Finite Group Databasebeta

← #103107

Subgroups — #103107

Order 512.

470 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 122 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
2#29219C211yesmaximal, characteristic
3#29141C211yesmaximal, characteristic
4#9870C211yesmaximal, characteristic
5#9692C211yesmaximal, characteristic
6#9691C211yesmaximal, characteristic
7#7015C211yesmaximal, characteristic
8#2868C2211yescharacteristic
9#2868C2211yescharacteristic
10C4×C4⋊Q8C2211yescharacteristic
11#2099—22no—
12#2024—22no—
13#1990—22no—
14#1990—22no—
15#1990—21no—
16#1988—22no—
17#1956—21no—
18#1842C2211yescharacteristic
19#1834—22no—
20#1834—21no—
21#1834—21no—
22#1803—22no—
23C4×C4⋊C8C2211yescharacteristic
24C4×C4⋊C8C2211yescharacteristic
25C42×C8C2211yescharacteristic
26C2×#333—22no—
27C2×#332—22no—
28#1279—22no—
29C2×C4⋊Q8—21no—
30C2×C4⋊Q8—21no—
31C2×C4⋊Q8—21no—
32C2×C4⋊Q8—21no—
33C2×C4⋊Q8—21no—
34C2×C4⋊Q8D411yescharacteristic
35C2×C4⋊Q8D412yes—
36C2×C4×Q8—21no—
37C2×C4⋊C8—21no—
38C2×C4⋊C8—21no—
39C2×C4⋊C8—21no—
40C2×C4⋊C8D412yes—
41C2×C4⋊C8D412yes—
42C2×#103—24no—
43C2×C4×C8—21no—
44C2×C4×C8—21no—
45C2×C4×C8—21no—
46C2×C4×C8—21no—
47C2×C4×C8D412yes—
48#391—21no—
49#389—21no—
50#389—21no—
51#384—21no—
52#384—21no—
53#384—21no—
54#384—21no—
55#384—21no—
56#384—21no—
57#384—21no—
58#384D411yescharacteristic
59#384D412yes—
60C4×C4⋊C4—21no—
61C4×C4⋊C4—21no—
62C43C2311yescharacteristic
63#340—24no—
64#340—24no—
65#340—28no—
66#340—24no—
67#336—21no—
68#336—21no—
69#333—44no—
70#332—44no—
71C22×Q8—21no—
72C22×Q8—22no—
73C22×Q8—41no—
74C22×C8—22no—
75C22×C8—22no—
76C22×C8—22no—
77C22×C8—41no—
78C4⋊Q8—21no—
79C4⋊Q8—21no—
80C4⋊Q8—21no—
81C4⋊Q8—44no—
82C4⋊Q8—42no—
83C4⋊Q8—42no—
84C4⋊Q8—42no—
85C4⋊Q8—42no—
86C4⋊Q8—42no—
87C4⋊Q8—42no—
88C4⋊Q8—42no—
89C4⋊Q8—42no—
90C4⋊Q8—82no—
91C4⋊Q8Q1612yes—
92C4×Q8—42no—
93C4×Q8—42no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—41no—
105C2×C4⋊C4—41no—
106C2×C4⋊C4—41no—
107C2×C4⋊C4—41no—
108C2×C4⋊C4—41no—
109C2×C4⋊C4—41no—
110C2×C4⋊C4—41no—
111C2×C4⋊C4—41no—
112C2×C42—41no—
113C2×C42C2×D411yescharacteristic
114C2×C42C2×D411yescharacteristic
115C2×C42C2×D411yescharacteristic
116C2×C42C2×D411yescharacteristic
117C2×C42C2×D411yescharacteristic
118C2×C42C2×D411yescharacteristic
119C2×C42#4111yescharacteristic
120C4⋊C8—22no—
121C4⋊C8—22no—
122C4⋊C8—22no—
123C4⋊C8—22no—
124C4⋊C8—42no—
125C4⋊C8—42no—
126C4⋊C8—42no—
127#103—84no—
128C4×C8—24no—
129C4×C8—22no—
130C4×C8—22no—
131C4×C8—42no—
132C4×C8—41no—
133C4×C8—41no—
134C4×C8—41no—
135C4×C8—41no—
136#95—41no—
137#95—41no—
138C2×Q8—22no—
139C2×Q8—41no—
140C2×Q8—42no—
141C2×Q8—42no—
142C2×Q8—44no—
143C2×Q8—82no—
144C2×Q8—82no—
145C2×Q8—81no—
146C2×Q8—81no—
147C2×Q8—81no—
148C2×Q8—81no—
149C2×Q8—81no—
150C2×Q8—81no—
151C22×C4—41no—
152C22×C4—41no—
153C22×C4—42no—
154C22×C4—42no—
155C22×C4#12011yescharacteristic
156C22×C4#12011yescharacteristic
157C22×C4#12011yescharacteristic
158C22×C4#12111yescharacteristic
159C22×C4#12111yescharacteristic
160C22×C4#12111yescharacteristic
161C22×C4#12711yescharacteristic
162C2×C8—24no—
163C2×C8—42no—
164C2×C8—42no—
165C2×C8—42no—
166C2×C8—42no—
167C2×C8—42no—
168C2×C8—42no—
169C2×C8—42no—
170C2×C8—42no—
171C2×C8—81no—
172C2×C8—81no—
173C2×C8—81no—
174C4⋊C4—22no—
175C4⋊C4—22no—
176C4⋊C4—44no—
177C4⋊C4—41no—
178C4⋊C4—41no—
179C4⋊C4—42no—
180C4⋊C4—44no—
181C4⋊C4—44no—
182C4⋊C4—44no—
183C4⋊C4—42no—
184C4⋊C4—44no—
185C4⋊C4—42no—
186C4⋊C4—82no—
187C4⋊C4—82no—
188C4⋊C4—82no—
189C4⋊C4—82no—
190C4⋊C4—82no—
191C4⋊C4—82no—
192C4⋊C4—82no—
193C4⋊C4—81no—
194C4⋊C4—81no—
195C42—22no—
196C42—22no—
197C42—22no—
198C42—81no—
199C42—81no—
200C42C2×Q1611yescharacteristic
201C42#13512yes—
202C42#13512yes—
203C42#13511yescharacteristic
204C42#13512yes—
205C42AGL(1,ℤ/8)12yes—
206C42AGL(1,ℤ/8)12yes—
207C42#13711yescharacteristic
208C42#13711yescharacteristic
209C42#13712yes—
210C23#39211yescharacteristic
211Q8—44no—
212Q8—82no—
213Q8—164no—
214Q8—161no—
215Q8—161no—
216Q8—161no—
217Q8—161no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—42no—
225C2×C4—81no—
226C2×C4—81no—
227C2×C4—82no—
228C2×C4—81no—
229C2×C4—81no—
230C2×C4—82no—
231C2×C4—82no—
232C2×C4—82no—
233C2×C4—82no—
234C2×C4—81no—
235C2×C4—82no—
236C2×C4#44912yes—
237C2×C4#44912yes—
238C2×C4#45111yescharacteristic
239C2×C4#45111yescharacteristic
240C2×C4#45212yes—
241C2×C4#45211yescharacteristic
242C2×C4#45211yescharacteristic
243C2×C4#45512yes—
244C2×C4#45612yes—
245C2×C4#46312yes—
246C2×C4#46312yes—
247C2×C4#46412yes—
248C2×C4#46711yescharacteristic
249C2×C4#47011yescharacteristic
250C2×C4#47112yes—
251C2×C4#49511yescharacteristic
252C2×C4#49511yescharacteristic
253C2×C4#49511yescharacteristic
254C2×C4#49611yescharacteristic
255C2×C4#49611yescharacteristic
256C2×C4#49711yescharacteristic
257C8—44no—
258C8—82no—
259C8—82no—
260C8—161no—
261C22#200111yescharacteristic
262C22#200411yescharacteristic
263C22#200511yescharacteristic
264C22#201211yescharacteristic
265C22#201211yescharacteristic
266C22#201511yescharacteristic
267C22#201911yescharacteristic
268C4—22no—
269C4—22no—
270C4—22no—
271C4—82no—
272C4—162no—
273C4—161no—
274C4—162no—
275C4#164412yes—
276C4#164612yes—
277C4#164812yes—
278C4#165012yes—
279C4#168412yes—
280C4#170412yes—
281C4#171111yescharacteristic
282C4#171711yescharacteristic
283C4#171911yescharacteristic
284C4#172011yescharacteristic
285C2#1007211yescharacteristic
286C2#1007711yescharacteristic
287C2#1016611yescharacteristic
288C2#1016911yescharacteristic
289C2#1027711yescharacteristic
290C2#1029811yescharacteristic
291C2#1031111yescharacteristic
subgroupG/H
center, socle, prime radicalC23#392
commutator, Frattini, cosocleC43C23
LCSG ▷ C43 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C43 ▷ C23 ▷ C1

Maximal subgroups

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

subgroupindex
#332892
#292192
#291412
#98702
#96922
#96912
#70152