Finite Group Databasebeta

← #215216

Subgroups — #215216

Order 512.

1014 conjugacy classes of subgroups; only the 230 normal subgroups are shown.

normal subgroups only

This group has 1014 conjugacy classes of subgroups — too many to draw. Only its 230 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#36639C211yesmaximal, characteristic
2#20777C212yesmaximal, complemented
3#9696C214yesmaximal, complemented
4#9693C214yesmaximal, complemented
5#8345C212yesmaximal, complemented
6#8344C212yesmaximal
7C2×#533C2211yescharacteristic
8C2×#532C2211yescharacteristic
9#2862C2212yes—
10#2861C414yes—
11#2858C2212yes—
12#2850—22no—
13#2850—22no—
14C4×#124—22no—
15C4×#124—22no—
16#2297C412yes—
17#2297C412yes—
18C2×C43C2211yescharacteristic
19#1839—22no—
20#1831C2214yes—
21#1817—22no—
22#1797C2214yes—
23C4×C4⋊C8C2214yes—
24C4×C22⋊C8—22no—
25#1744—22no—
26C2×#333C2214yes—
27C2×#332C2218yes—
28C2×#331C2214yes—
29C22×#41C2×C412yes—
30#533—44no—
31#532—44no—
32C2×C4⋊Q8C2311yescharacteristic
33C2×C4⋊Q8C2312yes—
34C2×#127C2311yescharacteristic
35C2×#125C2×C412yes—
36C2×#124—22no—
37C2×#124—22no—
38C2×#124C2×C412yes—
39C2×C22⋊Q8—22no—
40C2×#121—22no—
41C2×C4×Q8—22no—
42C2×C4×D4—22no—
43C2×#117C2×C412yes—
44C22×C42C2×C411yescharacteristic
45C22×C42C2×C411yescharacteristic
46C22×C42D412yes—
47C22×C42D412yes—
48C22×C42C2311yescharacteristic
49C2×C4⋊C8C2318yes—
50C2×C22⋊C8—24no—
51C2×C4×C8—24no—
52#391—22no—
53#391—22no—
54#391C2×C414yes—
55#391C2×C414yes—
56#390—22no—
57#390—22no—
58#390C2×C414yes—
59#390C2×C414yes—
60#388—24no—
61#388—24no—
62#388—24no—
63#388—24no—
64#386—24no—
65#386—22no—
66#386—22no—
67#382—22no—
68#382—22no—
69C4×C22⋊C4—24no—
70C4×C22⋊C4—22no—
71C4×C22⋊C4—22no—
72C43C2×C412yes—
73C43D412yes—
74C43D412yes—
75C43C2312yes—
76#336—24no—
77#336—24no—
78#336—24no—
79#336—24no—
80#333—44no—
81#332—48no—
82#331—44no—
83C2×#41—44no—
84C2×#41—44no—
85C2×#41—44no—
86C22×Q8—22no—
87C22×Q8C22×C412yes—
88C22×D4—22no—
89C22×D4—22no—
90C22×D4C22×C412yes—
91C23×C4—22no—
92C23×C4C22⋊C412yes—
93C23×C4C22⋊C411yescharacteristic
94C23×C4C22⋊C411yescharacteristic
95C23×C4C22×C411yescharacteristic
96C23×C4C2×D412yes—
97C22×C8—28no—
98C4⋊Q8—28no—
99C4⋊Q8—24no—
100#127—24no—
101#125—44no—
102#124—44no—
103#124—48no—
104#124—48no—
105C22⋊Q8—48no—
106#121—48no—
107C4×Q8—48no—
108C4×D4—48no—
109#117—44no—
110C2×C4⋊C4—24no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4C22×C414yes—
114C2×C22⋊C4—22no—
115C2×C22⋊C4—24no—
116C2×C22⋊C4—24no—
117C2×C22⋊C4—24no—
118C2×C22⋊C4—24no—
119C2×C22⋊C4—24no—
120C2×C42—28no—
121C2×C42—24no—
122C2×C42—22no—
123C2×C42—42no—
124C2×C42—44no—
125C2×C42—42no—
126C2×C42C22⋊C412yes—
127C2×C42C22⋊C412yes—
128C2×C42C22⋊C412yes—
129C2×C42C22⋊C412yes—
130C2×C42C22⋊C412yes—
131C2×C42C22⋊C411yescharacteristic
132C2×C42C22⋊C411yescharacteristic
133C2×C42C22×C411yescharacteristic
134C2×C42C22×C412yes—
135C2×C42C22×C412yes—
136C2×C42C2×D412yes—
137C2×C42C2×D412yes—
138C2×C42C2×D412yes—
139C2×C42C2×D412yes—
140C2×C42C2×D412yes—
141C2×C42C2411yescharacteristic
142C4⋊C8—28no—
143C4⋊C8—28no—
144C22⋊C8—48no—
145C4×C8—48no—
146#95—24no—
147#95—24no—
148#95—24no—
149#95—24no—
150C24—22no—
151C24C2×C22⋊C411yescharacteristic
152#41—816no—
153C2×Q8—24no—
154C2×Q8—42no—
155C2×Q8—42no—
156C2×Q8—44no—
157C2×Q8—44no—
158C2×Q8—44no—
159C2×D4—24no—
160C2×D4—42no—
161C2×D4—42no—
162C2×D4—44no—
163C2×D4—44no—
164C2×D4—44no—
165C2×D4—44no—
166C2×D4—44no—
167C2×D4—44no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—24no—
172C22×C4—24no—
173C22×C4—24no—
174C22×C4—24no—
175C22×C4—24no—
176C22×C4—24no—
177C22×C4—22no—
178C22×C4—24no—
179C22×C4—41no—
180C22×C4—42no—
181C22×C4—41no—
182C22×C4—42no—
183C22×C4—44no—
184C22×C4—44no—
185C22×C4—44no—
186C22×C4C2×C22⋊C412yes—
187C22×C4C2×C22⋊C412yes—
188C22×C4C2×C22⋊C412yes—
189C22×C4C2×C22⋊C411yescharacteristic
190C22×C4C2×C22⋊C412yes—
191C22×C4C2×C22⋊C412yes—
192C22×C4C23×C411yescharacteristic
193C22×C4C22×D412yes—
194C2×C8—48no—
195C2×C8—48no—
196C2×C8—48no—
197C4⋊C4—28no—
198C4⋊C4—48no—
199C4⋊C4—44no—
200C4⋊C4—44no—
201C22⋊C4—48no—
202C22⋊C4—48no—
203C22⋊C4—48no—
204C22⋊C4—44no—
205C22⋊C4—48no—
206C22⋊C4—48no—
207C42—22no—
208C42—24no—
209C42—24no—
210C42—24no—
211C42—24no—
212C42—24no—
213C42—24no—
214C42—24no—
215C42—41no—
216C42—41no—
217C42—42no—
218C42—42no—
219C42—42no—
220C42—42no—
221C42—44no—
222C42#13518yes—
223C42H(5,𝔽2)14yes—
224C23—21no—
225C23—21no—
226C23—22no—
227C23—22no—
228C23—42no—
229C23—42no—
230C23—42no—
231C23—44no—
232C23—44no—
233C23#41312yes—
234C23C22×C22⋊C411yescharacteristic
235Q8—84no—
236Q8—88no—
237D4—84no—
238D4—88no—
239D4—88no—
240C2×C4—28no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—22no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
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—44no—
261C2×C4—48no—
262C2×C4—44no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—44no—
267C2×C4—44no—
268C2×C4—88no—
269C2×C4#41312yes—
270C2×C4#41614yes—
271C2×C4#41614yes—
272C2×C4#51812yes—
273C2×C4#53514yes—
274C2×C4C2×#13514yes—
275C2×C4#57514yes—
276C8—88no—
277C22—22no—
278C22—24no—
279C22—41no—
280C22—42no—
281C22—41no—
282C22—82no—
283C22—82no—
284C22—84no—
285C22—84no—
286C22—82no—
287C22#230911yescharacteristic
288C22C2×#41311yescharacteristic
289C22C2×#41311yescharacteristic
290C22C2×#41612yes—
291C22#289612yes—
292C4—24no—
293C4—22no—
294C4—22no—
295C4—24no—
296C4—42no—
297C4—42no—
298C4—42no—
299C4—82no—
300C4—84no—
301C4#321218yes—
302C2—42no—
303C2—162no—
304C2#1770011yescharacteristic
305C2#1771312yes—
306C2#1775711yescharacteristic
307C2#2179811yescharacteristic
308C2#2180712yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20777C2semidirect
3. #9696C2semidirect
4. #9693C2semidirect
5. #8345C2semidirect

Maximal subgroups

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

subgroupindex
#366392
#207772
#96962
#96932
#83452
#83442