Finite Group Databasebeta

← #331621

Subgroups — #331621

Order 512.

576 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 576 conjugacy classes of subgroups — too many to draw. Only its 158 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#33263C211yesmaximal, characteristic
2#33232C211yesmaximal, characteristic
3#30307C212yesmaximal
4#30230C212yesmaximal
5#10686C211yesmaximal, characteristic
6#10685C211yesmaximal, characteristic
7#10682C212yesmaximal
8#10680C212yesmaximal
9#10676C211yesmaximal, characteristic
10#10674C211yesmaximal, characteristic
11#8134C211yesmaximal, characteristic
12C2×#491—22no—
13#2868C2211yescharacteristic
14#2859C2212yes—
15#2859C2211yescharacteristic
16#2100—24no—
17#2100—24no—
18#2085—24no—
19#2059—24no—
20#1990C2212yes—
21#1990C2212yes—
22#1990C2212yes—
23#1990C2212yes—
24#1960—22no—
25#1956C2211yescharacteristic
26#1956C2211yescharacteristic
27#1956C2211yescharacteristic
28#1956C2211yescharacteristic
29#1944—22no—
30#1943—22no—
31#1845C2212yes—
32#1845C2211yescharacteristic
33#1844C2212yes—
34#1844C2211yescharacteristic
35#1834C2212yes—
36#1834C2212yes—
37#1834C2212yes—
38#1744C2212yes—
39C4×#97C2211yescharacteristic
40#1273C2211yescharacteristic
41#1273C2211yescharacteristic
42#1273C2212yes—
43C2×C4⋊Q8—21no—
44C2×C4⋊Q8—21no—
45C2×C4⋊Q8C2311yescharacteristic
46C2×C4⋊Q8C2311yescharacteristic
47C2×#125—21no—
48C2×#125—22no—
49#491—48no—
50C2×#107—22no—
51C2×#107—24no—
52C2×#106—22no—
53C2×#106—24no—
54C2×#103—22no—
55C2×#103—22no—
56C2×#103—22no—
57C2×#103—22no—
58C2×#97D412yes—
59C2×#97D412yes—
60C2×C4×C8C2311yescharacteristic
61C2×C4×C8C2312yes—
62C2×C4×C8C2312yes—
63C2×C4×C8C2311yescharacteristic
64#389—22no—
65#389—22no—
66#389—22no—
67#389—22no—
68#389—22no—
69#389—22no—
70#384—22no—
71#384—22no—
72#384C2311yescharacteristic
73#384C2312yes—
74#384C2312yes—
75#384C2311yescharacteristic
76#382—22no—
77#382—21no—
78C43C2311yescharacteristic
79#340—24no—
80#340—24no—
81#340—24no—
82#340—24no—
83#340—24no—
84#340—24no—
85#340—24no—
86#336D414yes—
87#336D414yes—
88C22×Q8—21no—
89C22×Q8—21no—
90C22×C8C2×D414yes—
91C22×C8C2×D414yes—
92C22×C8C2×D412yes—
93C22×C8C2×D412yes—
94C4⋊Q8—22no—
95C4⋊Q8—22no—
96C4⋊Q8—21no—
97C4⋊Q8—21no—
98C4⋊Q8—21no—
99C4⋊Q8—21no—
100C4⋊Q8—44no—
101C4⋊Q8—44no—
102#125—44no—
103#125—84no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—41no—
117C2×C4⋊C4—41no—
118C2×C4⋊C4—42no—
119C2×C4⋊C4—42no—
120C2×C4⋊C4—41no—
121C2×C4⋊C4—42no—
122C2×C4⋊C4—41no—
123C2×C42C2×D411yescharacteristic
124C2×C42C2×D412yes—
125C2×C42C2×D411yescharacteristic
126C2×C42C2×D412yes—
127C2×C42C2411yescharacteristic
128#107—44no—
129#107—84no—
130#106—44no—
131#106—84no—
132#103—44no—
133#103—44no—
134#103—44no—
135#103—44no—
136#97—24no—
137#97—42no—
138C4×C8—22no—
139C4×C8—21no—
140C4×C8—21no—
141C4×C8—22no—
142C4×C8—22no—
143C4×C8—22no—
144C4×C8—22no—
145#95—42no—
146#95—42no—
147#95—42no—
148C2×Q8—41no—
149C2×Q8—41no—
150C2×Q8—41no—
151C2×Q8—41no—
152C2×Q8—42no—
153C2×Q8—42no—
154C2×Q8—42no—
155C2×Q8—42no—
156C22×C4—41no—
157C22×C4—41no—
158C22×C4—42no—
159C22×C4—42no—
160C22×C4—41no—
161C22×C4—41no—
162C22×C4#12711yescharacteristic
163C22×C4#12712yes—
164C22×C4#12711yescharacteristic
165C22×C4C22×D411yescharacteristic
166C22×C4C22×D412yes—
167C2×C8—22no—
168C2×C8—22no—
169C2×C8—22no—
170C2×C8—24no—
171C2×C8—24no—
172C2×C8—22no—
173C2×C8—22no—
174C2×C8—24no—
175C2×C8—24no—
176C2×C8—24no—
177C2×C8—24no—
178C2×C8AGL(1,ℤ/8)14yes—
179C4⋊C4—42no—
180C4⋊C4—42no—
181C4⋊C4—42no—
182C4⋊C4—42no—
183C4⋊C4—42no—
184C4⋊C4—42no—
185C4⋊C4—44no—
186C4⋊C4—44no—
187C4⋊C4—44no—
188C4⋊C4—44no—
189C4⋊C4—44no—
190C4⋊C4—44no—
191C4⋊C4—44no—
192C4⋊C4—44no—
193C4⋊C4—82no—
194C4⋊C4—82no—
195C4⋊C4—84no—
196C4⋊C4—84no—
197C4⋊C4—82no—
198C4⋊C4—82no—
199C4⋊C4—84no—
200C42—22no—
201C42—22no—
202C42—24no—
203C42—24no—
204C422−1+412yes—
205C422−1+411yescharacteristic
206C422−1+411yescharacteristic
207C23C2×#12711yescharacteristic
208Q8—82no—
209Q8—82no—
210Q8—82no—
211Q8—82no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—81no—
220C2×C4—81no—
221C2×C4—82no—
222C2×C4—81no—
223C2×C4—81no—
224C2×C4—82no—
225C2×C4—81no—
226C2×C4—81no—
227C2×C4—82no—
228C2×C4—82no—
229C2×C4—82no—
230C2×C4—82no—
231C2×C4—82no—
232C2×C4—82no—
233C2×C4—81no—
234C2×C4—81no—
235C2×C4#53611yescharacteristic
236C2×C4#53611yescharacteristic
237C2×C4#53612yes—
238C2×C4#53612yes—
239C2×C4C2×AGL(1,ℤ/8)12yes—
240C2×C4#57512yes—
241C2×C4#57714yes—
242C2×C4#57814yes—
243C8—24no—
244C8—42no—
245C8—44no—
246C8—44no—
247C22#265811yescharacteristic
248C22#315111yescharacteristic
249C22#315311yescharacteristic
250C22#315412yes—
251C22#315612yes—
252C4—41no—
253C4—42no—
254C4—41no—
255C4—161no—
256C4—162no—
257C4—162no—
258C4—161no—
259C4—161no—
260C4—161no—
261C4#323412yes—
262C4#323512yes—
263C4#324214yes—
264C4#324214yes—
265C2#1972111yescharacteristic
266C2#1972312yes—
267C2#1973111yescharacteristic
268C2#3028311yescharacteristic
269C2#3028812yes—
subgroupG/H
center, socle, prime radicalC23C2×#127
commutator, Frattini, cosocleC2×C42C24
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Maximal subgroups

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

subgroupindex
#332632
#332322
#303072
#302302
#106862
#106852
#106822
#106802
#106762
#106742
#81342