Finite Group Databasebeta

← #97022

Subgroups — #97022

Order 512.

1028 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 1028 conjugacy classes of subgroups — too many to draw. Only its 142 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×#2317C211yesmaximal, characteristic
2#8300C211yesmaximal, characteristic
3#8300C211yesmaximal, characteristic
4#7593C214yesmaximal, complemented
5C22×#117C411yescharacteristic
6C22×#117C2211yescharacteristic
7C23×C4⋊C4C411yescharacteristic
8#2317—28no—
9C2×#389C412yes—
10C2×#389C2212yes—
11C2×#385—21no—
12C2×#385—21no—
13C2×#384—21no—
14C2×#382—21no—
15#1817C2212yes—
16#1817C2212yes—
17C2×#324—21no—
18C2×#324—21no—
19C2×#323—21no—
20C2×#323—21no—
21#1271—28no—
22C24×C4—21no—
23C24×C4C2×C411yescharacteristic
24C2×#117—22no—
25C2×#117—24no—
26C2×#117—24no—
27C2×#117—24no—
28C2×#117—42no—
29C2×#117—42no—
30C2×#117C814yescomplemented
31C22×C4⋊C4—21no—
32C22×C4⋊C4—22no—
33C22×C4⋊C4—22no—
34C22×C4⋊C4—24no—
35C22×C4⋊C4—42no—
36C22×C4⋊C4—42no—
37C22×C4⋊C4C2×C412yes—
38C22×C4⋊C4C2×C412yes—
39C22×C4⋊C4D411yescharacteristic
40C22×C4⋊C4D411yescharacteristic
41C22×C22⋊C4—21no—
42C22×C22⋊C4D411yescharacteristic
43C22×C22⋊C4D411yescharacteristic
44C22×C42—21no—
45C22×C42C2×C411yescharacteristic
46C22×C42C2311yescharacteristic
47C2×C22⋊C8—22no—
48C2×C22⋊C8—22no—
49#389—24no—
50#389—24no—
51#389—24no—
52#389D418yes—
53#385—28no—
54#385—28no—
55#384—28no—
56#382—28no—
57C2×#95—22no—
58C2×#95—22no—
59#336—24no—
60#336—24no—
61#324—42no—
62#324—42no—
63#323—42no—
64#323—42no—
65C25C22⋊C411yescharacteristic
66C23×C4—22no—
67C23×C4—22no—
68C23×C4—22no—
69C23×C4—22no—
70C23×C4—22no—
71C23×C4—22no—
72C23×C4—22no—
73C23×C4—22no—
74C23×C4—24no—
75C23×C4—42no—
76C23×C4—42no—
77C23×C4—84no—
78C23×C4C22⋊C411yescharacteristic
79C23×C4C22⋊C411yescharacteristic
80C23×C4C22⋊C411yescharacteristic
81C23×C4C2×C812yes—
82C23×C4M1612yes—
83C23×C4C22×C411yescharacteristic
84C23×C4C2×D411yescharacteristic
85C23×C4C2×D411yescharacteristic
86C22×C8—42no—
87C22×C8—42no—
88#117—216no—
89#117—216no—
90#117—416no—
91C2×C4⋊C4—22no—
92C2×C4⋊C4—22no—
93C2×C4⋊C4—22no—
94C2×C4⋊C4—24no—
95C2×C4⋊C4—22no—
96C2×C4⋊C4—22no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—28no—
101C2×C4⋊C4—28no—
102C2×C4⋊C4—41no—
103C2×C4⋊C4—41no—
104C2×C4⋊C4—42no—
105C2×C4⋊C4—42no—
106C2×C4⋊C4—416no—
107C2×C4⋊C4—42no—
108C2×C4⋊C4—42no—
109C2×C4⋊C4—84no—
110C2×C4⋊C4—84no—
111C2×C4⋊C4C22⋊C414yes—
112C2×C4⋊C4C22⋊C414yes—
113C2×C4⋊C4C2×C814yes—
114C2×C4⋊C4M1614yes—
115C2×C22⋊C4—24no—
116C2×C22⋊C4—24no—
117C2×C22⋊C4—24no—
118C2×C22⋊C4—22no—
119C2×C22⋊C4—22no—
120C2×C22⋊C4—22no—
121C2×C22⋊C4—22no—
122C2×C22⋊C4—41no—
123C2×C22⋊C4—41no—
124C2×C42—24no—
125C2×C42—24no—
126C2×C42—24no—
127C2×C42—22no—
128C2×C42—41no—
129C2×C42—41no—
130C2×C42—41no—
131C2×C42—41no—
132C2×C42C2×D414yes—
133C22⋊C8—42no—
134C22⋊C8—42no—
135C22⋊C8—42no—
136C22⋊C8—42no—
137#95—28no—
138#95—28no—
139C24—21no—
140C24—21no—
141C24—21no—
142C24—21no—
143C24—42no—
144C24—42no—
145C24C22⋊C812yes—
146C24#9912yes—
147C24#10112yes—
148C24C2×C22⋊C411yescharacteristic
149C22×C4—28no—
150C22×C4—28no—
151C22×C4—28no—
152C22×C4—28no—
153C22×C4—24no—
154C22×C4—24no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—24no—
164C22×C4—24no—
165C22×C4—24no—
166C22×C4—24no—
167C22×C4—24no—
168C22×C4—24no—
169C22×C4—24no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—24no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—42no—
182C22×C4—42no—
183C22×C4—44no—
184C22×C4—44no—
185C22×C4—48no—
186C22×C4—44no—
187C22×C4—44no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4—42no—
192C22×C4—88no—
193C22×C4—88no—
194C22×C4—88no—
195C22×C4C22⋊C812yes—
196C22×C4#9912yes—
197C22×C4#10112yes—
198C22×C4C2×C22⋊C412yes—
199C22×C4#12012yes—
200C22×C4#12012yes—
201C22×C4C22×C811yescharacteristic
202C22×C4C2×M1612yes—
203C22×C4C2×M1611yescharacteristic
204C2×C8—82no—
205C2×C8—82no—
206C2×C8—84no—
207C2×C8—84no—
208C4⋊C4—24no—
209C4⋊C4—24no—
210C4⋊C4—24no—
211C4⋊C4—44no—
212C4⋊C4—44no—
213C4⋊C4—44no—
214C4⋊C4—44no—
215C4⋊C4—48no—
216C4⋊C4—88no—
217C4⋊C4—88no—
218C4⋊C4C22⋊C818yes—
219C22⋊C4—216no—
220C22⋊C4—44no—
221C22⋊C4—44no—
222C42—216no—
223C42—44no—
224C42—44no—
225C23—22no—
226C23—22no—
227C23—22no—
228C23—22no—
229C23—22no—
230C23—22no—
231C23—22no—
232C23—22no—
233C23—42no—
234C23—44no—
235C23—42no—
236C23—44no—
237C23—42no—
238C23—42no—
239C23—44no—
240C23—44no—
241C23—42no—
242C23—42no—
243C23#32414yes—
244C23#37911yescharacteristic
245C23C2×C22⋊C811yescharacteristic
246C23#40711yescharacteristic
247C23C2×#9911yescharacteristic
248C23C2×#9911yescharacteristic
249C23C2×#10111yescharacteristic
250C23C2×#10111yescharacteristic
251C2×C4—216no—
252C2×C4—216no—
253C2×C4—22no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—24no—
268C2×C4—24no—
269C2×C4—28no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—48no—
273C2×C4—48no—
274C2×C4—44no—
275C2×C4—44no—
276C2×C4—42no—
277C2×C4—44no—
278C2×C4—44no—
279C2×C4—42no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—44no—
283C2×C4—81no—
284C2×C4—81no—
285C2×C4—81no—
286C2×C4—81no—
287C2×C4—84no—
288C2×C4—84no—
289C2×C4—816no—
290C2×C4—81no—
291C2×C4—81no—
292C2×C4—81no—
293C2×C4—81no—
294C2×C4#32414yes—
295C2×C4C2×C22⋊C812yes—
296C2×C4#40712yes—
297C8—162no—
298C8—162no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—22no—
304C22—22no—
305C22—22no—
306C22—22no—
307C22—24no—
308C22—44no—
309C22—44no—
310C22—44no—
311C22—44no—
312C22—42no—
313C22—42no—
314C22—44no—
315C22—44no—
316C22C2×#32411yescharacteristic
317C22C2×#32411yescharacteristic
318C22#146311yescharacteristic
319C22#146311yescharacteristic
320C22#177911yescharacteristic
321C22#178111yescharacteristic
322C22#178411yescharacteristic
323C22#179512yes—
324C22#179612yes—
325C4—24no—
326C4—22no—
327C4—22no—
328C4—42no—
329C4—42no—
330C4—48no—
331C4—81no—
332C4—84no—
333C4—81no—
334C2—21no—
335C2—21no—
336C2—21no—
337C2—21no—
338C2—44no—
339C2#829911yescharacteristic
340C2#832412yes—
341C2#851611yescharacteristic
342C2#852011yescharacteristic
343C2#852512yes—
subgroupG/H
center, socleC23#379
commutatorC22×C4C22×C8
Frattini, cosocleC22×C42C23
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7593C2semidirect
30. C2×#117C8semidirect

Maximal subgroups

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

subgroupindex
C2×#23172
#83002
#75932