Finite Group Databasebeta

← #215162

Subgroups — #215162

Order 512.

1108 conjugacy classes of subgroups; only the 226 normal subgroups are shown.

normal subgroups only

This group has 1108 conjugacy classes of subgroups — too many to draw. Only its 226 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×#2583C211yesmaximal, characteristic
2#21904C212yesmaximal
3#9654C214yesmaximal, complemented
4#9650C214yesmaximal, complemented
5#8308C212yesmaximal
6#8299C212yesmaximal
7C22×C22⋊Q8C2211yescharacteristic
8C23×C4⋊C4C2211yescharacteristic
9#2583—24no—
10#2583—24no—
11C2×#398C412yes—
12C2×#395C412yes—
13C2×#389C2214yes—
14C2×#385C412yes—
15C2×#385C412yes—
16C2×C4×C22⋊C4C2211yescharacteristic
17#1924—24no—
18#1923—24no—
19#1797C2214yes—
20#1779C2214yes—
21#1749C2212yes—
22C4×C22⋊C8C2212yes—
23C2×#324C2212yes—
24C2×#324C2214yes—
25C2×#324C2212yes—
26C2×#324C2212yes—
27C2×#324C2214yes—
28C2×#324C2212yes—
29C23×Q8C2×C411yescharacteristic
30C24×C4C2×C411yescharacteristic
31C24×C4C2×C411yescharacteristic
32C24×C4C2311yescharacteristic
33C2×C22⋊Q8—24no—
34C2×C22⋊Q8—42no—
35C2×C22⋊Q8—42no—
36C22×C4⋊C4—24no—
37C22×C4⋊C4—42no—
38C22×C4⋊C4—42no—
39C22×C4⋊C4C2×C414yes—
40C22×C4⋊C4C2×C412yes—
41C22×C4⋊C4C2×C411yescharacteristic
42C22×C4⋊C4C2×C414yes—
43C22×C4⋊C4C2312yes—
44C22×C4⋊C4C2312yes—
45C22×C22⋊C4C2×C412yes—
46C22×C22⋊C4C2×C412yes—
47C22×C42C2312yes—
48C2×C4⋊C8—24no—
49C2×C22⋊C8—24no—
50C2×C22⋊C8—24no—
51C2×C22⋊C8C2314yes—
52C2×C22⋊C8C2314yes—
53C2×#97—22no—
54C2×C4×C8—22no—
55#398—22no—
56#398—24no—
57#398—22no—
58#395—22no—
59#395—24no—
60#395—22no—
61#389—24no—
62#389—24no—
63#389—24no—
64#389—24no—
65#385—22no—
66#385—24no—
67#385—22no—
68#385—22no—
69#385—24no—
70#385—22no—
71C4×C22⋊C4D414yes—
72C4×C22⋊C4D414yes—
73C2×#95C2×C414yes—
74C2×#95C2×C414yes—
75C2×#95C2×C412yes—
76#336—24no—
77#336—22no—
78#336—22no—
79#324—42no—
80#324—44no—
81#324—42no—
82#324—42no—
83#324—44no—
84#324—42no—
85C25C22×C411yescharacteristic
86C22×Q8—22no—
87C22×Q8—41no—
88C22×Q8—41no—
89C22×Q8—82no—
90C23×C4—22no—
91C23×C4—22no—
92C23×C4—22no—
93C23×C4—22no—
94C23×C4—24no—
95C23×C4—41no—
96C23×C4—41no—
97C23×C4—42no—
98C23×C4—41no—
99C23×C4—41no—
100C23×C4—82no—
101C23×C4—82no—
102C23×C4C22×C412yes—
103C23×C4C22×C412yes—
104C23×C4C22×C414yes—
105C23×C4C22×C412yes—
106C23×C4C22×C412yes—
107C23×C4C22×C411yescharacteristic
108C23×C4C2×D414yes—
109C23×C4C2411yescharacteristic
110C22×C8—24no—
111C22×C8—24no—
112C22⋊Q8—416no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—24no—
117C2×C4⋊C4—24no—
118C2×C4⋊C4—28no—
119C2×C4⋊C4—24no—
120C2×C4⋊C4—24no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—24no—
124C2×C4⋊C4—24no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—24no—
128C2×C4⋊C4—24no—
129C2×C4⋊C4—42no—
130C2×C4⋊C4—42no—
131C2×C4⋊C4—44no—
132C2×C4⋊C4—41no—
133C2×C4⋊C4—41no—
134C2×C4⋊C4—416no—
135C2×C4⋊C4—84no—
136C2×C22⋊C4—28no—
137C2×C22⋊C4—22no—
138C2×C22⋊C4—22no—
139C2×C22⋊C4—42no—
140C2×C22⋊C4—42no—
141C2×C22⋊C4C22⋊C412yes—
142C2×C22⋊C4C22⋊C414yes—
143C2×C22⋊C4C22⋊C412yes—
144C2×C42—82no—
145C2×C42C2×D412yes—
146C2×C42C2×D412yes—
147C2×C42C2×D414yes—
148C4⋊C8—84no—
149C22⋊C8—24no—
150C22⋊C8—24no—
151C22⋊C8—24no—
152C22⋊C8—24no—
153C22⋊C8—84no—
154C22⋊C8—84no—
155#97—82no—
156C4×C8—82no—
157#95—24no—
158#95—24no—
159#95—24no—
160#95—24no—
161#95C22⋊C412yes—
162#95C22⋊C414yes—
163#95C22⋊C412yes—
164C24—21no—
165C24—21no—
166C24—24no—
167C24—22no—
168C24—22no—
169C24—42no—
170C24C2×C22⋊C412yes—
171C24C23×C411yescharacteristic
172C2×Q8—44no—
173C2×Q8—84no—
174C2×Q8—84no—
175C2×Q8—84no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—24no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—24no—
185C22×C4—22no—
186C22×C4—28no—
187C22×C4—28no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—24no—
194C22×C4—24no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—44no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—44no—
203C22×C4—44no—
204C22×C4—44no—
205C22×C4—48no—
206C22×C4—48no—
207C22×C4—44no—
208C22×C4—44no—
209C22×C4—44no—
210C22×C4—44no—
211C22×C4—44no—
212C22×C4—84no—
213C22×C4—84no—
214C22×C4—84no—
215C22×C4—84no—
216C22×C4—84no—
217C22×C4C2×C22⋊C414yes—
218C22×C4C2×C22⋊C412yes—
219C22×C4C2×C22⋊C414yes—
220C22×C4#13114yes—
221C22×C4C22×D412yes—
222C22×C4H(5,𝔽2)12yes—
223C22×C4H(5,𝔽2)12yes—
224C2×C8—44no—
225C2×C8—44no—
226C2×C8—44no—
227C2×C8—44no—
228C2×C8—44no—
229C2×C8—44no—
230C4⋊C4—48no—
231C4⋊C4—48no—
232C4⋊C4—44no—
233C4⋊C4—44no—
234C4⋊C4—44no—
235C4⋊C4—44no—
236C4⋊C4—44no—
237C4⋊C4—44no—
238C4⋊C4—44no—
239C4⋊C4—84no—
240C4⋊C4—84no—
241C22⋊C4—24no—
242C22⋊C4—28no—
243C22⋊C4—44no—
244C22⋊C4—44no—
245C22⋊C4#10118yes—
246C42—42no—
247C42—42no—
248C42—44no—
249C23—22no—
250C23—22no—
251C23—24no—
252C23—24no—
253C23—22no—
254C23—24no—
255C23—24no—
256C23—22no—
257C23—48no—
258C23—48no—
259C23—42no—
260C23—42no—
261C23—42no—
262C23—41no—
263C23—41no—
264C23C2×#10114yes—
265C23C22×C22⋊C411yescharacteristic
266C23#51811yescharacteristic
267C23#51811yescharacteristic
268C23C2×#13112yes—
269C23#56812yes—
270Q8—88no—
271C2×C4—24no—
272C2×C4—24no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—24no—
277C2×C4—24no—
278C2×C4—24no—
279C2×C4—24no—
280C2×C4—24no—
281C2×C4—24no—
282C2×C4—24no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—48no—
286C2×C4—48no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—416no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—44no—
298C2×C4—44no—
299C2×C4—44no—
300C2×C4—44no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—88no—
304C2×C4—88no—
305C2×C4—82no—
306C2×C4—82no—
307C2×C4—82no—
308C2×C4—82no—
309C2×C4C2×#10118yes—
310C2×C4#53514yes—
311C8—84no—
312C8—84no—
313C22—24no—
314C22—24no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—21no—
319C22—28no—
320C22—24no—
321C22—48no—
322C22—44no—
323C22—44no—
324C22—44no—
325C22—42no—
326C22—42no—
327C22#230911yescharacteristic
328C22#288012yes—
329C22#288312yes—
330C22C22×#10112yes—
331C22#297612yes—
332C22#297711yescharacteristic
333C22#297811yescharacteristic
334C4—28no—
335C4—24no—
336C4—44no—
337C4—44no—
338C4—42no—
339C4—82no—
340C4—82no—
341C4—82no—
342C2—22no—
343C2—21no—
344C2—21no—
345C2—44no—
346C2#1742614yes—
347C2#2179211yescharacteristic
348C2#2179511yescharacteristic
349C2#2179911yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #9654C2semidirect
4. #9650C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25832
#219042
#96542
#96502
#83082
#82992