Finite Group Databasebeta

← #100826

Subgroups — #100826

Order 512.

382 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 72 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#29337C211yesmaximal, characteristic
2#21917C211yesmaximal, characteristic, complemented
3#7704C212yesmaximal
4#7646C211yesmaximal, characteristic, complemented
5#7046C212yesmaximal, complemented
6#2856C2211yescharacteristic
7#2296C2211yescharacteristic
8#2099C412yes—
9#2041C412yes—
10#1936—22no—
11#1917—24no—
12#1875—21no—
13#1875—21no—
14#1818—21no—
15#1817C2212yes—
16#1798—21no—
17#1749C2211yescharacteristic
18#1749C2212yes—
19#1271—22no—
20#1270—24no—
21C2×#127C2×C411yescharacteristic
22C2×#121—21no—
23C2×#121—22no—
24C2×#117D411yescharacteristic
25C2×#117D411yescharacteristic
26C22×C42C2311yescharacteristic
27C2×C4⋊C8—22no—
28C2×#102—22no—
29C2×#102—22no—
30C2×C22⋊C8—22no—
31C2×C22⋊C8—21no—
32C2×C22⋊C8—22no—
33C2×#97—22no—
34C2×#97—21no—
35#390—21no—
36#390—22no—
37#389C2×C412yes—
38#389D412yes—
39#385—22no—
40#384C2×C411yescharacteristic
41C4×C4⋊C4—22no—
42#340—22no—
43#340—22no—
44#336—22no—
45#336—24no—
46#336—22no—
47#336—22no—
48#336C2×C412yes—
49#336D414yes—
50C22×D4—21no—
51C22×D4—22no—
52C22×D4—41no—
53C23×C4C2×D411yescharacteristic
54C23×C4#4112yes—
55C22×C8—22no—
56C22×C8—24no—
57C22×C8—42no—
58#127—81no—
59#121—44no—
60#121—84no—
61#117—42no—
62#117—81no—
63C2×C4⋊C4—21no—
64C2×C4⋊C4—22no—
65C2×C4⋊C4—22no—
66C2×C4⋊C4—22no—
67C2×C4⋊C4C22⋊C412yes—
68C2×C4⋊C4C22⋊C412yes—
69C2×C22⋊C4—21no—
70C2×C22⋊C4—21no—
71C2×C22⋊C4—41no—
72C2×C22⋊C4—42no—
73C2×C42—21no—
74C2×C42—21no—
75C2×C42—81no—
76C2×C42C22×C411yescharacteristic
77C2×C42C2×D411yescharacteristic
78C2×C42C2×D412yes—
79C4⋊C8—82no—
80#102—44no—
81#102—82no—
82C22⋊C8—44no—
83C22⋊C8—42no—
84C22⋊C8—82no—
85#97—81no—
86#97—82no—
87#95—22no—
88#95—22no—
89C24—41no—
90C24#12311yescharacteristic
91C2×D4—42no—
92C2×D4—42no—
93C2×D4—44no—
94C2×D4—42no—
95C2×D4—82no—
96C2×D4—82no—
97C2×D4—82no—
98C2×D4—84no—
99C22×C4—22no—
100C22×C4—22no—
101C22×C4—22no—
102C22×C4—22no—
103C22×C4—41no—
104C22×C4—42no—
105C22×C4—42no—
106C22×C4—42no—
107C22×C4—41no—
108C22×C4C2×C22⋊C411yescharacteristic
109C22×C4C4×D412yes—
110C22×C4#12011yescharacteristic
111C22×C4#12112yes—
112C2×C8—42no—
113C2×C8—44no—
114C2×C8—42no—
115C2×C8—44no—
116C2×C8—42no—
117C2×C8—44no—
118C2×C8—82no—
119C2×C8—82no—
120C2×C8—82no—
121C4⋊C4—22no—
122C4⋊C4—22no—
123C4⋊C4—42no—
124C4⋊C4—42no—
125C4⋊C4—82no—
126C4⋊C4—82no—
127C4⋊C4—82no—
128C22⋊C4—42no—
129C22⋊C4—84no—
130C22⋊C4—81no—
131C22⋊C4—82no—
132C42—42no—
133C42—41no—
134C42—41no—
135C42—42no—
136C42—42no—
137C23—22no—
138C23—21no—
139C23—21no—
140C23—22no—
141C23—21no—
142C23—82no—
143C23—81no—
144C23—81no—
145C23—82no—
146C23—81no—
147C23#38611yescharacteristic
148D4—84no—
149D4—164no—
150D4—164no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—22no—
156C2×C4—22no—
157C2×C4—22no—
158C2×C4—22no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—44no—
162C2×C4—42no—
163C2×C4—42no—
164C2×C4—42no—
165C2×C4—42no—
166C2×C4—42no—
167C2×C4—42no—
168C2×C4—84no—
169C2×C4—81no—
170C2×C4—82no—
171C2×C4#42112yes—
172C2×C4#45412yes—
173C8—84no—
174C8—82no—
175C8—162no—
176C22—41no—
177C22—42no—
178C22—42no—
179C22—41no—
180C22—41no—
181C22—161no—
182C22—162no—
183C22—162no—
184C22—161no—
185C22—161no—
186C22#188112yes—
187C22#188811yescharacteristic
188C22#189711yescharacteristic
189C22#190312yes—
190C22#191311yescharacteristic
191C4—22no—
192C4—22no—
193C4—42no—
194C4—42no—
195C4—82no—
196C4—82no—
197C4—161no—
198C2—81no—
199C2—321no—
200C2#900811yescharacteristic
201C2#931312yes—
202C2#933212yes—
203C2#941411yescharacteristic
204C2#942411yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radical#2856C22
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21917C2semidirect
4. #7646C2semidirect
5. #7046C2semidirect

Maximal subgroups

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

subgroupindex
#293372
#219172
#77042
#76462
#70462