Finite Group Databasebeta

← #124274

Subgroups — #124274

Order 512.

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

normal subgroups only

This group has 390 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#21151C211yesmaximal, characteristic
2C4×C22⋊C16C211yesmaximal, characteristic
3#8595C211yesmaximal, characteristic
4#7121C212yesmaximal, complemented
5#7120C212yesmaximal, complemented
6C2×#427C411yescharacteristic
7C2×#427C411yescharacteristic
8C22×C4×C8C2211yescharacteristic
9#2296—21no—
10C2×C22⋊C16—21no—
11C2×C4×C16—21no—
12#1942C411yescharacteristic
13#1942C411yescharacteristic
14#1942C2211yescharacteristic
15#1942C2211yescharacteristic
16#1941—21no—
17#1941—21no—
18#1797—21no—
19#1735—21no—
20C2×#334—21no—
21#1374C2212yes—
22#1277C2212yes—
23C23×C8C2×C411yescharacteristic
24C23×C8D411yescharacteristic
25C2×#117—21no—
26C2×#117—21no—
27C22×C42D411yescharacteristic
28C22×C16—22no—
29#427—22no—
30#427—42no—
31#427C814yescomplemented
32C2×#107C2×C411yescharacteristic
33C2×#107C2×C411yescharacteristic
34C2×#107C2×C411yescharacteristic
35C2×#107C2×C411yescharacteristic
36C2×#106—21no—
37C2×#106—21no—
38C2×C4⋊C8—22no—
39C2×C22⋊C8—41no—
40C2×C4×C8—42no—
41C2×C4×C8C2×C411yescharacteristic
42C2×C4×C8D411yescharacteristic
43C2×C4×C8D411yescharacteristic
44C2×C4×C8C2311yescharacteristic
45#389—21no—
46#389—21no—
47#389—21no—
48#389—21no—
49#385—41no—
50#385—41no—
51C4×C4⋊C4—41no—
52C4×C4⋊C4—41no—
53C22⋊C16—24no—
54C4×C16—42no—
55#340—21no—
56#340—21no—
57#340—21no—
58#340—21no—
59#336—41no—
60#334—42no—
61C23×C4C22⋊C411yescharacteristic
62C23×C4D811yescharacteristic
63C23×C4SD1611yescharacteristic
64C22×C8—22no—
65C22×C8—22no—
66C22×C8—22no—
67C22×C8—22no—
68C22×C8—41no—
69C22×C8—42no—
70C22×C8C22⋊C411yescharacteristic
71C22×C8C22⋊C411yescharacteristic
72C22×C8C2×C812yes—
73C22×C8M1612yes—
74C22×C8C22×C411yescharacteristic
75C22×C8C2×D411yescharacteristic
76#117—24no—
77#117—42no—
78#117—42no—
79#117—42no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—41no—
85C2×C4⋊C4—41no—
86C2×C4⋊C4—41no—
87C2×C4⋊C4—41no—
88C2×C22⋊C4—41no—
89C2×C22⋊C4—41no—
90C2×C42—41no—
91C2×C42—41no—
92C2×C42—41no—
93C2×C42—41no—
94C2×C42C22⋊C411yescharacteristic
95C2×C42D811yescharacteristic
96C2×C42SD1611yescharacteristic
97C2×C42C2×D411yescharacteristic
98C2×C16—24no—
99C2×C16—42no—
100C2×C16—42no—
101#107—21no—
102#107—21no—
103#107—41no—
104#107—41no—
105#107C2×C812yes—
106#107C2×C812yes—
107#106—24no—
108#106—42no—
109C4⋊C8—42no—
110C4⋊C8—42no—
111C22⋊C8—82no—
112C4×C8—22no—
113C4×C8—22no—
114C4×C8—41no—
115C4×C8—41no—
116#95—41no—
117#95—41no—
118#95—41no—
119#95—41no—
120C24#10211yescharacteristic
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—22no—
126C22×C4—22no—
127C22×C4—22no—
128C22×C4—41no—
129C22×C4—41no—
130C22×C4—41no—
131C22×C4—41no—
132C22×C4—41no—
133C22×C4—41no—
134C22×C4—41no—
135C22×C4C22⋊C812yes—
136C22×C4#10211yescharacteristic
137C22×C4#10211yescharacteristic
138C22×C4#10211yescharacteristic
139C22×C4#10412yes—
140C22×C4C2×C22⋊C411yescharacteristic
141C22×C4C2×D811yescharacteristic
142C22×C4C2×SD1611yescharacteristic
143C2×C8—21no—
144C2×C8—21no—
145C2×C8—21no—
146C2×C8—21no—
147C2×C8—24no—
148C2×C8—24no—
149C2×C8—21no—
150C2×C8—21no—
151C2×C8—21no—
152C2×C8—44no—
153C2×C8—82no—
154C2×C8—82no—
155C2×C8—82no—
156C2×C8C22⋊C812yes—
157C2×C8#9912yes—
158C2×C8#10112yes—
159C2×C8C22×C811yescharacteristic
160C2×C8C2×M1611yescharacteristic
161C2×C8#13712yes—
162C4⋊C4—22no—
163C4⋊C4—22no—
164C4⋊C4—41no—
165C4⋊C4—41no—
166C4⋊C4—41no—
167C4⋊C4—41no—
168C4⋊C4—41no—
169C4⋊C4—41no—
170C4⋊C4—82no—
171C4⋊C4—82no—
172C4⋊C4—82no—
173C4⋊C4—82no—
174C22⋊C4—44no—
175C22⋊C4—82no—
176C42—21no—
177C42—21no—
178C42—21no—
179C42—21no—
180C42—41no—
181C42—41no—
182C42—44no—
183C42—82no—
184C16—44no—
185C23—21no—
186C23—21no—
187C23—21no—
188C23—21no—
189C23—21no—
190C23—21no—
191C23D4⋊C812yes—
192C23C2×#10211yescharacteristic
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—44no—
209C2×C4—44no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—81no—
213C2×C4—81no—
214C2×C4—81no—
215C2×C4—81no—
216C2×C4—81no—
217C2×C4—81no—
218C2×C4—81no—
219C2×C4—81no—
220C2×C4—81no—
221C2×C4—81no—
222C2×C4D4⋊C812yes—
223C2×C4#35912yes—
224C2×C4#36012yes—
225C2×C4C2×C22⋊C811yescharacteristic
226C2×C4C2×#9911yescharacteristic
227C2×C4C2×#10111yescharacteristic
228C2×C4#41911yescharacteristic
229C2×C4C2×#10411yescharacteristic
230C2×C4#42111yescharacteristic
231C2×C4#50812yes—
232C8—21no—
233C8—21no—
234C8—22no—
235C8—41no—
236C8—162no—
237C8#32414yes—
238C22—22no—
239C22—22no—
240C22—22no—
241C22—41no—
242C22—41no—
243C22—41no—
244C22—41no—
245C22C2×D4⋊C811yescharacteristic
246C22#151911yescharacteristic
247C22#154611yescharacteristic
248C22#213811yescharacteristic
249C22C2×#35911yescharacteristic
250C22C2×#36011yescharacteristic
251C22#214811yescharacteristic
252C4—22no—
253C4—22no—
254C4—22no—
255C4—22no—
256C4—41no—
257C4—82no—
258C4—82no—
259C4—161no—
260C4—161no—
261C4C2×#32411yescharacteristic
262C4#147911yescharacteristic
263C4#150811yescharacteristic
264C4#152811yescharacteristic
265C2—42no—
266C2#757311yescharacteristic
267C2#1159111yescharacteristic
268C2#1159811yescharacteristic
269C2#1188511yescharacteristic
270C2#1189311yescharacteristic
271C2#1196811yescharacteristic
272C2#1197311yescharacteristic
subgroupG/H
center, socleC23C2×#102
commutatorC2×C8C22×C8
Frattini, cosocleC2×C4×C8C23
prime radicalC24#102
LCSG ▷ C2×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. #7121C2semidirect
5. #7120C2semidirect
31. #427C8semidirect

Maximal subgroups

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

subgroupindex
#211512
C4×C22⋊C162
#85952
#71212
#71202