Finite Group Databasebeta

← #124963

Subgroups — #124963

Order 512.

342 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 90 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#21156C211yesmaximal, characteristic
2#12786C211yesmaximal, characteristic
3#8602C211yesmaximal, characteristic
4#7127C214yesmaximal, complemented
5C2×#427C411yescharacteristic
6C22×#107C411yescharacteristic
7C2×#405C2211yescharacteristic
8#2317—21no—
9#2150—21no—
10C2×#349—21no—
11#1944C412yes—
12#1944C2212yes—
13#1942—21no—
14#1942—21no—
15#1736—21no—
16#1564—21no—
17#1458—21no—
18#1377C2212yes—
19#1275C2212yes—
20C22×M16D411yescharacteristic
21C23×C8C2×C411yescharacteristic
22C2×#117—21no—
23C2×#117D411yescharacteristic
24C22×C4⋊C4—21no—
25C2×M32—22no—
26#431—22no—
27#427—24no—
28C2×#107—21no—
29C2×#107—21no—
30C2×#107—41no—
31C2×#107—41no—
32C2×#107C2×C412yes—
33C2×#106C2×C412yes—
34#405—41no—
35#405—41no—
36C2×#97C2×C411yescharacteristic
37C2×#97C2311yescharacteristic
38C2×C4×C8D411yescharacteristic
39C2×C4×C8D411yescharacteristic
40#389—22no—
41#389—22no—
42#385—41no—
43#385—41no—
44#384—41no—
45#382—41no—
46C4⋊C16—41no—
47C4⋊C16—41no—
48#349—42no—
49#340—22no—
50#340—22no—
51#335—41no—
52#334—41no—
53#324—41no—
54#323—41no—
55C23×C4—41no—
56C23×C4C22⋊C411yescharacteristic
57C2×M16—41no—
58C2×M16—41no—
59C2×M16SD1612yes—
60C2×M16Q1612yes—
61C22×C8—22no—
62C22×C8—21no—
63C22×C8—21no—
64C22×C8—22no—
65C22×C8C22⋊C411yescharacteristic
66C22×C8C22⋊C411yescharacteristic
67C22×C8C22×C411yescharacteristic
68C22×C8C2×D411yescharacteristic
69#117—24no—
70#117—41no—
71#117—41no—
72#117—42no—
73C2×C4⋊C4—22no—
74C2×C4⋊C4—22no—
75C2×C4⋊C4—42no—
76C2×C4⋊C4—42no—
77C2×C4⋊C4—41no—
78C2×C4⋊C4—42no—
79C2×C4⋊C4—42no—
80C2×C4⋊C4D811yescharacteristic
81C2×C4⋊C4SD1611yescharacteristic
82C2×C22⋊C4—41no—
83C2×C22⋊C4D811yescharacteristic
84C2×C22⋊C4SD1611yescharacteristic
85C2×C42—41no—
86C2×C42C22⋊C411yescharacteristic
87C2×C42C2×D411yescharacteristic
88M32—42no—
89M32—42no—
90C2×C16—42no—
91#107—24no—
92#107—41no—
93#107—41no—
94#107—42no—
95#106—22no—
96#106M1614yes—
97C4⋊C8—42no—
98C22⋊C8—42no—
99#97—41no—
100#97—41no—
101C4×C8—21no—
102C4×C8—21no—
103C4×C8—21no—
104C4×C8—21no—
105C4×C8—42no—
106#95—42no—
107#95—42no—
108C24#10211yescharacteristic
109C22×C4—21no—
110C22×C4—21no—
111C22×C4—21no—
112C22×C4—21no—
113C22×C4—42no—
114C22×C4—42no—
115C22×C4—82no—
116C22×C4—162no—
117C22×C4#10112yes—
118C22×C4#10211yescharacteristic
119C22×C4#10211yescharacteristic
120C22×C4#10211yescharacteristic
121C22×C4#10312yes—
122C22×C4C2×C22⋊C411yescharacteristic
123C22×C4C2×D811yescharacteristic
124C22×C4C2×SD1611yescharacteristic
125M16—42no—
126M16—42no—
127C2×C8—24no—
128C2×C8—24no—
129C2×C8—21no—
130C2×C8—21no—
131C2×C8—21no—
132C2×C8—21no—
133C2×C8—21no—
134C2×C8—21no—
135C2×C8—21no—
136C2×C8—21no—
137C2×C8—82no—
138C2×C8#9911yescharacteristic
139C2×C8#9911yescharacteristic
140C2×C8#10312yes—
141C2×C8C2×M1612yes—
142C2×C8C2×SD1611yescharacteristic
143C2×C8C2×Q1611yescharacteristic
144C4⋊C4—24no—
145C4⋊C4—41no—
146C4⋊C4—41no—
147C4⋊C4—42no—
148C4⋊C4—42no—
149C4⋊C4—42no—
150C4⋊C4—82no—
151C4⋊C4—162no—
152C4⋊C4—162no—
153C22⋊C4—41no—
154C22⋊C4—41no—
155C22⋊C4—44no—
156C42—21no—
157C42—21no—
158C42—21no—
159C42—21no—
160C42—44no—
161C16—82no—
162C23—21no—
163C23—21no—
164C23—41no—
165C23—41no—
166C23#33212yes—
167C23C2×#10211yescharacteristic
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—44no—
183C2×C4—82no—
184C2×C4—82no—
185C2×C4—82no—
186C2×C4—82no—
187C2×C4—82no—
188C2×C4—164no—
189C2×C4#33212yes—
190C2×C4#40711yescharacteristic
191C2×C4C2×#9911yescharacteristic
192C2×C4C2×#10111yescharacteristic
193C2×C4C2×#10311yescharacteristic
194C2×C4#42111yescharacteristic
195C2×C4#42111yescharacteristic
196C8—21no—
197C8—21no—
198C8—22no—
199C8—21no—
200C8—22no—
201C8—21no—
202C8—162no—
203C22—21no—
204C22—21no—
205C22—22no—
206C22—42no—
207C22—42no—
208C22—41no—
209C22—41no—
210C22#148611yescharacteristic
211C22#151911yescharacteristic
212C22C2×#33211yescharacteristic
213C4—22no—
214C4—41no—
215C4—41no—
216C4—41no—
217C4—41no—
218C4—84no—
219C4—162no—
220C4#146311yescharacteristic
221C4#148711yescharacteristic
222C4#152011yescharacteristic
223C4#153411yescharacteristic
224C2—21no—
225C2—21no—
226C2—42no—
227C2#764111yescharacteristic
228C2#1186811yescharacteristic
229C2#1187111yescharacteristic
subgroupG/H
center, socleC22#1519
commutatorC22×C8C22×C4
Frattini, cosocleC2×#97C23
prime radicalC24#102
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#405 ◁ G
LECSG ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7127C2semidirect

Maximal subgroups

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

subgroupindex
#211562
#127862
#86022
#71272