Finite Group Databasebeta

← #213747

Subgroups — #213747

Order 512.

875 conjugacy classes of subgroups.

The subgroup lattice has 357 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35453C211yesmaximal, characteristic
2#17368C212yesmaximal
3#9110C214yesmaximal, complemented
4#8968C214yesmaximal, complemented
5#8737C212yesmaximal, complemented
6#8306C212yesmaximal, complemented
7C22×#125C2211yescharacteristic
8C2×#520C2211yescharacteristic
9#2973C2214yes—
10#2579C412yes—
11#2486C412yes—
12#2484C411yescharacteristic
13#2484C411yescharacteristic
14#2475—22no—
15#2473—22no—
16#2462—22no—
17#2460—22no—
18#2406C411yescharacteristic
19#2406C411yescharacteristic
20#2322C2212yes—
21#2313C2211yescharacteristic
22C4×#123C2212yes—
23#1817C2214yes—
24#1467—22no—
25#1465—22no—
26#1464—22no—
27#1463—22no—
28#1459C2214yes—
29C2×#324C2214yes—
30C2×#324C2214yes—
31#1282C2218yes—
32C22×#41C2×C411yescharacteristic
33C2×#125—22no—
34C2×#125—22no—
35C2×#125—44no—
36C2×#123C2×C412yes—
37C2×#123C2×C412yes—
38#520—44no—
39C2×C4×Q8D412yes—
40C2×C4×D4C2×C412yes—
41C2×C4×D4D412yes—
42C2×C4×D4C2312yes—
43C2×#117C2×C411yescharacteristic
44C2×#117D412yes—
45C22×C4⋊C4C2×C411yescharacteristic
46C22×C4⋊C4C2×C411yescharacteristic
47C22×C4⋊C4C2×C411yescharacteristic
48C22×C4⋊C4C2×C411yescharacteristic
49C22×C4⋊C4D412yes—
50C22×C4⋊C4C2311yescharacteristic
51C22×C4⋊C4C2311yescharacteristic
52C22×C42C2311yescharacteristic
53#435—48no—
54#432—24no—
55#407—24no—
56C2×C22⋊C8C2318yes—
57#400—21no—
58#400—21no—
59#398—22no—
60#398—22no—
61#398—21no—
62#398—21no—
63#397—22no—
64#396—22no—
65#395—21no—
66#395—21no—
67#394—21no—
68#394—21no—
69#389—22no—
70#389—22no—
71#388—22no—
72#388—22no—
73#387—22no—
74#387—22no—
75#387—22no—
76#387—22no—
77#387—22no—
78#387—22no—
79#387—22no—
80#386C2×C412yes—
81#386C2×C412yes—
82#385—22no—
83#385—22no—
84#385—22no—
85#385—22no—
86#385C2×C412yes—
87#385C2×C412yes—
88#381C2×C411yescharacteristic
89#381C2×C411yescharacteristic
90#381C2×C412yes—
91#381C2×C412yes—
92#380—22no—
93#380C2×C412yes—
94C4×C4⋊C4—22no—
95C4×C22⋊C4—22no—
96C4×C22⋊C4C2312yes—
97#336—44no—
98#324—24no—
99#324—24no—
100#324—24no—
101#324—44no—
102#323—24no—
103C2×#41—42no—
104C2×#41—44no—
105C22×Q8C22⋊C411yescharacteristic
106C22×D4C22⋊C411yescharacteristic
107C22×D4C22×C412yes—
108C23×C4C22⋊C412yes—
109C23×C4C22⋊C412yes—
110C23×C4C22×C411yescharacteristic
111C23×C4C22×C411yescharacteristic
112C23×C4C22×C411yescharacteristic
113C23×C4C22×C411yescharacteristic
114C23×C4C22×C411yescharacteristic
115C23×C4C22×C411yescharacteristic
116C23×C4C2×D412yes—
117C23×C4C2411yescharacteristic
118C2×M16—44no—
119C22×C8—28no—
120#125—48no—
121#125—48no—
122#123—44no—
123#123—44no—
124C4×Q8—44no—
125C4×D4—28no—
126C4×D4—44no—
127C4×D4—44no—
128#117—44no—
129#117—42no—
130C2×C4⋊C4—24no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—44no—
145C2×C4⋊C4—42no—
146C2×C4⋊C4—42no—
147C2×C4⋊C4—42no—
148C2×C4⋊C4—42no—
149C2×C4⋊C4C22⋊C411yescharacteristic
150C2×C4⋊C4C22⋊C412yes—
151C2×C4⋊C4C22⋊C411yescharacteristic
152C2×C4⋊C4C2×D412yes—
153C2×C4⋊C4C2×D412yes—
154C2×C4⋊C4C2×D412yes—
155C2×C22⋊C4—22no—
156C2×C22⋊C4—22no—
157C2×C22⋊C4—22no—
158C2×C22⋊C4—22no—
159C2×C22⋊C4C22⋊C412yes—
160C2×C22⋊C4C22×C412yes—
161C2×C22⋊C4C22×C412yes—
162C2×C22⋊C4C22×C412yes—
163C2×C22⋊C4C2×D412yes—
164C2×C42—22no—
165C2×C42—24no—
166C2×C42—22no—
167C2×C42C22⋊C412yes—
168C2×C42C22⋊C412yes—
169C2×C42C2×D412yes—
170C4⋊C8—44no—
171C22⋊C8—28no—
172C22⋊C8—28no—
173C22⋊C8—44no—
174#97—44no—
175#95—22no—
176#95—22no—
177#95—22no—
178#95—22no—
179#95—22no—
180#95—22no—
181#95—21no—
182#95—21no—
183#95—21no—
184#95—21no—
185#95—22no—
186#95—22no—
187#95—22no—
188#95—22no—
189C24C2×C22⋊C412yes—
190C24C23×C411yescharacteristic
191#41—416no—
192C2×Q8—41no—
193C2×Q8—42no—
194C2×D4—24no—
195C2×D4—24no—
196C2×D4—24no—
197C2×D4—41no—
198C2×D4—42no—
199C22×C4—21no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—24no—
203C22×C4—22no—
204C22×C4—24no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—24no—
213C22×C4—24no—
214C22×C4—22no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—81no—
221C22×C4—81no—
222C22×C4—81no—
223C22×C4—82no—
224C22×C4—82no—
225C22×C4C2×C22⋊C411yescharacteristic
226C22×C4C2×C22⋊C412yes—
227C22×C4C2×C22⋊C411yescharacteristic
228C22×C4C2×C22⋊C412yes—
229C22×C4C2×C22⋊C412yes—
230C22×C4C2×C22⋊C412yes—
231C22×C4C22×D412yes—
232C22×C4H(5,𝔽2)14yes—
233M16—164no—
234C2×C8—48no—
235C2×C8—88no—
236C4⋊C4—24no—
237C4⋊C4—24no—
238C4⋊C4—24no—
239C4⋊C4—41no—
240C4⋊C4—42no—
241C4⋊C4—41no—
242C4⋊C4—44no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—44no—
246C4⋊C4—44no—
247C4⋊C4—42no—
248C4⋊C4—42no—
249C4⋊C4—44no—
250C4⋊C4—44no—
251C4⋊C4—44no—
252C4⋊C4—82no—
253C4⋊C4—82no—
254C22⋊C4—24no—
255C22⋊C4—24no—
256C22⋊C4—24no—
257C22⋊C4—24no—
258C22⋊C4—42no—
259C22⋊C4—44no—
260C22⋊C4—44no—
261C22⋊C4—44no—
262C22⋊C4—82no—
263C42—24no—
264C42—42no—
265C42—42no—
266C42—44no—
267C42—82no—
268C23—22no—
269C23—22no—
270C23—22no—
271C23—41no—
272C23—41no—
273C23—42no—
274C23—42no—
275C23C22×C22⋊C411yescharacteristic
276C23#51812yes—
277C23#56814yes—
278Q8—44no—
279D4—44no—
280D4—48no—
281C2×C4—22no—
282C2×C4—24no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—21no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—21no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—42no—
294C2×C4—44no—
295C2×C4—41no—
296C2×C4—42no—
297C2×C4—48no—
298C2×C4—41no—
299C2×C4—44no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—41no—
307C2×C4—42no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—44no—
317C2×C4—44no—
318C2×C4—84no—
319C2×C4—82no—
320C2×C4—82no—
321C2×C4—162no—
322C2×C4#53414yes—
323C8—88no—
324C22—21no—
325C22—22no—
326C22—21no—
327C22—22no—
328C22—44no—
329C22—44no—
330C22—42no—
331C22—42no—
332C22—42no—
333C22—82no—
334C22#230811yescharacteristic
335C22#287912yes—
336C22#297814yes—
337C4—24no—
338C4—42no—
339C4—42no—
340C4—42no—
341C4—42no—
342C4—41no—
343C4—42no—
344C4—41no—
345C4—82no—
346C4—82no—
347C4—81no—
348C4—81no—
349C4—81no—
350C2—22no—
351C2—24no—
352C2—82no—
353C2#2174311yescharacteristic
354C2#2177711yescharacteristic
355C2#2178011yescharacteristic
subgroupG/H
center, socleC22#2308
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#41C2×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #9110C2semidirect
4. #8968C2semidirect
5. #8737C2semidirect
6. #8306C2semidirect

Maximal subgroups

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

subgroupindex
#354532
#173682
#91102
#89682
#87372
#83062