Finite Group Databasebeta

← #511370

Subgroups — #511370

Order 512.

384 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 138 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#27992C212yesmaximal
2#13516C212yesmaximal
3#13514C212yesmaximal
4#13507C212yesmaximal
5#13178C212yesmaximal
6#13165C211yesmaximal, characteristic
7#13165C211yesmaximal, characteristic
8#13159C211yesmaximal, characteristic
9#13112C212yesmaximal
10#2508—22no—
11C2×#367C2212yes—
12C2×#366C2211yescharacteristic
13C2×#366C2211yescharacteristic
14C2×#366C2211yescharacteristic
15C2×#366C2211yescharacteristic
16C2×C4⋊C16C2212yes—
17C2×C4⋊C16C2211yescharacteristic
18C2×C4×C16C2211yescharacteristic
19#2101—22no—
20#2088—22no—
21#1942C2212yes—
22#1942C2212yes—
23#1942C2212yes—
24#1942C2212yes—
25#1920—22no—
26#1846C2212yes—
27C4×#107C2212yes—
28C8×C4⋊C4C2211yescharacteristic
29#1380C2212yes—
30#1380C2212yes—
31#1380C2212yes—
32#1380C2212yes—
33#1380C2212yes—
34#1374C2212yes—
35C22×C16C2312yes—
36C22×C16C2311yescharacteristic
37C22×C16C2311yescharacteristic
38C2×#107C2312yes—
39C2×#107C2312yes—
40C2×#107C2312yes—
41C2×#107C2312yes—
42C2×#106—22no—
43C2×C4⋊C8D411yescharacteristic
44C2×C4×C8C2312yes—
45C2×C4×C8C2311yescharacteristic
46#400—42no—
47#398—42no—
48#389—22no—
49#389—22no—
50#389—22no—
51#389—22no—
52#387—42no—
53#382—22no—
54#382—42no—
55C4×C4⋊C4—22no—
56C4×C4⋊C4D411yescharacteristic
57#367—42no—
58#366—22no—
59#366—41no—
60#366—41no—
61#366Q814yes—
62C4⋊C16—24no—
63C4⋊C16—41no—
64C4×C16—41no—
65#340—22no—
66#340—22no—
67#340—22no—
68#340—22no—
69#340—22no—
70#336D412yes—
71C22×C8C2×D411yescharacteristic
72C22×C8C2×D412yes—
73C22×C8C2411yescharacteristic
74C2×C4⋊C4—22no—
75C2×C4⋊C4—22no—
76C2×C4⋊C4—22no—
77C2×C4⋊C4—22no—
78C2×C4⋊C4—42no—
79C2×C4⋊C4—42no—
80C2×C4⋊C4—42no—
81C2×C4⋊C4—42no—
82C2×C4⋊C4D811yescharacteristic
83C2×C4⋊C4D811yescharacteristic
84C2×C42—42no—
85C2×C42C2×D412yes—
86C2×C42C2×D411yescharacteristic
87C2×C16—21no—
88C2×C16—21no—
89C2×C16—21no—
90C2×C16—21no—
91C2×C16—22no—
92C2×C16—22no—
93C2×C16—22no—
94C2×C16C2×Q812yes—
95C2×C16#4112yes—
96#107—22no—
97#107—22no—
98#107—22no—
99#107—22no—
100#107—22no—
101#107—22no—
102#107—22no—
103#107C2×Q814yes—
104#106—44no—
105C4⋊C8—41no—
106C4×C8—22no—
107C4×C8—21no—
108C4×C8—21no—
109C4×C8—22no—
110#95—42no—
111#95—42no—
112#95—42no—
113#95—42no—
114#95—42no—
115#95—42no—
116#95—42no—
117#95D812yes—
118C22×C4—42no—
119C22×C4—42no—
120C22×C4—42no—
121C22×C4—42no—
122C22×C4C2×D812yes—
123C22×C4C2×D811yescharacteristic
124C22×C4C2×D812yes—
125C22×C4C2×D811yescharacteristic
126C22×C4C22×D411yescharacteristic
127C2×C8—21no—
128C2×C8—22no—
129C2×C8—22no—
130C2×C8—22no—
131C2×C8—21no—
132C2×C8—21no—
133C2×C8C22×Q811yescharacteristic
134C2×C8C2×#4111yescharacteristic
135C2×C8H(5,𝔽2)11yescharacteristic
136C2×C82−1+411yescharacteristic
137C2×C82−1+411yescharacteristic
138C2×C82−1+411yescharacteristic
139C4⋊C4—22no—
140C4⋊C4—24no—
141C4⋊C4—41no—
142C4⋊C4—42no—
143C4⋊C4—42no—
144C4⋊C4—42no—
145C4⋊C4—42no—
146C4⋊C4—42no—
147C4⋊C4—42no—
148C4⋊C4—42no—
149C4⋊C4—84no—
150C4⋊C4—162no—
151C4⋊C4—162no—
152C4⋊C4—162no—
153C42—21no—
154C42—22no—
155C42—21no—
156C42—22no—
157C42—84no—
158C16—22no—
159C16—22no—
160C16—42no—
161C23C22×D811yescharacteristic
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—22no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—22no—
168C2×C4—21no—
169C2×C4—22no—
170C2×C4—21no—
171C2×C4—22no—
172C2×C4—21no—
173C2×C4—44no—
174C2×C4—82no—
175C2×C4—82no—
176C2×C4—82no—
177C2×C4—82no—
178C2×C4—82no—
179C2×C4—82no—
180C2×C4—82no—
181C2×C4—82no—
182C2×C4—82no—
183C2×C4—82no—
184C2×C4—82no—
185C2×C4#50812yes—
186C2×C4#53711yescharacteristic
187C2×C4#54711yescharacteristic
188C2×C4D4×Q811yescharacteristic
189C2×C4#57511yescharacteristic
190C2×C4#57811yescharacteristic
191C2×C4#57811yescharacteristic
192C8—42no—
193C8—41no—
194C8#53911yescharacteristic
195C8#54411yescharacteristic
196C8#55711yescharacteristic
197C8#56411yescharacteristic
198C22#324511yescharacteristic
199C22#333411yescharacteristic
200C22Q8×D811yescharacteristic
201C22C2×#50811yescharacteristic
202C22#341411yescharacteristic
203C22#341511yescharacteristic
204C22#341711yescharacteristic
205C4—22no—
206C4—41no—
207C4—42no—
208C4—42no—
209C4—84no—
210C4—162no—
211C4—162no—
212C4—162no—
213C4#272611yescharacteristic
214C4#326111yescharacteristic
215C4#335211yescharacteristic
216C4#339011yescharacteristic
217C2#3250411yescharacteristic
218C2#3381611yescharacteristic
219C2#3382811yescharacteristic
220C2#3390811yescharacteristic
221C2#3391911yescharacteristic
222C2#3395011yescharacteristic
223C2#3395611yescharacteristic
subgroupG/H
center, socle, prime radicalC23C22×D8
commutator, Frattini, cosocleC22×C8C24
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ C8×C4⋊C4 ◁ 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
#279922
#135162
#135142
#135072
#131782
#131652
#131592
#131122