Finite Group Databasebeta

← #124693

Subgroups — #124693

Order 512.

371 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 371 conjugacy classes of subgroups — too many to draw. Only its 84 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#21154C211yesmaximal, characteristic
2#12846C211yesmaximal, characteristic
3#7861C211yesmaximal, characteristic
4#7127C212yesmaximal, complemented
5#7126C212yesmaximal, complemented
6#2920—21no—
7C2×#427C411yescharacteristic
8C2×#427C411yescharacteristic
9C22×#97C2211yescharacteristic
10#2296—21no—
11#2112—21no—
12#1944C411yescharacteristic
13#1944C411yescharacteristic
14#1944C2211yescharacteristic
15#1944C2211yescharacteristic
16#1943—21no—
17#1943—21no—
18#1735—21no—
19#1377—21no—
20#1377C2212yes—
21#1275—21no—
22#1275C2212yes—
23C23×C8C2×C411yescharacteristic
24C23×C8D411yescharacteristic
25C2×#117—21no—
26C2×#117—21no—
27C22×C42D411yescharacteristic
28C2×M32—22no—
29#432—41no—
30#432—41no—
31#431—22no—
32#431—22no—
33#427—22no—
34#427—22no—
35#427—41no—
36#427—41no—
37C2×#107—21no—
38C2×#107C2×C411yescharacteristic
39C2×#107C2×C411yescharacteristic
40C2×#106—21no—
41C2×#106C2×C411yescharacteristic
42C2×#106C2×C411yescharacteristic
43#423—41no—
44#407—41no—
45C4×M16—22no—
46C2×#97—41no—
47C2×#97—41no—
48C2×#97C2×C411yescharacteristic
49C2×#97D411yescharacteristic
50C2×#97D411yescharacteristic
51C2×#97C2311yescharacteristic
52#389—21no—
53#389—21no—
54#389—21no—
55#389—21no—
56#385—41no—
57#385—41no—
58C4×C4⋊C4—41no—
59C4×C4⋊C4—41no—
60C22⋊C16—41no—
61C22⋊C16—41no—
62#340—21no—
63#340—21no—
64#340—21no—
65#340—21no—
66C23×C4C22⋊C411yescharacteristic
67C23×C4D811yescharacteristic
68C23×C4SD1611yescharacteristic
69C2×M16—42no—
70C22×C8—21no—
71C22×C8—21no—
72C22×C8—22no—
73C22×C8—21no—
74C22×C8—21no—
75C22×C8—21no—
76C22×C8—21no—
77C22×C8—22no—
78C22×C8—41no—
79C22×C8C22⋊C411yescharacteristic
80C22×C8C22⋊C411yescharacteristic
81C22×C8C22×C411yescharacteristic
82C22×C8C2×D411yescharacteristic
83#117—24no—
84#117—42no—
85#117—41no—
86#117—41no—
87#117—41no—
88#117—41no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—41no—
94C2×C4⋊C4—41no—
95C2×C4⋊C4—41no—
96C2×C4⋊C4—41no—
97C2×C22⋊C4—41no—
98C2×C22⋊C4—41no—
99C2×C42—21no—
100C2×C42—21no—
101C2×C42—21no—
102C2×C42—21no—
103C2×C42—41no—
104C2×C42—41no—
105C2×C42C22⋊C411yescharacteristic
106C2×C42D811yescharacteristic
107C2×C42SD1611yescharacteristic
108C2×C42C2×D411yescharacteristic
109M32—42no—
110M32—42no—
111C2×C16—42no—
112#107—24no—
113#107—41no—
114#107—41no—
115#106—21no—
116#106—21no—
117#106—42no—
118#106M1612yescomplemented
119#106M1612yescomplemented
120C4⋊C8—42no—
121C4⋊C8—42no—
122C22⋊C8—42no—
123C22⋊C8—42no—
124#97—41no—
125#97—41no—
126#97—41no—
127#97—41no—
128#97—42no—
129C4×C8—24no—
130#95—41no—
131#95—41no—
132#95—41no—
133#95—41no—
134C24#10211yescharacteristic
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—41no—
144C22×C4—41no—
145C22×C4—41no—
146C22×C4—41no—
147C22×C4—81no—
148C22×C4—81no—
149C22×C4#9911yescharacteristic
150C22×C4#9911yescharacteristic
151C22×C4#10211yescharacteristic
152C22×C4#10211yescharacteristic
153C22×C4#10211yescharacteristic
154C22×C4#10411yescharacteristic
155C22×C4#10411yescharacteristic
156C22×C4C2×C22⋊C411yescharacteristic
157C22×C4C2×D811yescharacteristic
158C22×C4C2×SD1611yescharacteristic
159M16—84no—
160C2×C8—22no—
161C2×C8—22no—
162C2×C8—22no—
163C2×C8—22no—
164C2×C8—21no—
165C2×C8—21no—
166C2×C8—21no—
167C2×C8—21no—
168C2×C8—21no—
169C2×C8—21no—
170C2×C8—21no—
171C2×C8—21no—
172C2×C8—21no—
173C2×C8—42no—
174C2×C8—42no—
175C2×C8—44no—
176C2×C8#10112yes—
177C2×C8C2×M1611yescharacteristic
178C2×C8C2×M1611yescharacteristic
179C2×C8#13712yes—
180C4⋊C4—22no—
181C4⋊C4—22no—
182C4⋊C4—41no—
183C4⋊C4—41no—
184C4⋊C4—41no—
185C4⋊C4—41no—
186C4⋊C4—41no—
187C4⋊C4—41no—
188C4⋊C4—82no—
189C4⋊C4—82no—
190C4⋊C4—82no—
191C4⋊C4—82no—
192C22⋊C4—44no—
193C22⋊C4—81no—
194C22⋊C4—81no—
195C42—21no—
196C42—21no—
197C42—21no—
198C42—21no—
199C42—21no—
200C42—21no—
201C42—21no—
202C42—21no—
203C42—44no—
204C42—81no—
205C42—81no—
206C16—82no—
207C23—21no—
208C23—21no—
209C23—41no—
210C23—41no—
211C23#32711yescharacteristic
212C23#32711yescharacteristic
213C23C2×#10211yescharacteristic
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—81no—
237C2×C4—81no—
238C2×C4—81no—
239C2×C4—81no—
240C2×C4—81no—
241C2×C4—81no—
242C2×C4—81no—
243C2×C4—81no—
244C2×C4—81no—
245C2×C4—81no—
246C2×C4—81no—
247C2×C4—81no—
248C2×C4#32711yescharacteristic
249C2×C4#32711yescharacteristic
250C2×C4#40711yescharacteristic
251C2×C4C2×#9911yescharacteristic
252C2×C4C2×#10111yescharacteristic
253C2×C4#41911yescharacteristic
254C2×C4C2×#10411yescharacteristic
255C2×C4#42111yescharacteristic
256C8—22no—
257C8—21no—
258C8—21no—
259C8—22no—
260C8—41no—
261C8—84no—
262C22—21no—
263C22—21no—
264C22—21no—
265C22—21no—
266C22—21no—
267C22—21no—
268C22—21no—
269C22—21no—
270C22—42no—
271C22—42no—
272C22#148611yescharacteristic
273C22C2×#32711yescharacteristic
274C22#154611yescharacteristic
275C4—21no—
276C4—21no—
277C4—41no—
278C4—41no—
279C4—41no—
280C4—41no—
281C4—82no—
282C4—82no—
283C4—161no—
284C4—161no—
285C4#146311yescharacteristic
286C4#149111yescharacteristic
287C4#151111yescharacteristic
288C4#152811yescharacteristic
289C2—21no—
290C2—21no—
291C2—41no—
292C2—41no—
293C2#761011yescharacteristic
294C2#1168511yescharacteristic
295C2#1168811yescharacteristic
subgroupG/H
center, socleC22C2×#327
commutatorC22×C8C22×C4
Frattini, cosocleC2×#97C23
prime radicalC24#102
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×#97 ◁ G
LECSG ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7127C2semidirect
5. #7126C2semidirect
118. #106M16semidirect
119. #106M16semidirect

Maximal subgroups

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

subgroupindex
#211542
#128462
#78612
#71272
#71262