Finite Group Databasebeta

← #349943

Subgroups — #349943

Order 512.

1126 conjugacy classes of subgroups; only the 192 normal subgroups are shown.

normal subgroups only

This group has 1126 conjugacy classes of subgroups — too many to draw. Only its 192 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×#2369C211yesmaximal, characteristic, complemented
2#33106C212yesmaximal
3#25718C211yesmaximal, characteristic
4#9063C214yesmaximal, complemented
5#8929C214yesmaximal, complemented
6#8706C212yesmaximal, complemented
7#8342C211yesmaximal, characteristic, complemented
8C22×C4⋊Q8C2211yescharacteristic
9C2×#492C2212yes—
10C2×#480—22no—
11C2×#448—22no—
12#2856C2211yescharacteristic
13#2369—24no—
14#2369—22no—
15#2369C2214yes—
16C2×#391—22no—
17C2×#389—22no—
18C2×C4×C4⋊C4—22no—
19C2×C4×C4⋊C4C2212yes—
20#2013—22no—
21#1982C2214yes—
22#1956C2212yes—
23#1947C2214yes—
24#1925—24no—
25#1874—22no—
26#1797C2212yes—
27C4×C22⋊C8C2211yescharacteristic
28C2×#333C2214yes—
29#1279C2218yes—
30C23×Q8D412yes—
31C22×SD16—24no—
32C2×C4⋊Q8—28no—
33C2×C4⋊Q8—22no—
34C2×C4⋊Q8D412yes—
35C2×C4⋊Q8C2312yes—
36C2×#127C2311yescharacteristic
37C2×#121—22no—
38C2×#121—21no—
39C22×C4⋊C4—22no—
40C22×C4⋊C4—22no—
41C22×C4⋊C4D412yes—
42C22×C4⋊C4D412yes—
43C22×C42—22no—
44C22×C42—22no—
45C22×C42C2311yescharacteristic
46#492—28no—
47#480—48no—
48#448—48no—
49C2×#107—24no—
50C2×C4⋊C8C2314yes—
51C2×#103—24no—
52C2×#102—24no—
53C2×#102—24no—
54C2×C22⋊C8—21no—
55C2×C22⋊C8—22no—
56C2×C4×C8C2312yes—
57#391—28no—
58#391—28no—
59#390—22no—
60#390—21no—
61#389—28no—
62#389—28no—
63#384C2311yescharacteristic
64C4×C4⋊C4—28no—
65C4×C4⋊C4—28no—
66C4×C4⋊C4—22no—
67C4×C4⋊C4—22no—
68C4×C4⋊C4D414yes—
69C4×C4⋊C4C2314yes—
70C2×#95—24no—
71C2×#95—22no—
72#340—24no—
73#336—22no—
74#336—21no—
75#333—28no—
76C22×Q8—28no—
77C22×Q8—22no—
78C22×Q8—44no—
79C22×Q8—82no—
80C22×Q8C2×D414yes—
81C22×D4—21no—
82C22×D4—22no—
83C22×D4—41no—
84C23×C4—22no—
85C23×C4—22no—
86C23×C4—22no—
87C23×C4C2×D411yescharacteristic
88C23×C4C2×D412yes—
89C2×SD16—48no—
90C2×SD16—48no—
91C2×SD16—84no—
92C22×C8—22no—
93C22×C8—24no—
94C4⋊Q8—216no—
95C4⋊Q8—28no—
96C4⋊Q8—24no—
97C4⋊Q8#4118yes—
98#127—24no—
99#121—44no—
100#121—44no—
101#121—44no—
102C2×C4⋊C4—216no—
103C2×C4⋊C4—24no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—24no—
106C2×C4⋊C4—28no—
107C2×C4⋊C4—24no—
108C2×C4⋊C4—24no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—28no—
113C2×C4⋊C4—84no—
114C2×C4⋊C4C2×D414yes—
115C2×C4⋊C4C2×D414yes—
116C2×C22⋊C4—41no—
117C2×C22⋊C4—42no—
118C2×C42—24no—
119C2×C42—24no—
120C2×C42—24no—
121C2×C42—28no—
122C2×C42—44no—
123C2×C42—44no—
124C2×C42—84no—
125C2×C42C2×D412yes—
126C2×C42C2×D411yescharacteristic
127C2×C42C2411yescharacteristic
128#107—48no—
129C4⋊C8—28no—
130#103—48no—
131#102—48no—
132#102—48no—
133C22⋊C8—24no—
134C22⋊C8—44no—
135C4×C8—24no—
136#95—216no—
137#95—24no—
138#95—24no—
139C24—41no—
140C24#12011yescharacteristic
141C2×Q8—28no—
142C2×Q8—24no—
143C2×Q8—44no—
144C2×Q8—44no—
145C2×Q8—82no—
146C2×Q8—82no—
147C2×Q8—816no—
148C2×Q8#12318yes—
149C2×D4—42no—
150C2×D4—42no—
151C2×D4—42no—
152C2×D4—44no—
153C2×D4—42no—
154C2×D4—42no—
155C2×D4—44no—
156C2×D4—84no—
157C2×D4—82no—
158C22×C4—24no—
159C22×C4—24no—
160C22×C4—28no—
161C22×C4—28no—
162C22×C4—24no—
163C22×C4—24no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—24no—
167C22×C4—41no—
168C22×C4—44no—
169C22×C4—44no—
170C22×C4—44no—
171C22×C4—44no—
172C22×C4—44no—
173C22×C4—44no—
174C22×C4—44no—
175C22×C4—41no—
176C22×C4#12011yescharacteristic
177C22×C4#12012yes—
178C22×C4#13514yes—
179C22×C4AGL(1,ℤ/8)14yes—
180C22×C4C22×D411yescharacteristic
181C22×C4C22×D412yes—
182SD16—816no—
183C2×C8—24no—
184C2×C8—42no—
185C2×C8—42no—
186C2×C8—44no—
187C2×C8—44no—
188C2×C8—44no—
189C4⋊C4—28no—
190C4⋊C4—216no—
191C4⋊C4—216no—
192C4⋊C4—24no—
193C4⋊C4—24no—
194C4⋊C4—24no—
195C4⋊C4—42no—
196C4⋊C4—44no—
197C4⋊C4—42no—
198C4⋊C4—42no—
199C4⋊C4—88no—
200C4⋊C4#13718yes—
201C22⋊C4—82no—
202C22⋊C4—84no—
203C42—22no—
204C42—24no—
205C42—44no—
206C42—44no—
207C42—48no—
208C42—88no—
209C42C2×#4114yes—
210C23—21no—
211C23—22no—
212C23—21no—
213C23—81no—
214C23—82no—
215C23—82no—
216C23—81no—
217C23—81no—
218C23#44914yes—
219C23#45312yes—
220C23C2×#12011yescharacteristic
221Q8—48no—
222Q8—84no—
223Q8—88no—
224D4—84no—
225D4—84no—
226D4—84no—
227D4—164no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—28no—
233C2×C4—28no—
234C2×C4—21no—
235C2×C4—28no—
236C2×C4—28no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—44no—
240C2×C4—416no—
241C2×C4—416no—
242C2×C4—44no—
243C2×C4—44no—
244C2×C4—44no—
245C2×C4—44no—
246C2×C4—44no—
247C2×C4—44no—
248C2×C4—44no—
249C2×C4—88no—
250C2×C4—82no—
251C2×C4—81no—
252C2×C4#44914yes—
253C2×C4C2×#12312yes—
254C2×C4#52514yes—
255C2×C4C2×#13512yes—
256C2×C4C2×AGL(1,ℤ/8)12yes—
257C2×C4C2×#13714yes—
258C8—44no—
259C8—84no—
260C22—24no—
261C22—22no—
262C22—22no—
263C22—24no—
264C22—162no—
265C22—161no—
266C22—161no—
267C22—161no—
268C22—162no—
269C22#162414yes—
270C22#284911yescharacteristic
271C22C2×#44912yes—
272C22#300412yes—
273C22C2×#45311yescharacteristic
274C22#301911yescharacteristic
275C4—22no—
276C4—24no—
277C4—48no—
278C4—41no—
279C4—48no—
280C4—84no—
281C4—161no—
282C4#310814yes—
283C4#311214yes—
284C2—24no—
285C2—321no—
286C2C2×#162411yescharacteristic
287C2#1854611yescharacteristic
288C2#1875212yes—
289C2#3299011yescharacteristic
290C2#3300312yes—
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radical#2856C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2369C2semidirect
4. #9063C2semidirect
5. #8929C2semidirect
6. #8706C2semidirect
7. #8342C2semidirect

Maximal subgroups

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

subgroupindex
C2×#23692
#331062
#257182
#90632
#89292
#87062
#83422