Finite Group Databasebeta

← #135201

Subgroups — #135201

Order 512.

572 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 110 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#20385C211yesmaximal, characteristic
2#12679C212yesmaximal
3#7186C212yesmaximal, complemented
4#7186C212yesmaximal, complemented
5C24×C8C2211yescharacteristic
6C2×#407C411yescharacteristic
7C2×#405C411yescharacteristic
8C2×#405C411yescharacteristic
9#2280C411yescharacteristic
10C2×C22⋊C16—22no—
11C2×C22⋊C16—22no—
12C2×C22⋊C16C2214yes—
13#1752—22no—
14C8×C22⋊C4—22no—
15#1729C2211yescharacteristic
16#1729C2211yescharacteristic
17C24×C4C2×C411yescharacteristic
18C22×M16C2×C411yescharacteristic
19C22×M16C2×C411yescharacteristic
20C23×C8—22no—
21C23×C8—22no—
22C23×C8—42no—
23C23×C8—42no—
24C23×C8C2×C411yescharacteristic
25C23×C8D412yes—
26C23×C8D412yes—
27C23×C8C2311yescharacteristic
28C2×#117C2×C411yescharacteristic
29C2×#117C2×C411yescharacteristic
30C22×C16—24no—
31#407—22no—
32#407—22no—
33C2×C22⋊C8—22no—
34#405—42no—
35#405—42no—
36C2×#97—21no—
37C2×#97—21no—
38C2×C4×C8—21no—
39C2×C4×C8—21no—
40#381—21no—
41#380C812yes—
42#380C812yes—
43#379—21no—
44C4×C22⋊C4—22no—
45C22⋊C16—24no—
46C22⋊C16—24no—
47C22⋊C16—44no—
48C22⋊C16—44no—
49#336—22no—
50#336—22no—
51C25M1611yescharacteristic
52C23×C4—21no—
53C23×C4—21no—
54C23×C4—21no—
55C23×C4—21no—
56C23×C4—41no—
57C23×C4—41no—
58C23×C4—41no—
59C23×C4—41no—
60C23×C4C22⋊C412yes—
61C23×C4C2×C812yes—
62C23×C4M1611yescharacteristic
63C23×C4C22×C411yescharacteristic
64C2×M16—21no—
65C2×M16—21no—
66C2×M16—41no—
67C2×M16—41no—
68C2×M16—41no—
69C2×M16—41no—
70C2×M16M1612yes—
71C2×M16M1612yes—
72C22×C8—21no—
73C22×C8—21no—
74C22×C8—22no—
75C22×C8—22no—
76C22×C8—21no—
77C22×C8—21no—
78C22×C8—24no—
79C22×C8—24no—
80C22×C8—24no—
81C22×C8—24no—
82C22×C8—42no—
83C22×C8—42no—
84C22×C8—41no—
85C22×C8—41no—
86C22×C8—48no—
87C22×C8—48no—
88C22×C8C22⋊C411yescharacteristic
89C22×C8C22⋊C411yescharacteristic
90C22×C8C2×D412yes—
91C22×C8#4112yes—
92C22×C8#4112yes—
93#117—41no—
94#117—41no—
95#117—41no—
96#117—41no—
97C2×C4⋊C4C2×C811yescharacteristic
98C2×C4⋊C4C2×C812yes—
99C2×C4⋊C4C2×C811yescharacteristic
100C2×C22⋊C4—22no—
101C2×C22⋊C4—22no—
102C2×C22⋊C4—21no—
103C2×C22⋊C4—21no—
104C2×C42—21no—
105C2×C42—21no—
106C2×C16—44no—
107C2×C16—44no—
108C2×C16—44no—
109C22⋊C8—28no—
110#97—42no—
111#97—42no—
112C4×C8—42no—
113C4×C8—42no—
114#95—22no—
115#95—22no—
116C24—24no—
117C24—21no—
118C24—21no—
119C24—41no—
120C24—41no—
121C24—42no—
122C24C22⋊C812yes—
123C24C2×M1611yescharacteristic
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—22no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—22no—
136C22×C4—22no—
137C22×C4—22no—
138C22×C4—21no—
139C22×C4—22no—
140C22×C4—21no—
141C22×C4—42no—
142C22×C4—42no—
143C22×C4—41no—
144C22×C4—41no—
145C22×C4—42no—
146C22×C4—42no—
147C22×C4—42no—
148C22×C4—42no—
149C22×C4—42no—
150C22×C4—42no—
151C22×C4—42no—
152C22×C4—42no—
153C22×C4C22⋊C812yes—
154C22×C4C2×C22⋊C411yescharacteristic
155C22×C4#11711yescharacteristic
156C22×C4#11711yescharacteristic
157C22×C4C22×C811yescharacteristic
158C22×C4C2×M1611yescharacteristic
159C22×C4C2×M1611yescharacteristic
160M16—42no—
161M16—42no—
162M16—42no—
163M16—42no—
164C2×C8—24no—
165C2×C8—24no—
166C2×C8—21no—
167C2×C8—21no—
168C2×C8—21no—
169C2×C8—21no—
170C2×C8—22no—
171C2×C8—22no—
172C2×C8—22no—
173C2×C8—22no—
174C2×C8—42no—
175C2×C8—42no—
176C2×C8—48no—
177C2×C8—48no—
178C2×C8—41no—
179C2×C8—41no—
180C2×C8—41no—
181C2×C8—41no—
182C2×C8#9912yes—
183C2×C8#9912yes—
184C2×C8#12314yes—
185C4⋊C4—41no—
186C4⋊C4—41no—
187C4⋊C4—42no—
188C22⋊C4—44no—
189C22⋊C4—42no—
190C22⋊C4—42no—
191C22⋊C4—44no—
192C42—41no—
193C42—41no—
194C42—41no—
195C42—41no—
196C16—84no—
197C23—22no—
198C23—22no—
199C23—22no—
200C23—22no—
201C23—22no—
202C23—22no—
203C23—21no—
204C23—21no—
205C23—22no—
206C23—22no—
207C23—44no—
208C23—44no—
209C23—44no—
210C23—42no—
211C23—42no—
212C23—48no—
213C23—44no—
214C23#35014yes—
215C23C2×C22⋊C811yescharacteristic
216C23#43211yescharacteristic
217C23#43211yescharacteristic
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—44no—
232C2×C4—44no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—42no—
246C2×C4#38111yescharacteristic
247C2×C4#40711yescharacteristic
248C2×C4C2×#9911yescharacteristic
249C2×C4C2×#9911yescharacteristic
250C2×C4#41311yescharacteristic
251C2×C4#41311yescharacteristic
252C2×C4#43111yescharacteristic
253C2×C4#43111yescharacteristic
254C8—24no—
255C8—22no—
256C8—22no—
257C8—42no—
258C8—42no—
259C22—24no—
260C22—22no—
261C22—22no—
262C22—24no—
263C22—24no—
264C22—21no—
265C22—21no—
266C22—48no—
267C22—48no—
268C22—42no—
269C22—41no—
270C22—42no—
271C22—41no—
272C22—42no—
273C22—42no—
274C22—42no—
275C22#145911yescharacteristic
276C22#145911yescharacteristic
277C22#181711yescharacteristic
278C22C2×#35012yes—
279C22#211712yes—
280C4—21no—
281C4—21no—
282C4—42no—
283C4—81no—
284C4—81no—
285C4—82no—
286C4#146511yescharacteristic
287C4#146511yescharacteristic
288C4#181911yescharacteristic
289C4#182211yescharacteristic
290C2—44no—
291C2—41no—
292C2—41no—
293C2#870011yescharacteristic
294C2#1146811yescharacteristic
295C2#1146911yescharacteristic
296C2#1147012yes—
297C2#1284812yes—
subgroupG/H
centerC22×C4C2×C22⋊C4
commutatorC22×C4C22×C8
Frattini, cosocleC23×C8C23
socleC23C2×C22⋊C8
prime radicalC25M16
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C24×C8 ◁ G
LECSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #7186C2semidirect
4. #7186C2semidirect

Maximal subgroups

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

subgroupindex
#203852
#126792
#71862