Finite Group Databasebeta

← #241185

Subgroups — #241185

Order 512.

950 conjugacy classes of subgroups; only the 210 normal subgroups are shown.

normal subgroups only

This group has 950 conjugacy classes of subgroups — too many to draw. Only its 210 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
1#35167C211yesmaximal, characteristic
2#21986C212yesmaximal
3#14298C211yesmaximal, characteristic
4#10570C214yesmaximal
5#10569C214yesmaximal
6#8018C212yesmaximal
7#7912C211yesmaximal, characteristic
8C2×#544—21no—
9C2×#432C2212yes—
10#2857—22no—
11#2659C2214yes—
12#2552—22no—
13#2551—22no—
14#2550—24no—
15#2316—21no—
16#2285C2212yes—
17C2×#383C2211yescharacteristic
18C2×#382C2211yescharacteristic
19C2×C4×C4⋊C4C2211yescharacteristic
20#1949—24no—
21#1853C2214yes—
22#1852C2214yes—
23#1847—22no—
24#1836—22no—
25#1444—21no—
26#1444C2212yes—
27#1442—21no—
28#1442C2212yes—
29#1436—22no—
30#1294C2218yes—
31#1280C2214yes—
32#544—48no—
33C2×C4⋊Q8—24no—
34C2×C4⋊Q8—22no—
35C2×#125—24no—
36C2×C22⋊Q8—24no—
37C2×#117—22no—
38C22×C4⋊C4—22no—
39C22×C4⋊C4D411yescharacteristic
40C22×C4⋊C4D411yescharacteristic
41C22×C4⋊C4C2311yescharacteristic
42C22×C4⋊C4C2311yescharacteristic
43C22×C42D412yes—
44C22×C42C2311yescharacteristic
45#432—24no—
46#432Q818yescomplemented
47C2×#107—24no—
48C2×#106—24no—
49C2×C4⋊C8—22no—
50C2×C22⋊C8C2314yes—
51C2×#97—22no—
52#398—28no—
53#395—28no—
54#391—24no—
55#389—24no—
56#389—24no—
57#389—24no—
58#389—24no—
59#384—22no—
60#384D412yes—
61#384D412yes—
62#383—22no—
63#383Q814yescomplemented
64#382—22no—
65#382—22no—
66#382—22no—
67#380—24no—
68#380—22no—
69#380C2314yes—
70C4×C4⋊C4—22no—
71C4×C4⋊C4—22no—
72C4×C4⋊C4—22no—
73C4×C4⋊C4D414yes—
74C2×#95—21no—
75C2×#95—21no—
76C2×#95—21no—
77C2×#95—22no—
78C2×#95—21no—
79C2×#95—22no—
80C2×#95—22no—
81C2×#95C2312yes—
82C2×#95C2312yes—
83#340—24no—
84#340—24no—
85C22×Q8—24no—
86C23×C4—22no—
87C23×C4—21no—
88C23×C4—21no—
89C23×C4C2×D411yescharacteristic
90C23×C4C2×D412yes—
91C23×C4C2×D412yes—
92C23×C4C2×D411yescharacteristic
93C23×C4C2411yescharacteristic
94C22×C8—24no—
95C4⋊Q8—416no—
96C4⋊Q8—84no—
97#125—416no—
98C22⋊Q8—48no—
99C22⋊Q8—48no—
100#117—48no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—24no—
104C2×C4⋊C4—24no—
105C2×C4⋊C4—24no—
106C2×C4⋊C4—28no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—24no—
114C2×C4⋊C4—24no—
115C2×C4⋊C4—44no—
116C2×C4⋊C4—81no—
117C2×C4⋊C4—81no—
118C2×C4⋊C4—82no—
119C2×C4⋊C4—82no—
120C2×C4⋊C4C2×D414yes—
121C2×C4⋊C4C2×D414yes—
122C2×C22⋊C4—24no—
123C2×C42—24no—
124C2×C42—22no—
125C2×C42—22no—
126C2×C42—82no—
127C2×C42C2×D412yes—
128C2×C42C2×D412yes—
129C2×C42C2×Q814yes—
130C2×C42C2×Q812yes—
131C2×C42#4112yes—
132#107—84no—
133#106—84no—
134C4⋊C8—28no—
135C22⋊C8—24no—
136C22⋊C8C2×Q818yes—
137#97—28no—
138#95—28no—
139#95—24no—
140#95—24no—
141#95—24no—
142#95—24no—
143#95—24no—
144#95—22no—
145#95—41no—
146#95—41no—
147#95—42no—
148#95—44no—
149#95—82no—
150#95—82no—
151#95C2×Q814yes—
152C24C22×D411yescharacteristic
153C2×Q8—48no—
154C2×Q8—84no—
155C2×Q8—84no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—24no—
163C22×C4—24no—
164C22×C4—24no—
165C22×C4—41no—
166C22×C4—41no—
167C22×C4—41no—
168C22×C4—41no—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4—42no—
173C22×C4—42no—
174C22×C4—42no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4—41no—
180C22×C4—42no—
181C22×C4—41no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—41no—
185C22×C4—41no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4#12011yescharacteristic
189C22×C4#12012yes—
190C22×C4#12011yescharacteristic
191C22×C4C22⋊Q812yes—
192C22×C4C22⋊Q812yes—
193C22×C4C4⋊Q812yes—
194C22×C4C4⋊Q812yes—
195C22×C4AGL(1,ℤ/8)14yes—
196C22×C4#13714yes—
197C22×C4C22×D412yes—
198C22×C4C22×Q812yes—
199C22×C4C22×Q811yescharacteristic
200C22×C4C2×#4111yescharacteristic
201C2×C8—28no—
202C2×C8—44no—
203C2×C8—44no—
204C4⋊C4—28no—
205C4⋊C4—28no—
206C4⋊C4—44no—
207C4⋊C4—44no—
208C4⋊C4—44no—
209C4⋊C4—44no—
210C4⋊C4—44no—
211C4⋊C4—48no—
212C4⋊C4—416no—
213C4⋊C4—44no—
214C4⋊C4—44no—
215C4⋊C4—44no—
216C4⋊C4—84no—
217C4⋊C4—84no—
218C4⋊C4—88no—
219C22⋊C4—48no—
220C42—44no—
221C42—48no—
222C42—82no—
223C42C22⋊Q814yes—
224C42C22⋊Q814yes—
225C42C4⋊Q814yes—
226C42C4⋊Q814yes—
227C23—21no—
228C23—21no—
229C23—21no—
230C23—21no—
231C23C2×#12011yescharacteristic
232C23C2×C22⋊Q811yescharacteristic
233C23C2×C4⋊Q811yescharacteristic
234C23C2×AGL(1,ℤ/8)12yes—
235C23C2×#13712yes—
236Q8—168no—
237C2×C4—24no—
238C2×C4—24no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—24no—
248C2×C4—24no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—44no—
259C2×C4—44no—
260C2×C4—44no—
261C2×C4—44no—
262C2×C4—44no—
263C2×C4—44no—
264C2×C4—44no—
265C2×C4—44no—
266C2×C4—44no—
267C2×C4—44no—
268C2×C4—44no—
269C2×C4—82no—
270C2×C4—82no—
271C2×C4—82no—
272C2×C4—82no—
273C2×C4—82no—
274C2×C4—82no—
275C2×C4—82no—
276C2×C4—82no—
277C2×C4—84no—
278C2×C4—81no—
279C2×C4—81no—
280C2×C4—81no—
281C2×C4—81no—
282C2×C4—81no—
283C2×C4—81no—
284C2×C4#39514yes—
285C2×C4#39512yes—
286C2×C4#39512yes—
287C2×C4C2×C22⋊Q812yes—
288C2×C4C2×C4⋊Q812yes—
289C8—48no—
290C22—21no—
291C22—21no—
292C22—21no—
293C22—21no—
294C22—22no—
295C22—22no—
296C22—22no—
297C22—22no—
298C22C2×#39511yescharacteristic
299C22#300311yescharacteristic
300C22#300411yescharacteristic
301C22#301911yescharacteristic
302C22#302011yescharacteristic
303C22#302711yescharacteristic
304C22#302811yescharacteristic
305C22#307912yes—
306C22#316512yes—
307C4—24no—
308C4—22no—
309C4—22no—
310C4—22no—
311C4—22no—
312C4—44no—
313C4—44no—
314C4—84no—
315C4—84no—
316C4—84no—
317C4—161no—
318C4—161no—
319C2—22no—
320C2—22no—
321C2#1835911yescharacteristic
322C2#1836311yescharacteristic
323C2#1836811yescharacteristic
324C2#1837411yescharacteristic
325C2#2355211yescharacteristic
326C2#2356111yescharacteristic
327C2#2356611yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
46. #432Q8semidirect
63. #383Q8semidirect

Maximal subgroups

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

subgroupindex
#351672
#219862
#142982
#105702
#105692
#80182
#79122