Finite Group Databasebeta

← #103154

Subgroups — #103154

Order 512.

592 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 592 conjugacy classes of subgroups — too many to draw. Only its 130 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#33289C211yesmaximal, characteristic, complemented
2#29139C211yesmaximal, characteristic
3#28438C211yesmaximal, characteristic
4#9898C211yesmaximal, characteristic, complemented
5#9691C211yesmaximal, characteristic, complemented
6#9638C211yesmaximal, characteristic
7#7015C211yesmaximal, characteristic, complemented
8C2×#492—21no—
9C2×#461—22no—
10#2868C2211yescharacteristic
11#2868C2211yescharacteristic
12C4×#127C2211yescharacteristic
13#2023—22no—
14#2013—22no—
15#1990—22no—
16#1990—22no—
17#1957—21no—
18#1956—21no—
19#1842C2211yescharacteristic
20#1834—22no—
21#1834—21no—
22#1802—22no—
23C4×C4⋊C8C2211yescharacteristic
24C4×C4⋊C8C2211yescharacteristic
25C42×C8C2211yescharacteristic
26C2×#333—22no—
27C2×D4⋊C8—22no—
28#1296—22no—
29C22×SD16—24no—
30C2×C4⋊Q8—21no—
31C2×C4⋊Q8—21no—
32C2×C4⋊Q8—21no—
33C2×C4⋊Q8—21no—
34C2×C4⋊Q8D412yes—
35C2×C4⋊Q8D412yes—
36C2×#127D411yescharacteristic
37C2×C4×D4—21no—
38C2×C4×D4—21no—
39C2×C4×D4—21no—
40#492—42no—
41#492—42no—
42#461—44no—
43#461—44no—
44C2×C4⋊C8—21no—
45C2×C4⋊C8—21no—
46C2×C4⋊C8—21no—
47C2×C4⋊C8D412yes—
48C2×C4⋊C8D412yes—
49C2×#103—24no—
50C2×#102—24no—
51C2×C4×C8—21no—
52C2×C4×C8—21no—
53C2×C4×C8—21no—
54C2×C4×C8—21no—
55C2×C4×C8D412yes—
56#390—21no—
57#390—21no—
58#390—21no—
59#384—21no—
60#384—21no—
61#384—21no—
62#384—21no—
63#384—21no—
64#384—21no—
65#384—21no—
66#384—21no—
67#384D411yescharacteristic
68C43C2311yescharacteristic
69#340—24no—
70#340—28no—
71#336—21no—
72#336—21no—
73#333—44no—
74D4⋊C8—44no—
75C22×Q8—22no—
76C22×Q8—22no—
77C22×D4—21no—
78C22×D4—21no—
79C22×D4—21no—
80C23×C4—41no—
81C2×SD16—48no—
82C2×SD16—48no—
83C2×SD16—84no—
84C22×C8—22no—
85C22×C8—22no—
86C22×C8—22no—
87C22×C8—41no—
88C4⋊Q8—44no—
89C4⋊Q8—44no—
90C4⋊Q8—42no—
91C4⋊Q8—42no—
92C4⋊Q8—41no—
93C4⋊Q8—41no—
94C4⋊Q8—42no—
95C4⋊Q8—42no—
96C4⋊Q8—41no—
97C4⋊Q8—41no—
98C4⋊Q8—82no—
99#127—21no—
100#127—21no—
101#127—21no—
102#127SD1612yes—
103C4×D4—42no—
104C4×D4—42no—
105C4×D4—42no—
106C4×D4—42no—
107C4×D4—42no—
108C4×D4—42no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—41no—
117C2×C4⋊C4—41no—
118C2×C4⋊C4—41no—
119C2×C4⋊C4—41no—
120C2×C4⋊C4—41no—
121C2×C4⋊C4—41no—
122C2×C4⋊C4—41no—
123C2×C4⋊C4—41no—
124C2×C22⋊C4—41no—
125C2×C22⋊C4—41no—
126C2×C22⋊C4—41no—
127C2×C42C2×D411yescharacteristic
128C2×C42C2×D411yescharacteristic
129C2×C42C2×D411yescharacteristic
130C2×C42C2×D411yescharacteristic
131C2×C42C2×D411yescharacteristic
132C2×C42C2×D411yescharacteristic
133C2×C42#4111yescharacteristic
134C4⋊C8—22no—
135C4⋊C8—22no—
136C4⋊C8—22no—
137C4⋊C8—42no—
138C4⋊C8—42no—
139C4⋊C8—42no—
140C4⋊C8D814yes—
141#103—48no—
142#102—84no—
143C4×C8—22no—
144C4×C8—42no—
145C4×C8—42no—
146C4×C8—41no—
147C4×C8—41no—
148C4×C8—41no—
149C4×C8—41no—
150C4×C8SD1614yescomplemented
151C24—41no—
152C2×Q8—44no—
153C2×Q8—42no—
154C2×Q8—42no—
155C2×Q8—44no—
156C2×Q8—44no—
157C2×Q8—82no—
158C2×Q8—82no—
159C2×D4—22no—
160C2×D4—22no—
161C2×D4—22no—
162C2×D4—42no—
163C2×D4—41no—
164C2×D4—42no—
165C2×D4—41no—
166C2×D4—42no—
167C2×D4—42no—
168C2×D4—42no—
169C2×D4—41no—
170C2×D4—42no—
171C22×C4—41no—
172C22×C4—42no—
173C22×C4—42no—
174C22×C4—81no—
175C22×C4—81no—
176C22×C4—81no—
177C22×C4—81no—
178C22×C4—81no—
179C22×C4—81no—
180C22×C4#12011yescharacteristic
181C22×C4#12011yescharacteristic
182C22×C4#12011yescharacteristic
183C22×C4#12111yescharacteristic
184C22×C4#12111yescharacteristic
185C22×C4#12111yescharacteristic
186C22×C4#12711yescharacteristic
187SD16—816no—
188C2×C8—24no—
189C2×C8—24no—
190C2×C8—42no—
191C2×C8—42no—
192C2×C8—42no—
193C2×C8—42no—
194C2×C8—42no—
195C2×C8—42no—
196C2×C8—42no—
197C2×C8—81no—
198C2×C8—81no—
199C2×C8—81no—
200C4⋊C4—44no—
201C4⋊C4—44no—
202C4⋊C4—42no—
203C4⋊C4—42no—
204C4⋊C4—44no—
205C4⋊C4—42no—
206C4⋊C4—82no—
207C4⋊C4—82no—
208C4⋊C4—82no—
209C4⋊C4—81no—
210C4⋊C4—81no—
211C4⋊C4—82no—
212C4⋊C4—81no—
213C4⋊C4—81no—
214C4⋊C4—82no—
215C4⋊C4—81no—
216C4⋊C4—81no—
217C4⋊C4—81no—
218C4⋊C4—81no—
219C22⋊C4—82no—
220C22⋊C4—82no—
221C22⋊C4—82no—
222C42—22no—
223C42—22no—
224C42—22no—
225C42C2×D812yes—
226C42C2×SD1611yescharacteristic
227C42C2×SD1612yes—
228C42#13511yescharacteristic
229C42#13512yes—
230C42AGL(1,ℤ/8)12yes—
231C42AGL(1,ℤ/8)11yescharacteristic
232C42#13711yescharacteristic
233C42#13712yes—
234C42#13712yes—
235C23—42no—
236C23—81no—
237C23—81no—
238C23—81no—
239C23—81no—
240C23—81no—
241C23—81no—
242C23#39211yescharacteristic
243Q8—84no—
244Q8—84no—
245Q8—164no—
246D4—44no—
247D4—44no—
248D4—44no—
249D4—82no—
250D4—82no—
251D4—82no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—82no—
259C2×C4—82no—
260C2×C4—82no—
261C2×C4—81no—
262C2×C4—82no—
263C2×C4—82no—
264C2×C4—81no—
265C2×C4—81no—
266C2×C4—82no—
267C2×C4—161no—
268C2×C4—161no—
269C2×C4—161no—
270C2×C4—161no—
271C2×C4#44712yes—
272C2×C4#44812yes—
273C2×C4#44811yescharacteristic
274C2×C4#44911yescharacteristic
275C2×C4#45011yescharacteristic
276C2×C4#45211yescharacteristic
277C2×C4#45212yes—
278C2×C4#45512yes—
279C2×C4#45512yes—
280C2×C4#45912yes—
281C2×C4#46112yes—
282C2×C4#46412yes—
283C2×C4#46511yescharacteristic
284C2×C4#46811yescharacteristic
285C2×C4#47112yes—
286C2×C4#49211yescharacteristic
287C2×C4#49311yescharacteristic
288C2×C4#49511yescharacteristic
289C2×C4#49611yescharacteristic
290C2×C4#49711yescharacteristic
291C2×C4#49711yescharacteristic
292C8—44no—
293C8—44no—
294C8—82no—
295C8—161no—
296C22—82no—
297C22—82no—
298C22—82no—
299C22—161no—
300C22—161no—
301C22—161no—
302C22—161no—
303C22#200011yescharacteristic
304C22#200411yescharacteristic
305C22#200411yescharacteristic
306C22#201111yescharacteristic
307C22#201311yescharacteristic
308C22#201511yescharacteristic
309C22#201911yescharacteristic
310C4—22no—
311C4—22no—
312C4—22no—
313C4—161no—
314C4—162no—
315C4—162no—
316C4#163912yes—
317C4#164112yes—
318C4#164312yes—
319C4#164612yes—
320C4#167112yes—
321C4#169112yes—
322C4#170611yescharacteristic
323C4#171411yescharacteristic
324C4#171911yescharacteristic
325C4#172211yescharacteristic
326C2—162no—
327C2#1006411yescharacteristic
328C2#1007011yescharacteristic
329C2#1016211yescharacteristic
330C2#1016811yescharacteristic
331C2#1027111yescharacteristic
332C2#1028711yescharacteristic
333C2#1030811yescharacteristic
subgroupG/H
center, socleC23#392
commutator, Frattini, cosocleC43C23
prime radicalC2×#127D4
LCSG ▷ C43 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C43 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33289C2semidirect
4. #9898C2semidirect
5. #9691C2semidirect
7. #7015C2semidirect
150. C4×C8SD16semidirect

Maximal subgroups

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

subgroupindex
#332892
#291392
#284382
#98982
#96912
#96382
#70152