Finite Group Databasebeta

← #213628

Subgroups — #213628

Order 512.

1104 conjugacy classes of subgroups; only the 214 normal subgroups are shown.

normal subgroups only

This group has 1104 conjugacy classes of subgroups — too many to draw. Only its 214 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#36625C211yesmaximal, characteristic
2#20715C212yesmaximal
3#9125C218yesmaximal, complemented
4#8721C212yesmaximal, complemented
5#8338C212yesmaximal, complemented
6C2×#533C2211yescharacteristic
7C23×C42C2211yescharacteristic
8C2×#432C2214yes—
9#2857C412yes—
10#2857C412yes—
11#2855—22no—
12#2851C2214yes—
13#2848—22no—
14C4×C22⋊Q8—22no—
15C4×C22⋊Q8—22no—
16#2296C412yes—
17#2285C2211yescharacteristic
18#2280C412yes—
19#1817—22no—
20#1797—22no—
21#1797—22no—
22#1779—22no—
23#1749C2214yes—
24#1749C2214yes—
25C2×#330C2218yescomplemented
26#1295C2218yescomplemented
27C24×C4C2×C411yescharacteristic
28C24×C4D412yes—
29#533—44no—
30C2×C4⋊Q8C2×C412yes—
31C2×#125C2312yes—
32C2×C22⋊Q8—22no—
33C2×C22⋊Q8—22no—
34C2×C22⋊Q8—22no—
35C2×C4×Q8—22no—
36C2×#117C2×C412yes—
37C2×#117C2×C412yes—
38C22×C42—44no—
39C22×C42C2×C412yes—
40C22×C42C2×C411yescharacteristic
41C22×C42D412yes—
42C22×C42D412yes—
43C22×C42D412yes—
44C22×C42C2312yes—
45C22×C42C2311yescharacteristic
46#432—48no—
47C2×C4⋊C8—24no—
48C2×C22⋊C8—24no—
49C2×C22⋊C8—24no—
50C2×C22⋊C8—24no—
51C2×#97C2318yes—
52#391—22no—
53#391—22no—
54#389—22no—
55#389—22no—
56#389C2×C414yes—
57#389C2×C414yes—
58#387—24no—
59#387—24no—
60#385—24no—
61#385—24no—
62#385—24no—
63#384C2×C411yescharacteristic
64#384C2×C411yescharacteristic
65#382C2312yes—
66#381—22no—
67#380—22no—
68#380—22no—
69#380—22no—
70#380C2×C414yes—
71#380C2×C414yes—
72#379—22no—
73C4×C4⋊C4—22no—
74C4×C4⋊C4—22no—
75C4×C4⋊C4—22no—
76C4×C22⋊C4—24no—
77#336—24no—
78#336—24no—
79#330—48no—
80C25C22⋊C411yescharacteristic
81C22×Q8—22no—
82C23×C4—24no—
83C23×C4—42no—
84C23×C4—42no—
85C23×C4—44no—
86C23×C4—44no—
87C23×C4—44no—
88C23×C4C22⋊C412yes—
89C23×C4C22⋊C412yes—
90C23×C4C22⋊C412yes—
91C23×C4C22⋊C412yes—
92C23×C4C22⋊C411yescharacteristic
93C23×C4C22×C411yescharacteristic
94C23×C4C22×C412yes—
95C23×C4C2×D412yes—
96C23×C4C2×D412yes—
97C23×C4C2×D412yes—
98C22×C8—28no—
99C4⋊Q8—44no—
100#125—28no—
101C22⋊Q8—48no—
102C22⋊Q8—48no—
103C22⋊Q8—48no—
104C4×Q8—48no—
105#117—44no—
106#117—44no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—24no—
111C2×C4⋊C4—24no—
112C2×C4⋊C4—24no—
113C2×C4⋊C4C22×C414yes—
114C2×C4⋊C4C22×C412yes—
115C2×C4⋊C4C22×C412yes—
116C2×C22⋊C4—22no—
117C2×C22⋊C4—22no—
118C2×C22⋊C4—24no—
119C2×C22⋊C4—24no—
120C2×C42—24no—
121C2×C42—28no—
122C2×C42—22no—
123C2×C42—22no—
124C2×C42—42no—
125C2×C42—44no—
126C2×C42—44no—
127C2×C42—44no—
128C2×C42—44no—
129C2×C42C22⋊C411yescharacteristic
130C2×C42C22⋊C411yescharacteristic
131C2×C42C22⋊C412yes—
132C2×C42C22⋊C412yes—
133C2×C42C22×C411yescharacteristic
134C2×C42C22×C412yes—
135C2×C42C2×D412yes—
136C2×C42C2×D412yes—
137C2×C42C2×D412yes—
138C2×C42C2411yescharacteristic
139C4⋊C8—48no—
140C22⋊C8—48no—
141C22⋊C8—48no—
142C22⋊C8—48no—
143#97—216no—
144#95—24no—
145#95—24no—
146#95—24no—
147#95—24no—
148#95—24no—
149C24—22no—
150C24—24no—
151C24—41no—
152C24—41no—
153C24—42no—
154C24C2×C22⋊C411yescharacteristic
155C24C2×C22⋊C412yes—
156C2×Q8—44no—
157C2×Q8—44no—
158C2×Q8—44no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—22no—
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—22no—
175C22×C4—22no—
176C22×C4—24no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4—42no—
180C22×C4—41no—
181C22×C4—42no—
182C22×C4—41no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4—416no—
192C22×C4—432no—
193C22×C4#10114yes—
194C22×C4C2×C22⋊C412yes—
195C22×C4C2×C22⋊C412yes—
196C22×C4C2×C22⋊C412yes—
197C22×C4C2×C22⋊C411yescharacteristic
198C22×C4C2×C22⋊C412yes—
199C22×C4C23×C411yescharacteristic
200C22×C4C22×D412yes—
201C2×C8—48no—
202C2×C8—48no—
203C2×C8—48no—
204C4⋊C4—28no—
205C4⋊C4—42no—
206C4⋊C4—42no—
207C4⋊C4—44no—
208C4⋊C4—44no—
209C4⋊C4—48no—
210C4⋊C4—48no—
211C4⋊C4—48no—
212C4⋊C4—44no—
213C22⋊C4—44no—
214C22⋊C4—44no—
215C22⋊C4—48no—
216C22⋊C4—48no—
217C42—24no—
218C42—22no—
219C42—24no—
220C42—24no—
221C42—24no—
222C42—41no—
223C42—41no—
224C42—42no—
225C42—42no—
226C42—44no—
227C42—44no—
228C42H(5,𝔽2)14yes—
229C23—21no—
230C23—22no—
231C23—22no—
232C23—21no—
233C23—22no—
234C23—22no—
235C23—22no—
236C23—22no—
237C23—22no—
238C23—22no—
239C23—48no—
240C23—44no—
241C23—48no—
242C23—44no—
243C23—44no—
244C23C2×#10112yes—
245C23#42114yes—
246C23C22×C22⋊C411yescharacteristic
247Q8—88no—
248C2×C4—28no—
249C2×C4—21no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—44no—
266C2×C4—48no—
267C2×C4—48no—
268C2×C4—48no—
269C2×C4—48no—
270C2×C4—48no—
271C2×C4—48no—
272C2×C4—48no—
273C2×C4—48no—
274C2×C4—44no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—44no—
278C2×C4—44no—
279C2×C4—44no—
280C2×C4—44no—
281C2×C4C2×#10114yes—
282C2×C4#41312yes—
283C2×C4#42114yes—
284C2×C4#51812yes—
285C2×C4#53414yes—
286C8—88no—
287C22—24no—
288C22—24no—
289C22—24no—
290C22—24no—
291C22—22no—
292C22—416no—
293C22—41no—
294C22—42no—
295C22—42no—
296C22—41no—
297C22—42no—
298C22—42no—
299C22—42no—
300C22#230811yescharacteristic
301C22C22×#10111yescharacteristic
302C22C2×#41311yescharacteristic
303C22C2×#42112yes—
304C22C2×#42112yes—
305C4—24no—
306C4—24no—
307C4—22no—
308C4—22no—
309C4—24no—
310C4—42no—
311C4—42no—
312C4—82no—
313C4—82no—
314C4—84no—
315C2—42no—
316C2—44no—
317C2#1766511yescharacteristic
318C2#1767212yes—
319C2#1767811yescharacteristic
320C2#2175311yescharacteristic
321C2#2177412yes—
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. #9125C2semidirect
4. #8721C2semidirect
5. #8338C2semidirect
25. C2×#330C22semidirect
26. #1295C22semidirect

Maximal subgroups

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

subgroupindex
#366252
#207152
#91252
#87212
#83382