Finite Group Databasebeta

← #507697

Subgroups — #507697

Order 512.

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

normal subgroups only

This group has 508 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#33242C211yesmaximal, characteristic
2C4×#499C211yesmaximal, characteristic
3#13483C212yesmaximal
4#13366C212yesmaximal
5#13276C212yesmaximal
6#13253C212yesmaximal
7#13150C214yesmaximal
8#12653C211yesmaximal, characteristic
9C2×#500C2211yescharacteristic
10C2×#499C2211yescharacteristic
11C2×#494C2211yescharacteristic
12C2×#487—21no—
13#2862—21no—
14C4×#125—21no—
15C2×C4⋊C16C2212yes—
16C2×#358C2214yes—
17C2×#358C2214yes—
18C2×#346C2212yes—
19#1991—21no—
20#1969—21no—
21#1969—21no—
22#1969C2212yes—
23#1969C2212yes—
24#1942C2212yes—
25#1941—21no—
26#1846C2211yescharacteristic
27C4×#107C2212yes—
28C4×#106—21no—
29C42×C8C2211yescharacteristic
30#1380C2214yes—
31#1380C2214yes—
32#1374C2212yes—
33C22×Q16—21no—
34C22×Q16C2312yes—
35C2×C4⋊Q8—21no—
36C2×C4⋊Q8—21no—
37C2×#125—21no—
38C22×C16C2314yes—
39#500—24no—
40#499—24no—
41#494—24no—
42#487—44no—
43C2×#107—21no—
44C2×#107C2312yes—
45C2×#107C2312yes—
46C2×#107C2312yes—
47C2×#106—21no—
48C2×#106—21no—
49C2×#103—22no—
50C2×#103—22no—
51C2×C4×C8D411yescharacteristic
52C2×C4×C8D411yescharacteristic
53C2×C4×C8D411yescharacteristic
54C2×C4×C8C2311yescharacteristic
55C2×C4×C8C2311yescharacteristic
56C2×C4×C8C2311yescharacteristic
57#391—22no—
58#391—22no—
59#391—42no—
60#389—22no—
61#387—42no—
62#382—21no—
63#382—42no—
64C4×C4⋊C4—22no—
65C4×C4⋊C4—42no—
66C43D411yescharacteristic
67C4⋊C16—24no—
68#358—28no—
69#358—44no—
70#346—42no—
71#340—22no—
72#340—22no—
73C22×Q8—22no—
74C22×Q8—41no—
75C2×Q16—24no—
76C2×Q16—24no—
77C2×Q16—22no—
78C2×Q16—22no—
79C2×Q16—42no—
80C2×Q16—42no—
81C2×Q16—42no—
82C22×C8C2×D411yescharacteristic
83C22×C8C2×D411yescharacteristic
84C22×C8C2×D411yescharacteristic
85C22×C8C2411yescharacteristic
86C4⋊Q8—42no—
87C4⋊Q8—42no—
88C4⋊Q8—42no—
89C4⋊Q8—42no—
90#125—24no—
91#125—42no—
92C2×C4⋊C4—22no—
93C2×C4⋊C4—22no—
94C2×C4⋊C4—22no—
95C2×C4⋊C4—42no—
96C2×C4⋊C4—41no—
97C2×C4⋊C4—41no—
98C2×C4⋊C4—41no—
99C2×C4⋊C4—42no—
100C2×C42—42no—
101C2×C42D812yes—
102C2×C42D812yes—
103C2×C42C2×D411yescharacteristic
104C2×C42C2×D411yescharacteristic
105C2×C42C2×D411yescharacteristic
106C2×C16—24no—
107C2×C16—24no—
108C2×C16—24no—
109#107—22no—
110#107—24no—
111#107—22no—
112#107—22no—
113#107—42no—
114#107#4114yes—
115#106—24no—
116#106—42no—
117#103—44no—
118#103—44no—
119C4×C8—22no—
120C4×C8—21no—
121C4×C8—21no—
122C4×C8—41no—
123C4×C8—41no—
124C4×C8SD1614yes—
125C4×C8#4114yes—
126#95—42no—
127#95—42no—
128#95—42no—
129#95—44no—
130#95—44no—
131C2×Q8—42no—
132C2×Q8—42no—
133C2×Q8—42no—
134C2×Q8—42no—
135C2×Q8—42no—
136C2×Q8—42no—
137C2×Q8—84no—
138C2×Q8—82no—
139C22×C4—42no—
140C22×C4—42no—
141C22×C4—42no—
142C22×C4—42no—
143C22×C4C2×D812yes—
144C22×C4C2×D812yes—
145C22×C4C2×D811yescharacteristic
146C22×C4C2×D811yescharacteristic
147C22×C4C22×D411yescharacteristic
148Q16—44no—
149Q16—44no—
150Q16—84no—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—21no—
154C2×C8—21no—
155C2×C8—21no—
156C2×C8—21no—
157C2×C8#12412yes—
158C2×C8C2×SD1612yes—
159C2×C8C2×SD1612yes—
160C2×C8C2×#4112yes—
161C2×C8C2×#4112yes—
162C2×C8H(5,𝔽2)11yescharacteristic
163C2×C82−1+411yescharacteristic
164C4⋊C4—24no—
165C4⋊C4—48no—
166C4⋊C4—42no—
167C4⋊C4—42no—
168C4⋊C4—42no—
169C4⋊C4—42no—
170C4⋊C4—42no—
171C4⋊C4—84no—
172C4⋊C4—82no—
173C4⋊C4—82no—
174C4⋊C4—82no—
175C42—21no—
176C42—21no—
177C42—21no—
178C42—21no—
179C42—24no—
180C42—42no—
181C42—84no—
182C42#12412yes—
183C42C2×SD1612yes—
184C16—44no—
185C23C22×D811yescharacteristic
186Q8—82no—
187Q8—82no—
188Q8—82no—
189Q8—82no—
190Q8—164no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—44no—
199C2×C4—82no—
200C2×C4—82no—
201C2×C4—82no—
202C2×C4—82no—
203C2×C4—82no—
204C2×C4—82no—
205C2×C4—82no—
206C2×C4—84no—
207C2×C4—82no—
208C2×C4—82no—
209C2×C4#48612yes—
210C2×C4#48612yes—
211C2×C4#51014yes—
212C2×C4#52511yescharacteristic
213C2×C4C2×#12411yescharacteristic
214C2×C4#53711yescharacteristic
215C2×C4C22×SD1611yescharacteristic
216C2×C4#57511yescharacteristic
217C2×C4#57511yescharacteristic
218C8—22no—
219C8—22no—
220C8—22no—
221C8#55612yes—
222C8#55912yes—
223C22#310011yescharacteristic
224C22C2×#48611yescharacteristic
225C22#324511yescharacteristic
226C22C2×#51012yes—
227C22#341412yes—
228C4—22no—
229C4—42no—
230C4—42no—
231C4—84no—
232C4—162no—
233C4—162no—
234C4—162no—
235C4#224514yes—
236C4#259211yescharacteristic
237C4#310311yescharacteristic
238C4#314011yescharacteristic
239C4#324611yescharacteristic
240C2#2934611yescharacteristic
241C2#3367612yes—
242C2C2×#224511yescharacteristic
243C2#3370611yescharacteristic
244C2#3381812yes—
subgroupG/H
center, socle, prime radicalC23C22×D8
commutator, Frattini, cosocleC22×C8C24
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ C42×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#332422
C4×#4992
#134832
#133662
#132762
#132532
#131502
#126532