Finite Group Databasebeta

← #103121

Subgroups — #103121

Order 512.

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

normal subgroups only

This group has 578 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#28509C211yesmaximal, characteristic
3#28508C211yesmaximal, characteristic
4#9896C211yesmaximal, characteristic, complemented
5#9692C211yesmaximal, characteristic, complemented
6#9638C211yesmaximal, characteristic
7#7015C211yesmaximal, characteristic, complemented
8C2×#476—22no—
9C2×#474—22no—
10#2868C2211yescharacteristic
11#2868C2211yescharacteristic
12C4×#127C2211yescharacteristic
13#2094—22no—
14#2093—22no—
15#1990—22no—
16#1990—22no—
17#1957—21no—
18#1957—21no—
19#1845—21no—
20#1844—21no—
21#1842C2211yescharacteristic
22#1834—22no—
23C4×C4⋊C8C2211yescharacteristic
24C4×C4⋊C8C2211yescharacteristic
25C42×C8C2211yescharacteristic
26C2×#332—22no—
27C2×D4⋊C8—22no—
28#1296—22no—
29C2×C4⋊Q8—21no—
30C2×C4⋊Q8—21no—
31C2×C4⋊Q8—21no—
32C2×C4⋊Q8—21no—
33C2×C4⋊Q8—21no—
34C2×C4⋊Q8—21no—
35C2×#127D411yescharacteristic
36C2×C4×D4—21no—
37C2×C4×D4—21no—
38C2×C4×D4—21no—
39#476—44no—
40#476—44no—
41#474—44no—
42#474—44no—
43C2×#107—24no—
44C2×#106—24no—
45C2×C4⋊C8—21no—
46C2×C4⋊C8—21no—
47C2×C4⋊C8—21no—
48C2×C4⋊C8D412yes—
49C2×C4⋊C8D412yes—
50C2×#102—24no—
51C2×#102—24no—
52C2×C4×C8—21no—
53C2×C4×C8—21no—
54C2×C4×C8—21no—
55C2×C4×C8—21no—
56C2×C4×C8D412yes—
57#390—21no—
58#390—21no—
59#390—21no—
60#384—21no—
61#384—21no—
62#384—21no—
63#384—21no—
64#384—21no—
65#384—21no—
66#384D411yescharacteristic
67#384D412yes—
68#384D412yes—
69C43C2311yescharacteristic
70#340—28no—
71#336—21no—
72#336—21no—
73#332—44no—
74D4⋊C8—44no—
75C22×Q8—41no—
76C22×Q8—41no—
77C22×D4—21no—
78C22×D4—21no—
79C22×D4—21no—
80C23×C4—41no—
81C22×C8—22no—
82C22×C8—22no—
83C22×C8—22no—
84C22×C8—41no—
85C4⋊Q8—44no—
86C4⋊Q8—42no—
87C4⋊Q8—42no—
88C4⋊Q8—42no—
89C4⋊Q8—42no—
90C4⋊Q8—42no—
91C4⋊Q8—42no—
92C4⋊Q8—42no—
93C4⋊Q8—42no—
94C4⋊Q8—82no—
95#127—21no—
96#127—21no—
97#127—21no—
98#127Q1612yes—
99C4×D4—42no—
100C4×D4—42no—
101C4×D4—42no—
102C4×D4—42no—
103C4×D4—42no—
104C4×D4—42no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—41no—
115C2×C4⋊C4—41no—
116C2×C4⋊C4—41no—
117C2×C4⋊C4—41no—
118C2×C4⋊C4—41no—
119C2×C4⋊C4—41no—
120C2×C22⋊C4—41no—
121C2×C22⋊C4—41no—
122C2×C22⋊C4—41no—
123C2×C42C2×D411yescharacteristic
124C2×C42C2×D411yescharacteristic
125C2×C42C2×D411yescharacteristic
126C2×C42C2×D411yescharacteristic
127C2×C42C2×D411yescharacteristic
128C2×C42C2×D411yescharacteristic
129C2×C42#4111yescharacteristic
130#107—48no—
131#106—48no—
132C4⋊C8—22no—
133C4⋊C8—22no—
134C4⋊C8—42no—
135C4⋊C8—42no—
136C4⋊C8—42no—
137C4⋊C8D814yes—
138C4⋊C8SD1614yes—
139#102—48no—
140#102—48no—
141C4×C8—24no—
142C4×C8—22no—
143C4×C8—22no—
144C4×C8—42no—
145C4×C8—41no—
146C4×C8—41no—
147C4×C8—41no—
148C4×C8—41no—
149C24—41no—
150C2×Q8—82no—
151C2×Q8—82no—
152C2×Q8—82no—
153C2×Q8—81no—
154C2×Q8—81no—
155C2×Q8—81no—
156C2×Q8—81no—
157C2×Q8—81no—
158C2×Q8—81no—
159C2×D4—22no—
160C2×D4—22no—
161C2×D4—22no—
162C2×D4—42no—
163C2×D4—42no—
164C2×D4—41no—
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
187C2×C8—24no—
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—81no—
197C2×C8—81no—
198C2×C8—81no—
199C4⋊C4—44no—
200C4⋊C4—44no—
201C4⋊C4—42no—
202C4⋊C4—44no—
203C4⋊C4—44no—
204C4⋊C4—42no—
205C4⋊C4—44no—
206C4⋊C4—42no—
207C4⋊C4—82no—
208C4⋊C4—82no—
209C4⋊C4—82no—
210C4⋊C4—82no—
211C4⋊C4—82no—
212C4⋊C4—82no—
213C4⋊C4—81no—
214C4⋊C4—81no—
215C22⋊C4—82no—
216C22⋊C4—82no—
217C22⋊C4—82no—
218C42—22no—
219C42—22no—
220C42—22no—
221C42C2×D812yes—
222C42C2×SD1612yes—
223C42C2×Q1611yescharacteristic
224C42#13512yes—
225C42#13511yescharacteristic
226C42AGL(1,ℤ/8)12yes—
227C42AGL(1,ℤ/8)12yes—
228C42AGL(1,ℤ/8)12yes—
229C42#13711yescharacteristic
230C42#13711yescharacteristic
231C23—42no—
232C23—81no—
233C23—81no—
234C23—81no—
235C23—81no—
236C23—81no—
237C23—81no—
238C23#39211yescharacteristic
239Q8—164no—
240Q8—161no—
241Q8—161no—
242Q8—161no—
243Q8—161no—
244D4—44no—
245D4—44no—
246D4—44no—
247D4—82no—
248D4—82no—
249D4—82no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—82no—
257C2×C4—81no—
258C2×C4—82no—
259C2×C4—82no—
260C2×C4—81no—
261C2×C4—82no—
262C2×C4—81no—
263C2×C4—82no—
264C2×C4—82no—
265C2×C4—161no—
266C2×C4—161no—
267C2×C4—161no—
268C2×C4—161no—
269C2×C4#44712yes—
270C2×C4#44912yes—
271C2×C4#45012yes—
272C2×C4#45111yescharacteristic
273C2×C4#45111yescharacteristic
274C2×C4#45211yescharacteristic
275C2×C4#45211yescharacteristic
276C2×C4#45412yes—
277C2×C4#45612yes—
278C2×C4#45912yes—
279C2×C4#46012yes—
280C2×C4#46312yes—
281C2×C4#46711yescharacteristic
282C2×C4#47011yescharacteristic
283C2×C4#47112yes—
284C2×C4#49211yescharacteristic
285C2×C4#49311yescharacteristic
286C2×C4#49511yescharacteristic
287C2×C4#49611yescharacteristic
288C2×C4#49611yescharacteristic
289C2×C4#49611yescharacteristic
290C8—44no—
291C8—44no—
292C8—44no—
293C8—161no—
294C22—82no—
295C22—82no—
296C22—82no—
297C22—161no—
298C22—161no—
299C22—161no—
300C22—161no—
301C22#200111yescharacteristic
302C22#200311yescharacteristic
303C22#200511yescharacteristic
304C22#201111yescharacteristic
305C22#201211yescharacteristic
306C22#201411yescharacteristic
307C22#201911yescharacteristic
308C4—22no—
309C4—22no—
310C4—22no—
311C4—161no—
312C4—162no—
313C4—162no—
314C4#163512yes—
315C4#163712yes—
316C4#164712yes—
317C4#164912yes—
318C4#168312yes—
319C4#170312yes—
320C4#170611yescharacteristic
321C4#171511yescharacteristic
322C4#171711yescharacteristic
323C4#172111yescharacteristic
324C2—162no—
325C2#1006011yescharacteristic
326C2#1007611yescharacteristic
327C2#1015011yescharacteristic
328C2#1017211yescharacteristic
329C2#1027511yescharacteristic
330C2#1028611yescharacteristic
331C2#1030511yescharacteristic
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. #9896C2semidirect
5. #9692C2semidirect
7. #7015C2semidirect

Maximal subgroups

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

subgroupindex
#332892
#285092
#285082
#98962
#96922
#96382
#70152