Finite Group Databasebeta

← #103968

Subgroups — #103968

Order 512.

400 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 90 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#30230C211yesmaximal, characteristic
2#29223C211yesmaximal, characteristic
3#29208C211yesmaximal, characteristic
4#9903C211yesmaximal, characteristic
5#7660C211yesmaximal, characteristic
6#7590C211yesmaximal, characteristic
7#7027C211yesmaximal, characteristic
8#2868C2211yescharacteristic
9#2859C2211yescharacteristic
10C4×C4⋊Q8C2211yescharacteristic
11#2100—22no—
12#2100—21no—
13#2100—22no—
14#2099—21no—
15#2057—22no—
16#1993—21no—
17#1990—21no—
18#1990—21no—
19#1990—21no—
20#1989—22no—
21#1947—22no—
22#1845—21no—
23#1836—21no—
24#1831C2211yescharacteristic
25#1767C2211yescharacteristic
26C4×C4⋊C8C2211yescharacteristic
27#1744C2211yescharacteristic
28#1296—22no—
29#1272—22no—
30#1271—22no—
31C2×C4⋊Q8—21no—
32C2×C4⋊Q8—21no—
33C2×C4⋊Q8—21no—
34C2×C4⋊Q8D411yescharacteristic
35C2×#125—21no—
36C2×C4×Q8—21no—
37C2×#107—24no—
38C2×C4⋊C8—21no—
39C2×C4⋊C8—21no—
40C2×C4⋊C8D412yes—
41C2×#97—21no—
42C2×C4×C8—21no—
43C2×C4×C8—21no—
44#391—21no—
45#389—21no—
46#389—21no—
47#389—21no—
48#389—21no—
49#389—21no—
50#389—21no—
51#384—21no—
52#384—21no—
53#384—21no—
54#384D411yescharacteristic
55#384D411yescharacteristic
56#384D411yescharacteristic
57#384D412yes—
58#382—21no—
59C4×C4⋊C4—21no—
60C4×C4⋊C4—21no—
61C43C2311yescharacteristic
62#340—24no—
63#340—22no—
64#340—22no—
65#340—22no—
66#340—24no—
67#340—24no—
68#340—22no—
69#336—21no—
70#336—21no—
71#336—21no—
72#336—21no—
73#336D412yes—
74#336D412yes—
75C22×Q8—21no—
76C22×Q8—41no—
77C22×C8—22no—
78C22×C8—22no—
79C22×C8—22no—
80C22×C8—41no—
81C4⋊Q8—21no—
82C4⋊Q8—21no—
83C4⋊Q8—21no—
84C4⋊Q8—44no—
85C4⋊Q8—82no—
86C4⋊Q8—82no—
87C4⋊Q8Q1612yes—
88#125—42no—
89#125—42no—
90C4×Q8—82no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—22no—
96C2×C4⋊C4—22no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—21no—
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×C42—41no—
112C2×C42C2×D411yescharacteristic
113C2×C42C2×D411yescharacteristic
114C2×C42C2×D411yescharacteristic
115C2×C42C2×D411yescharacteristic
116C2×C42C2×D411yescharacteristic
117C2×C42C2×D411yescharacteristic
118C2×C42#4111yescharacteristic
119#107—84no—
120C4⋊C8—24no—
121C4⋊C8—22no—
122C4⋊C8—22no—
123C4⋊C8—42no—
124#97—41no—
125#97—41no—
126C4×C8—42no—
127C4×C8—41no—
128C4×C8—41no—
129#95—41no—
130#95—41no—
131#95—41no—
132#95—41no—
133C2×Q8—22no—
134C2×Q8—41no—
135C2×Q8—82no—
136C2×Q8—82no—
137C2×Q8—81no—
138C2×Q8—81no—
139C2×Q8—82no—
140C22×C4—41no—
141C22×C4—41no—
142C22×C4—41no—
143C22×C4—41no—
144C22×C4—42no—
145C22×C4#12011yescharacteristic
146C22×C4#12011yescharacteristic
147C22×C4#12011yescharacteristic
148C22×C4#12111yescharacteristic
149C22×C4#12111yescharacteristic
150C22×C4#12111yescharacteristic
151C22×C4#12711yescharacteristic
152C2×C8—24no—
153C2×C8—42no—
154C2×C8—42no—
155C2×C8—42no—
156C2×C8—42no—
157C2×C8—42no—
158C2×C8—42no—
159C2×C8—42no—
160C2×C8—42no—
161C2×C8—81no—
162C2×C8—81no—
163C2×C8—81no—
164C4⋊C4—22no—
165C4⋊C4—22no—
166C4⋊C4—42no—
167C4⋊C4—44no—
168C4⋊C4—42no—
169C4⋊C4—44no—
170C4⋊C4—41no—
171C4⋊C4—41no—
172C4⋊C4—42no—
173C4⋊C4—42no—
174C4⋊C4—42no—
175C4⋊C4—42no—
176C4⋊C4—42no—
177C4⋊C4—81no—
178C4⋊C4—81no—
179C4⋊C4—82no—
180C4⋊C4—82no—
181C4⋊C4—82no—
182C4⋊C4—82no—
183C4⋊C4—82no—
184C4⋊C4—82no—
185C4⋊C4—81no—
186C4⋊C4—81no—
187C42—22no—
188C42—22no—
189C42—22no—
190C42—41no—
191C42—41no—
192C42—82no—
193C42C2×Q1611yescharacteristic
194C42#13511yescharacteristic
195C42#13512yes—
196C42#13711yescharacteristic
197C42#13711yescharacteristic
198C42#13712yes—
199C23#39211yescharacteristic
200Q8—82no—
201Q8—82no—
202Q8—164no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—42no—
216C2×C4—81no—
217C2×C4—81no—
218C2×C4—82no—
219C2×C4—81no—
220C2×C4—81no—
221C2×C4—82no—
222C2×C4—81no—
223C2×C4—81no—
224C2×C4—81no—
225C2×C4—81no—
226C2×C4—81no—
227C2×C4—81no—
228C2×C4—81no—
229C2×C4—82no—
230C2×C4#45111yescharacteristic
231C2×C4#45111yescharacteristic
232C2×C4#45211yescharacteristic
233C2×C4#45212yes—
234C2×C4#45211yescharacteristic
235C2×C4#45412yes—
236C2×C4#45612yes—
237C2×C4#46412yes—
238C2×C4#46711yescharacteristic
239C2×C4#47011yescharacteristic
240C2×C4#47212yes—
241C2×C4#49511yescharacteristic
242C2×C4#49711yescharacteristic
243C8—44no—
244C8—82no—
245C8—82no—
246C8—161no—
247C22#200111yescharacteristic
248C22#200311yescharacteristic
249C22#200511yescharacteristic
250C22#200811yescharacteristic
251C22#200911yescharacteristic
252C22#201511yescharacteristic
253C22#201811yescharacteristic
254C4—21no—
255C4—21no—
256C4—22no—
257C4—22no—
258C4—22no—
259C4—21no—
260C4—21no—
261C4—21no—
262C4—21no—
263C4—82no—
264C4—161no—
265C4—162no—
266C4—161no—
267C4—161no—
268C4#163612yes—
269C4#163812yes—
270C2#1006111yescharacteristic
271C2#1010311yescharacteristic
272C2#1013011yescharacteristic
273C2#1016411yescharacteristic
274C2#1019011yescharacteristic
275C2#1022311yescharacteristic
276C2#1025311yescharacteristic
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
#302302
#292232
#292082
#99032
#76602
#75902
#70272