Finite Group Databasebeta

← #110561

Subgroups — #110561

Order 512.

1148 conjugacy classes of subgroups; only the 178 normal subgroups are shown.

normal subgroups only

This group has 1148 conjugacy classes of subgroups — too many to draw. Only its 178 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×#1508C212yesmaximal
2#14159C211yesmaximal, characteristic
3#13749C211yesmaximal, characteristic
4C2×#1297C212yesmaximal
5#7909C211yesmaximal, characteristic
6C22×C22⋊Q8C2211yescharacteristic
7C2×#407C2212yes—
8C2×C4×C22⋊C4—22no—
9C2×C4×C22⋊C4C2211yescharacteristic
10C2×C4×C22⋊C4C2212yes—
11C22×#95C2211yescharacteristic
12#1508—24no—
13#1508C418yescomplemented
14C2×#328—24no—
15#1446C412yescomplemented
16#1445C412yescomplemented
17#1436—22no—
18C22⋊#95—22no—
19C4×#95—24no—
20#1297—24no—
21#1280—22no—
22C23×Q8C2×C411yescharacteristic
23C24×C4D412yes—
24C24×C4C2311yescharacteristic
25C22×M16—22no—
26C2×C22⋊Q8—22no—
27C2×C22⋊Q8—22no—
28C2×C22⋊Q8—22no—
29C2×C22⋊Q8—22no—
30C2×C22⋊Q8—22no—
31C2×C22⋊Q8C2×C414yes—
32C22×C4⋊C4—21no—
33C22×C4⋊C4C2×C411yescharacteristic
34C22×C22⋊C4—22no—
35C22×C22⋊C4—21no—
36C22×C22⋊C4—21no—
37C22×C42—24no—
38C22×C42—22no—
39C22×C42—21no—
40#407—22no—
41#407—22no—
42#407—22no—
43#407C2×C414yes—
44C2×C22⋊C8—22no—
45C2×C22⋊C8—22no—
46C4×C22⋊C4—216no—
47C4×C22⋊C4—22no—
48C4×C22⋊C4—22no—
49C4×C22⋊C4—22no—
50C4×C22⋊C4—22no—
51C4×C22⋊C4C2×C414yes—
52C4×C22⋊C4D414yes—
53C2×#95—24no—
54C2×#95—22no—
55C2×#95—22no—
56C2×#95—24no—
57C2×#95—24no—
58C2×#95C2×C412yes—
59C2×#95C2×C412yes—
60C2×#95D412yes—
61C2×#95D414yes—
62#328—216no—
63C25C2×D411yescharacteristic
64C22×Q8—21no—
65C22×Q8—22no—
66C22×Q8—22no—
67C22×Q8—82no—
68C22×Q8C4212yescomplemented
69C23×C4—24no—
70C23×C4—22no—
71C23×C4—22no—
72C23×C4—28no—
73C23×C4—21no—
74C23×C4—21no—
75C23×C4—22no—
76C23×C4—44no—
77C23×C4—82no—
78C23×C4—82no—
79C23×C4C22⋊C414yes—
80C23×C4C22×C411yescharacteristic
81C23×C4C22×C412yes—
82C23×C4C2×D412yes—
83C23×C4C2×D411yescharacteristic
84C23×C4C2×D412yes—
85C23×C4#4112yes—
86C23×C4#4112yes—
87C23×C4#4112yes—
88C2×M16—24no—
89C2×M16—42no—
90C2×M16—84no—
91C22×C8—42no—
92C22⋊Q8—24no—
93C22⋊Q8—24no—
94C22⋊Q8—24no—
95C22⋊Q8—24no—
96C22⋊Q8—42no—
97C22⋊Q8—42no—
98C22⋊Q8—42no—
99C22⋊Q8—42no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—24no—
104C2×C4⋊C4—42no—
105C2×C4⋊C4—41no—
106C2×C4⋊C4—41no—
107C2×C4⋊C4C4212yescomplemented
108C2×C22⋊C4—216no—
109C2×C22⋊C4—24no—
110C2×C22⋊C4—24no—
111C2×C22⋊C4—24no—
112C2×C22⋊C4—24no—
113C2×C22⋊C4—42no—
114C2×C22⋊C4—41no—
115C2×C22⋊C4—42no—
116C2×C22⋊C4—42no—
117C2×C22⋊C4—41no—
118C2×C42—216no—
119C2×C42—24no—
120C2×C42—24no—
121C2×C42—41no—
122C2×C42—41no—
123C2×C42—84no—
124C2×C42—84no—
125C2×C42—84no—
126C2×C42—82no—
127C22⋊C8—24no—
128C22⋊C8—24no—
129C22⋊C8—42no—
130C22⋊C8—42no—
131#95—216no—
132#95—24no—
133#95—24no—
134#95—24no—
135#95—24no—
136#95—48no—
137#95—84no—
138#95C22⋊C418yes—
139C24—24no—
140C24—22no—
141C24—22no—
142C24—42no—
143C24C2×C22⋊C411yescharacteristic
144C24C4×D412yes—
145C24#12011yescharacteristic
146C24#12112yes—
147C24#12311yescharacteristic
148C2×Q8—24no—
149C2×Q8—42no—
150C2×Q8—84no—
151C2×Q8—82no—
152C2×Q8—82no—
153C2×Q8—84no—
154C22×C4—24no—
155C22×C4—216no—
156C22×C4—24no—
157C22×C4—28no—
158C22×C4—24no—
159C22×C4—28no—
160C22×C4—24no—
161C22×C4—24no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—42no—
165C22×C4—44no—
166C22×C4—44no—
167C22×C4—42no—
168C22×C4—48no—
169C22×C4—44no—
170C22×C4—44no—
171C22×C4—44no—
172C22×C4—44no—
173C22×C4—44no—
174C22×C4—44no—
175C22×C4—44no—
176C22×C4—44no—
177C22×C4—42no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4—42no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—44no—
184C22×C4—44no—
185C22×C4—82no—
186C22×C4—82no—
187C22×C4—82no—
188C22×C4—82no—
189C22×C4—816no—
190C22×C4—82no—
191C22×C4—82no—
192C22×C4#10418yes—
193C22×C4C2×C4211yescharacteristic
194C22×C4C2×C22⋊C412yes—
195C22×C4#11712yes—
196C22×C4#11711yescharacteristic
197C22×C4C4×D412yes—
198C22×C4C4×D412yes—
199C22×C4C4×D412yes—
200C22×C4C4×D412yes—
201C22×C4C4×D412yes—
202C22×C4#12112yes—
203C22×C4#12112yes—
204C22×C4#12312yes—
205C22×C4#12411yescharacteristic
206C22×C4#12412yes—
207C22×C4#12612yes—
208C22×C4#12711yescharacteristic
209M16—84no—
210M16—84no—
211C2×C8—44no—
212C2×C8—84no—
213C4⋊C4—24no—
214C4⋊C4—42no—
215C4⋊C4—44no—
216C4⋊C4—44no—
217C22⋊C4—216no—
218C22⋊C4—216no—
219C22⋊C4—48no—
220C22⋊C4—44no—
221C22⋊C4—44no—
222C42—84no—
223C42—84no—
224C42—816no—
225C42—82no—
226C42—82no—
227C23—21no—
228C23—21no—
229C23—22no—
230C23—22no—
231C23—22no—
232C23—24no—
233C23—24no—
234C23—21no—
235C23—21no—
236C23—22no—
237C23—22no—
238C23—24no—
239C23—24no—
240C23—42no—
241C23—42no—
242C23—48no—
243C23—48no—
244C23C4×C22⋊C411yescharacteristic
245C23#38612yes—
246C23#38611yescharacteristic
247C23#38812yes—
248C23#39011yescharacteristic
249C23C2×#10414yes—
250C23#45512yes—
251C23#45612yes—
252Q8—84no—
253Q8—84no—
254C2×C4—24no—
255C2×C4—24no—
256C2×C4—216no—
257C2×C4—28no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—41no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—41no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—48no—
273C2×C4—416no—
274C2×C4—48no—
275C2×C4—416no—
276C2×C4—48no—
277C2×C4—48no—
278C2×C4—44no—
279C2×C4—44no—
280C2×C4—48no—
281C2×C4—44no—
282C2×C4—48no—
283C2×C4—44no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—44no—
287C2×C4—84no—
288C2×C4—88no—
289C2×C4—82no—
290C2×C4—84no—
291C2×C4—82no—
292C2×C4—82no—
293C2×C4—82no—
294C2×C4—82no—
295C2×C4—82no—
296C2×C4C4×C22⋊C412yes—
297C2×C4#38812yes—
298C2×C4#38812yes—
299C2×C4#39012yes—
300C8—84no—
301C22—22no—
302C22—21no—
303C22—22no—
304C22—21no—
305C22—24no—
306C22—24no—
307C22—24no—
308C22—24no—
309C22—21no—
310C22—21no—
311C22—22no—
312C22—22no—
313C22—22no—
314C22—48no—
315C22—48no—
316C22—42no—
317C22—42no—
318C22#143711yescharacteristic
319C22C4×#10412yes—
320C22#188214yes—
321C22#188511yescharacteristic
322C22#188611yescharacteristic
323C22#189612yes—
324C22#191312yes—
325C22#196212yes—
326C4—22no—
327C4—22no—
328C4—22no—
329C4—22no—
330C4—416no—
331C4—48no—
332C4—84no—
333C4—82no—
334C4—82no—
335C2—24no—
336C2—22no—
337C2—22no—
338C2—42no—
339C2#940712yes—
340C2#940812yes—
341C2#1092311yescharacteristic
342C2#1093811yescharacteristic
343C2#1095711yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC22×C4C2×C42
Frattini, cosocleC24×C4C23
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
13. #1508C4semidirect
15. #1446C4semidirect
16. #1445C4semidirect
68. C22×Q8C42semidirect
107. C2×C4⋊C4C42semidirect

Maximal subgroups

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

subgroupindex
C2×#15082
#141592
#137492
C2×#12972
#79092