Finite Group Databasebeta

← #511367

Subgroups — #511367

Order 512.

392 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 146 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#27995C212yesmaximal
2#13515C212yesmaximal
3#13515C212yesmaximal
4#13505C212yesmaximal
5#13176C212yesmaximal
6#13165C211yesmaximal, characteristic
7#13165C211yesmaximal, characteristic
8#13159C211yesmaximal, characteristic
9#13112C212yesmaximal
10#2507—22no—
11C2×#366C2211yescharacteristic
12C2×#366C2211yescharacteristic
13C2×#366C2211yescharacteristic
14C2×#366C2212yes—
15C2×#366C2211yescharacteristic
16C2×C4⋊C16C2212yes—
17C2×C4⋊C16C2211yescharacteristic
18C2×C4×C16C2211yescharacteristic
19#2100—22no—
20#2087—22no—
21#1942C2212yes—
22#1942C2212yes—
23#1942C2212yes—
24#1942C2212yes—
25#1921—22no—
26#1845C2212yes—
27C4×#107C2212yes—
28C8×C4⋊C4C2211yescharacteristic
29#1380C2212yes—
30#1380C2212yes—
31#1380C2212yes—
32#1380C2212yes—
33#1380C2212yes—
34#1374C2212yes—
35C22×C16C2312yes—
36C22×C16C2311yescharacteristic
37C22×C16C2311yescharacteristic
38C2×#107—22no—
39C2×#107C2312yes—
40C2×#107C2312yes—
41C2×#107C2312yes—
42C2×#107C2312yes—
43C2×C4⋊C8D411yescharacteristic
44C2×C4×C8C2312yes—
45C2×C4×C8C2311yescharacteristic
46#398—42no—
47#398—42no—
48#389—22no—
49#389—22no—
50#389—22no—
51#389—22no—
52#389—42no—
53#387—42no—
54#384—22no—
55C4×C4⋊C4—22no—
56C4×C4⋊C4D411yescharacteristic
57#366—22no—
58#366—41no—
59#366—41no—
60#366—42no—
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⋊C4—42no—
83C2×C4⋊C4—42no—
84C2×C4⋊C4D811yescharacteristic
85C2×C4⋊C4D811yescharacteristic
86C2×C42—42no—
87C2×C42C2×D412yes—
88C2×C42C2×D411yescharacteristic
89C2×C16—21no—
90C2×C16—21no—
91C2×C16—21no—
92C2×C16—21no—
93C2×C16—22no—
94C2×C16—22no—
95C2×C16—22no—
96C2×C16C2×Q812yes—
97C2×C16#4112yes—
98#107—22no—
99#107—22no—
100#107—22no—
101#107—22no—
102#107—22no—
103#107—22no—
104#107—22no—
105#107—44no—
106#107C2×Q814yes—
107C4⋊C8—41no—
108C4×C8—22no—
109C4×C8—21no—
110C4×C8—21no—
111C4×C8—22no—
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×D811yescharacteristic
125C22×C4C2×D812yes—
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—24no—
140C4⋊C4—41no—
141C4⋊C4—42no—
142C4⋊C4—42no—
143C4⋊C4—42no—
144C4⋊C4—42no—
145C4⋊C4—42no—
146C4⋊C4—42no—
147C4⋊C4—42no—
148C4⋊C4—84no—
149C4⋊C4—162no—
150C4⋊C4—162no—
151C4⋊C4—162no—
152C4⋊C4—162no—
153C4⋊C4—162no—
154C4⋊C4Q3214yes—
155C42—21no—
156C42—22no—
157C42—21no—
158C42—22no—
159C42—162no—
160C16—22no—
161C16—22no—
162C16—42no—
163C23C22×D811yescharacteristic
164C2×C4—22no—
165C2×C4—21no—
166C2×C4—22no—
167C2×C4—21no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—22no—
173C2×C4—21no—
174C2×C4—44no—
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—82no—
186C2×C4C2×Q3212yes—
187C2×C4C2×Q3214yes—
188C2×C4#53711yescharacteristic
189C2×C4#54711yescharacteristic
190C2×C4D4×Q811yescharacteristic
191C2×C4#57511yescharacteristic
192C2×C4#57811yescharacteristic
193C2×C4#57811yescharacteristic
194C8—42no—
195C8—41no—
196C8#53911yescharacteristic
197C8#54411yescharacteristic
198C8#55711yescharacteristic
199C8#56411yescharacteristic
200C22#324511yescharacteristic
201C22#333411yescharacteristic
202C22Q8×D811yescharacteristic
203C22C22×Q3211yescharacteristic
204C22#341411yescharacteristic
205C22#341511yescharacteristic
206C22#341511yescharacteristic
207C4—22no—
208C4—24no—
209C4—41no—
210C4—42no—
211C4—84no—
212C4—162no—
213C4—162no—
214C4—162no—
215C4#272611yescharacteristic
216C4#326111yescharacteristic
217C4#335211yescharacteristic
218C4#339011yescharacteristic
219C2#3250411yescharacteristic
220C2#3381511yescharacteristic
221C2#3382511yescharacteristic
222C2#3390511yescharacteristic
223C2#3392211yescharacteristic
224C2Q8×Q3211yescharacteristic
225C2#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
#279952
#135152
#135052
#131762
#131652
#131592
#131122