Finite Group Databasebeta

← #507653

Subgroups — #507653

Order 512.

434 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 166 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#33261C211yesmaximal, characteristic
2#21725C211yesmaximal, characteristic
3#13507C212yesmaximal
4#13452C212yesmaximal
5#13178C212yesmaximal
6#13177C212yesmaximal
7#13167C212yesmaximal
8#13166C212yesmaximal
9#12654C211yesmaximal, characteristic
10C2×#500C2211yescharacteristic
11#2647—21no—
12#2331—21no—
13C2×#367C2212yes—
14C2×#367C2212yes—
15C2×#367C2212yes—
16C2×#366C2212yes—
17C2×C4⋊C16C2212yes—
18C2×#346C2211yescharacteristic
19C2×C4×C16C2211yescharacteristic
20#1993—21no—
21#1944—21no—
22#1943—21no—
23#1943—21no—
24#1942C2212yes—
25#1942C2212yes—
26#1942C2212yes—
27#1846C2211yescharacteristic
28#1846C2211yescharacteristic
29#1845C2211yescharacteristic
30#1836—21no—
31#1776—21no—
32C4×#107C2212yes—
33#1726C2211yescharacteristic
34#1380C2212yes—
35#1380C2212yes—
36#1380C2212yes—
37#1380C2212yes—
38#1374C2212yes—
39C2×C4⋊Q8—21no—
40C22×C16C2312yes—
41C22×C16C2312yes—
42#500—24no—
43C2×#107—21no—
44C2×#107—21no—
45C2×#107C2312yes—
46C2×#107C2312yes—
47C2×#107C2312yes—
48C2×#107C2312yes—
49C2×#106—21no—
50C2×#106—21no—
51C2×#97D411yescharacteristic
52C2×#97D411yescharacteristic
53C2×#97D411yescharacteristic
54C2×C4×C8C2311yescharacteristic
55C2×C4×C8C2311yescharacteristic
56C2×C4×C8C2311yescharacteristic
57#400—42no—
58#398—42no—
59#391—42no—
60#389—22no—
61#389—22no—
62#389—22no—
63#387—42no—
64#384—21no—
65#382—21no—
66#382—21no—
67C4×C4⋊C4—22no—
68#376D411yescharacteristic
69#367—24no—
70#367—24no—
71#367—42no—
72#366—42no—
73C4⋊C16Q818yescomplemented
74#346—41no—
75C4×C16—41no—
76#340—22no—
77#340—22no—
78#340—22no—
79#340—22no—
80C22×Q8—41no—
81C22×C8C2×D411yescharacteristic
82C22×C8C2×D411yescharacteristic
83C22×C8C2×D411yescharacteristic
84C22×C8C2411yescharacteristic
85C4⋊Q8—24no—
86C4⋊Q8—42no—
87C2×C4⋊C4—22no—
88C2×C4⋊C4—22no—
89C2×C4⋊C4—22no—
90C2×C4⋊C4—22no—
91C2×C4⋊C4—41no—
92C2×C4⋊C4—42no—
93C2×C4⋊C4—42no—
94C2×C4⋊C4—42no—
95C2×C4⋊C4—42no—
96C2×C42—42no—
97C2×C42C2×D411yescharacteristic
98C2×C42C2×D411yescharacteristic
99C2×C42C2×D411yescharacteristic
100C2×C16—22no—
101C2×C16—22no—
102C2×C16—22no—
103C2×C16—22no—
104C2×C16C2×Q814yes—
105C2×C16C2×Q814yes—
106#107—24no—
107#107—22no—
108#107—22no—
109#107—22no—
110#107—22no—
111#107—22no—
112#107—22no—
113#107—22no—
114#107—42no—
115#107#4114yes—
116#106—42no—
117#106—42no—
118#97—41no—
119#97—41no—
120#97Q1614yes—
121C4×C8—22no—
122C4×C8—21no—
123C4×C8—21no—
124C4×C8C2×Q814yes—
125#95—42no—
126#95—41no—
127#95—41no—
128#95—42no—
129#95—42no—
130#95—42no—
131#95—42no—
132#95D812yes—
133#95D812yes—
134C2×Q8—44no—
135C2×Q8—162no—
136C22×C4—42no—
137C22×C4—42no—
138C22×C4—42no—
139C22×C4—42no—
140C22×C4C2×D811yescharacteristic
141C22×C4C2×D812yes—
142C22×C4C2×D811yescharacteristic
143C22×C4C2×D812yes—
144C22×C4C22×D411yescharacteristic
145C2×C8—21no—
146C2×C8—21no—
147C2×C8—21no—
148C2×C8—21no—
149C2×C8—21no—
150C2×C8—21no—
151C2×C8C4⋊Q812yes—
152C2×C8C2×Q1612yes—
153C2×C8C2×Q1612yes—
154C2×C8C22×Q812yes—
155C2×C8C2×#4112yes—
156C2×C8H(5,𝔽2)11yescharacteristic
157C2×C82−1+411yescharacteristic
158C4⋊C4—24no—
159C4⋊C4—44no—
160C4⋊C4—42no—
161C4⋊C4—42no—
162C4⋊C4—42no—
163C4⋊C4—42no—
164C4⋊C4—42no—
165C4⋊C4—42no—
166C4⋊C4—42no—
167C4⋊C4—162no—
168C4⋊C4—162no—
169C4⋊C4—162no—
170C4⋊C4—162no—
171C42—21no—
172C42—21no—
173C42—21no—
174C42—21no—
175C42—162no—
176C42C4⋊Q812yes—
177C42C2×Q1612yes—
178C16—24no—
179C16—24no—
180C23C22×D811yescharacteristic
181Q8—164no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—44no—
192C2×C4—82no—
193C2×C4—82no—
194C2×C4—82no—
195C2×C4—82no—
196C2×C4—82no—
197C2×C4—82no—
198C2×C4—82no—
199C2×C4—82no—
200C2×C4—82no—
201C2×C4—82no—
202C2×C4—82no—
203C2×C4#50014yes—
204C2×C4#52511yescharacteristic
205C2×C4C2×C4⋊Q811yescharacteristic
206C2×C4#53711yescharacteristic
207C2×C4C22×Q1611yescharacteristic
208C2×C4#57511yescharacteristic
209C2×C4#57511yescharacteristic
210C8—22no—
211C8—22no—
212C8—22no—
213C8#55412yes—
214C8#55712yes—
215C22#310011yescharacteristic
216C22C2×#50011yescharacteristic
217C22#324511yescharacteristic
218C22#341612yes—
219C22#341612yes—
220C4—24no—
221C4—42no—
222C4—41no—
223C4—41no—
224C4—84no—
225C4—162no—
226C4—162no—
227C4—162no—
228C4#258411yescharacteristic
229C4#310411yescharacteristic
230C4#316611yescharacteristic
231C4#324911yescharacteristic
232C2#2931211yescharacteristic
233C2#3368311yescharacteristic
234C2#3368611yescharacteristic
235C2#3373012yes—
236C2#3382912yes—
subgroupG/H
center, socle, prime radicalC23C22×D8
commutator, Frattini, cosocleC22×C8C24
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #376 ◁ #1726 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
73. C4⋊C16Q8semidirect

Maximal subgroups

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

subgroupindex
#332612
#217252
#135072
#134522
#131782
#131772
#131672
#131662
#126542