Finite Group Databasebeta

← #172001

Subgroups — #172001

Order 512.

858 conjugacy classes of subgroups; only the 168 normal subgroups are shown.

normal subgroups only

This group has 858 conjugacy classes of subgroups — too many to draw. Only its 168 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
2#29277C212yesmaximal
3#23801C211yesmaximal, characteristic
4#7829C214yesmaximal
5#7757C214yesmaximal
6#7653C211yesmaximal, characteristic
7#7591C212yesmaximal
8C2×Q82—21no—
9C2×C4×Q16—22no—
10#2734C2214yes—
11#2582—21no—
12#2551—22no—
13#2546—22no—
14#2451—24no—
15#2365—22no—
16C4×C4⋊Q8C2212yes—
17C4×C4⋊Q8C2211yescharacteristic
18C4×C4⋊Q8C2211yescharacteristic
19#2100C2214yes—
20#2095—21no—
21#2095C2212yes—
22#2057C2214yes—
23#2016—21no—
24#2016C2212yes—
25#1987—22no—
26#1947—22no—
27#1921—24no—
28#1831C2212yes—
29#1744C2211yescharacteristic
30#1272C2214yes—
31#1271C2218yes—
32C22×Q16—22no—
33Q82—48no—
34C2×C4⋊Q8—24no—
35C2×C4⋊Q8—24no—
36C2×C4⋊Q8—22no—
37C2×C4⋊Q8D411yescharacteristic
38C2×C4⋊Q8D412yes—
39C2×C4⋊Q8C2311yescharacteristic
40C2×C4×Q8—24no—
41C2×C4×Q8—22no—
42C2×C4×Q8—21no—
43C2×C4×Q8D412yes—
44C4×Q16—48no—
45C2×#107—24no—
46C2×#107—22no—
47C2×C4⋊C8—22no—
48C2×#103—22no—
49C2×#103—22no—
50C2×C4×C8—21no—
51#398—24no—
52#398—24no—
53#395—28no—
54#391—24no—
55#391—24no—
56#391—22no—
57#391—21no—
58#391C2312yes—
59#389—24no—
60#389—24no—
61#389—22no—
62#389—22no—
63#389—22no—
64#389C2314yes—
65#387—28no—
66#384D411yescharacteristic
67#384D412yes—
68#384C2311yescharacteristic
69C4×C4⋊C4—22no—
70C4×C4⋊C4—22no—
71C4×C4⋊C4—22no—
72C4×C4⋊C4—22no—
73C4×C4⋊C4D414yes—
74C43C2311yescharacteristic
75#340—24no—
76#340—24no—
77#340—24no—
78#336—22no—
79#336—21no—
80#336C2314yes—
81#336C2312yes—
82C22×Q8—24no—
83C22×Q8—22no—
84C22×Q8—21no—
85C22×Q8C2×D412yes—
86C2×Q16—44no—
87C2×Q16—44no—
88C2×Q16—44no—
89C22×C8—22no—
90C22×C8—24no—
91C4⋊Q8—24no—
92C4⋊Q8—24no—
93C4⋊Q8—42no—
94C4⋊Q8—416no—
95C4⋊Q8—48no—
96C4⋊Q8—48no—
97C4⋊Q8—84no—
98C4⋊Q8Q1618yescomplemented
99C4×Q8—48no—
100C4×Q8—44no—
101C4×Q8—44no—
102C4×Q8—416no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—28no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—24no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—24no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—41no—
116C2×C4⋊C4C2×D414yes—
117C2×C4⋊C4C2×D412yes—
118C2×C4⋊C4C2×D414yes—
119C2×C42—24no—
120C2×C42—24no—
121C2×C42—22no—
122C2×C42—41no—
123C2×C42C2×D411yescharacteristic
124C2×C42C2×D411yescharacteristic
125C2×C42C2×D412yes—
126C2×C42C2×D412yes—
127C2×C42C2411yescharacteristic
128#107—48no—
129#107—44no—
130C4⋊C8—44no—
131#103—44no—
132#103—44no—
133C4×C8—24no—
134#95—24no—
135#95—28no—
136#95—24no—
137#95—24no—
138#95—24no—
139#95—42no—
140C2×Q8—22no—
141C2×Q8—28no—
142C2×Q8—22no—
143C2×Q8—44no—
144C2×Q8—44no—
145C2×Q8—41no—
146C2×Q8—48no—
147C2×Q8—42no—
148C2×Q8—42no—
149C2×Q8—42no—
150C2×Q8—42no—
151C2×Q8—84no—
152C2×Q8—84no—
153C2×Q8AGL(1,ℤ/8)14yes—
154C22×C4—24no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—24no—
158C22×C4—41no—
159C22×C4—41no—
160C22×C4#12011yescharacteristic
161C22×C4#12012yes—
162C22×C4#12011yescharacteristic
163C22×C4C22×D412yes—
164C22×C4C22×D411yescharacteristic
165Q16—416no—
166C2×C8—24no—
167C2×C8—42no—
168C2×C8—42no—
169C2×C8—44no—
170C2×C8—44no—
171C2×C8—44no—
172C4⋊C4—28no—
173C4⋊C4—24no—
174C4⋊C4—28no—
175C4⋊C4—24no—
176C4⋊C4—42no—
177C4⋊C4—44no—
178C4⋊C4—44no—
179C4⋊C4—44no—
180C4⋊C4—416no—
181C4⋊C4—42no—
182C4⋊C4—42no—
183C4⋊C4—44no—
184C4⋊C4—48no—
185C4⋊C4—44no—
186C4⋊C4—82no—
187C4⋊C4—82no—
188C4⋊C4—84no—
189C4⋊C4C2×Q1618yes—
190C42—22no—
191C42—22no—
192C42—42no—
193C42—41no—
194C42—48no—
195C42—48no—
196C42—44no—
197C42—82no—
198C42C2×Q1614yes—
199C42#13714yes—
200C23C2×#12011yescharacteristic
201Q8—44no—
202Q8—416no—
203Q8—44no—
204Q8—82no—
205Q8—82no—
206Q8—168no—
207C2×C4—22no—
208C2×C4—21no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—21no—
213C2×C4—28no—
214C2×C4—24no—
215C2×C4—21no—
216C2×C4—22no—
217C2×C4—44no—
218C2×C4—42no—
219C2×C4—44no—
220C2×C4—42no—
221C2×C4—44no—
222C2×C4—42no—
223C2×C4—44no—
224C2×C4—44no—
225C2×C4—42no—
226C2×C4—44no—
227C2×C4—81no—
228C2×C4—81no—
229C2×C4—81no—
230C2×C4—82no—
231C2×C4#45114yes—
232C2×C4#45114yes—
233C2×C4#45312yes—
234C2×C4C22×Q1612yes—
235C2×C4C2×AGL(1,ℤ/8)12yes—
236C2×C4C2×#13712yes—
237C8—44no—
238C8—84no—
239C22C2×#45112yes—
240C22#300311yescharacteristic
241C22C2×#45311yescharacteristic
242C22#302012yes—
243C22#302511yescharacteristic
244C4—22no—
245C4—22no—
246C4—22no—
247C4—24no—
248C4—44no—
249C4—48no—
250C4—41no—
251C4—84no—
252C4—82no—
253C4—82no—
254C4—161no—
255C4#163314yes—
256C2#1817811yescharacteristic
257C2#1822112yes—
258C2C2×#163311yescharacteristic
259C2#1875912yes—
260C2#1905411yescharacteristic
subgroupG/H
center, socle, prime radicalC23C2×#120
commutator, Frattini, cosocleC2×C42C24
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
98. C4⋊Q8Q16semidirect

Maximal subgroups

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

subgroupindex
#598322
#292772
#238012
#78292
#77572
#76532
#75912