Finite Group Databasebeta

← #103089

Subgroups — #103089

Order 512.

578 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 578 conjugacy classes of subgroups — too many to draw. Only its 126 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#33225C211yesmaximal, characteristic, complemented
2#29139C211yesmaximal, characteristic
3#28509C211yesmaximal, characteristic
4#9896C211yesmaximal, characteristic, complemented
5#9896C211yesmaximal, characteristic, complemented
6#9638C211yesmaximal, characteristic
7#7015C211yesmaximal, characteristic, complemented
8C2×#487—22no—
9C2×#476—22no—
10#2868C2211yescharacteristic
11#2868C2211yescharacteristic
12C4×#127C2211yescharacteristic
13#2094—22no—
14#2023—22no—
15#1957—21no—
16#1957—21no—
17#1956—22no—
18#1956—22no—
19#1844—21no—
20#1842C2211yescharacteristic
21#1834—21no—
22#1802—22no—
23C4×C4⋊C8C2211yescharacteristic
24C4×C4⋊C8C2211yescharacteristic
25C42×C8C2211yescharacteristic
26C2×D4⋊C8—22no—
27#1296—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⋊Q8D412yes—
35C2×#127D411yescharacteristic
36C2×C4×D4—21no—
37C2×C4×D4—21no—
38C2×C4×D4—21no—
39#487—48no—
40#476—44no—
41#476—44no—
42C2×#106—24no—
43C2×C4⋊C8—21no—
44C2×C4⋊C8—21no—
45C2×C4⋊C8—21no—
46C2×C4⋊C8D412yes—
47C2×C4⋊C8D412yes—
48C2×#103—28no—
49C2×#102—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#384D411yescharacteristic
67#384D412yes—
68C43C2311yescharacteristic
69#340—24no—
70#336—21no—
71#336—21no—
72D4⋊C8—44no—
73C22×Q8—22no—
74C22×Q8—41no—
75C22×D4—21no—
76C22×D4—21no—
77C22×D4—21no—
78C23×C4—41no—
79C22×C8—22no—
80C22×C8—22no—
81C22×C8—22no—
82C22×C8—41no—
83C4⋊Q8—44no—
84C4⋊Q8—44no—
85C4⋊Q8—44no—
86C4⋊Q8—42no—
87C4⋊Q8—42no—
88C4⋊Q8—42no—
89C4⋊Q8—42no—
90C4⋊Q8—82no—
91#127—21no—
92#127—21no—
93#127—21no—
94#127Q1612yes—
95C4×D4—42no—
96C4×D4—42no—
97C4×D4—42no—
98C4×D4—42no—
99C4×D4—42no—
100C4×D4—42no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—41no—
110C2×C4⋊C4—41no—
111C2×C4⋊C4—41no—
112C2×C4⋊C4—41no—
113C2×C4⋊C4—41no—
114C2×C4⋊C4—41no—
115C2×C4⋊C4—41no—
116C2×C22⋊C4—41no—
117C2×C22⋊C4—41no—
118C2×C22⋊C4—41no—
119C2×C42C2×D411yescharacteristic
120C2×C42C2×D411yescharacteristic
121C2×C42C2×D411yescharacteristic
122C2×C42C2×D411yescharacteristic
123C2×C42C2×D411yescharacteristic
124C2×C42C2×D411yescharacteristic
125C2×C42#4111yescharacteristic
126#106—48no—
127C4⋊C8—22no—
128C4⋊C8—22no—
129C4⋊C8—22no—
130C4⋊C8—42no—
131C4⋊C8—42no—
132C4⋊C8—42no—
133C4⋊C8SD1614yes—
134#103—416no—
135#102—48no—
136#102—84no—
137C4×C8—24no—
138C4×C8—22no—
139C4×C8—22no—
140C4×C8—42no—
141C4×C8—41no—
142C4×C8—41no—
143C4×C8—41no—
144C4×C8—41no—
145C24—41no—
146C2×Q8—44no—
147C2×Q8—44no—
148C2×Q8—44no—
149C2×Q8—82no—
150C2×Q8—82no—
151C2×Q8—82no—
152C2×D4—22no—
153C2×D4—22no—
154C2×D4—22no—
155C2×D4—42no—
156C2×D4—42no—
157C2×D4—41no—
158C2×D4—41no—
159C2×D4—42no—
160C2×D4—42no—
161C2×D4—42no—
162C2×D4—41no—
163C2×D4—42no—
164C22×C4—41no—
165C22×C4—42no—
166C22×C4—42no—
167C22×C4—81no—
168C22×C4—81no—
169C22×C4—81no—
170C22×C4—81no—
171C22×C4—81no—
172C22×C4—81no—
173C22×C4#12011yescharacteristic
174C22×C4#12011yescharacteristic
175C22×C4#12011yescharacteristic
176C22×C4#12111yescharacteristic
177C22×C4#12111yescharacteristic
178C22×C4#12111yescharacteristic
179C22×C4#12711yescharacteristic
180C2×C8—24no—
181C2×C8—24no—
182C2×C8—42no—
183C2×C8—42no—
184C2×C8—42no—
185C2×C8—42no—
186C2×C8—42no—
187C2×C8—42no—
188C2×C8—42no—
189C2×C8—81no—
190C2×C8—81no—
191C2×C8—81no—
192C4⋊C4—44no—
193C4⋊C4—44no—
194C4⋊C4—42no—
195C4⋊C4—42no—
196C4⋊C4—44no—
197C4⋊C4—42no—
198C4⋊C4—82no—
199C4⋊C4—82no—
200C4⋊C4—82no—
201C4⋊C4—82no—
202C4⋊C4—82no—
203C4⋊C4—82no—
204C4⋊C4—82no—
205C4⋊C4—81no—
206C4⋊C4—81no—
207C4⋊C4—81no—
208C4⋊C4—81no—
209C22⋊C4—82no—
210C22⋊C4—82no—
211C22⋊C4—82no—
212C42—22no—
213C42—22no—
214C42—22no—
215C42C2×SD1612yes—
216C42C2×Q1611yescharacteristic
217C42#13512yes—
218C42#13511yescharacteristic
219C42#13512yes—
220C42AGL(1,ℤ/8)12yes—
221C42AGL(1,ℤ/8)12yes—
222C42#13711yescharacteristic
223C42#13711yescharacteristic
224C42#13712yes—
225C23—42no—
226C23—81no—
227C23—81no—
228C23—81no—
229C23—81no—
230C23—81no—
231C23—81no—
232C23#39211yescharacteristic
233Q8—88no—
234Q8—164no—
235D4—44no—
236D4—44no—
237D4—44no—
238D4—82no—
239D4—82no—
240D4—82no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—82no—
248C2×C4—82no—
249C2×C4—82no—
250C2×C4—81no—
251C2×C4—82no—
252C2×C4—81no—
253C2×C4—82no—
254C2×C4—81no—
255C2×C4—82no—
256C2×C4—161no—
257C2×C4—161no—
258C2×C4—161no—
259C2×C4—161no—
260C2×C4#44912yes—
261C2×C4#45012yes—
262C2×C4#45111yescharacteristic
263C2×C4#45111yescharacteristic
264C2×C4#45212yes—
265C2×C4#45211yescharacteristic
266C2×C4#45211yescharacteristic
267C2×C4#45312yes—
268C2×C4#45412yes—
269C2×C4#46012yes—
270C2×C4#46312yes—
271C2×C4#46412yes—
272C2×C4#46711yescharacteristic
273C2×C4#47011yescharacteristic
274C2×C4#47112yes—
275C2×C4#49211yescharacteristic
276C2×C4#49511yescharacteristic
277C2×C4#49511yescharacteristic
278C2×C4#49611yescharacteristic
279C2×C4#49611yescharacteristic
280C2×C4#49711yescharacteristic
281C8—44no—
282C8—44no—
283C8—82no—
284C8—161no—
285C22—82no—
286C22—82no—
287C22—82no—
288C22—161no—
289C22—161no—
290C22—161no—
291C22—161no—
292C22#200111yescharacteristic
293C22#200211yescharacteristic
294C22#200311yescharacteristic
295C22#201211yescharacteristic
296C22#201411yescharacteristic
297C22#201511yescharacteristic
298C22#201911yescharacteristic
299C4—22no—
300C4—22no—
301C4—22no—
302C4—162no—
303C4—161no—
304C4—162no—
305C4#163212yes—
306C4#163412yes—
307C4#163512yes—
308C4#163712yes—
309C4#168312yes—
310C4#170312yes—
311C4#170811yescharacteristic
312C4#171411yescharacteristic
313C4#171711yescharacteristic
314C4#171911yescharacteristic
315C2—162no—
316C2#1005611yescharacteristic
317C2#1006011yescharacteristic
318C2#1014611yescharacteristic
319C2#1015611yescharacteristic
320C2#1027511yescharacteristic
321C2#1028211yescharacteristic
322C2#1031011yescharacteristic
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. #33225C2semidirect
4. #9896C2semidirect
5. #9896C2semidirect
7. #7015C2semidirect

Maximal subgroups

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

subgroupindex
#332252
#291392
#285092
#98962
#96382
#70152