Finite Group Databasebeta

← #125337

Subgroups — #125337

Order 512.

410 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 114 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
2C4×C22⋊C16C211yesmaximal, characteristic
3#7874C211yesmaximal, characteristic
4#7120C214yesmaximal, complemented
5C2×#500C411yescharacteristic
6C2×#423—21no—
7C22×C4×C8C2211yescharacteristic
8#2857—21no—
9C2×C22⋊C16—21no—
10C2×C4×C16—21no—
11#1990—21no—
12#1942—21no—
13#1942C2212yes—
14#1845C411yescharacteristic
15#1808—21no—
16#1808C412yes—
17#1374C2212yes—
18#1277—21no—
19#1277C2212yes—
20C22×M16—41no—
21C23×C8C2×C411yescharacteristic
22C23×C8D411yescharacteristic
23C2×C4⋊Q8—21no—
24C2×C22⋊Q8—41no—
25C22×C42D411yescharacteristic
26C22×C16—22no—
27#500—42no—
28C2×#107—21no—
29C2×#107—21no—
30C2×#107C2×C412yes—
31C2×#107C2×C412yes—
32#423—24no—
33#423—42no—
34C2×C4⋊C8—22no—
35C2×C4⋊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—
47C22⋊C16—24no—
48C4×C16—42no—
49#340—22no—
50#340—22no—
51C22×Q8—41no—
52C23×C4C22⋊C411yescharacteristic
53C23×C4D811yescharacteristic
54C23×C4SD1611yescharacteristic
55C2×M16—44no—
56C2×M16—82no—
57C2×M16—82no—
58C22×C8—22no—
59C22×C8—22no—
60C22×C8—22no—
61C22×C8—22no—
62C22×C8—41no—
63C22×C8—41no—
64C22×C8—42no—
65C22×C8C22⋊C411yescharacteristic
66C22×C8C22⋊C411yescharacteristic
67C22×C8C22×C411yescharacteristic
68C22×C8C2×D411yescharacteristic
69C4⋊Q8—42no—
70C4⋊Q8—42no—
71C22⋊Q8—82no—
72C22⋊Q8—82no—
73C2×C4⋊C4—22no—
74C2×C4⋊C4—22no—
75C2×C4⋊C4—41no—
76C2×C4⋊C4—41no—
77C2×C4⋊C4—41no—
78C2×C4⋊C4—42no—
79C2×C22⋊C4—42no—
80C2×C42—22no—
81C2×C42—22no—
82C2×C42C22⋊C411yescharacteristic
83C2×C42D811yescharacteristic
84C2×C42SD1611yescharacteristic
85C2×C42C2×D411yescharacteristic
86C2×C16—24no—
87C2×C16—42no—
88C2×C16—42no—
89#107—24no—
90#107—42no—
91#107—42no—
92#107—42no—
93C4⋊C8—24no—
94C4⋊C8—42no—
95C4⋊C8—44no—
96C4×C8—22no—
97C4×C8—22no—
98C4×C8—41no—
99C4×C8—41no—
100#95—42no—
101#95—42no—
102C24#10211yescharacteristic
103C2×Q8—82no—
104C2×Q8—82no—
105C2×Q8—82no—
106C22×C4—22no—
107C22×C4—22no—
108C22×C4—22no—
109C22×C4—22no—
110C22×C4—22no—
111C22×C4—21no—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—21no—
116C22×C4—21no—
117C22×C4—42no—
118C22×C4—42no—
119C22×C4#10211yescharacteristic
120C22×C4#10211yescharacteristic
121C22×C4#10211yescharacteristic
122C22×C4D1612yes—
123C22×C4SD3212yes—
124C22×C4C2×C22⋊C411yescharacteristic
125C22×C4C2×D811yescharacteristic
126C22×C4C2×SD1611yescharacteristic
127M16—88no—
128C2×C8—21no—
129C2×C8—21no—
130C2×C8—21no—
131C2×C8—21no—
132C2×C8—21no—
133C2×C8—21no—
134C2×C8—21no—
135C2×C8—21no—
136C2×C8—44no—
137C2×C8—44no—
138C2×C8—44no—
139C2×C8—82no—
140C2×C8—82no—
141C2×C8#10112yes—
142C2×C8#10412yes—
143C2×C8#13112yes—
144C2×C8#13712yes—
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—22no—
156C42—22no—
157C42—21no—
158C42—21no—
159C42#9912yes—
160C42#10412yes—
161C16—44no—
162C23—21no—
163C23—21no—
164C23—21no—
165C23—21no—
166C23—21no—
167C23—21no—
168C23#35712yes—
169C23C2×#10211yescharacteristic
170Q8—164no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—24no—
174C2×C4—24no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—44no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—82no—
187C2×C4—82no—
188C2×C4—82no—
189C2×C4—82no—
190C2×C4—82no—
191C2×C4—82no—
192C2×C4#32712yes—
193C2×C4#32712yes—
194C2×C4#35712yes—
195C2×C4#35912yes—
196C2×C4#36012yes—
197C2×C4#36212yes—
198C2×C4#40811yescharacteristic
199C2×C4C2×#9911yescharacteristic
200C2×C4C2×#10111yescharacteristic
201C2×C4#41911yescharacteristic
202C2×C4C2×#10411yescharacteristic
203C2×C4C2×#10411yescharacteristic
204C2×C4C2×D1611yescharacteristic
205C2×C4C2×SD3211yescharacteristic
206C8—22no—
207C8—22no—
208C8—22no—
209C8—22no—
210C8—84no—
211C22—22no—
212C22—22no—
213C22—22no—
214C22—41no—
215C22—41no—
216C22—41no—
217C22—41no—
218C22#149211yescharacteristic
219C22C2×#32711yescharacteristic
220C22#154611yescharacteristic
221C22C2×#35711yescharacteristic
222C22C2×#35911yescharacteristic
223C22C2×#36011yescharacteristic
224C22C2×#36211yescharacteristic
225C4—21no—
226C4—21no—
227C4—41no—
228C4—41no—
229C4—162no—
230C4—162no—
231C4#134714yes—
232C4#135014yes—
233C4#147211yescharacteristic
234C4#148511yescharacteristic
235C4#150811yescharacteristic
236C4#152611yescharacteristic
237C2—42no—
238C2#768511yescharacteristic
239C2#1167111yescharacteristic
240C2#1168111yescharacteristic
241C2C2×#134711yescharacteristic
242C2C2×#135011yescharacteristic
243C2#1196811yescharacteristic
244C2#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
C4×C22⋊C162
#78742
#71202