Finite Group Databasebeta

← #323596

Subgroups — #323596

Order 512.

636 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 636 conjugacy classes of subgroups — too many to draw. Only its 174 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#29220C212yesmaximal
3#29179C212yesmaximal
4#21691C211yesmaximal, characteristic
5#10686C211yesmaximal, characteristic
6#10686C211yesmaximal, characteristic
7#10676C211yesmaximal, characteristic
8#10674C211yesmaximal, characteristic
9#9835C212yesmaximal
10#8821C212yesmaximal
11#8263C211yesmaximal, characteristic
12C2×#494C2211yescharacteristic
13C2×#441—22no—
14#2859C2212yes—
15C4×C4⋊Q8C2211yescharacteristic
16C4×C4⋊Q8C2211yescharacteristic
17#2101—24no—
18#2087—24no—
19#1990C2212yes—
20#1990C2212yes—
21#1990C2212yes—
22#1990C2212yes—
23#1989—24no—
24#1971—22no—
25#1956C2211yescharacteristic
26#1956C2211yescharacteristic
27#1944—22no—
28#1943—22no—
29#1921—24no—
30#1845C2212yes—
31#1845C2211yescharacteristic
32#1844C2211yescharacteristic
33#1844C2211yescharacteristic
34#1834C2212yes—
35#1834C2212yes—
36#1834C2212yes—
37#1834C2212yes—
38C4×C4⋊C8C2212yes—
39C4×#97C2211yescharacteristic
40C2×#322C2212yes—
41#1273C2211yescharacteristic
42#1273C2211yescharacteristic
43C22×Q16—22no—
44C2×C4⋊Q8C2311yescharacteristic
45C2×C4⋊Q8C2311yescharacteristic
46C2×#125—22no—
47C2×C4×Q8—21no—
48C2×C4×Q8—21no—
49#494—22no—
50#494Q814yes—
51#441—48no—
52C2×#107—24no—
53C2×#107—22no—
54C2×#106—22no—
55C2×#106—22no—
56C2×C4⋊C8D414yes—
57C2×#103—22no—
58C2×#103—22no—
59C2×#97D412yes—
60C2×#97D412yes—
61C2×C4×C8C2312yes—
62C2×C4×C8C2312yes—
63C2×C4×C8C2311yescharacteristic
64C2×C4×C8C2311yescharacteristic
65#391—21no—
66#391—21no—
67#389—22no—
68#389—22no—
69#389—22no—
70#389—22no—
71#389—22no—
72#389—22no—
73#384C2311yescharacteristic
74#384C2311yescharacteristic
75#384C2312yes—
76#384C2312yes—
77#382—22no—
78C4×C4⋊C4—22no—
79C4×C4⋊C4—22no—
80C43C2311yescharacteristic
81#340—24no—
82#340—24no—
83#340—24no—
84#340—24no—
85#340—24no—
86#340—24no—
87#340—24no—
88#340—24no—
89#336D414yes—
90#322—24no—
91C22×Q8—21no—
92C22×Q8—21no—
93C2×Q16—28no—
94C2×Q16—44no—
95C2×Q16—44no—
96C22×C8C2×D414yes—
97C22×C8C2×D414yes—
98C22×C8C2×D412yes—
99C22×C8C2×D412yes—
100C4⋊Q8—21no—
101C4⋊Q8—21no—
102C4⋊Q8—22no—
103C4⋊Q8—22no—
104C4⋊Q8C2×Q812yes—
105C4⋊Q8C2×Q812yes—
106#125—48no—
107C4×Q8—44no—
108C4×Q8—44no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—41no—
122C2×C4⋊C4—41no—
123C2×C4⋊C4—42no—
124C2×C4⋊C4—42no—
125C2×C42—41no—
126C2×C42—41no—
127C2×C42C2×D411yescharacteristic
128C2×C42C2×D412yes—
129C2×C42C2×D412yes—
130C2×C42C2×D411yescharacteristic
131C2×C42C2411yescharacteristic
132#107—48no—
133#107—44no—
134#106—44no—
135#106—44no—
136C4⋊C8—44no—
137#103—44no—
138#103—44no—
139#97—24no—
140#97—42no—
141C4×C8—22no—
142C4×C8—22no—
143C4×C8—22no—
144C4×C8—22no—
145C4×C8—22no—
146C4×C8C2×Q812yes—
147C4×C8#4112yes—
148#95—42no—
149#95—42no—
150#95—42no—
151#95—42no—
152C2×Q8—22no—
153C2×Q8—22no—
154C2×Q8—41no—
155C2×Q8—41no—
156C2×Q8—42no—
157C2×Q8—42no—
158C2×Q8—42no—
159C2×Q8—42no—
160C22×C4—41no—
161C22×C4—41no—
162C22×C4—41no—
163C22×C4—41no—
164C22×C4—42no—
165C22×C4—42no—
166C22×C4#12712yes—
167C22×C4#12711yescharacteristic
168C22×C4#12711yescharacteristic
169C22×C4C22×D411yescharacteristic
170C22×C4C22×D412yes—
171Q16—416no—
172C2×C8—22no—
173C2×C8—22no—
174C2×C8—22no—
175C2×C8—22no—
176C2×C8—24no—
177C2×C8—24no—
178C2×C8—24no—
179C2×C8—24no—
180C2×C8—24no—
181C2×C8—24no—
182C2×C8C22⋊Q814yes—
183C2×C8AGL(1,ℤ/8)14yes—
184C4⋊C4—24no—
185C4⋊C4—24no—
186C4⋊C4—42no—
187C4⋊C4—42no—
188C4⋊C4—42no—
189C4⋊C4—42no—
190C4⋊C4—44no—
191C4⋊C4—44no—
192C4⋊C4—44no—
193C4⋊C4—44no—
194C4⋊C4—44no—
195C4⋊C4—44no—
196C4⋊C4—44no—
197C4⋊C4—44no—
198C4⋊C4—82no—
199C4⋊C4—82no—
200C4⋊C4—84no—
201C4⋊C4—84no—
202C42—22no—
203C42—22no—
204C42—24no—
205C42—24no—
206C42—82no—
207C42—82no—
208C42C22×Q811yescharacteristic
209C42C2×#4111yescharacteristic
210C422−1+412yes—
211C23C2×#12711yescharacteristic
212Q8—44no—
213Q8—44no—
214Q8—82no—
215Q8—82no—
216C2×C4—22no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—81no—
226C2×C4—81no—
227C2×C4—81no—
228C2×C4—81no—
229C2×C4—82no—
230C2×C4—81no—
231C2×C4—81no—
232C2×C4—82no—
233C2×C4—82no—
234C2×C4—82no—
235C2×C4—82no—
236C2×C4—82no—
237C2×C4—82no—
238C2×C4—82no—
239C2×C4C2×C22⋊Q811yescharacteristic
240C2×C4#53611yescharacteristic
241C2×C4#54712yes—
242C2×C4D4×Q812yes—
243C2×C4C2×AGL(1,ℤ/8)12yes—
244C2×C4#57512yes—
245C2×C4#57514yes—
246C2×C4#57814yes—
247C8—24no—
248C8—24no—
249C8—44no—
250C8—44no—
251C22#254211yescharacteristic
252C22#315011yescharacteristic
253C22#315111yescharacteristic
254C22#315312yes—
255C22#315612yes—
256C4—41no—
257C4—41no—
258C4—42no—
259C4—82no—
260C4—82no—
261C4—161no—
262C4—161no—
263C4—162no—
264C4—162no—
265C4#307712yes—
266C4#323412yes—
267C4#334914yes—
268C4#339014yes—
269C2#1969612yes—
270C2#1973111yescharacteristic
271C2#1975011yescharacteristic
272C2#2851611yescharacteristic
273C2#2854612yes—
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
#292202
#291792
#216912
#106862
#106762
#106742
#98352
#88212
#82632