Finite Group Databasebeta

← #129257

Subgroups — #129257

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
2C2×#1326C214yesmaximal
3#8300C211yesmaximal, characteristic
4#8300C211yesmaximal, characteristic
5C22×#117C411yescharacteristic
6C22×#117C2211yescharacteristic
7C23×C4⋊C4C411yescharacteristic
8#2317—28no—
9C2×#389—21no—
10C2×#389—21no—
11C2×#385C412yes—
12C2×#385C2212yes—
13C2×#384—21no—
14C2×#382—21no—
15#1817—21no—
16#1817—21no—
17C2×#324—21no—
18C2×#324—21no—
19C2×#323C2212yes—
20C2×#323C2212yes—
21#1326—28no—
22C24×C4C2×C411yescharacteristic
23C24×C4C2×C412yes—
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—24no—
33C22×C4⋊C4—21no—
34C22×C4⋊C4—21no—
35C22×C4⋊C4—22no—
36C22×C4⋊C4—22no—
37C22×C4⋊C4—42no—
38C22×C4⋊C4—42no—
39C22×C4⋊C4D411yescharacteristic
40C22×C4⋊C4D411yescharacteristic
41C22×C22⋊C4C2×C412yes—
42C22×C22⋊C4C2×C411yescharacteristic
43C22×C22⋊C4C2311yescharacteristic
44C22×C42—21no—
45C22×C42D411yescharacteristic
46C22×C42D411yescharacteristic
47C2×C22⋊C8—22no—
48C2×C22⋊C8—22no—
49#389—28no—
50#389—28no—
51#385—24no—
52#385—24no—
53#385—24no—
54#385D418yes—
55#384—28no—
56#382—28no—
57C2×#95—22no—
58C2×#95—22no—
59#336—42no—
60#336—42no—
61#324—42no—
62#324—42no—
63#323—24no—
64#323—24no—
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—22no—
75C23×C4—42no—
76C23×C4—42no—
77C23×C4—84no—
78C23×C4C22⋊C411yescharacteristic
79C23×C4C22⋊C411yescharacteristic
80C23×C4C22⋊C411yescharacteristic
81C23×C4C22⋊C414yes—
82C23×C4C22⋊C411yescharacteristic
83C23×C4C2×C812yes—
84C23×C4M1612yes—
85C23×C4C2×D411yescharacteristic
86C23×C4C2×D411yescharacteristic
87C22×C8—42no—
88C22×C8—42no—
89#117—216no—
90#117—216no—
91#117—416no—
92C2×C4⋊C4—24no—
93C2×C4⋊C4—24no—
94C2×C4⋊C4—24no—
95C2×C4⋊C4—24no—
96C2×C4⋊C4—24no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—28no—
102C2×C4⋊C4—28no—
103C2×C4⋊C4—41no—
104C2×C4⋊C4—41no—
105C2×C4⋊C4—416no—
106C2×C4⋊C4—42no—
107C2×C4⋊C4—42no—
108C2×C4⋊C4—42no—
109C2×C4⋊C4—42no—
110C2×C4⋊C4—84no—
111C2×C4⋊C4—84no—
112C2×C22⋊C4—22no—
113C2×C22⋊C4—22no—
114C2×C22⋊C4—22no—
115C2×C22⋊C4—22no—
116C2×C22⋊C4—41no—
117C2×C22⋊C4—41no—
118C2×C22⋊C4—41no—
119C2×C22⋊C4—41no—
120C2×C22⋊C4C22⋊C414yes—
121C2×C22⋊C4C2×C814yes—
122C2×C22⋊C4M1614yes—
123C2×C22⋊C4C2×D414yes—
124C2×C42—24no—
125C2×C42—24no—
126C2×C42—24no—
127C2×C42—22no—
128C2×C42—22no—
129C2×C42—22no—
130C2×C42—22no—
131C2×C42—41no—
132C2×C42—41no—
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—
145C24C2×C22⋊C411yescharacteristic
146C24C2×C22⋊C412yes—
147C24C22×C811yescharacteristic
148C24C2×M1611yescharacteristic
149C24C2×M1612yes—
150C22×C4—28no—
151C22×C4—28no—
152C22×C4—28no—
153C22×C4—28no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—24no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—24no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—24no—
165C22×C4—24no—
166C22×C4—24no—
167C22×C4—24no—
168C22×C4—24no—
169C22×C4—24no—
170C22×C4—24no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—24no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—44no—
183C22×C4—44no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—44no—
189C22×C4—44no—
190C22×C4—44no—
191C22×C4—42no—
192C22×C4—42no—
193C22×C4—44no—
194C22×C4—84no—
195C22×C4—84no—
196C22×C4—88no—
197C22×C4—88no—
198C22×C4C22⋊C812yes—
199C22×C4C22⋊C812yes—
200C22×C4#9912yes—
201C22×C4#9912yes—
202C22×C4#10112yes—
203C22×C4#10112yes—
204C22×C4#12012yes—
205C22×C4#12012yes—
206C2×C8—82no—
207C2×C8—82no—
208C2×C8—84no—
209C2×C8—84no—
210C4⋊C4—216no—
211C4⋊C4—44no—
212C4⋊C4—44no—
213C4⋊C4—48no—
214C4⋊C4—48no—
215C4⋊C4—88no—
216C4⋊C4—88no—
217C22⋊C4—24no—
218C22⋊C4—24no—
219C22⋊C4—24no—
220C22⋊C4—44no—
221C22⋊C4—44no—
222C22⋊C4C22⋊C818yes—
223C42—216no—
224C42—44no—
225C42—44no—
226C23—22no—
227C23—22no—
228C23—21no—
229C23—21no—
230C23—21no—
231C23—21no—
232C23—22no—
233C23—22no—
234C23—22no—
235C23—22no—
236C23—42no—
237C23—42no—
238C23—42no—
239C23—44no—
240C23—42no—
241C23—42no—
242C23—44no—
243C23—42no—
244C23—44no—
245C23—42no—
246C23—42no—
247C23#37911yescharacteristic
248C23C2×C22⋊C811yescharacteristic
249C23C2×C22⋊C812yes—
250C23#40711yescharacteristic
251C23#40712yes—
252C23C2×#9911yescharacteristic
253C23C2×#9911yescharacteristic
254C23C2×#10111yescharacteristic
255C23C2×#10111yescharacteristic
256C2×C4—216no—
257C2×C4—216no—
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—22no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—28no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—44no—
274C2×C4—48no—
275C2×C4—44no—
276C2×C4—48no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—44no—
283C2×C4—44no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—44no—
290C2×C4—81no—
291C2×C4—81no—
292C2×C4—84no—
293C2×C4—88no—
294C2×C4—84no—
295C2×C4—81no—
296C2×C4—81no—
297C2×C4—88no—
298C2×C4#32414yes—
299C2×C4#32414yes—
300C8—162no—
301C8—162no—
302C22—22no—
303C22—22no—
304C22—21no—
305C22—21no—
306C22—21no—
307C22—21no—
308C22—22no—
309C22—22no—
310C22—22no—
311C22—22no—
312C22—42no—
313C22—42no—
314C22—42no—
315C22—42no—
316C22—44no—
317C22—44no—
318C22—44no—
319C22—42no—
320C22—42no—
321C22—42no—
322C22—42no—
323C22C2×#32411yescharacteristic
324C22C2×#32411yescharacteristic
325C22#146311yescharacteristic
326C22#146311yescharacteristic
327C22#177911yescharacteristic
328C22#178111yescharacteristic
329C22#178411yescharacteristic
330C22#212212yes—
331C22#212312yes—
332C4—24no—
333C4—24no—
334C4—48no—
335C4—42no—
336C4—42no—
337C4—81no—
338C4—84no—
339C4—81no—
340C2—21no—
341C2—21no—
342C2—21no—
343C2—21no—
344C2—42no—
345C2—42no—
346C2#829911yescharacteristic
347C2#1145412yes—
348C2#1152012yes—
349C2#1269111yescharacteristic
350C2#1269211yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC24C22×C8
Frattini, cosocleC22×C22⋊C4C23
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#117 ◁ G
LECSG ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
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
C2×#13262
#83002