Finite Group Databasebeta

← #134238

Subgroups — #134238

Order 512.

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

normal subgroups only

This group has 674 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
1C4×#407C211yesmaximal, characteristic
2#12785C212yesmaximal, complemented
3C2×#1320C212yesmaximal
4#7191C212yesmaximal, complemented
5C23×C42C411yescharacteristic
6C2×#407C411yescharacteristic
7C2×#407C2211yescharacteristic
8C2×C4×M16C411yescharacteristic
9C2×C4×M16C2211yescharacteristic
10#2150—22no—
11C2×#349C2214yes—
12#1749—22no—
13C4×C22⋊C8—22no—
14#1728C411yescharacteristic
15#1728C2211yescharacteristic
16#1377—22no—
17#1320—24no—
18C24×C4C812yes—
19C24×C4C2×C411yescharacteristic
20C22×M16C2×C411yescharacteristic
21C22×M16C2×C411yescharacteristic
22C22×M16C2×C411yescharacteristic
23C22×M16C2311yescharacteristic
24C22×C42—21no—
25C22×C42—21no—
26C22×C42—22no—
27C22×C42—42no—
28C22×C42—41no—
29C22×C42—41no—
30C22×C42C812yes—
31C22×C42C2×C411yescharacteristic
32C22×C42C2×C411yescharacteristic
33C2×M32—24no—
34#407—22no—
35#407—22no—
36#407—22no—
37#407D412yes—
38#407Q812yes—
39C2×C22⋊C8—22no—
40C2×C22⋊C8—22no—
41C4×M16—42no—
42C4×M16—41no—
43C4×M16—41no—
44C2×#97—21no—
45C2×#97D412yes—
46C2×C4×C8—21no—
47C2×C4×C8D412yes—
48C4⋊C16—44no—
49#349—24no—
50#349—24no—
51#336—22no—
52#336—22no—
53C25C2×C811yescharacteristic
54C23×C4—22no—
55C23×C4—21no—
56C23×C4—21no—
57C23×C4—21no—
58C23×C4—21no—
59C23×C4—22no—
60C23×C4—48no—
61C23×C4—42no—
62C23×C4—42no—
63C23×C4—41no—
64C23×C4—41no—
65C23×C4—41no—
66C23×C4—41no—
67C23×C4C4⋊C412yes—
68C23×C4C2×C812yes—
69C23×C4C2×C811yescharacteristic
70C23×C4C2×C812yes—
71C23×C4M1612yes—
72C23×C4C22×C411yescharacteristic
73C2×M16—21no—
74C2×M16—21no—
75C2×M16—21no—
76C2×M16—21no—
77C2×M16—41no—
78C2×M16—41no—
79C2×M16—41no—
80C2×M16—41no—
81C2×M16—41no—
82C2×M16—41no—
83C2×M16—41no—
84C2×M16—41no—
85C2×M16C4⋊C412yes—
86C2×M16M1612yes—
87C2×M16C2×D411yescharacteristic
88C2×M16C2×Q811yescharacteristic
89C2×M16#4112yes—
90C22×C8—21no—
91C22×C8—21no—
92C22×C8C22⋊C412yes—
93C22×C8C2×D412yes—
94C2×C42—22no—
95C2×C42—21no—
96C2×C42—21no—
97C2×C42—22no—
98C2×C42—22no—
99C2×C42—21no—
100C2×C42—21no—
101C2×C42—44no—
102C2×C42—44no—
103C2×C42—42no—
104C2×C42—42no—
105C2×C42—42no—
106C2×C42—41no—
107C2×C42—41no—
108C2×C42—42no—
109C2×C42—42no—
110C2×C42—41no—
111C2×C42—41no—
112C2×C42C22⋊C411yescharacteristic
113C2×C42C22⋊C411yescharacteristic
114C2×C42M1612yes—
115M32—44no—
116M32—44no—
117C2×C16—44no—
118C22⋊C8—28no—
119C22⋊C8—24no—
120C22⋊C8—24no—
121#97—42no—
122#97—42no—
123C4×C8—24no—
124C4×C8—41no—
125C4×C8—41no—
126C24—21no—
127C24—21no—
128C24—41no—
129C24—44no—
130C24—41no—
131C24C4⋊C812yes—
132C24C22×C811yescharacteristic
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—28no—
138C22×C4—22no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4—22no—
146C22×C4—22no—
147C22×C4—42no—
148C22×C4—44no—
149C22×C4—42no—
150C22×C4—42no—
151C22×C4—42no—
152C22×C4—44no—
153C22×C4—42no—
154C22×C4—42no—
155C22×C4—44no—
156C22×C4—42no—
157C22×C4—42no—
158C22×C4—44no—
159C22×C4—44no—
160C22×C4—42no—
161C22×C4—42no—
162C22×C4—44no—
163C22×C4—48no—
164C22×C4—48no—
165C22×C4—48no—
166C22×C4—42no—
167C22×C4—42no—
168C22×C4—41no—
169C22×C4—41no—
170C22×C4—41no—
171C22×C4—41no—
172C22×C4—41no—
173C22×C4—42no—
174C22×C4—41no—
175C22×C4—48no—
176C22×C4C22⋊C812yes—
177C22×C4C22⋊C812yes—
178C22×C4C4⋊C812yes—
179C22×C4C2×C22⋊C411yescharacteristic
180C22×C4C2×C4⋊C411yescharacteristic
181C22×C4#11711yescharacteristic
182C22×C4C2×M1611yescharacteristic
183C22×C4C2×M1611yescharacteristic
184C22×C4C2×M1611yescharacteristic
185M16—41no—
186M16—41no—
187M16—42no—
188M16—42no—
189M16—42no—
190M16—42no—
191M16—42no—
192M16—42no—
193M16—42no—
194C2×C8—22no—
195C2×C8—22no—
196C2×C8—22no—
197C2×C8—22no—
198C2×C8—22no—
199C2×C8—22no—
200C2×C8—22no—
201C2×C8—41no—
202C2×C8—41no—
203C2×C8—41no—
204C2×C8—41no—
205C2×C8#12112yes—
206C2×C8C22⋊Q812yes—
207C42—22no—
208C42—22no—
209C42—21no—
210C42—21no—
211C42—42no—
212C42—44no—
213C42—42no—
214C42—42no—
215C42—41no—
216C42—41no—
217C42#9912yes—
218C42#10011yescharacteristic
219C42#10011yescharacteristic
220C16—84no—
221C23—24no—
222C23—22no—
223C23—21no—
224C23—21no—
225C23—42no—
226C23—42no—
227C23—44no—
228C23—42no—
229C23—42no—
230C23—44no—
231C23—44no—
232C23—44no—
233C23—48no—
234C23—41no—
235C23—41no—
236C23C2×C22⋊C811yescharacteristic
237C23C2×C4⋊C811yescharacteristic
238C23#43111yescharacteristic
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—24no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—24no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—44no—
254C2×C4—44no—
255C2×C4—48no—
256C2×C4—44no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—416no—
260C2×C4—44no—
261C2×C4—41no—
262C2×C4—42no—
263C2×C4—41no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—48no—
267C2×C4—44no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—44no—
273C2×C4#32312yes—
274C2×C4#32312yes—
275C2×C4#38011yescharacteristic
276C2×C4#40711yescharacteristic
277C2×C4C2×#9911yescharacteristic
278C2×C4#41011yescharacteristic
279C2×C4C2×#10011yescharacteristic
280C2×C4#41311yescharacteristic
281C2×C4#42311yescharacteristic
282C2×C4#43211yescharacteristic
283C8—22no—
284C8—22no—
285C8—42no—
286C8—42no—
287C8—42no—
288C22—22no—
289C22—24no—
290C22—21no—
291C22—21no—
292C22—41no—
293C22—41no—
294C22—42no—
295C22—44no—
296C22—42no—
297C22—44no—
298C22—44no—
299C22—48no—
300C22—42no—
301C22—44no—
302C22—42no—
303C22C2×#32311yescharacteristic
304C22#145811yescharacteristic
305C22#179711yescharacteristic
306C4—21no—
307C4—21no—
308C4—22no—
309C4—42no—
310C4—44no—
311C4—42no—
312C4#132014yescomplemented
313C4#146511yescharacteristic
314C4#146611yescharacteristic
315C4#179911yescharacteristic
316C4#180011yescharacteristic
317C2—21no—
318C2—21no—
319C2—44no—
320C2—41no—
321C2—41no—
322C2#855711yescharacteristic
323C2C2×#132012yes—
subgroupG/H
centerC2×C4C2×#99
commutatorC24C22×C8
Frattini, cosocleC22×M16C23
socleC22C2×#323
prime radicalC25C2×C8
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×M16 ◁ G
LECSG ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #12785C2semidirect
4. #7191C2semidirect
312. C4#1320semidirect

Maximal subgroups

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

subgroupindex
C4×#4072
#127852
C2×#13202
#71912