Finite Group Databasebeta

← #215118

Subgroups — #215118

Order 512.

1346 conjugacy classes of subgroups; only the 258 normal subgroups are shown.

normal subgroups only

This group has 1346 conjugacy classes of subgroups — too many to draw. Only its 258 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×#2857C211yesmaximal, characteristic
2#20714C212yesmaximal
3#9650C218yesmaximal, complemented
4#8308C214yesmaximal
5C22×C4⋊Q8C411yescharacteristic
6C22×C22⋊Q8C2212yes—
7C23×C42C2211yescharacteristic
8C2×#431—22no—
9#2857—22no—
10#2857—22no—
11#2857—22no—
12#2857—22no—
13C2×#391C412yes—
14C2×#389C2214yes—
15C2×#384C411yescharacteristic
16C2×#380C414yes—
17#1817—22no—
18#1797—22no—
19#1797C2214yes—
20#1779C2214yes—
21C4×C22⋊C8—22no—
22C4×C22⋊C8C2214yes—
23C2×#324C22116yes—
24C23×Q8C2×C412yes—
25C24×C4C2×C412yes—
26C24×C4C2311yescharacteristic
27C2×C4⋊Q8—21no—
28C2×C4⋊Q8—22no—
29C2×C4⋊Q8—21no—
30C2×C4⋊Q8—44no—
31C2×C22⋊Q8—24no—
32C2×C22⋊Q8—24no—
33C2×C22⋊Q8—48no—
34C22×C4⋊C4C2×C414yes—
35C22×C4⋊C4C2×C412yes—
36C22×C4⋊C4C2×C414yes—
37C22×C4⋊C4C2314yes—
38C22×C22⋊C4C2×C414yes—
39C22×C42—44no—
40C22×C42C2×C411yescharacteristic
41C22×C42C2×C411yescharacteristic
42C22×C42D412yes—
43C22×C42D412yes—
44C22×C42D412yes—
45C22×C42D412yes—
46C22×C42C2312yes—
47#431—48no—
48C2×C4⋊C8—24no—
49C2×C22⋊C8—24no—
50C2×C22⋊C8—24no—
51C2×C22⋊C8C2318yes—
52C2×C4×C8—24no—
53#391—24no—
54#391—22no—
55#391—22no—
56#389—24no—
57#389—24no—
58#389—24no—
59#389—24no—
60#384—21no—
61#384—22no—
62#384—21no—
63#380—24no—
64#380—24no—
65#380—24no—
66#380—24no—
67C2×#95C2×C418yes—
68#336—24no—
69#336—24no—
70#324—416no—
71C25C22×C411yescharacteristic
72C22×Q8—22no—
73C22×Q8—22no—
74C22×Q8—44no—
75C22×Q8—44no—
76C22×Q8—44no—
77C23×C4—28no—
78C23×C4—28no—
79C23×C4—24no—
80C23×C4—44no—
81C23×C4—42no—
82C23×C4—42no—
83C23×C4C22⋊C412yes—
84C23×C4C22⋊C412yes—
85C23×C4C22⋊C412yes—
86C23×C4C22⋊C412yes—
87C23×C4C22×C414yes—
88C23×C4C22×C411yescharacteristic
89C23×C4C22×C412yes—
90C23×C4C22×C412yes—
91C23×C4C22×C414yes—
92C23×C4C2×D412yes—
93C23×C4C2×D412yes—
94C23×C4C2411yescharacteristic
95C22×C8—28no—
96C4⋊Q8—44no—
97C4⋊Q8—48no—
98C4⋊Q8—44no—
99C22⋊Q8—416no—
100C22⋊Q8—416no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—24no—
104C2×C4⋊C4—24no—
105C2×C4⋊C4—24no—
106C2×C4⋊C4—24no—
107C2×C4⋊C4—216no—
108C2×C4⋊C4—28no—
109C2×C4⋊C4—216no—
110C2×C4⋊C4—44no—
111C2×C4⋊C4—48no—
112C2×C22⋊C4—28no—
113C2×C22⋊C4—48no—
114C2×C42—24no—
115C2×C42—28no—
116C2×C42—42no—
117C2×C42—44no—
118C2×C42—44no—
119C2×C42—44no—
120C2×C42—44no—
121C2×C42C22⋊C411yescharacteristic
122C2×C42C22⋊C412yes—
123C2×C42C22⋊C412yes—
124C2×C42C22⋊C411yescharacteristic
125C2×C42C22⋊C411yescharacteristic
126C2×C42C22⋊C411yescharacteristic
127C2×C42C2×D412yes—
128C2×C42C2×D412yes—
129C2×C42C2×D412yes—
130C2×C42C2×D412yes—
131C4⋊C8—48no—
132C22⋊C8—216no—
133C22⋊C8—48no—
134C22⋊C8—48no—
135C4×C8—48no—
136#95—28no—
137#95—28no—
138C24—22no—
139C24—24no—
140C24—24no—
141C24—41no—
142C24—41no—
143C24C2×C22⋊C412yes—
144C24C23×C411yescharacteristic
145C2×Q8—44no—
146C2×Q8—44no—
147C2×Q8—44no—
148C2×Q8—44no—
149C2×Q8—44no—
150C2×Q8—44no—
151C2×Q8—816no—
152C22×C4—22no—
153C22×C4—24no—
154C22×C4—24no—
155C22×C4—24no—
156C22×C4—22no—
157C22×C4—22no—
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—24no—
172C22×C4—24no—
173C22×C4—24no—
174C22×C4—28no—
175C22×C4—216no—
176C22×C4—28no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4—42no—
180C22×C4—42no—
181C22×C4—41no—
182C22×C4—42no—
183C22×C4—41no—
184C22×C4—432no—
185C22×C4—416no—
186C22×C4—44no—
187C22×C4—44no—
188C22×C4#10118yes—
189C22×C4C2×C22⋊C412yes—
190C22×C4C2×C22⋊C412yes—
191C22×C4C2×C22⋊C411yescharacteristic
192C22×C4C2×C22⋊C411yescharacteristic
193C22×C4C2×C22⋊C412yes—
194C22×C4C2×C22⋊C412yes—
195C22×C4#13118yes—
196C22×C4C22×D412yes—
197C22×C4H(5,𝔽2)14yes—
198C2×C8—48no—
199C2×C8—48no—
200C2×C8—48no—
201C4⋊C4—44no—
202C4⋊C4—44no—
203C4⋊C4—48no—
204C4⋊C4—416no—
205C4⋊C4—48no—
206C4⋊C4—48no—
207C4⋊C4—48no—
208C22⋊C4—416no—
209C42—24no—
210C42—24no—
211C42—24no—
212C42—22no—
213C42—24no—
214C42—41no—
215C42—41no—
216C42—42no—
217C42—42no—
218C42#9914yes—
219C23—21no—
220C23—21no—
221C23—22no—
222C23—28no—
223C23—22no—
224C23—22no—
225C23—22no—
226C23—22no—
227C23—22no—
228C23—22no—
229C23—48no—
230C23—48no—
231C23—44no—
232C23—44no—
233C23C2×#10114yes—
234C23C22×C22⋊C411yescharacteristic
235C23#51812yes—
236C23C2×#13114yes—
237Q8—88no—
238Q8—88no—
239C2×C4—28no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—21no—
247C2×C4—22no—
248C2×C4—21no—
249C2×C4—22no—
250C2×C4—216no—
251C2×C4—216no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—44no—
256C2×C4—48no—
257C2×C4—48no—
258C2×C4—48no—
259C2×C4—48no—
260C2×C4—48no—
261C2×C4—48no—
262C2×C4—44no—
263C2×C4—44no—
264C2×C4—48no—
265C2×C4—44no—
266C2×C4—44no—
267C2×C4—416no—
268C2×C4—88no—
269C2×C4#40814yes—
270C2×C4#40814yes—
271C2×C4C2×#9912yes—
272C2×C4C2×#9914yes—
273C2×C4#41012yes—
274C2×C4C2×#10114yes—
275C2×C4C2×#10114yes—
276C2×C4#53514yes—
277C8—88no—
278C22—22no—
279C22—22no—
280C22—24no—
281C22—24no—
282C22—24no—
283C22—24no—
284C22—24no—
285C22—416no—
286C22—42no—
287C22—42no—
288C22—42no—
289C22—42no—
290C22#230911yescharacteristic
291C22C2×#40812yes—
292C22C22×#9911yescharacteristic
293C22C2×#41011yescharacteristic
294C22C22×#10112yes—
295C22#297914yes—
296C4—24no—
297C4—22no—
298C4—22no—
299C4—24no—
300C4—24no—
301C4—42no—
302C4—84no—
303C4—84no—
304C4#147218yes—
305C2—24no—
306C2—44no—
307C2C2×#147212yes—
308C2#1738512yes—
309C2#2178711yescharacteristic
310C2#2179311yescharacteristic
311C2#2179911yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #9650C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28572
#207142
#96502
#83082