Finite Group Databasebeta

← #213562

Subgroups — #213562

Order 512.

1175 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 1175 conjugacy classes of subgroups — too many to draw. Only its 202 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×#2636C211yesmaximal, characteristic
2#21742C212yesmaximal, complemented
3#9085C214yesmaximal, complemented
4#9083C214yesmaximal, complemented
5#8703C212yesmaximal, complemented
6#8309C212yesmaximal, complemented
7C22×#126C2211yescharacteristic
8C22×C4×D4C2211yescharacteristic
9C2×#407C2214yes—
10#2636—28no—
11C2×#400C411yescharacteristic
12C2×#399C411yescharacteristic
13C2×#396C412yes—
14C2×#394C412yes—
15C2×#394C411yescharacteristic
16C2×#392C411yescharacteristic
17C2×#386C2212yes—
18C2×#385C2212yes—
19C2×#382C2211yescharacteristic
20#1870—28no—
21#1817C2214yes—
22#1779C2214yes—
23C2×#324C2214yes—
24C2×#323C2214yes—
25#1280C2218yes—
26C23×D4C2×C412yes—
27C23×D4C2×C411yescharacteristic
28C24×C4C2312yes—
29C22×M16—24no—
30C2×#126—24no—
31C2×#126—44no—
32C2×C4×D4—44no—
33C2×C4×D4D418yes—
34C22×C4⋊C4C2×C411yescharacteristic
35C22×C4⋊C4C2×C411yescharacteristic
36C22×C4⋊C4C2×C412yes—
37C22×C4⋊C4C2×C412yes—
38C22×C4⋊C4C2×C411yescharacteristic
39C22×C4⋊C4C2311yescharacteristic
40C22×C22⋊C4C2×C411yescharacteristic
41C22×C22⋊C4C2×C411yescharacteristic
42C22×C22⋊C4C2×C412yes—
43C22×C22⋊C4C2×C412yes—
44C22×C22⋊C4C2×C412yes—
45C22×C22⋊C4C2×C412yes—
46C22×C22⋊C4C2×C412yes—
47C22×C22⋊C4C2311yescharacteristic
48C22×C42C2311yescharacteristic
49#407—48no—
50C2×C22⋊C8—24no—
51C2×C22⋊C8—24no—
52C2×C22⋊C8—24no—
53C2×C22⋊C8C2318yes—
54#400—24no—
55#399—24no—
56#396—28no—
57#394—24no—
58#394—28no—
59#392—24no—
60#386—24no—
61#386—24no—
62#385—24no—
63#385—24no—
64#382—24no—
65C2×#95C2×C411yescharacteristic
66C2×#95C2×C411yescharacteristic
67C2×#95C2×C412yes—
68C2×#95C2×C412yes—
69C2×#95C2312yes—
70#336—28no—
71#324—44no—
72#323—44no—
73C25C22×C411yescharacteristic
74C25C22×C412yes—
75C22×D4—24no—
76C22×D4—42no—
77C22×D4—44no—
78C22×D4—44no—
79C22×D4—82no—
80C22×D4—82no—
81C22×D4C22⋊C414yes—
82C23×C4—28no—
83C23×C4—82no—
84C23×C4—82no—
85C23×C4C22×C411yescharacteristic
86C23×C4C22×C411yescharacteristic
87C23×C4C22×C411yescharacteristic
88C23×C4C22×C412yes—
89C23×C4C22×C411yescharacteristic
90C23×C4C22×C412yes—
91C23×C4C22×C411yescharacteristic
92C23×C4C22×C412yes—
93C23×C4C2×D414yes—
94C23×C4C2×D414yes—
95C23×C4C2411yescharacteristic
96C2×M16—48no—
97C2×M16—84no—
98C2×M16—84no—
99C22×C8—28no—
100#126—416no—
101C4×D4—48no—
102C4×D4—48no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—24no—
107C2×C4⋊C4—24no—
108C2×C4⋊C4—24no—
109C2×C4⋊C4—42no—
110C2×C4⋊C4—42no—
111C2×C4⋊C4—42no—
112C2×C4⋊C4—82no—
113C2×C4⋊C4—82no—
114C2×C4⋊C4C22⋊C414yes—
115C2×C22⋊C4—24no—
116C2×C22⋊C4—22no—
117C2×C22⋊C4—22no—
118C2×C22⋊C4—22no—
119C2×C22⋊C4—24no—
120C2×C22⋊C4—24no—
121C2×C22⋊C4—24no—
122C2×C22⋊C4—24no—
123C2×C22⋊C4—44no—
124C2×C22⋊C4—44no—
125C2×C22⋊C4—42no—
126C2×C22⋊C4—42no—
127C2×C22⋊C4—82no—
128C2×C22⋊C4—82no—
129C2×C22⋊C4—82no—
130C2×C22⋊C4C22⋊C418yes—
131C2×C42—24no—
132C2×C42C2×D414yes—
133C22⋊C8—28no—
134C22⋊C8—28no—
135C22⋊C8—48no—
136C22⋊C8—84no—
137C22⋊C8—84no—
138#95—22no—
139#95—22no—
140#95—24no—
141#95—24no—
142#95—22no—
143#95—22no—
144C24—24no—
145C24—24no—
146C24—24no—
147C24—21no—
148C24—22no—
149C24—24no—
150C24—42no—
151C24—42no—
152C24—42no—
153C24—42no—
154C24—42no—
155C24—41no—
156C24—41no—
157C24C2×C22⋊C414yes—
158C24C23×C411yescharacteristic
159C2×D4—44no—
160C2×D4—48no—
161C2×D4—416no—
162C2×D4—48no—
163C2×D4—84no—
164C2×D4—84no—
165C2×D4—816no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—24no—
169C22×C4—24no—
170C22×C4—24no—
171C22×C4—24no—
172C22×C4—28no—
173C22×C4—22no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—22no—
177C22×C4—21no—
178C22×C4—22no—
179C22×C4—24no—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—24no—
183C22×C4—24no—
184C22×C4—41no—
185C22×C4—44no—
186C22×C4—44no—
187C22×C4—42no—
188C22×C4—41no—
189C22×C4—44no—
190C22×C4—44no—
191C22×C4—42no—
192C22×C4—42no—
193C22×C4—41no—
194C22×C4—42no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—41no—
200C22×C4—816no—
201C22×C4C2×C22⋊C412yes—
202C22×C4C2×C22⋊C414yes—
203C22×C4C2×C22⋊C412yes—
204C22×C4C22×D412yes—
205C22×C4H(5,𝔽2)14yes—
206M16—816no—
207C2×C8—48no—
208C2×C8—48no—
209C2×C8—48no—
210C4⋊C4—44no—
211C4⋊C4—44no—
212C4⋊C4—44no—
213C4⋊C4—44no—
214C4⋊C4—84no—
215C4⋊C4—84no—
216C22⋊C4—48no—
217C22⋊C4—44no—
218C22⋊C4—44no—
219C22⋊C4—48no—
220C22⋊C4—44no—
221C22⋊C4—84no—
222C22⋊C4—84no—
223C22⋊C4—84no—
224C42—28no—
225C23—22no—
226C23—21no—
227C23—21no—
228C23—24no—
229C23—24no—
230C23—24no—
231C23—24no—
232C23—24no—
233C23—24no—
234C23—24no—
235C23—48no—
236C23—48no—
237C23—48no—
238C23—41no—
239C23—42no—
240C23—42no—
241C23—41no—
242C23—41no—
243C23—44no—
244C23—84no—
245C23—84no—
246C23—84no—
247C23—84no—
248C23—84no—
249C23—82no—
250C23—82no—
251C23—84no—
252C23C22×C22⋊C411yescharacteristic
253C23#51812yes—
254C23#56814yes—
255D4—416no—
256D4—816no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—24no—
262C2×C4—24no—
263C2×C4—24no—
264C2×C4—24no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—44no—
268C2×C4—48no—
269C2×C4—48no—
270C2×C4—44no—
271C2×C4—44no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—41no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—41no—
285C2×C4—44no—
286C2×C4—88no—
287C2×C4—84no—
288C2×C4—82no—
289C2×C4—84no—
290C2×C4—82no—
291C2×C4—84no—
292C2×C4#53414yes—
293C8—88no—
294C22—22no—
295C22—24no—
296C22—22no—
297C22—24no—
298C22—44no—
299C22—44no—
300C22—44no—
301C22—44no—
302C22—44no—
303C22—44no—
304C22—44no—
305C22—816no—
306C22—81no—
307C22—81no—
308C22—82no—
309C22—81no—
310C22—82no—
311C22—164no—
312C22#230811yescharacteristic
313C22#287912yes—
314C22#288312yes—
315C22#288811yescharacteristic
316C22#288911yescharacteristic
317C22#297814yes—
318C4—24no—
319C4—42no—
320C4—44no—
321C4—42no—
322C4—81no—
323C4—81no—
324C4—82no—
325C4—82no—
326C4—81no—
327C2—24no—
328C2—84no—
329C2—161no—
330C2#1740512yes—
331C2#1740612yes—
332C2#2174311yescharacteristic
333C2#2175011yescharacteristic
334C2#2175711yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#392C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21742C2semidirect
3. #9085C2semidirect
4. #9083C2semidirect
5. #8703C2semidirect
6. #8309C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26362
#217422
#90852
#90832
#87032
#83092