Finite Group Databasebeta

← #173937

Subgroups — #173937

Order 512.

1021 conjugacy classes of subgroups; only the 192 normal subgroups are shown.

normal subgroups only

This group has 1021 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
1#59832C211yesmaximal, characteristic, complemented
2#28749C212yesmaximal
3#28441C211yesmaximal, characteristic
4#7741C218yesmaximal, complemented
5#7571C211yesmaximal, characteristic, complemented
6#7570C212yesmaximal, complemented
7C2×Q82—21no—
8C2×#492—21no—
9C2×#486—22no—
10C2×#460—22no—
11#2867C2211yescharacteristic
12#2734C2214yes—
13#2582—21no—
14#2551—22no—
15#2546—22no—
16#2451—24no—
17#2365—22no—
18C4×C4⋊Q8C2212yes—
19C4×C4⋊Q8C2211yescharacteristic
20#2041C2218yes—
21#2014C2214yes—
22#1982—24no—
23#1957—21no—
24#1957—22no—
25C4×C4⋊C8C2211yescharacteristic
26C4×C4⋊C8C2212yes—
27#1272C2214yes—
28#1271C2218yes—
29C22×SD16—24no—
30Q82—44no—
31Q82—44no—
32C2×C4⋊Q8—24no—
33C2×C4⋊Q8—24no—
34C2×C4⋊Q8—22no—
35C2×C4⋊Q8D411yescharacteristic
36C2×C4⋊Q8D412yes—
37C2×#127—22no—
38C2×#127—21no—
39C2×#127—22no—
40C2×#127—21no—
41C2×#127C2312yes—
42C2×C4×Q8—24no—
43C2×C4×Q8—22no—
44C2×C4×Q8D412yes—
45#492—44no—
46#486—48no—
47#460—48no—
48C2×C4⋊C8—21no—
49C2×C4⋊C8—22no—
50C2×#102—28no—
51C2×#102—24no—
52C2×#102—28no—
53C2×C4×C8—22no—
54C2×C4×C8—21no—
55#398—24no—
56#398—24no—
57#395—28no—
58#391—24no—
59#391—24no—
60#391—22no—
61#391C2312yes—
62#389—24no—
63#389—24no—
64#389—22no—
65#389—22no—
66#389C2314yes—
67#387—28no—
68#384D411yescharacteristic
69#384D412yes—
70C4×C4⋊C4—22no—
71C4×C4⋊C4—22no—
72C4×C4⋊C4—22no—
73C4×C4⋊C4D414yes—
74C43C2311yescharacteristic
75#336C2312yes—
76#336C2314yes—
77C22×Q8—24no—
78C22×Q8—22no—
79C22×Q8C2×D412yes—
80C22×D4—22no—
81C22×D4—24no—
82C22×D4—41no—
83C22×D4—42no—
84C22×D4—41no—
85C2×SD16—48no—
86C2×SD16—48no—
87C2×SD16—84no—
88C22×C8—22no—
89C22×C8—24no—
90C4⋊Q8—22no—
91C4⋊Q8—24no—
92C4⋊Q8—416no—
93C4⋊Q8—48no—
94C4⋊Q8—48no—
95C4⋊Q8—84no—
96C4⋊Q8D814yes—
97C4⋊Q8SD1618yescomplemented
98#127—48no—
99#127—44no—
100#127—42no—
101#127—44no—
102#127—42no—
103#127—44no—
104#127—82no—
105C4×Q8—44no—
106C4×Q8—416no—
107C4×Q8—44no—
108C4×Q8—44no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—28no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—24no—
116C2×C4⋊C4—24no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4C2×D414yes—
119C2×C4⋊C4C2×D412yes—
120C2×C4⋊C4C2×D414yes—
121C2×C42—24no—
122C2×C42—24no—
123C2×C42—22no—
124C2×C42C2×D411yescharacteristic
125C2×C42C2×D412yes—
126C2×C42C2×D411yescharacteristic
127C2×C42C2×D412yes—
128C2×C42C2411yescharacteristic
129C4⋊C8—24no—
130C4⋊C8—44no—
131#102—48no—
132#102—416no—
133#102—88no—
134C4×C8—42no—
135C4×C8—44no—
136#95—24no—
137#95—28no—
138#95—24no—
139#95—24no—
140#95—24no—
141C24—42no—
142C2×Q8—22no—
143C2×Q8—28no—
144C2×Q8—48no—
145C2×Q8—42no—
146C2×Q8—42no—
147C2×Q8—44no—
148C2×Q8—44no—
149C2×Q8—84no—
150C2×Q8—84no—
151C2×Q8C2×D814yes—
152C2×D4—44no—
153C2×D4—48no—
154C2×D4—48no—
155C2×D4—44no—
156C2×D4—42no—
157C2×D4—44no—
158C2×D4—42no—
159C2×D4—44no—
160C2×D4—82no—
161C2×D4—84no—
162C2×D4—84no—
163C2×D4—84no—
164C2×D4—84no—
165C2×D4—82no—
166C2×D4—84no—
167C22×C4—24no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—24no—
171C22×C4#12011yescharacteristic
172C22×C4#12012yes—
173C22×C4#12011yescharacteristic
174C22×C4C22×D411yescharacteristic
175C22×C4C22×D412yes—
176SD16—816no—
177C2×C8—24no—
178C2×C8—42no—
179C2×C8—42no—
180C2×C8—44no—
181C2×C8—44no—
182C2×C8—44no—
183C4⋊C4—24no—
184C4⋊C4—28no—
185C4⋊C4—28no—
186C4⋊C4—42no—
187C4⋊C4—44no—
188C4⋊C4—416no—
189C4⋊C4—44no—
190C4⋊C4—44no—
191C4⋊C4—44no—
192C4⋊C4—44no—
193C4⋊C4—44no—
194C4⋊C4—82no—
195C4⋊C4—84no—
196C4⋊C4C2×SD1618yes—
197C42—22no—
198C42—24no—
199C42—41no—
200C42—48no—
201C42—48no—
202C42—44no—
203C42C2×D812yes—
204C42C2×SD1614yes—
205C42AGL(1,ℤ/8)12yes—
206C42#13714yes—
207C23—82no—
208C23—84no—
209C23—82no—
210C23—84no—
211C23—82no—
212C23C2×#12011yescharacteristic
213Q8—44no—
214Q8—416no—
215Q8—82no—
216Q8—168no—
217D4—84no—
218D4—84no—
219D4—88no—
220D4—88no—
221D4—164no—
222D4—164no—
223D4—168no—
224C2×C4—21no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—21no—
228C2×C4—24no—
229C2×C4—28no—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—44no—
233C2×C4—42no—
234C2×C4—44no—
235C2×C4—42no—
236C2×C4—44no—
237C2×C4—42no—
238C2×C4—44no—
239C2×C4—44no—
240C2×C4—44no—
241C2×C4#44714yes—
242C2×C4#44814yes—
243C2×C4#44814yes—
244C2×C4#45312yes—
245C2×C4#45414yes—
246C2×C4C22×D811yescharacteristic
247C2×C4C22×SD1612yes—
248C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
249C2×C4C2×#13712yes—
250C8—44no—
251C8—84no—
252C22—164no—
253C22—162no—
254C22—162no—
255C22—162no—
256C22—164no—
257C22C2×#44711yescharacteristic
258C22C2×#44812yes—
259C22C2×#45311yescharacteristic
260C22C2×#45412yes—
261C22#302711yescharacteristic
262C4—21no—
263C4—21no—
264C4—22no—
265C4—22no—
266C4—48no—
267C4—41no—
268C4—44no—
269C4—82no—
270C4—84no—
271C4#162314yes—
272C4#162718yes—
273C2—322no—
274C2#1838711yescharacteristic
275C2#1840312yes—
276C2C2×#162712yes—
277C2C2×#162311yescharacteristic
278C2#1907711yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radical#2867C22
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #59832C2semidirect
4. #7741C2semidirect
5. #7571C2semidirect
6. #7570C2semidirect
97. C4⋊Q8SD16semidirect

Maximal subgroups

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

subgroupindex
#598322
#287492
#284412
#77412
#75712
#75702