Finite Group Databasebeta

← #213607

Subgroups — #213607

Order 512.

1258 conjugacy classes of subgroups; only the 234 normal subgroups are shown.

normal subgroups only

This group has 1258 conjugacy classes of subgroups — too many to draw. Only its 234 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
1C2×#2851C211yesmaximal, characteristic
2C4×#407C212yesmaximal
3#9122C218yesmaximal, complemented
4#8719C212yesmaximal, complemented
5#8335C212yesmaximal, complemented
6C22×#125C2211yescharacteristic
7C23×C42C2211yescharacteristic
8C2×#407—22no—
9C2×C4×M16C2214yes—
10#2851—22no—
11#2851—22no—
12#2851C414yes—
13#2851C2214yes—
14C2×#389C412yes—
15C2×#387—22no—
16C2×#385—22no—
17C2×#382C2211yescharacteristic
18C2×#380C412yes—
19#1749C2214yes—
20#1749C2214yes—
21C4×C22⋊C8—22no—
22C4×C22⋊C8—22no—
23#1728—22no—
24C2×#330C2218yescomplemented
25#1295C2218yescomplemented
26C24×C4C2×C411yescharacteristic
27C24×C4D412yes—
28C22×M16—24no—
29C2×#125—22no—
30C2×#125—44no—
31C2×#125C2×C412yes—
32C2×#125C2312yes—
33C22×C4⋊C4—22no—
34C22×C4⋊C4—22no—
35C22×C4⋊C4C2×C412yes—
36C22×C4⋊C4C2×C412yes—
37C22×C22⋊C4—22no—
38C22×C42—44no—
39C22×C42C2×C412yes—
40C22×C42C2×C411yescharacteristic
41C22×C42D412yes—
42C22×C42D412yes—
43C22×C42D412yes—
44C22×C42C2312yes—
45C22×C42C2311yescharacteristic
46#407—216no—
47C2×C22⋊C8—24no—
48C2×C22⋊C8—24no—
49C4×M16—48no—
50C2×#97C2318yes—
51C2×C4×C8—24no—
52#389—22no—
53#389—22no—
54#389C2×C414yes—
55#389C2×C414yes—
56#387—24no—
57#387—24no—
58#387—24no—
59#387—24no—
60#385—24no—
61#385—24no—
62#385—24no—
63#385—24no—
64#382—22no—
65#382C2×C412yes—
66#382C2312yes—
67#380—22no—
68#380—22no—
69#380C2×C414yes—
70#380C2×C414yes—
71C2×#95—22no—
72C2×#95—22no—
73C2×#95—22no—
74#336—24no—
75#336—24no—
76#330—48no—
77C25C22⋊C411yescharacteristic
78C23×C4—24no—
79C23×C4—28no—
80C23×C4—22no—
81C23×C4—22no—
82C23×C4—42no—
83C23×C4—42no—
84C23×C4—44no—
85C23×C4—44no—
86C23×C4C22⋊C412yes—
87C23×C4C22⋊C412yes—
88C23×C4C22⋊C412yes—
89C23×C4C22⋊C411yescharacteristic
90C23×C4C22⋊C412yes—
91C23×C4C22×C411yescharacteristic
92C23×C4C22×C412yes—
93C23×C4C2×D412yes—
94C23×C4C2×D412yes—
95C23×C4C2×D412yes—
96C2×M16—216no—
97C2×M16—48no—
98C2×M16—48no—
99C22×C8—28no—
100#125—28no—
101#125—44no—
102#125—48no—
103C2×C4⋊C4—24no—
104C2×C4⋊C4—24no—
105C2×C4⋊C4—24no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—24no—
109C2×C4⋊C4—44no—
110C2×C4⋊C4—48no—
111C2×C4⋊C4—48no—
112C2×C4⋊C4—44no—
113C2×C4⋊C4C22×C414yes—
114C2×C4⋊C4C22×C414yes—
115C2×C22⋊C4—24no—
116C2×C22⋊C4—24no—
117C2×C22⋊C4—48no—
118C2×C42—24no—
119C2×C42—28no—
120C2×C42—42no—
121C2×C42—44no—
122C2×C42—44no—
123C2×C42—44no—
124C2×C42—44no—
125C2×C42C22⋊C412yes—
126C2×C42C22⋊C412yes—
127C2×C42C22⋊C411yescharacteristic
128C2×C42C22⋊C411yescharacteristic
129C2×C42C22×C411yescharacteristic
130C2×C42C22×C412yes—
131C2×C42C2×D412yes—
132C2×C42C2×D412yes—
133C2×C42C2×D412yes—
134C2×C42C2411yescharacteristic
135C22⋊C8—216no—
136C22⋊C8—216no—
137#97—216no—
138C4×C8—48no—
139#95—24no—
140#95—24no—
141#95—24no—
142#95—24no—
143#95—24no—
144#95—24no—
145C24—22no—
146C24—24no—
147C24—24no—
148C24—41no—
149C24—41no—
150C24C2×C22⋊C412yes—
151C24C2×C22⋊C411yescharacteristic
152C22×C4—22no—
153C22×C4—24no—
154C22×C4—22no—
155C22×C4—24no—
156C22×C4—22no—
157C22×C4—24no—
158C22×C4—24no—
159C22×C4—24no—
160C22×C4—24no—
161C22×C4—24no—
162C22×C4—24no—
163C22×C4—24no—
164C22×C4—24no—
165C22×C4—24no—
166C22×C4—24no—
167C22×C4—24no—
168C22×C4—24no—
169C22×C4—24no—
170C22×C4—24no—
171C22×C4—42no—
172C22×C4—41no—
173C22×C4—42no—
174C22×C4—42no—
175C22×C4—42no—
176C22×C4—41no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4—42no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4—432no—
183C22×C4—416no—
184C22×C4—44no—
185C22×C4—44no—
186C22×C4—44no—
187C22×C4—44no—
188C22×C4—44no—
189C22×C4—44no—
190C22×C4#10114yes—
191C22×C4#10418yes—
192C22×C4C2×C22⋊C412yes—
193C22×C4C2×C22⋊C412yes—
194C22×C4C2×C22⋊C411yescharacteristic
195C22×C4C2×C22⋊C412yes—
196C22×C4C2×C22⋊C412yes—
197C22×C4C23×C411yescharacteristic
198C22×C4C22×D412yes—
199M16—432no—
200C2×C8—216no—
201C2×C8—48no—
202C2×C8—48no—
203C4⋊C4—28no—
204C4⋊C4—44no—
205C4⋊C4—44no—
206C4⋊C4—48no—
207C4⋊C4—48no—
208C4⋊C4—44no—
209C4⋊C4—48no—
210C4⋊C4—48no—
211C22⋊C4—48no—
212C22⋊C4—48no—
213C42—24no—
214C42—22no—
215C42—24no—
216C42—24no—
217C42—24no—
218C42—41no—
219C42—41no—
220C42—42no—
221C42—42no—
222C42H(5,𝔽2)14yes—
223C23—21no—
224C23—21no—
225C23—22no—
226C23—28no—
227C23—22no—
228C23—22no—
229C23—22no—
230C23—22no—
231C23—22no—
232C23—22no—
233C23—48no—
234C23—48no—
235C23—44no—
236C23—44no—
237C23C2×#10112yes—
238C23C2×#10414yes—
239C23#42114yes—
240C23C22×C22⋊C411yescharacteristic
241C2×C4—216no—
242C2×C4—28no—
243C2×C4—21no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—44no—
259C2×C4—48no—
260C2×C4—48no—
261C2×C4—48no—
262C2×C4—48no—
263C2×C4—48no—
264C2×C4—48no—
265C2×C4—48no—
266C2×C4—44no—
267C2×C4—44no—
268C2×C4—44no—
269C2×C4—44no—
270C2×C4—44no—
271C2×C4—44no—
272C2×C4—88no—
273C2×C4—88no—
274C2×C4C2×#10114yes—
275C2×C4#41312yes—
276C2×C4C2×#10414yes—
277C2×C4C2×#10414yes—
278C2×C4#51812yes—
279C2×C4#53414yes—
280C8—416no—
281C22—22no—
282C22—24no—
283C22—24no—
284C22—22no—
285C22—24no—
286C22—24no—
287C22—24no—
288C22—416no—
289C22—42no—
290C22—42no—
291C22—42no—
292C22—42no—
293C22#152614yes—
294C22#230811yescharacteristic
295C22C22×#10111yescharacteristic
296C22C2×#41311yescharacteristic
297C22C22×#10412yes—
298C22C2×#42112yes—
299C4—24no—
300C4—24no—
301C4—24no—
302C4—22no—
303C4—22no—
304C4—24no—
305C4—42no—
306C4—84no—
307C4—84no—
308C2—24no—
309C2—44no—
310C2C2×#152611yescharacteristic
311C2#1764212yes—
312C2#1766711yescharacteristic
313C2#2175311yescharacteristic
314C2#2177312yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #9122C2semidirect
4. #8719C2semidirect
5. #8335C2semidirect
24. C2×#330C22semidirect
25. #1295C22semidirect

Maximal subgroups

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

subgroupindex
C2×#28512
C4×#4072
#91222
#87192
#83352