Finite Group Databasebeta

← #214025

Subgroups — #214025

Order 512.

845 conjugacy classes of subgroups; only the 190 normal subgroups are shown.

normal subgroups only

This group has 845 conjugacy classes of subgroups — too many to draw. Only its 190 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#35678C211yesmaximal, characteristic
2#17371C212yesmaximal, complemented
3#9458C214yesmaximal, complemented
4#9450C214yesmaximal, complemented
5#8745C212yesmaximal, complemented
6#8526C212yesmaximal
7C2×#540C2211yescharacteristic
8C2×#519C2211yescharacteristic
9#2983C2214yes—
10#2565C2212yes—
11#2548C2212yes—
12#2500C412yes—
13#2476—22no—
14#2476—22no—
15#2458—22no—
16#2458—22no—
17#2436C412yes—
18#2326C412yes—
19C4×#124C412yes—
20#2298C2211yescharacteristic
21#1555C2214yes—
22#1534C2214yes—
23#1467—22no—
24#1467C2214yes—
25#1464—22no—
26#1464C2214yes—
27#1461—22no—
28#1458—22no—
29#1285C2218yes—
30C23×Q8C2×C411yescharacteristic
31#540—44no—
32C2×#127C2311yescharacteristic
33C2×#125C2311yescharacteristic
34C2×#124C2×C412yes—
35C2×#124C2×C412yes—
36C2×#123—22no—
37C2×#121—22no—
38C2×#121C2×C412yes—
39#519—44no—
40C2×C4×Q8C2×C412yes—
41C2×C4×Q8D412yes—
42C2×C4×Q8D412yes—
43C2×C4×Q8C2312yes—
44C2×C4×D4C2×C412yes—
45C2×#117C2×C411yescharacteristic
46C2×#117D412yes—
47C2×#117D412yes—
48C2×#117C2311yescharacteristic
49#446—48no—
50#432—24no—
51#432C2318yes—
52#431—24no—
53#398—22no—
54#398—22no—
55#394—22no—
56#394—22no—
57#394—22no—
58#394—22no—
59#393—22no—
60#393—22no—
61#391C2×C412yes—
62#391C2×C412yes—
63#390—22no—
64#390—22no—
65#390C2×C412yes—
66#390C2×C412yes—
67#388—22no—
68#388—22no—
69#388—22no—
70#388—22no—
71#388—22no—
72#388—22no—
73#388—22no—
74#387—22no—
75#387—22no—
76#387—22no—
77#386—22no—
78#386—22no—
79#385—22no—
80#385—22no—
81#384C2×C412yes—
82#382C2×C412yes—
83C4×C4⋊C4C2312yes—
84C4×C22⋊C4—22no—
85C4×C22⋊C4—22no—
86C4×C22⋊C4C2×C412yes—
87#376C2×C411yescharacteristic
88C43C2×C411yescharacteristic
89#331—44no—
90#330—44no—
91#324—24no—
92#324—24no—
93#324—24no—
94#323—24no—
95C22×Q8—42no—
96C22×Q8—44no—
97C22×Q8C22⋊C412yes—
98C22×Q8C22×C412yes—
99C22×D4—22no—
100C22×D4—22no—
101C22×D4C22×C412yes—
102C23×C4—22no—
103C23×C4C22⋊C411yescharacteristic
104C23×C4C22⋊C411yescharacteristic
105C23×C4C22×C411yescharacteristic
106#127—24no—
107#125—24no—
108#124—44no—
109#124—44no—
110#123—48no—
111#121—48no—
112#121—44no—
113C4×Q8—28no—
114C4×Q8—44no—
115C4×Q8—44no—
116C4×Q8—44no—
117C4×D4—44no—
118#117—24no—
119#117—44no—
120#117—44no—
121#117—42no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4C22⋊C412yes—
127C2×C4⋊C4C22⋊C412yes—
128C2×C4⋊C4C22×C412yes—
129C2×C4⋊C4C22×C412yes—
130C2×C4⋊C4C2×D412yes—
131C2×C4⋊C4C2×D412yes—
132C2×C4⋊C4C2×D412yes—
133C2×C22⋊C4—22no—
134C2×C22⋊C4—22no—
135C2×C22⋊C4—22no—
136C2×C22⋊C4—22no—
137C2×C22⋊C4—22no—
138C2×C22⋊C4—22no—
139C2×C22⋊C4—22no—
140C2×C22⋊C4—22no—
141C2×C22⋊C4—22no—
142C2×C22⋊C4—22no—
143C2×C22⋊C4—22no—
144C2×C22⋊C4C22⋊C412yes—
145C2×C22⋊C4C2×D412yes—
146C2×C42—22no—
147C2×C42C22⋊C411yescharacteristic
148C2×C42C22⋊C411yescharacteristic
149C2×C42C22⋊C412yes—
150C2×C42C22⋊C412yes—
151C2×C42C22×C411yescharacteristic
152C2×C42C22×C412yes—
153C2×C42C22×C412yes—
154C2×C42C2×D412yes—
155C2×C42C2×D412yes—
156C2×C42C2411yescharacteristic
157C4⋊C8—28no—
158C4⋊C8—44no—
159C4⋊C8—44no—
160C22⋊C8—28no—
161#97—28no—
162#97—44no—
163C4×C8—44no—
164#95—22no—
165#95—22no—
166#95—22no—
167#95—22no—
168#95—22no—
169#95—22no—
170#95—22no—
171#95—22no—
172#95—22no—
173C24—22no—
174C24C2×C22⋊C411yescharacteristic
175C2×Q8—24no—
176C2×Q8—24no—
177C2×Q8—24no—
178C2×Q8—42no—
179C2×Q8—42no—
180C2×Q8—42no—
181C2×Q8—416no—
182C2×D4—24no—
183C2×D4—42no—
184C2×D4—42no—
185C2×D4—44no—
186C2×D4—44no—
187C2×D4—82no—
188C2×D4—82no—
189C2×D4—82no—
190C2×D4—82no—
191C22×C4—22no—
192C22×C4—24no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—41no—
197C22×C4—41no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4—44no—
201C22×C4—82no—
202C22×C4—82no—
203C22×C4C2×C22⋊C411yescharacteristic
204C22×C4C2×C22⋊C412yes—
205C22×C4C2×C22⋊C412yes—
206C22×C4C2×C22⋊C412yes—
207C22×C4C2×C22⋊C412yes—
208C22×C4C2×C22⋊C412yes—
209C22×C4C23×C411yescharacteristic
210C22×C4C22×D412yes—
211C2×C8—48no—
212C4⋊C4—24no—
213C4⋊C4—24no—
214C4⋊C4—24no—
215C4⋊C4—24no—
216C4⋊C4—24no—
217C4⋊C4—42no—
218C4⋊C4—42no—
219C4⋊C4—44no—
220C4⋊C4—82no—
221C4⋊C4—82no—
222C4⋊C4—82no—
223C22⋊C4—24no—
224C22⋊C4—42no—
225C22⋊C4—44no—
226C22⋊C4—44no—
227C22⋊C4—44no—
228C22⋊C4—44no—
229C22⋊C4—44no—
230C22⋊C4—44no—
231C22⋊C4—82no—
232C22⋊C4—82no—
233C22⋊C4—82no—
234C22⋊C4—82no—
235C22⋊C4—82no—
236C42—22no—
237C42—24no—
238C42—24no—
239C42—42no—
240C42—42no—
241C42—41no—
242C42—41no—
243C42—42no—
244C42—42no—
245C42—82no—
246C42H(5,𝔽2)14yes—
247C23—21no—
248C23—22no—
249C23—42no—
250C23—42no—
251C23—42no—
252C23—42no—
253C23—42no—
254C23—42no—
255C23—42no—
256C23—42no—
257C23C22×C22⋊C411yescharacteristic
258Q8—48no—
259Q8—48no—
260D4—84no—
261D4—164no—
262D4—164no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—21no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—42no—
275C2×C4—44no—
276C2×C4—44no—
277C2×C4—44no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—164no—
291C2×C4#41314yes—
292C2×C4#51812yes—
293C2×C4#53414yes—
294C8—88no—
295C22—22no—
296C22—42no—
297C22—41no—
298C22—82no—
299C22—82no—
300C22—82no—
301C22—82no—
302C22—82no—
303C22—82no—
304C22—82no—
305C22—82no—
306C22#230811yescharacteristic
307C22C2×#41312yes—
308C4—24no—
309C4—22no—
310C4—24no—
311C4—42no—
312C4—42no—
313C4—42no—
314C4—42no—
315C4—41no—
316C4—82no—
317C4—82no—
318C4—82no—
319C2—22no—
320C2—81no—
321C2—162no—
322C2#2175411yescharacteristic
323C2#2178512yes—
subgroupG/H
center, socleC22#2308
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#540C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #17371C2semidirect
3. #9458C2semidirect
4. #9450C2semidirect
5. #8745C2semidirect

Maximal subgroups

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

subgroupindex
#356782
#173712
#94582
#94502
#87452
#85262