Finite Group Databasebeta

← #125182

Subgroups — #125182

Order 512.

398 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 118 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
2C2×#1374C211yesmaximal, characteristic
3#8596C211yesmaximal, characteristic
4#7120C214yesmaximal, complemented
5C23×C16—21no—
6C2×#500C411yescharacteristic
7C22×C4×C8C2211yescharacteristic
8#2857—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—24no—
26C22×C16—22no—
27C22×C16—42no—
28C22×C16—42no—
29#500—42no—
30C2×#107—21no—
31C2×#107—21no—
32C2×#107C2×C412yes—
33C2×#107C2×C412yes—
34C2×C4⋊C8—22no—
35C2×C22⋊C8—41no—
36C2×C4×C8—42no—
37C2×C4×C8C2×C411yescharacteristic
38C2×C4×C8D411yescharacteristic
39C2×C4×C8D411yescharacteristic
40C2×C4×C8C2311yescharacteristic
41#391—41no—
42#389—22no—
43#389—41no—
44#384—21no—
45#380—22no—
46#380—41no—
47#340—22no—
48#340—22no—
49#336—41no—
50#335—42no—
51C22×Q8—41no—
52C23×C4C22⋊C411yescharacteristic
53C23×C4D811yescharacteristic
54C23×C4SD1611yescharacteristic
55C22×C8—22no—
56C22×C8—22no—
57C22×C8—22no—
58C22×C8—22no—
59C22×C8—41no—
60C22×C8—42no—
61C22×C8C22⋊C411yescharacteristic
62C22×C8C22⋊C411yescharacteristic
63C22×C8SD1612yes—
64C22×C8Q1612yes—
65C22×C8C22×C411yescharacteristic
66C22×C8C2×D411yescharacteristic
67C4⋊Q8—42no—
68C4⋊Q8—42no—
69C22⋊Q8—82no—
70C22⋊Q8—82no—
71C2×C4⋊C4—22no—
72C2×C4⋊C4—22no—
73C2×C4⋊C4—41no—
74C2×C4⋊C4—41no—
75C2×C4⋊C4—41no—
76C2×C4⋊C4—42no—
77C2×C22⋊C4—42no—
78C2×C42—41no—
79C2×C42—41no—
80C2×C42C22⋊C411yescharacteristic
81C2×C42D811yescharacteristic
82C2×C42SD1611yescharacteristic
83C2×C42C2×D411yescharacteristic
84C2×C16—24no—
85C2×C16—42no—
86C2×C16—48no—
87C2×C16—42no—
88#107—24no—
89#107—42no—
90#107—42no—
91#107—42no—
92C4⋊C8—42no—
93C4⋊C8—42no—
94C22⋊C8—82no—
95C4×C8—22no—
96C4×C8—22no—
97C4×C8—41no—
98C4×C8—41no—
99#95—42no—
100#95—42no—
101C24#10211yescharacteristic
102C2×Q8—82no—
103C2×Q8—82no—
104C2×Q8—82no—
105C22×C4—21no—
106C22×C4—21no—
107C22×C4—21no—
108C22×C4—21no—
109C22×C4—22no—
110C22×C4—22no—
111C22×C4—41no—
112C22×C4—41no—
113C22×C4—41no—
114C22×C4—42no—
115C22×C4—42no—
116C22×C4#10112yes—
117C22×C4#10211yescharacteristic
118C22×C4#10211yescharacteristic
119C22×C4#10211yescharacteristic
120C22×C4#10312yes—
121C22×C4D1612yes—
122C22×C4SD3212yes—
123C22×C4C2×C22⋊C411yescharacteristic
124C22×C4C2×D811yescharacteristic
125C22×C4C2×SD1611yescharacteristic
126C2×C8—21no—
127C2×C8—21no—
128C2×C8—21no—
129C2×C8—21no—
130C2×C8—24no—
131C2×C8—24no—
132C2×C8—21no—
133C2×C8—21no—
134C2×C8—21no—
135C2×C8—44no—
136C2×C8—82no—
137C2×C8—82no—
138C2×C8—82no—
139C2×C8#9912yes—
140C2×C8#10312yes—
141C2×C8#10412yes—
142C2×C8#13112yes—
143C2×C8C2×SD1611yescharacteristic
144C2×C8C2×Q1611yescharacteristic
145C4⋊C4—42no—
146C4⋊C4—42no—
147C4⋊C4—42no—
148C4⋊C4—42no—
149C4⋊C4—82no—
150C4⋊C4—82no—
151C4⋊C4—82no—
152C4⋊C4—82no—
153C4⋊C4—82no—
154C22⋊C4—84no—
155C42—21no—
156C42—21no—
157C42—21no—
158C42—21no—
159C42—41no—
160C42—41no—
161C16—44no—
162C23—21no—
163C23—21no—
164C23—21no—
165C23—21no—
166C23#33212yes—
167C23#35712yes—
168C23#36212yes—
169C23C2×#10211yescharacteristic
170Q8—164no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—44no—
189C2×C4—82no—
190C2×C4—82no—
191C2×C4—82no—
192C2×C4—82no—
193C2×C4—82no—
194C2×C4—82no—
195C2×C4#33212yes—
196C2×C4#35712yes—
197C2×C4#35912yes—
198C2×C4#40811yescharacteristic
199C2×C4C2×#9911yescharacteristic
200C2×C4C2×#10111yescharacteristic
201C2×C4C2×#10311yescharacteristic
202C2×C4C2×#10411yescharacteristic
203C2×C4#42111yescharacteristic
204C2×C4C2×D1611yescharacteristic
205C2×C4C2×SD3211yescharacteristic
206C8—22no—
207C8—21no—
208C8—21no—
209C8—41no—
210C8—162no—
211C8#32814yes—
212C22—22no—
213C22—22no—
214C22—22no—
215C22—22no—
216C22—22no—
217C22—22no—
218C22#136414yes—
219C22#149211yescharacteristic
220C22#151911yescharacteristic
221C22C2×#33211yescharacteristic
222C22C2×#35711yescharacteristic
223C22C2×#35911yescharacteristic
224C22#214511yescharacteristic
225C22C2×#36211yescharacteristic
226C4—21no—
227C4—22no—
228C4—21no—
229C4—22no—
230C4—22no—
231C4—41no—
232C4—162no—
233C4—162no—
234C4#147211yescharacteristic
235C4#148111yescharacteristic
236C4C2×#32811yescharacteristic
237C4#153111yescharacteristic
238C2—24no—
239C2#766511yescharacteristic
240C2#1167111yescharacteristic
241C2#1167911yescharacteristic
242C2#1188611yescharacteristic
243C2#1189211yescharacteristic
244C2#1193411yescharacteristic
245C2C2×#136411yescharacteristic
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
C2×#13742
#85962
#71202