Finite Group Databasebeta

← #149937

Subgroups — #149937

Order 512.

259 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 259 conjugacy classes of subgroups — too many to draw. Only its 54 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#27957C211yesmaximal, characteristic
2#11516C211yesmaximal, characteristic
3#11448C211yesmaximal, characteristic
4#7371C212yesmaximal, complemented
5#7358C212yesmaximal, complemented
6#2629C411yescharacteristic
7#2317C2211yescharacteristic
8#2085—21no—
9#2084—21no—
10#1992C2211yescharacteristic
11#1991C411yescharacteristic
12#1854—21no—
13#1853—21no—
14#1847C411yescharacteristic
15#1844C2211yescharacteristic
16#1750C411yescharacteristic
17#1464—21no—
18C2×#324—21no—
19#1325—21no—
20#1325C2212yes—
21#1325C2212yes—
22#1324—21no—
23C2×#125C2×C411yescharacteristic
24C2×#117C2×C411yescharacteristic
25C2×#117D411yescharacteristic
26C22×C4⋊C4D411yescharacteristic
27#432—41no—
28C2×#107—21no—
29C2×#106—21no—
30C2×#106—21no—
31C2×#106—21no—
32C2×C22⋊C8—21no—
33C2×C22⋊C8—41no—
34C2×#97C2×C411yescharacteristic
35C2×C4×C8C2×C411yescharacteristic
36#398—41no—
37#396—41no—
38#389—21no—
39#389—41no—
40#385—41no—
41#384C2×C411yescharacteristic
42#384D411yescharacteristic
43#384C2311yescharacteristic
44#382C2×C411yescharacteristic
45#382D411yescharacteristic
46#380—21no—
47#340—21no—
48#340—21no—
49#340—21no—
50#340—21no—
51#336—21no—
52#324—22no—
53#324—22no—
54#324—41no—
55#323—41no—
56C23×C4—21no—
57C23×C4C22⋊C411yescharacteristic
58C22×C8—21no—
59C22×C8—21no—
60C22×C8—81no—
61#125—22no—
62#125—41no—
63#117—22no—
64#117—41no—
65#117—81no—
66C2×C4⋊C4—21no—
67C2×C4⋊C4—21no—
68C2×C4⋊C4—21no—
69C2×C4⋊C4—21no—
70C2×C4⋊C4—41no—
71C2×C4⋊C4—41no—
72C2×C4⋊C4—41no—
73C2×C4⋊C4—41no—
74C2×C4⋊C4—41no—
75C2×C4⋊C4—81no—
76C2×C4⋊C4C22⋊C411yescharacteristic
77C2×C4⋊C4C22⋊C411yescharacteristic
78C2×C4⋊C4C2×D411yescharacteristic
79C2×C4⋊C4C2×D411yescharacteristic
80C2×C22⋊C4—21no—
81C2×C22⋊C4—21no—
82C2×C22⋊C4—41no—
83C2×C42—21no—
84C2×C42C22⋊C411yescharacteristic
85C2×C42C22×C411yescharacteristic
86#107—81no—
87#106—81no—
88#106—81no—
89#106—81no—
90C4⋊C8—81no—
91C22⋊C8—24no—
92C22⋊C8—42no—
93C22⋊C8—42no—
94C22⋊C8—81no—
95#97—41no—
96#97—81no—
97C4×C8—22no—
98#95—41no—
99#95—41no—
100#95—41no—
101#95—41no—
102C24#9911yescharacteristic
103C22×C4—22no—
104C22×C4—21no—
105C22×C4—21no—
106C22×C4—21no—
107C22×C4—21no—
108C22×C4—21no—
109C22×C4—21no—
110C22×C4—21no—
111C22×C4—21no—
112C22×C4—41no—
113C22×C4—41no—
114C22×C4—41no—
115C22×C4—41no—
116C22×C4—41no—
117C22×C4—161no—
118C22×C4#9911yescharacteristic
119C22×C4C2×C22⋊C411yescharacteristic
120C2×C8—22no—
121C2×C8—22no—
122C2×C8—41no—
123C2×C8—41no—
124C2×C8—41no—
125C2×C8—41no—
126C2×C8—82no—
127C2×C8—82no—
128C2×C8—162no—
129C4⋊C4—42no—
130C4⋊C4—41no—
131C4⋊C4—41no—
132C4⋊C4—41no—
133C4⋊C4—41no—
134C4⋊C4—41no—
135C4⋊C4—41no—
136C4⋊C4—41no—
137C4⋊C4—41no—
138C4⋊C4—81no—
139C4⋊C4—82no—
140C4⋊C4—161no—
141C4⋊C4—161no—
142C4⋊C4—161no—
143C4⋊C4—161no—
144C22⋊C4—42no—
145C22⋊C4—81no—
146C22⋊C4—161no—
147C42—21no—
148C42—21no—
149C42—21no—
150C42—41no—
151C42—41no—
152C42#13112yes—
153C23—21no—
154C23—21no—
155C23—41no—
156C23—41no—
157C23#35612yes—
158C23C2×#9911yescharacteristic
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—22no—
164C2×C4—21no—
165C2×C4—42no—
166C2×C4—42no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—82no—
177C2×C4—81no—
178C2×C4—81no—
179C2×C4—81no—
180C2×C4—81no—
181C2×C4—81no—
182C2×C4—81no—
183C2×C4—81no—
184C2×C4—81no—
185C2×C4—81no—
186C2×C4—81no—
187C2×C4—161no—
188C2×C4#40811yescharacteristic
189C2×C4#41311yescharacteristic
190C8—42no—
191C8—42no—
192C8—162no—
193C8—162no—
194C22—21no—
195C22—21no—
196C22—22no—
197C22—42no—
198C22—41no—
199C22—41no—
200C22—81no—
201C22#146811yescharacteristic
202C22#212711yescharacteristic
203C22C2×#35611yescharacteristic
204C4—21no—
205C4—21no—
206C4—42no—
207C4—41no—
208C4—81no—
209C4—81no—
210C4—81no—
211C4—81no—
212C4—81no—
213C4—161no—
214C4—161no—
215C2—21no—
216C2—21no—
217C2—42no—
218C2#1155511yescharacteristic
219C2#1361011yescharacteristic
220C2#1361111yescharacteristic
subgroupG/H
center, socleC22#2127
commutatorC2×C42C22×C4
Frattini, cosocle#384C23
prime radicalC24#99
LCSG ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23 ◁ C23×C4 ◁ #2317 ◁ G
LECSG ▷ #384 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7371C2semidirect
5. #7358C2semidirect

Maximal subgroups

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

subgroupindex
#279572
#115162
#114482
#73712
#73582