Finite Group Databasebeta

← #125574

Subgroups — #125574

Order 512.

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

normal subgroups only

This group has 396 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#33172C211yesmaximal, characteristic
2#12844C211yesmaximal, characteristic
3#8595C211yesmaximal, characteristic
4#7113C214yesmaximal, complemented
5C2×#494C411yescharacteristic
6C2×#467—21no—
7C2×#467C412yes—
8C22×C4×C8C2211yescharacteristic
9#2857—21no—
10C2×C22⋊C16—21no—
11#1969—21no—
12#1969C2212yes—
13#1956—21no—
14#1845C411yescharacteristic
15#1797—21no—
16C2×#334—21no—
17#1374—21no—
18#1374C2212yes—
19#1277C2212yes—
20C22×Q16—21no—
21C22×Q16C2×C412yes—
22C23×C8C2×C411yescharacteristic
23C23×C8D411yescharacteristic
24C2×C4⋊Q8—21no—
25C2×C22⋊Q8—22no—
26C22×C42D411yescharacteristic
27C22×C16—22no—
28#494—42no—
29#467—44no—
30#467—44no—
31C2×#107—21no—
32C2×#107C2×C412yes—
33C2×C4⋊C8—22no—
34C2×#103—22no—
35C2×#103—22no—
36C2×C22⋊C8—41no—
37C2×C4×C8—42no—
38C2×C4×C8C2×C411yescharacteristic
39C2×C4×C8D411yescharacteristic
40C2×C4×C8D411yescharacteristic
41C2×C4×C8C2311yescharacteristic
42#391—22no—
43#389—41no—
44#389—41no—
45#384—21no—
46#380—41no—
47#380—41no—
48C22⋊C16—42no—
49#336—41no—
50#334—42no—
51C22×Q8—22no—
52C23×C4C22⋊C411yescharacteristic
53C23×C4D811yescharacteristic
54C23×C4SD1611yescharacteristic
55C2×Q16—24no—
56C2×Q16—42no—
57C2×Q16—42no—
58C2×Q16—42no—
59C2×Q16—42no—
60C2×Q16—42no—
61C22×C8—22no—
62C22×C8—22no—
63C22×C8—22no—
64C22×C8—22no—
65C22×C8—41no—
66C22×C8—41no—
67C22×C8—42no—
68C22×C8C22⋊C411yescharacteristic
69C22×C8C22⋊C411yescharacteristic
70C22×C8C22×C411yescharacteristic
71C22×C8C2×D411yescharacteristic
72C4⋊Q8—42no—
73C4⋊Q8—42no—
74C22⋊Q8—44no—
75C22⋊Q8—44no—
76C2×C4⋊C4—22no—
77C2×C4⋊C4—42no—
78C2×C4⋊C4—41no—
79C2×C4⋊C4—41no—
80C2×C4⋊C4—41no—
81C2×C4⋊C4—41no—
82C2×C22⋊C4—42no—
83C2×C42—41no—
84C2×C42—41no—
85C2×C42C22⋊C411yescharacteristic
86C2×C42D811yescharacteristic
87C2×C42SD1611yescharacteristic
88C2×C42C2×D411yescharacteristic
89C2×C16—42no—
90C2×C16—42no—
91C2×C16—42no—
92#107—42no—
93#107—42no—
94C4⋊C8—42no—
95C4⋊C8—42no—
96#103—44no—
97#103—44no—
98C22⋊C8—82no—
99C4×C8—22no—
100C4×C8—22no—
101C4×C8—41no—
102C4×C8—41no—
103#95—42no—
104#95—42no—
105C24#10211yescharacteristic
106C2×Q8—42no—
107C2×Q8—42no—
108C2×Q8—42no—
109C2×Q8—42no—
110C2×Q8—42no—
111C2×Q8—42no—
112C22×C4—22no—
113C22×C4—22no—
114C22×C4—22no—
115C22×C4—21no—
116C22×C4—21no—
117C22×C4—21no—
118C22×C4—21no—
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—22no—
122C22×C4—41no—
123C22×C4—41no—
124C22×C4—42no—
125C22×C4—42no—
126C22×C4#10211yescharacteristic
127C22×C4#10211yescharacteristic
128C22×C4#10211yescharacteristic
129C22×C4C2×C22⋊C411yescharacteristic
130C22×C4C2×D811yescharacteristic
131C22×C4C2×SD1611yescharacteristic
132Q16—84no—
133Q16—84no—
134C2×C8—21no—
135C2×C8—21no—
136C2×C8—21no—
137C2×C8—21no—
138C2×C8—21no—
139C2×C8—21no—
140C2×C8—21no—
141C2×C8—21no—
142C2×C8—44no—
143C2×C8—44no—
144C2×C8—82no—
145C2×C8—82no—
146C2×C8—82no—
147C2×C8#9912yes—
148C2×C8#10012yes—
149C2×C8#13112yes—
150C2×C8AGL(1,ℤ/8)12yes—
151C4⋊C4—42no—
152C4⋊C4—42no—
153C4⋊C4—84no—
154C4⋊C4—82no—
155C4⋊C4—82no—
156C4⋊C4—82no—
157C4⋊C4—82no—
158C22⋊C4—84no—
159C42—21no—
160C42—21no—
161C42—21no—
162C42—21no—
163C42—41no—
164C42—41no—
165C16—82no—
166C23—21no—
167C23—21no—
168C23—21no—
169C23—21no—
170C23—21no—
171C23—21no—
172C23#36212yes—
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—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—44no—
191C2×C4—44no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—82no—
197C2×C4—82no—
198C2×C4—82no—
199C2×C4—82no—
200C2×C4—82no—
201C2×C4—82no—
202C2×C4#35912yes—
203C2×C4#36012yes—
204C2×C4#40811yescharacteristic
205C2×C4C2×#9911yescharacteristic
206C2×C4C2×#10011yescharacteristic
207C2×C4#41811yescharacteristic
208C2×C4#42111yescharacteristic
209C2×C4#42111yescharacteristic
210C2×C4#51012yes—
211C8—22no—
212C8—22no—
213C8—22no—
214C8—22no—
215C8—162no—
216C22—22no—
217C22—22no—
218C22—22no—
219C22—41no—
220C22—41no—
221C22—41no—
222C22—41no—
223C22#148011yescharacteristic
224C22#151911yescharacteristic
225C22#154511yescharacteristic
226C22#214111yescharacteristic
227C22C2×#35911yescharacteristic
228C22C2×#36011yescharacteristic
229C22C2×#36211yescharacteristic
230C4—22no—
231C4—22no—
232C4—22no—
233C4—41no—
234C4—41no—
235C4—162no—
236C4—162no—
237C4#147111yescharacteristic
238C4#149611yescharacteristic
239C4#151011yescharacteristic
240C4#153611yescharacteristic
241C2—42no—
242C2#772411yescharacteristic
243C2#1165011yescharacteristic
244C2#1165511yescharacteristic
245C2#1188911yescharacteristic
246C2#1189211yescharacteristic
247C2#1195811yescharacteristic
248C2#1196511yescharacteristic
subgroupG/H
center, socleC23C2×#102
commutatorC22×C8C22×C4
Frattini, cosocleC2×C4×C8C23
prime radicalC24#102
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G
LECSG ▷ C2×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7113C2semidirect

Maximal subgroups

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

subgroupindex
#331722
#128442
#85952
#71132