Finite Group Databasebeta

← #125565

Subgroups — #125565

Order 512.

348 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 74 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#33180C211yesmaximal, characteristic
2#12847C211yesmaximal, characteristic
3#8602C211yesmaximal, characteristic
4#7111C214yesmaximal, complemented
5C2×#494C411yescharacteristic
6C2×#470C412yes—
7C2×#467—21no—
8C2×#405C2211yescharacteristic
9#2621—21no—
10#2112—21no—
11#1971C2212yes—
12#1969—21no—
13#1960—21no—
14#1846C411yescharacteristic
15#1564—21no—
16#1458—21no—
17#1377—21no—
18#1377C2212yes—
19#1275C2212yes—
20C22×Q16—21no—
21C22×Q16C2×C412yes—
22C22×M16C2×C411yescharacteristic
23C23×C8D411yescharacteristic
24C2×C4⋊Q8—21no—
25C2×C22⋊Q8—22no—
26C2×#117D411yescharacteristic
27C2×M32—22no—
28#494—41no—
29#494—41no—
30#470—44no—
31#467—44no—
32#431—22no—
33C2×#107—21no—
34C2×#106C2×C412yes—
35C2×#103—22no—
36C2×#103—22no—
37#405—41no—
38#405—41no—
39C2×#97D411yescharacteristic
40C2×#97C2311yescharacteristic
41C2×C4×C8C2×C411yescharacteristic
42C2×C4×C8D411yescharacteristic
43#398—41no—
44#398—41no—
45#397—41no—
46#397—41no—
47#391—22no—
48#382—21no—
49C22⋊C16—41no—
50C22⋊C16—41no—
51#335—41no—
52#334—41no—
53#324—41no—
54#323—41no—
55C22×Q8—22no—
56C23×C4C22⋊C411yescharacteristic
57C2×Q16—24no—
58C2×Q16—42no—
59C2×Q16—42no—
60C2×Q16—42no—
61C2×Q16—42no—
62C2×Q16—41no—
63C2×Q16—41no—
64C2×M16—22no—
65C2×M16—22no—
66C2×M16—41no—
67C22×C8—21no—
68C22×C8—21no—
69C22×C8—22no—
70C22×C8—41no—
71C22×C8C22⋊C411yescharacteristic
72C22×C8C22⋊C411yescharacteristic
73C22×C8C22×C411yescharacteristic
74C22×C8C2×D411yescharacteristic
75C4⋊Q8—42no—
76C4⋊Q8—42no—
77C22⋊Q8—44no—
78C22⋊Q8—44no—
79#117—41no—
80#117—41no—
81C2×C4⋊C4—22no—
82C2×C4⋊C4—42no—
83C2×C4⋊C4—41no—
84C2×C4⋊C4—41no—
85C2×C4⋊C4D811yescharacteristic
86C2×C4⋊C4SD1611yescharacteristic
87C2×C22⋊C4—42no—
88C2×C22⋊C4D811yescharacteristic
89C2×C22⋊C4SD1611yescharacteristic
90C2×C42C22⋊C411yescharacteristic
91C2×C42C2×D411yescharacteristic
92M32—42no—
93M32—42no—
94C2×C16—42no—
95#107—42no—
96#106—42no—
97C4⋊C8—42no—
98#103—44no—
99#103—44no—
100C22⋊C8—42no—
101#97—41no—
102#97—41no—
103C4×C8—21no—
104C4×C8—21no—
105C4×C8—21no—
106C4×C8—21no—
107C4×C8—42no—
108#95—41no—
109#95—41no—
110#95—42no—
111#95—42no—
112C24#10211yescharacteristic
113C2×Q8—42no—
114C2×Q8—42no—
115C2×Q8—42no—
116C2×Q8—42no—
117C2×Q8—42no—
118C2×Q8—42no—
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—42no—
126C22×C4—42no—
127C22×C4#10211yescharacteristic
128C22×C4#10211yescharacteristic
129C22×C4#10211yescharacteristic
130C22×C4C2×C22⋊C411yescharacteristic
131C22×C4C2×D811yescharacteristic
132C22×C4C2×SD1611yescharacteristic
133Q16—84no—
134Q16—82no—
135Q16—82no—
136M16—44no—
137C2×C8—21no—
138C2×C8—21no—
139C2×C8—21no—
140C2×C8—21no—
141C2×C8—21no—
142C2×C8—21no—
143C2×C8—21no—
144C2×C8—21no—
145C2×C8—42no—
146C2×C8—42no—
147C2×C8—82no—
148C2×C8#9911yescharacteristic
149C2×C8#9911yescharacteristic
150C2×C8#10012yes—
151C2×C8#13112yes—
152C2×C8AGL(1,ℤ/8)12yes—
153C4⋊C4—41no—
154C4⋊C4—41no—
155C4⋊C4—42no—
156C4⋊C4—42no—
157C4⋊C4—84no—
158C4⋊C4—82no—
159C4⋊C4—82no—
160C22⋊C4—41no—
161C22⋊C4—41no—
162C22⋊C4—84no—
163C42—21no—
164C42—21no—
165C42—21no—
166C42—21no—
167C16—82no—
168C23—21no—
169C23—21no—
170C23—21no—
171C23—41no—
172C23—41no—
173C23C2×#10211yescharacteristic
174Q8—82no—
175Q8—82no—
176Q8—82no—
177Q8—82no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—82no—
194C2×C4—82no—
195C2×C4—82no—
196C2×C4—82no—
197C2×C4—82no—
198C2×C4—82no—
199C2×C4#40811yescharacteristic
200C2×C4C2×#9911yescharacteristic
201C2×C4C2×#10011yescharacteristic
202C2×C4#41811yescharacteristic
203C2×C4#42111yescharacteristic
204C2×C4#42111yescharacteristic
205C8—22no—
206C8—22no—
207C8—22no—
208C8—21no—
209C8—21no—
210C8—162no—
211C22—21no—
212C22—21no—
213C22—42no—
214C22—41no—
215C22—41no—
216C22—41no—
217C22—81no—
218C22#148011yescharacteristic
219C22#151911yescharacteristic
220C22#154511yescharacteristic
221C4—41no—
222C4—41no—
223C4—41no—
224C4—41no—
225C4—41no—
226C4—162no—
227C4—162no—
228C4#147111yescharacteristic
229C4#149611yescharacteristic
230C4#151011yescharacteristic
231C4#153611yescharacteristic
232C2—21no—
233C2—21no—
234C2—81no—
235C2#772411yescharacteristic
236C2#1186911yescharacteristic
237C2#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. #7111C2semidirect

Maximal subgroups

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

subgroupindex
#331802
#128472
#86022
#71112