Finite Group Databasebeta

← #102952

Subgroups — #102952

Order 512.

529 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 529 conjugacy classes of subgroups — too many to draw. Only its 90 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#29290C211yesmaximal, characteristic, complemented
2#28157C212yesmaximal
3#9644C211yesmaximal, characteristic
4#7591C212yesmaximal, complemented
5#7040C211yesmaximal, characteristic, complemented
6C2×#461—22no—
7C4×C4⋊Q8C2212yes—
8C4×#127C2211yescharacteristic
9#2087—22no—
10#2072—22no—
11#2014—22no—
12#1987—22no—
13#1947—22no—
14#1880—24no—
15#1842C2211yescharacteristic
16#1831C2212yes—
17#1802—22no—
18#1744C2211yescharacteristic
19C2×D4⋊C8—22no—
20#1271—24no—
21C22×SD16—24no—
22C2×C4⋊Q8D412yes—
23C2×#127D411yescharacteristic
24C2×C4×Q8—22no—
25C2×C4×D4—22no—
26C2×C4×D4—21no—
27#461—44no—
28#461—44no—
29C2×#107—22no—
30C2×C4⋊C8—22no—
31C2×C4⋊C8—22no—
32C2×C4⋊C8D412yes—
33C2×#103—24no—
34C2×#102—24no—
35C2×C4×C8—21no—
36#391—22no—
37#390—22no—
38#390—21no—
39#389—22no—
40#389—22no—
41#384D411yescharacteristic
42#384D412yes—
43C4×C4⋊C4—22no—
44C4×C4⋊C4—22no—
45C43C2311yescharacteristic
46#340—24no—
47#340—24no—
48#340—22no—
49#336—22no—
50#336—22no—
51#336D414yes—
52D4⋊C8—44no—
53C22×Q8—22no—
54C22×D4—22no—
55C22×D4—21no—
56C23×C4—41no—
57C2×SD16—48no—
58C2×SD16—48no—
59C2×SD16—84no—
60C22×C8—24no—
61C22×C8—22no—
62C22×C8—41no—
63C4⋊Q8—22no—
64C4⋊Q8—22no—
65C4⋊Q8—22no—
66C4⋊Q8SD1614yescomplemented
67#127—42no—
68C4×Q8—84no—
69C4×D4—42no—
70C4×D4—42no—
71C4×D4—42no—
72C4×D4—42no—
73C4×D4—42no—
74C4×D4—42no—
75C2×C4⋊C4—22no—
76C2×C4⋊C4—22no—
77C2×C4⋊C4—22no—
78C2×C4⋊C4—22no—
79C2×C4⋊C4—22no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—22no—
82C2×C4⋊C4—42no—
83C2×C22⋊C4—42no—
84C2×C22⋊C4—41no—
85C2×C42—42no—
86C2×C42C2×D411yescharacteristic
87C2×C42C2×D411yescharacteristic
88C2×C42C2×D412yes—
89C2×C42C2×D412yes—
90C2×C42#4111yescharacteristic
91#107—82no—
92C4⋊C8—42no—
93C4⋊C8—42no—
94C4⋊C8—42no—
95C4⋊C8—42no—
96C4⋊C8—42no—
97#103—48no—
98#102—84no—
99C4×C8—22no—
100C4×C8—41no—
101#95—42no—
102#95—42no—
103C24—41no—
104C2×Q8—24no—
105C2×Q8—44no—
106C2×Q8—44no—
107C2×Q8—42no—
108C2×D4—24no—
109C2×D4—22no—
110C2×D4—42no—
111C2×D4—42no—
112C2×D4—42no—
113C2×D4—42no—
114C2×D4—41no—
115C2×D4—42no—
116C2×D4—42no—
117C2×D4—42no—
118C22×C4—42no—
119C22×C4—42no—
120C22×C4—41no—
121C22×C4—82no—
122C22×C4—82no—
123C22×C4—81no—
124C22×C4—81no—
125C22×C4#12012yes—
126C22×C4#12011yescharacteristic
127C22×C4#12112yes—
128C22×C4#12111yescharacteristic
129C22×C4#12711yescharacteristic
130SD16—816no—
131C2×C8—42no—
132C2×C8—44no—
133C2×C8—44no—
134C2×C8—44no—
135C2×C8—44no—
136C2×C8—42no—
137C2×C8—82no—
138C4⋊C4—24no—
139C4⋊C4—24no—
140C4⋊C4—44no—
141C4⋊C4—42no—
142C4⋊C4—42no—
143C4⋊C4—42no—
144C4⋊C4—42no—
145C4⋊C4—42no—
146C4⋊C4—42no—
147C4⋊C4—42no—
148C4⋊C4—82no—
149C4⋊C4—84no—
150C22⋊C4—82no—
151C22⋊C4—82no—
152C22⋊C4—82no—
153C42—21no—
154C42—21no—
155C42—22no—
156C42—42no—
157C42—41no—
158C42—84no—
159C42C2×SD1612yes—
160C42#13512yes—
161C42AGL(1,ℤ/8)12yes—
162C42#13712yes—
163C23—42no—
164C23—82no—
165C23—81no—
166C23—82no—
167C23—81no—
168C23#39211yescharacteristic
169Q8—48no—
170Q8—84no—
171D4—44no—
172D4—44no—
173D4—44no—
174D4—82no—
175D4—82no—
176D4—82no—
177C2×C4—21no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—21no—
182C2×C4—22no—
183C2×C4—21no—
184C2×C4—22no—
185C2×C4—44no—
186C2×C4—82no—
187C2×C4—82no—
188C2×C4—82no—
189C2×C4—81no—
190C2×C4—82no—
191C2×C4—82no—
192C2×C4—82no—
193C2×C4—161no—
194C2×C4—161no—
195C2×C4—162no—
196C2×C4#44812yes—
197C2×C4#44912yes—
198C2×C4#45012yes—
199C2×C4#45212yes—
200C2×C4#45311yescharacteristic
201C2×C4#45411yescharacteristic
202C2×C4#46512yes—
203C2×C4#46812yes—
204C2×C4#47312yes—
205C8—84no—
206C8—82no—
207C8—82no—
208C22—84no—
209C22—82no—
210C22—162no—
211C22—161no—
212C22—161no—
213C22#200012yes—
214C22#200611yescharacteristic
215C22#200812yes—
216C22#201011yescharacteristic
217C22#202011yescharacteristic
218C4—22no—
219C4—22no—
220C4—22no—
221C4—22no—
222C4—22no—
223C4—41no—
224C4—84no—
225C4—162no—
226C4—161no—
227C4#162011yescharacteristic
228C4#162311yescharacteristic
229C4#162612yes—
230C2—162no—
231C2#1000012yes—
232C2#1003011yescharacteristic
233C2#1018112yes—
234C2#1021211yescharacteristic
235C2#1023011yescharacteristic
subgroupG/H
center, socleC23#392
commutator, Frattini, cosocleC43C23
prime radicalC2×#127D4
LCSG ▷ C43 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C43 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #29290C2semidirect
4. #7591C2semidirect
5. #7040C2semidirect
66. C4⋊Q8SD16semidirect

Maximal subgroups

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

subgroupindex
#292902
#281572
#96442
#75912
#70402