Finite Group Databasebeta

← #182021

Subgroups — #182021

Order 512.

1104 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2314C211yesmaximal, characteristic
2#20534C212yesmaximal, complemented
3C2×#1464C212yesmaximal
4#8155C214yesmaximal
5#8154C214yesmaximal, complemented
6#7912C212yesmaximal, complemented
7C22×#117C2211yescharacteristic
8C2×#432C2214yes—
9#2314—28no—
10C2×#388C412yes—
11C2×#388C2212yes—
12C2×#387C412yes—
13C2×#387C2212yes—
14C2×#384C411yescharacteristic
15C2×#382C411yescharacteristic
16C2×#380C411yescharacteristic
17C2×#380C411yescharacteristic
18C2×C4×C4⋊C4C2211yescharacteristic
19C2×C4×C4⋊C4C2211yescharacteristic
20#1771—28no—
21#1749C2214yes—
22C4×C22⋊C8C2214yes—
23#1464C4116yescomplemented
24C2×#324C2214yes—
25C2×#323C2214yes—
26#1280C2218yes—
27C24×C4C2×C411yescharacteristic
28C2×#117—24no—
29C2×#117—24no—
30C2×#117C2×C418yes—
31C22×C4⋊C4C2×C411yescharacteristic
32C22×C4⋊C4C2×C411yescharacteristic
33C22×C4⋊C4C2×C411yescharacteristic
34C22×C4⋊C4C2×C412yes—
35C22×C4⋊C4C2×C412yes—
36C22×C4⋊C4C2×C411yescharacteristic
37C22×C4⋊C4C2×C411yescharacteristic
38C22×C4⋊C4C2311yescharacteristic
39C22×C22⋊C4C2×C411yescharacteristic
40C22×C22⋊C4C2×C412yes—
41C22×C22⋊C4C2×C412yes—
42C22×C22⋊C4C2311yescharacteristic
43C22×C42C2×C411yescharacteristic
44C22×C42C2×C412yes—
45C22×C42C2312yes—
46C22×C42C2311yescharacteristic
47#432—28no—
48#432C2×C4116yes—
49C2×C4⋊C8—24no—
50C2×C22⋊C8C2318yes—
51C2×#97—24no—
52C2×#97—24no—
53C2×C4×C8—24no—
54#388—24no—
55#388—24no—
56#388—28no—
57#387—24no—
58#387—24no—
59#387—28no—
60#384—24no—
61#382—24no—
62#380—24no—
63#380—24no—
64C4×C4⋊C4—24no—
65C4×C4⋊C4D418yes—
66C2×#95C2×C412yes—
67C2×#95C2×C412yes—
68C2×#95C2×C412yes—
69C2×#95C2×C412yes—
70C2×#95C2×C412yes—
71C2×#95C2312yes—
72#336—24no—
73#336—24no—
74#324C2×C4116yes—
75#323C2×C4116yes—
76C25C22×C411yescharacteristic
77C23×C4—22no—
78C23×C4—22no—
79C23×C4—42no—
80C23×C4—42no—
81C23×C4C4214yes—
82C23×C4C22×C411yescharacteristic
83C23×C4C22×C411yescharacteristic
84C23×C4C22×C412yes—
85C23×C4C22×C412yes—
86C23×C4C22×C411yescharacteristic
87C23×C4C22×C411yescharacteristic
88C23×C4C22×C412yes—
89C23×C4C22×C412yes—
90C23×C4C22×C411yescharacteristic
91C23×C4C2411yescharacteristic
92C22×C8—28no—
93#117—216no—
94#117—216no—
95C2×C4⋊C4—22no—
96C2×C4⋊C4—22no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—24no—
102C2×C4⋊C4—24no—
103C2×C4⋊C4—42no—
104C2×C4⋊C4—42no—
105C2×C4⋊C4—42no—
106C2×C4⋊C4—42no—
107C2×C4⋊C4—82no—
108C2×C4⋊C4—82no—
109C2×C4⋊C4C4214yes—
110C2×C4⋊C4C22⋊C414yes—
111C2×C4⋊C4C22⋊C414yes—
112C2×C4⋊C4C22×C414yes—
113C2×C22⋊C4—22no—
114C2×C22⋊C4—22no—
115C2×C22⋊C4—22no—
116C2×C22⋊C4—22no—
117C2×C22⋊C4—24no—
118C2×C22⋊C4—24no—
119C2×C22⋊C4—82no—
120C2×C22⋊C4—82no—
121C2×C22⋊C4C4214yes—
122C2×C22⋊C4C22×C414yes—
123C2×C42—22no—
124C2×C42—22no—
125C2×C42—24no—
126C2×C42—82no—
127C2×C42—82no—
128C2×C42C4214yes—
129C2×C42C22×C414yes—
130C2×C42C2×D414yes—
131C2×C42C2×D414yes—
132C2×C42C2×D414yes—
133C2×C42#4114yes—
134C4⋊C8—216no—
135C22⋊C8—28no—
136C22⋊C8C22×C4116yes—
137#97—216no—
138#97—84no—
139C4×C8—84no—
140#95—24no—
141#95—24no—
142#95—24no—
143#95—24no—
144#95—22no—
145#95—22no—
146#95C22⋊C418yes—
147C24—21no—
148C24—21no—
149C24—22no—
150C24—22no—
151C24—41no—
152C24—41no—
153C24—41no—
154C24—41no—
155C24C2×C4212yes—
156C24C23×C411yescharacteristic
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—24no—
168C22×C4—24no—
169C22×C4—28no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—24no—
173C22×C4—24no—
174C22×C4—41no—
175C22×C4—41no—
176C22×C4—48no—
177C22×C4—41no—
178C22×C4—41no—
179C22×C4—42no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4—42no—
183C22×C4—44no—
184C22×C4—44no—
185C22×C4—48no—
186C22×C4—41no—
187C22×C4—41no—
188C22×C4—41no—
189C22×C4—41no—
190C22×C4—42no—
191C22×C4—42no—
192C22×C4—42no—
193C22×C4—42no—
194C22×C4C2×C4212yes—
195C22×C4C2×C4212yes—
196C22×C4C2×C4212yes—
197C22×C4C2×C4212yes—
198C22×C4C2×C4212yes—
199C22×C4C2×C22⋊C414yes—
200C22×C4C2×C22⋊C414yes—
201C22×C4C2×C22⋊C412yes—
202C22×C4C2×C22⋊C412yes—
203C22×C4#11712yes—
204C22×C4#11712yes—
205C22×C4C23×C412yes—
206C22×C4C22×D412yes—
207C22×C4C2×#4112yes—
208C2×C8—216no—
209C2×C8—48no—
210C2×C8—48no—
211C4⋊C4—28no—
212C4⋊C4—28no—
213C4⋊C4—44no—
214C4⋊C4—44no—
215C4⋊C4—44no—
216C4⋊C4—44no—
217C4⋊C4—84no—
218C4⋊C4—84no—
219C22⋊C4—28no—
220C22⋊C4—28no—
221C22⋊C4—84no—
222C22⋊C4—84no—
223C42—28no—
224C42—44no—
225C42—44no—
226C42—84no—
227C42C4×D418yes—
228C42C4×D418yes—
229C23—21no—
230C23—21no—
231C23—21no—
232C23—21no—
233C23—22no—
234C23—24no—
235C23—42no—
236C23—42no—
237C23—42no—
238C23—42no—
239C23—41no—
240C23—41no—
241C23—41no—
242C23—41no—
243C23—41no—
244C23—41no—
245C23—42no—
246C23—82no—
247C23—82no—
248C23—82no—
249C23—82no—
250C23C22×C4211yescharacteristic
251C23C22×C22⋊C411yescharacteristic
252C23C2×#11711yescharacteristic
253C23#56812yes—
254C23#56812yes—
255C2×C4—24no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—24no—
263C2×C4—24no—
264C2×C4—24no—
265C2×C4—24no—
266C2×C4—24no—
267C2×C4—24no—
268C2×C4—24no—
269C2×C4—24no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—416no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—84no—
296C2×C4—84no—
297C2×C4—84no—
298C2×C4—84no—
299C2×C4C4×C22⋊C414yes—
300C2×C4C4×C22⋊C414yes—
301C2×C4C2×C4×D414yes—
302C8—416no—
303C22—22no—
304C22—22no—
305C22—22no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—22no—
311C22—42no—
312C22—42no—
313C22—42no—
314C22—84no—
315C22—84no—
316C22—81no—
317C22—81no—
318C22—81no—
319C22—81no—
320C22C2×C4×C22⋊C411yescharacteristic
321C22#287312yes—
322C22#287911yescharacteristic
323C22#287911yescharacteristic
324C22#288311yescharacteristic
325C22#288411yescharacteristic
326C22#288811yescharacteristic
327C22#288911yescharacteristic
328C22#292112yes—
329C4—24no—
330C4—24no—
331C4—24no—
332C4—44no—
333C4—42no—
334C4—42no—
335C4—42no—
336C4—44no—
337C4—82no—
338C4—82no—
339C2—22no—
340C2—22no—
341C2—82no—
342C2#1740711yescharacteristic
343C2#1740811yescharacteristic
344C2#1740911yescharacteristic
345C2#1741011yescharacteristic
346C2#2037711yescharacteristic
347C2#2039311yescharacteristic
348C2#2039711yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC23C22×C42
Frattini, cosocleC23×C4C24
prime radicalC25C22×C4
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20534C2semidirect
5. #8154C2semidirect
6. #7912C2semidirect
23. #1464C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23142
#205342
C2×#14642
#81552
#81542
#79122