Finite Group Databasebeta

← #124139

Subgroups — #124139

Order 512.

636 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 636 conjugacy classes of subgroups — too many to draw. Only its 106 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#13853C211yesmaximal, characteristic
2C4×#323C212yesmaximal
3#7132C212yesmaximal, complemented
4#7104C212yesmaximal, complemented
5C22×#117C411yescharacteristic
6C2×C4×C22⋊C4C411yescharacteristic
7C2×C4×C22⋊C4C2211yescharacteristic
8C4×C22⋊C8—22no—
9C2×#323—22no—
10#1445C411yescharacteristic
11#1445C411yescharacteristic
12#1445C2211yescharacteristic
13#1445C2211yescharacteristic
14#1440—22no—
15C4×#95—22no—
16#1280C2214yes—
17C24×C4C2×C411yescharacteristic
18C2×#117—22no—
19C2×#117—41no—
20C2×#117—41no—
21C2×#117—42no—
22C2×#117C814yescomplemented
23C22×C4⋊C4C2×C411yescharacteristic
24C22×C4⋊C4C2×C411yescharacteristic
25C22×C22⋊C4—21no—
26C22×C22⋊C4—21no—
27C22×C22⋊C4D412yes—
28C22×C42—21no—
29C22×C42—21no—
30C22×C42D412yes—
31C2×C22⋊C8—24no—
32C2×C4×C8—42no—
33C4×C22⋊C4—41no—
34C4×C22⋊C4—41no—
35C4×C22⋊C4—42no—
36C2×#95—22no—
37C2×#95—22no—
38C2×#95—22no—
39C2×#95—22no—
40C2×#95C2×C411yescharacteristic
41C2×#95C2×C411yescharacteristic
42C2×#95C2×C411yescharacteristic
43C2×#95C2311yescharacteristic
44#336—42no—
45#323—28no—
46C25C22⋊C411yescharacteristic
47C23×C4—21no—
48C23×C4—21no—
49C23×C4—22no—
50C23×C4—21no—
51C23×C4—21no—
52C23×C4—21no—
53C23×C4—21no—
54C23×C4—44no—
55C23×C4C22⋊C412yes—
56C23×C4C22⋊C411yescharacteristic
57C23×C4C2×C812yes—
58C23×C4M1612yes—
59C23×C4C22×C411yescharacteristic
60C23×C4C2×D412yes—
61C22×C8—44no—
62#117—44no—
63#117—44no—
64#117—44no—
65#117—44no—
66C2×C4⋊C4—21no—
67C2×C4⋊C4—21no—
68C2×C4⋊C4—41no—
69C2×C4⋊C4—41no—
70C2×C4⋊C4—41no—
71C2×C4⋊C4—41no—
72C2×C4⋊C4C2×C812yes—
73C2×C4⋊C4C2×C812yes—
74C2×C22⋊C4—24no—
75C2×C22⋊C4—24no—
76C2×C22⋊C4—24no—
77C2×C22⋊C4—42no—
78C2×C22⋊C4—42no—
79C2×C22⋊C4—42no—
80C2×C22⋊C4—42no—
81C2×C22⋊C4—42no—
82C2×C22⋊C4—42no—
83C2×C42—24no—
84C2×C42—22no—
85C2×C42—22no—
86C2×C42—41no—
87C2×C42—41no—
88C2×C42—42no—
89C2×C42—44no—
90C2×C42—44no—
91C2×C42—42no—
92C22⋊C8—28no—
93C22⋊C8—44no—
94C4×C8—84no—
95#95—41no—
96#95—41no—
97#95—41no—
98#95—41no—
99#95—44no—
100#95—44no—
101#95—44no—
102#95—44no—
103C24—21no—
104C24—21no—
105C24—22no—
106C24—22no—
107C24—41no—
108C24—41no—
109C24—42no—
110C24C22⋊C812yes—
111C24C2×C22⋊C411yescharacteristic
112C22×C4—24no—
113C22×C4—21no—
114C22×C4—22no—
115C22×C4—22no—
116C22×C4—21no—
117C22×C4—22no—
118C22×C4—22no—
119C22×C4—22no—
120C22×C4—22no—
121C22×C4—22no—
122C22×C4—22no—
123C22×C4—22no—
124C22×C4—22no—
125C22×C4—22no—
126C22×C4—22no—
127C22×C4—44no—
128C22×C4—48no—
129C22×C4—44no—
130C22×C4—44no—
131C22×C4—44no—
132C22×C4—42no—
133C22×C4—42no—
134C22×C4—42no—
135C22×C4—42no—
136C22×C4—42no—
137C22×C4—42no—
138C22×C4—42no—
139C22×C4—42no—
140C22×C4—42no—
141C22×C4—41no—
142C22×C4—41no—
143C22×C4—41no—
144C22×C4—41no—
145C22×C4—41no—
146C22×C4—41no—
147C22×C4—41no—
148C22×C4—41no—
149C22×C4—42no—
150C22×C4—42no—
151C22×C4—42no—
152C22×C4—42no—
153C22×C4C22⋊C812yes—
154C22×C4#9911yescharacteristic
155C22×C4#9911yescharacteristic
156C22×C4#10012yes—
157C22×C4#10414yes—
158C22×C4C22×C811yescharacteristic
159C22×C4C2×M1611yescharacteristic
160C22×C4#13514yes—
161C2×C8—48no—
162C2×C8—84no—
163C2×C8—84no—
164C4⋊C4—42no—
165C4⋊C4—42no—
166C4⋊C4—42no—
167C4⋊C4—42no—
168C22⋊C4—48no—
169C22⋊C4—48no—
170C22⋊C4—48no—
171C42—44no—
172C42—44no—
173C42—44no—
174C42—82no—
175C42—84no—
176C23—22no—
177C23—21no—
178C23—21no—
179C23—22no—
180C23—24no—
181C23—42no—
182C23—42no—
183C23—42no—
184C23—42no—
185C23—41no—
186C23—42no—
187C23—41no—
188C23—41no—
189C23—42no—
190C23—41no—
191C23—82no—
192C23—82no—
193C23—82no—
194C23—82no—
195C23C2×C22⋊C811yescharacteristic
196C23C2×#9911yescharacteristic
197C23C2×#10011yescharacteristic
198C23#41612yes—
199C23C2×#10412yes—
200C2×C4—22no—
201C2×C4—24no—
202C2×C4—24no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—24no—
206C2×C4—24no—
207C2×C4—44no—
208C2×C4—44no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—44no—
216C2×C4—44no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—44no—
220C2×C4—44no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—44no—
224C2×C4—44no—
225C2×C4—44no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—44no—
229C2×C4—82no—
230C2×C4—82no—
231C2×C4—81no—
232C2×C4—81no—
233C2×C4—82no—
234C2×C4—82no—
235C2×C4—82no—
236C2×C4—81no—
237C2×C4—81no—
238C2×C4—81no—
239C2×C4—81no—
240C2×C4—82no—
241C2×C4—81no—
242C2×C4—81no—
243C2×C4#32312yes—
244C2×C4#32312yes—
245C2×C4#35214yes—
246C8—88no—
247C22—21no—
248C22—21no—
249C22—21no—
250C22—21no—
251C22—22no—
252C22—22no—
253C22—22no—
254C22—22no—
255C22—42no—
256C22—42no—
257C22—42no—
258C22—84no—
259C22—82no—
260C22—82no—
261C22—82no—
262C22—82no—
263C22C2×#32311yescharacteristic
264C22#147612yes—
265C22#150112yes—
266C22#152511yescharacteristic
267C22#154311yescharacteristic
268C22C2×#35212yes—
269C22#212611yescharacteristic
270C22#213211yescharacteristic
271C4—22no—
272C4—41no—
273C4—41no—
274C4—44no—
275C4—44no—
276C4—82no—
277C4—82no—
278C4—82no—
279C4—82no—
280C4#131714yes—
281C2—21no—
282C2—21no—
283C2—22no—
284C2—82no—
285C2#756311yescharacteristic
286C2C2×#131711yescharacteristic
287C2#1143111yescharacteristic
288C2#1172712yes—
289C2#1173512yes—
subgroupG/H
centerC22×C4#99
commutatorC22×C4C22×C8
Frattini, cosocleC2×#95C23
socleC23C2×#99
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C4 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C2×#95 ▷ C24 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #7132C2semidirect
4. #7104C2semidirect
22. C2×#117C8semidirect

Maximal subgroups

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

subgroupindex
#138532
C4×#3232
#71322
#71042