yesmaximal, characteristic
3#7872C211yesmaximal, characteristic
4#7121C212yesmaximal, complemented
5#7113C212yesmaximal, complemented
6C2×#467C411yescharacteristic
7C2×#431—21no—
8C22×C4×C8C2211yescharacteristic
9#2855—21no—
10C2×C22⋊C16—21no—
11#1969C411yescharacteristic
12#1969C2211yescharacteristic
13#1942C411yescharacteristic
14#1942C2211yescharacteristic
15#1933—21no—
16#1933—21no—
17#1892—21no—
18#1808C411yescharacteristic
19#1374—21no—
20#1374C2212yes—
21#1277—21no—
22#1277C2212yes—
23C22×Q16C2×C411yescharacteristic
24C23×C8C2×C411yescharacteristic
25C23×C8D411yescharacteristic
26C2×C22⋊Q8—21no—
27C22×C42D411yescharacteristic
28C22×C16—22no—
29#467—42no—
30#431—24no—
31#431—42no—
32C2×#107C2×C411yescharacteristic
33C2×#107C2×C411yescharacteristic
34C2×#107C2×C411yescharacteristic
35C2×C4⋊C8—22no—
36C2×#103—21no—
37C2×#103—21no—
38C2×#103—21no—
39C2×C22⋊C8—41no—
40C2×C4×C8—42no—
41C2×C4×C8—41no—
42C2×C4×C8C2×C411yescharacteristic
43C2×C4×C8D411yescharacteristic
44C2×C4×C8D411yescharacteristic
45C2×C4×C8C2311yescharacteristic
46#391—21no—
47#391—21no—
48#389—21no—
49#389—21no—
50#387—41no—
51#387—41no—
52#385—41no—
53#381—41no—
54#380—21no—
55C22⋊C16—42no—
56#340—21no—
57#340—21no—
58#340—21no—
59C22×Q8—21no—
60C23×C4C22⋊C411yescharacteristic
61C23×C4D811yescharacteristic
62C23×C4SD1611yescharacteristic
63C2×Q16—22no—
64C2×Q16—41no—
65C2×Q16—41no—
66C22×C8—22no—
67C22×C8—22no—
68C22×C8—22no—
69C22×C8—22no—
70C22×C8—41no—
71C22×C8—41no—
72C22×C8—42no—
73C22×C8C22⋊C411yescharacteristic
74C22×C8C22⋊C411yescharacteristic
75C22×C8C22×C411yescharacteristic
76C22×C8C2×D411yescharacteristic
77C22⋊Q8—42no—
78C22⋊Q8—42no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—41no—
83C2×C4⋊C4—41no—
84C2×C4⋊C4—41no—
85C2×C22⋊C4—41no—
86C2×C22⋊C4—41no—
87C2×C42—22no—
88C2×C42—22no—
89C2×C42C22⋊C411yescharacteristic
90C2×C42D811yescharacteristic
91C2×C42SD1611yescharacteristic
92C2×C42C2×D411yescharacteristic
93C2×C16—42no—
94C2×C16—42no—
95C2×C16—42no—
96#107—22no—
97#107—41no—
98#107—41no—
99C4⋊C8—24no—
100C4⋊C8—42no—
101#103—42no—
102#103—42no—
103#103—42no—
104C22⋊C8—44no—
105C4×C8—41no—
106C4×C8—41no—
107C4×C8—41no—
108C4×C8—41no—
109C4×C8—44no—
110#95—41no—
111#95—41no—
112#95—41no—
113#95—41no—
114#95—41no—
115#95—41no—
116#95—41no—
117C24#10211yescharacteristic
118C2×Q8—41no—
119C2×Q8—41no—
120C2×Q8—41no—
121C2×Q8—41no—
122C2×Q8—41no—
123C2×Q8—41no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—22no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—22no—
132C22×C4—41no—
133C22×C4—41no—
134C22×C4—41no—
135C22×C4—41no—
136C22×C4—41no—
137C22×C4—41no—
138C22×C4#9912yes—
139C22×C4#10211yescharacteristic
140C22×C4#10211yescharacteristic
141C22×C4#10211yescharacteristic
142C22×C4#10412yes—
143C22×C4C2×C22⋊C411yescharacteristic
144C22×C4C2×D811yescharacteristic
145C22×C4C2×SD1611yescharacteristic
146Q16—82no—
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—21no—
156C2×C8—21no—
157C2×C8—44no—
158C2×C8—44no—
159C2×C8—44no—
160C2×C8—82no—
161C2×C8—82no—
162C2×C8#13112yes—
163C2×C8#13512yes—
164C4⋊C4—41no—
165C4⋊C4—41no—
166C4⋊C4—41no—
167C4⋊C4—41no—
168C4⋊C4—41no—
169C4⋊C4—41no—
170C4⋊C4—82no—
171C4⋊C4—82no—
172C4⋊C4—82no—
173C22⋊C4—82no—
174C22⋊C4—82no—
175C42—22no—
176C42—22no—
177C42—21no—
178C42—21no—
179C42—21no—
180C42—21no—
181C16—82no—
182C23—21no—
183C23—21no—
184C23—21no—
185C23—21no—
186C23—21no—
187C23—21no—
188C23#32712yes—
189C23C2×#10211yescharacteristic
190Q8—81no—
191Q8—81no—
192Q8—81no—
193Q8—81no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—44no—
211C2×C4—44no—
212C2×C4—81no—
213C2×C4—81no—
214C2×C4—81no—
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#32712yes—
225C2×C4#35912yes—
226C2×C4#40811yescharacteristic
227C2×C4C2×#9911yescharacteristic
228C2×C4#41311yescharacteristic
229C2×C4#41611yescharacteristic
230C2×C4C2×#10411yescharacteristic
231C2×C4#42111yescharacteristic
232C2×C4#51012yes—
233C8—22no—
234C8—22no—
235C8—41no—
236C8—41no—
237C8—84no—
238C22—22no—
239C22—22no—
240C22—22no—
241C22—41no—
242C22—41no—
243C22—41no—
244C22—41no—
245C22#148011yescharacteristic
246C22C2×#32711yescharacteristic
247C22#154011yescharacteristic
248C22#214111yescharacteristic
249C22C2×#35911yescharacteristic
250C22#214511yescharacteristic
251C22#214811yescharacteristic
252C4—22no—
253C4—22no—
254C4—22no—
255C4—41no—
256C4—41no—
257C4—161no—
258C4—161no—
259C4—161no—
260C4—161no—
261C4#146811yescharacteristic
262C4#149411yescharacteristic
263C4#151611yescharacteristic
264C4#153211yescharacteristic
265C2—42no—
266C2#765411yescharacteristic
267C2#1165011yescharacteristic
268C2#1165611yescharacteristic
269C2#1170611yescharacteristic
270C2#1170911yescharacteristic
271C2#1194311yescharacteristic
272C2#1195311yescharacteristic
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. #7113C2semidirect

Maximal subgroups

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

subgroupindex
#331302
#128442
#78722
#71212
#71132