Finite Group Databasebeta

← #511268

Subgroups — #511268

Order 512.

412 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 130 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#27837C212yesmaximal
2#13516C212yesmaximal, complemented
3#13507C212yesmaximal, complemented
4#13493C211yesmaximal, characteristic
5#13492C211yesmaximal, characteristic
6#13178C212yesmaximal, complemented
7#13111C212yesmaximal, complemented
8#12912C211yesmaximal, characteristic
9#12835C212yesmaximal
10C23×C16C2211yescharacteristic
11#2494—22no—
12C2×#367C2212yes—
13C2×#367C2212yes—
14C2×#366C2211yescharacteristic
15C2×#366C2211yescharacteristic
16C2×C4⋊C16C2212yes—
17C2×C22⋊C16C2211yescharacteristic
18#2101—22no—
19#2077—22no—
20#1942C2212yes—
21#1942C2212yes—
22#1919—22no—
23#1846C2212yes—
24#1808C2212yes—
25#1808C2212yes—
26C4×#107C2212yes—
27C8×C22⋊C4C2211yescharacteristic
28#1380C2212yes—
29#1380C2212yes—
30#1380C2211yescharacteristic
31#1380C2212yes—
32#1380C2212yes—
33#1380C2211yescharacteristic
34#1374C2212yes—
35C23×C8C2311yescharacteristic
36C22×C16—24no—
37C22×C16—41no—
38C22×C16C2311yescharacteristic
39C22×C16C2311yescharacteristic
40C22×C16C2312yes—
41C2×#107C2312yes—
42C2×#107C2312yes—
43C2×#107C2312yes—
44C2×#107C2312yes—
45C2×#106—22no—
46C2×C22⋊C8D411yescharacteristic
47C2×C4×C8C2312yes—
48#401—42no—
49#397—42no—
50#389—22no—
51#389—22no—
52#388—42no—
53#387—42no—
54#382—22no—
55#380—22no—
56#380—22no—
57C4×C4⋊C4—22no—
58C4×C22⋊C4D411yescharacteristic
59#367—24no—
60#367—42no—
61#366—41no—
62#366—41no—
63C4⋊C16—24no—
64C22⋊C16—41no—
65#340—22no—
66#340—22no—
67#340—22no—
68#340—22no—
69#340—22no—
70#336D412yes—
71C23×C4C2×D411yescharacteristic
72C22×C8—22no—
73C22×C8—21no—
74C22×C8—21no—
75C22×C8—22no—
76C22×C8C2×D411yescharacteristic
77C22×C8C2×D412yes—
78C22×C8C2411yescharacteristic
79C2×C4⋊C4—22no—
80C2×C4⋊C4—22no—
81C2×C4⋊C4—22no—
82C2×C4⋊C4—22no—
83C2×C4⋊C4—42no—
84C2×C4⋊C4—42no—
85C2×C22⋊C4—42no—
86C2×C22⋊C4—42no—
87C2×C22⋊C4D811yescharacteristic
88C2×C22⋊C4D811yescharacteristic
89C2×C42—42no—
90C2×C42C2×D412yes—
91C2×C16—22no—
92C2×C16—22no—
93C2×C16—22no—
94C2×C16—21no—
95C2×C16—21no—
96C2×C16—22no—
97C2×C16—44no—
98C2×C16#4114yes—
99#107—22no—
100#107—22no—
101#107—22no—
102#107—22no—
103#107—22no—
104#107—22no—
105#107—22no—
106#107—22no—
107#106—44no—
108C22⋊C8—41no—
109C4×C8—22no—
110C4×C8—22no—
111#95—42no—
112#95—42no—
113#95—42no—
114#95—42no—
115#95—42no—
116#95—42no—
117#95—42no—
118#95D812yes—
119C24C2×D811yescharacteristic
120C22×C4—21no—
121C22×C4—22no—
122C22×C4—21no—
123C22×C4—22no—
124C22×C4—42no—
125C22×C4—42no—
126C22×C4—42no—
127C22×C4—42no—
128C22×C4C2×D811yescharacteristic
129C22×C4C2×D812yes—
130C22×C4C2×D812yes—
131C22×C4C22×D411yescharacteristic
132C2×C8—21no—
133C2×C8—22no—
134C2×C8—22no—
135C2×C8—22no—
136C2×C8—22no—
137C2×C8—42no—
138C2×C8—41no—
139C2×C8—41no—
140C2×C8C2×#4112yes—
141C2×C8H(5,𝔽2)11yescharacteristic
142C2×C82−1+411yescharacteristic
143C2×C82−1+412yes—
144C4⋊C4—42no—
145C4⋊C4—42no—
146C4⋊C4—42no—
147C4⋊C4—42no—
148C4⋊C4—42no—
149C4⋊C4—42no—
150C4⋊C4—42no—
151C4⋊C4—42no—
152C4⋊C4—84no—
153C4⋊C4—162no—
154C22⋊C4—22no—
155C22⋊C4—41no—
156C22⋊C4—84no—
157C22⋊C4—162no—
158C42—22no—
159C42—22no—
160C42—162no—
161C16—24no—
162C16—41no—
163C16—41no—
164C23—22no—
165C23—21no—
166C23—22no—
167C23—21no—
168C23#50812yes—
169C23C22×D811yescharacteristic
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—21no—
178C2×C4—42no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—82no—
182C2×C4—82no—
183C2×C4—82no—
184C2×C4—82no—
185C2×C4—82no—
186C2×C4—82no—
187C2×C4—82no—
188C2×C4—82no—
189C2×C4—82no—
190C2×C4—82no—
191C2×C4—82no—
192C2×C4—82no—
193C2×C4#53711yescharacteristic
194C2×C4#54712yes—
195C2×C4#57511yescharacteristic
196C2×C4#57812yes—
197C8—42no—
198C8—41no—
199C8#54112yes—
200C8#55111yescharacteristic
201C8#56311yescharacteristic
202C22—22no—
203C22—24no—
204C22—41no—
205C22—42no—
206C22—41no—
207C22#324511yescharacteristic
208C22#333412yes—
209C22C2×#50811yescharacteristic
210C22#341411yescharacteristic
211C22#341612yes—
212C4—42no—
213C4—42no—
214C4—41no—
215C4—162no—
216C4—162no—
217C4—162no—
218C4—162no—
219C4#272211yescharacteristic
220C4#326111yescharacteristic
221C4#334912yes—
222C2—42no—
223C2#3240611yescharacteristic
224C2#3381611yescharacteristic
225C2#3382911yescharacteristic
226C2#3390412yes—
227C2#3391712yes—
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC24C2×D8
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ C8×C22⋊C4 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #13516C2semidirect
3. #13507C2semidirect
6. #13178C2semidirect
7. #13111C2semidirect

Maximal subgroups

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

subgroupindex
#278372
#135162
#135072
#134932
#134922
#131782
#131112
#129122
#128352