Finite Group Databasebeta

← #125607

Subgroups — #125607

Order 512.

352 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 352 conjugacy classes of subgroups — too many to draw. Only its 94 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#33209C211yesmaximal, characteristic
2#12844C211yesmaximal, characteristic
3#8596C211yesmaximal, characteristic
4#7120C214yesmaximal, complemented
5C2×#500C411yescharacteristic
6C22×C4×C8C2211yescharacteristic
7#2857—21no—
8C2×C22⋊C16—21no—
9#1990—21no—
10#1942—21no—
11#1942C2212yes—
12#1845C411yescharacteristic
13#1808—21no—
14#1808C412yes—
15#1797—21no—
16C2×#335—21no—
17#1374—21no—
18#1374C2212yes—
19#1277C2212yes—
20C23×C8C2×C411yescharacteristic
21C23×C8D411yescharacteristic
22C2×C4⋊Q8—21no—
23C2×C22⋊Q8—41no—
24C22×C42D411yescharacteristic
25C22×C16—22no—
26#500—42no—
27C2×#107—21no—
28C2×#107—21no—
29C2×#107C2×C412yes—
30C2×#107C2×C412yes—
31C2×C4⋊C8—22no—
32C2×C22⋊C8—41no—
33C2×C4×C8—42no—
34C2×C4×C8C2×C411yescharacteristic
35C2×C4×C8D411yescharacteristic
36C2×C4×C8D411yescharacteristic
37C2×C4×C8C2311yescharacteristic
38#391—41no—
39#389—22no—
40#389—41no—
41#384—21no—
42#380—22no—
43#380—41no—
44C22⋊C16—42no—
45#340—22no—
46#340—22no—
47#336—41no—
48#335—42no—
49C22×Q8—41no—
50C23×C4C22⋊C411yescharacteristic
51C23×C4D811yescharacteristic
52C23×C4SD1611yescharacteristic
53C22×C8—22no—
54C22×C8—22no—
55C22×C8—22no—
56C22×C8—22no—
57C22×C8—41no—
58C22×C8—41no—
59C22×C8—42no—
60C22×C8C22⋊C411yescharacteristic
61C22×C8C22⋊C411yescharacteristic
62C22×C8C22×C411yescharacteristic
63C22×C8C2×D411yescharacteristic
64C4⋊Q8—42no—
65C4⋊Q8—42no—
66C22⋊Q8—82no—
67C22⋊Q8—82no—
68C2×C4⋊C4—22no—
69C2×C4⋊C4—22no—
70C2×C4⋊C4—41no—
71C2×C4⋊C4—41no—
72C2×C4⋊C4—41no—
73C2×C4⋊C4—42no—
74C2×C22⋊C4—42no—
75C2×C42—41no—
76C2×C42—41no—
77C2×C42C22⋊C411yescharacteristic
78C2×C42D811yescharacteristic
79C2×C42SD1611yescharacteristic
80C2×C42C2×D411yescharacteristic
81C2×C16—42no—
82C2×C16—42no—
83C2×C16—42no—
84#107—24no—
85#107—42no—
86#107—42no—
87#107—42no—
88C4⋊C8—42no—
89C4⋊C8—42no—
90C22⋊C8—82no—
91C4×C8—22no—
92C4×C8—22no—
93C4×C8—41no—
94C4×C8—41no—
95#95—42no—
96#95—42no—
97C24#10211yescharacteristic
98C2×Q8—82no—
99C2×Q8—82no—
100C2×Q8—82no—
101C22×C4—22no—
102C22×C4—21no—
103C22×C4—21no—
104C22×C4—21no—
105C22×C4—21no—
106C22×C4—21no—
107C22×C4—21no—
108C22×C4—41no—
109C22×C4—41no—
110C22×C4—41no—
111C22×C4—41no—
112C22×C4—42no—
113C22×C4—42no—
114C22×C4#10211yescharacteristic
115C22×C4#10211yescharacteristic
116C22×C4#10211yescharacteristic
117C22×C4D1612yes—
118C22×C4SD3212yes—
119C22×C4C2×C22⋊C411yescharacteristic
120C22×C4C2×D811yescharacteristic
121C22×C4C2×SD1611yescharacteristic
122C2×C8—21no—
123C2×C8—21no—
124C2×C8—21no—
125C2×C8—21no—
126C2×C8—21no—
127C2×C8—21no—
128C2×C8—21no—
129C2×C8—21no—
130C2×C8—44no—
131C2×C8—44no—
132C2×C8—82no—
133C2×C8—82no—
134C2×C8—82no—
135C2×C8#9912yes—
136C2×C8#10112yes—
137C2×C8#13112yes—
138C2×C8#13712yes—
139C4⋊C4—42no—
140C4⋊C4—42no—
141C4⋊C4—42no—
142C4⋊C4—42no—
143C4⋊C4—82no—
144C4⋊C4—82no—
145C4⋊C4—82no—
146C4⋊C4—82no—
147C4⋊C4—82no—
148C22⋊C4—84no—
149C42—21no—
150C42—21no—
151C42—21no—
152C42—21no—
153C42—41no—
154C42—41no—
155C16—82no—
156C23—21no—
157C23—21no—
158C23—21no—
159C23—21no—
160C23—21no—
161C23—21no—
162C23#35712yes—
163C23C2×#10211yescharacteristic
164Q8—164no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—42no—
178C2×C4—42no—
179C2×C4—44no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—82no—
185C2×C4—82no—
186C2×C4—82no—
187C2×C4—82no—
188C2×C4—82no—
189C2×C4—82no—
190C2×C4#35712yes—
191C2×C4#36212yes—
192C2×C4#40811yescharacteristic
193C2×C4C2×#9911yescharacteristic
194C2×C4C2×#10111yescharacteristic
195C2×C4#41911yescharacteristic
196C2×C4#42111yescharacteristic
197C2×C4#42111yescharacteristic
198C2×C4C2×D1611yescharacteristic
199C2×C4C2×SD3211yescharacteristic
200C8—22no—
201C8—22no—
202C8—22no—
203C8—22no—
204C8—162no—
205C22—22no—
206C22—22no—
207C22—22no—
208C22—41no—
209C22—41no—
210C22—41no—
211C22—41no—
212C22#148011yescharacteristic
213C22#151911yescharacteristic
214C22#154611yescharacteristic
215C22C2×#35711yescharacteristic
216C22#214311yescharacteristic
217C22#214511yescharacteristic
218C22C2×#36211yescharacteristic
219C4—21no—
220C4—22no—
221C4—21no—
222C4—22no—
223C4—41no—
224C4—41no—
225C4—162no—
226C4—162no—
227C4#147211yescharacteristic
228C4#149711yescharacteristic
229C4#151111yescharacteristic
230C4#153711yescharacteristic
231C2—42no—
232C2#772811yescharacteristic
233C2#1164511yescharacteristic
234C2#1165311yescharacteristic
235C2#1188611yescharacteristic
236C2#1189611yescharacteristic
237C2#1197011yescharacteristic
238C2#1197111yescharacteristic
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. #7120C2semidirect

Maximal subgroups

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

subgroupindex
#332092
#128442
#85962
#71202