Finite Group Databasebeta

← #149811

Subgroups — #149811

Order 512.

356 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 62 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#20797C211yesmaximal, characteristic
2#11513C212yesmaximal
3#7371C212yesmaximal, complemented
4#7370C212yesmaximal, complemented
5C2×#427—22no—
6C22×C4×C8C411yescharacteristic
7#2285C411yescharacteristic
8#2285C2211yescharacteristic
9#1846—22no—
10#1845C411yescharacteristic
11#1845C2211yescharacteristic
12#1844C411yescharacteristic
13#1844C2211yescharacteristic
14#1464—22no—
15#1325C2214yes—
16#1324—22no—
17C23×C8—21no—
18C2×#117—21no—
19C2×#117D412yes—
20C22×C42C2×C411yescharacteristic
21#432—42no—
22#427—44no—
23#427—44no—
24C2×#107—22no—
25C2×#107—22no—
26C2×#106—22no—
27C2×#106—22no—
28C2×C4×C8—21no—
29C2×C4×C8—22no—
30C2×C4×C8—22no—
31C2×C4×C8C2×C411yescharacteristic
32C2×C4×C8C2×C411yescharacteristic
33#384C2×C411yescharacteristic
34#384C2×C411yescharacteristic
35#384C2×C411yescharacteristic
36#384C2311yescharacteristic
37#382—21no—
38#382D412yes—
39#380—41no—
40#380—41no—
41C4×C4⋊C4—41no—
42C4×C4⋊C4—41no—
43#324—24no—
44#323—42no—
45C23×C4—21no—
46C23×C4C22⋊C411yescharacteristic
47C22×C8—24no—
48C22×C8—24no—
49C22×C8—22no—
50C22×C8—22no—
51C22×C8—21no—
52C22×C8—21no—
53#117—42no—
54#117—42no—
55#117—42no—
56#117—42no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4—21no—
59C2×C4⋊C4—41no—
60C2×C4⋊C4—41no—
61C2×C4⋊C4—41no—
62C2×C4⋊C4—41no—
63C2×C4⋊C4C22⋊C412yes—
64C2×C4⋊C4C2×D412yes—
65C2×C22⋊C4—22no—
66C2×C22⋊C4—41no—
67C2×C42—21no—
68C2×C42—22no—
69C2×C42—21no—
70C2×C42—21no—
71C2×C42—22no—
72C2×C42—41no—
73C2×C42C22⋊C411yescharacteristic
74C2×C42C22×C411yescharacteristic
75#107—44no—
76#107—44no—
77#106—44no—
78#106—44no—
79C4⋊C8—82no—
80C22⋊C8—44no—
81#97—82no—
82C4×C8—24no—
83C4×C8—21no—
84C4×C8—21no—
85C4×C8M1612yes—
86C4×C8M1612yes—
87#95—42no—
88#95—42no—
89C24#9911yescharacteristic
90C22×C4—22no—
91C22×C4—21no—
92C22×C4—21no—
93C22×C4—21no—
94C22×C4—21no—
95C22×C4—21no—
96C22×C4—22no—
97C22×C4—22no—
98C22×C4—21no—
99C22×C4—22no—
100C22×C4—22no—
101C22×C4—42no—
102C22×C4—42no—
103C22×C4—41no—
104C22×C4—41no—
105C22×C4#9911yescharacteristic
106C22×C4C2×C22⋊C411yescharacteristic
107C2×C8—22no—
108C2×C8—22no—
109C2×C8—28no—
110C2×C8—28no—
111C2×C8—22no—
112C2×C8—22no—
113C2×C8—22no—
114C2×C8—22no—
115C2×C8—84no—
116C4⋊C4—42no—
117C4⋊C4—42no—
118C4⋊C4—42no—
119C4⋊C4—42no—
120C4⋊C4—82no—
121C4⋊C4—82no—
122C4⋊C4—82no—
123C4⋊C4—82no—
124C22⋊C4—82no—
125C22⋊C4—82no—
126C42—22no—
127C42—22no—
128C42—21no—
129C42—21no—
130C42—42no—
131C42—42no—
132C42—81no—
133C42—81no—
134C42#10112yes—
135C42C2×M1611yescharacteristic
136C42C2×M1611yescharacteristic
137C23—21no—
138C23—21no—
139C23—41no—
140C23—41no—
141C23#35612yes—
142C23C2×#9911yescharacteristic
143C2×C4—22no—
144C2×C4—22no—
145C2×C4—22no—
146C2×C4—22no—
147C2×C4—22no—
148C2×C4—22no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—22no—
156C2×C4—44no—
157C2×C4—44no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—42no—
161C2×C4—42no—
162C2×C4—82no—
163C2×C4—82no—
164C2×C4—82no—
165C2×C4—82no—
166C2×C4—81no—
167C2×C4—81no—
168C2×C4#35311yescharacteristic
169C2×C4#35311yescharacteristic
170C2×C4#40711yescharacteristic
171C2×C4C2×#10111yescharacteristic
172C8—24no—
173C8—24no—
174C8—164no—
175C22—21no—
176C22—21no—
177C22—22no—
178C22—42no—
179C22—42no—
180C22—41no—
181C22—41no—
182C22#146311yescharacteristic
183C22C2×#35311yescharacteristic
184C22C2×#35611yescharacteristic
185C4—22no—
186C4—22no—
187C4—21no—
188C4—21no—
189C4—21no—
190C4—21no—
191C4—42no—
192C4—82no—
193C4—82no—
194C4—161no—
195C4—161no—
196C2—21no—
197C2—21no—
198C2—42no—
199C2#1151211yescharacteristic
200C2#1358311yescharacteristic
201C2#1358911yescharacteristic
subgroupG/H
center, socleC22C2×#353
commutatorC2×C42C22×C4
Frattini, cosocle#384C23
prime radicalC24#99
LCSG ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ #2285 ◁ G
LECSG ▷ #384 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #7371C2semidirect
4. #7370C2semidirect

Maximal subgroups

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

subgroupindex
#207972
#115132
#73712
#73702