Finite Group Databasebeta

← #190022

Subgroups — #190022

Order 512.

1782 conjugacy classes of subgroups; only the 286 normal subgroups are shown.

normal subgroups only

This group has 1782 conjugacy classes of subgroups — too many to draw. Only its 286 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×#2273C211yesmaximal, characteristic
2C2×#1462C212yesmaximal
3#8734C218yesmaximal, complemented
4#7929C214yesmaximal, complemented
5C23×C22⋊C4C412yes—
6C23×C22⋊C4C2211yescharacteristic
7C2×#407C2214yes—
8#2273—22no—
9#2273—22no—
10#2273C414yes—
11#2273C2214yes—
12C2×C4×C22⋊C4—24no—
13C2×C4×C22⋊C4C412yes—
14C2×C4×C22⋊C4C2212yes—
15#1462—24no—
16#1462—22no—
17#1462—22no—
18C2×#323C2218yes—
19#1282C22116yescomplemented
20C26C2×C411yescharacteristic
21C24×C4—22no—
22C24×C4C2×C411yescharacteristic
23C24×C4C2311yescharacteristic
24C22×M16—24no—
25C22×C22⋊C4—22no—
26C22×C22⋊C4—22no—
27C22×C22⋊C4—22no—
28C22×C22⋊C4—24no—
29C22×C22⋊C4—24no—
30C22×C22⋊C4—216no—
31C22×C22⋊C4C2×C414yes—
32C22×C22⋊C4C2×C414yes—
33C22×C22⋊C4C2×C412yes—
34C22×C22⋊C4C2×C414yes—
35C22×C22⋊C4D414yes—
36C22×C22⋊C4D412yes—
37C22×C22⋊C4D412yes—
38C22×C22⋊C4C2312yes—
39C22×C42—22no—
40C22×C42—22no—
41#407—24no—
42#407—24no—
43#407—24no—
44#407—24no—
45C2×C22⋊C8—24no—
46C2×C22⋊C8C2318yes—
47C4×C22⋊C4—24no—
48C4×C22⋊C4—22no—
49C4×C22⋊C4—22no—
50C4×C22⋊C4—28no—
51C4×C22⋊C4—28no—
52C4×C22⋊C4—28no—
53C4×C22⋊C4—28no—
54C4×C22⋊C4C2×C418yes—
55C4×C22⋊C4C2×C414yes—
56C4×C22⋊C4C2314yes—
57C2×#95—22no—
58C2×#95—22no—
59#323—28no—
60#323—28no—
61C25—22no—
62C25—22no—
63C25—28no—
64C25—82no—
65C25—82no—
66C25C22⋊C412yes—
67C25C22⋊C411yescharacteristic
68C25C22⋊C411yescharacteristic
69C25C22×C411yescharacteristic
70C25C22×C412yes—
71C23×C4—24no—
72C23×C4—216no—
73C23×C4—24no—
74C23×C4—24no—
75C23×C4—22no—
76C23×C4—22no—
77C23×C4—82no—
78C23×C4—84no—
79C23×C4—84no—
80C23×C4—82no—
81C23×C4C22⋊C412yes—
82C23×C4C22⋊C412yes—
83C23×C4C22×C412yes—
84C23×C4C22×C411yescharacteristic
85C23×C4C2×D412yes—
86C23×C4C2×D412yes—
87C23×C4#4114yes—
88C23×C4#4114yes—
89C23×C4C2411yescharacteristic
90C2×M16—44no—
91C2×M16—44no—
92C2×M16—88no—
93C22×C8—28no—
94C2×C22⋊C4—216no—
95C2×C22⋊C4—216no—
96C2×C22⋊C4—216no—
97C2×C22⋊C4—24no—
98C2×C22⋊C4—24no—
99C2×C22⋊C4—24no—
100C2×C22⋊C4—28no—
101C2×C22⋊C4—28no—
102C2×C22⋊C4—28no—
103C2×C22⋊C4—28no—
104C2×C22⋊C4—24no—
105C2×C22⋊C4—24no—
106C2×C22⋊C4—24no—
107C2×C22⋊C4—24no—
108C2×C22⋊C4—28no—
109C2×C22⋊C4—216no—
110C2×C22⋊C4—216no—
111C2×C22⋊C4C22⋊C414yes—
112C2×C22⋊C4C22⋊C414yes—
113C2×C22⋊C4M16116yescomplemented
114C2×C22⋊C4C22×C418yes—
115C2×C22⋊C4C2×D414yes—
116C2×C22⋊C4C2×D414yes—
117C2×C42—28no—
118C2×C42—24no—
119C2×C42—24no—
120C2×C42—84no—
121C2×C42—84no—
122C22⋊C8—28no—
123C22⋊C8—28no—
124C22⋊C8—44no—
125C22⋊C8—44no—
126#95—28no—
127#95—24no—
128#95—24no—
129C24—216no—
130C24—216no—
131C24—24no—
132C24—22no—
133C24—22no—
134C24—24no—
135C24—24no—
136C24—24no—
137C24—24no—
138C24—22no—
139C24—22no—
140C24—24no—
141C24—24no—
142C24—42no—
143C24—42no—
144C24—44no—
145C24—44no—
146C24—84no—
147C24—84no—
148C24—81no—
149C24—81no—
150C24—81no—
151C24—81no—
152C24—82no—
153C24—82no—
154C24—84no—
155C24—84no—
156C24—832no—
157C24C2×C22⋊C411yescharacteristic
158C24C2×C22⋊C411yescharacteristic
159C24C2×C22⋊C412yes—
160C24C2×C22⋊C412yes—
161C24#11712yes—
162C24#11712yes—
163C24C2×M1618yes—
164C24C23×C411yescharacteristic
165C22×C4—216no—
166C22×C4—28no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—216no—
170C22×C4—24no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—216no—
174C22×C4—44no—
175C22×C4—48no—
176C22×C4—48no—
177C22×C4—48no—
178C22×C4—44no—
179C22×C4—81no—
180C22×C4—82no—
181C22×C4—84no—
182C22×C4—81no—
183C22×C4—84no—
184C22×C4—84no—
185C22×C4—84no—
186C22×C4—84no—
187C22×C4—84no—
188C22×C4—84no—
189C22×C4—832no—
190C22×C4—82no—
191C22×C4—82no—
192C22×C4C2×C22⋊C414yes—
193C22×C4C2×C22⋊C412yes—
194C22×C4#11714yes—
195C22×C4#12314yes—
196C22×C4#12314yes—
197C22×C4#12314yes—
198C22×C4#12314yes—
199C22×C4C22×D412yes—
200C22×C4C2×#4114yes—
201M16—88no—
202M16—88no—
203C2×C8—48no—
204C2×C8—88no—
205C22⋊C4—216no—
206C22⋊C4—216no—
207C22⋊C4—28no—
208C22⋊C4—216no—
209C22⋊C4—28no—
210C22⋊C4—28no—
211C22⋊C4—28no—
212C22⋊C4—28no—
213C22⋊C4C2×M16116yes—
214C42—84no—
215C42—84no—
216C42—88no—
217C23—22no—
218C23—22no—
219C23—22no—
220C23—28no—
221C23—28no—
222C23—28no—
223C23—28no—
224C23—24no—
225C23—24no—
226C23—22no—
227C23—22no—
228C23—22no—
229C23—24no—
230C23—24no—
231C23—22no—
232C23—22no—
233C23—216no—
234C23—41no—
235C23—41no—
236C23—42no—
237C23—42no—
238C23—41no—
239C23—41no—
240C23—42no—
241C23—42no—
242C23—42no—
243C23—42no—
244C23—416no—
245C23—416no—
246C23—416no—
247C23—44no—
248C23—44no—
249C23—44no—
250C23—44no—
251C23—84no—
252C23—84no—
253C23—82no—
254C23—82no—
255C23—82no—
256C23—82no—
257C23—88no—
258C23—88no—
259C23—832no—
260C23—832no—
261C23—88no—
262C23—88no—
263C23#38112yes—
264C23#38111yescharacteristic
265C23#38111yescharacteristic
266C23#40714yes—
267C23#40714yes—
268C23C22×C22⋊C411yescharacteristic
269C23C2×#11712yes—
270C23C22×M1614yes—
271C2×C4—28no—
272C2×C4—216no—
273C2×C4—24no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—44no—
277C2×C4—44no—
278C2×C4—44no—
279C2×C4—44no—
280C2×C4—44no—
281C2×C4—48no—
282C2×C4—48no—
283C2×C4—432no—
284C2×C4—44no—
285C2×C4—44no—
286C2×C4—44no—
287C2×C4—84no—
288C2×C4—82no—
289C2×C4—82no—
290C2×C4—816no—
291C2×C4—816no—
292C2×C4#38114yes—
293C2×C4C2×#12314yes—
294C8—88no—
295C22—28no—
296C22—21no—
297C22—21no—
298C22—21no—
299C22—21no—
300C22—22no—
301C22—21no—
302C22—21no—
303C22—22no—
304C22—22no—
305C22—24no—
306C22—24no—
307C22—22no—
308C22—22no—
309C22—44no—
310C22—48no—
311C22—48no—
312C22—44no—
313C22—44no—
314C22—44no—
315C22—44no—
316C22—44no—
317C22—416no—
318C22—416no—
319C22—42no—
320C22—42no—
321C22—42no—
322C22—42no—
323C22—816no—
324C22—88no—
325C22—88no—
326C22#182812yes—
327C22#182912yes—
328C22C2×#38111yescharacteristic
329C22C2×#40712yes—
330C22#292014yes—
331C4—24no—
332C4—24no—
333C4—416no—
334C4—44no—
335C2—22no—
336C2—22no—
337C2—22no—
338C2—24no—
339C2—42no—
340C2—44no—
341C2—44no—
342C2#2068211yescharacteristic
343C2C2×#182811yescharacteristic
344C2C2×#182911yescharacteristic
subgroupG/H
center, socleC22C2×#381
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC26C2×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #8734C2semidirect
4. #7929C2semidirect
19. #1282C22semidirect
113. C2×C22⋊C4M16semidirect

Maximal subgroups

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

subgroupindex
C2×#22732
C2×#14622
#87342
#79292