Finite Group Databasebeta

← #102954

Subgroups — #102954

Order 512.

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

normal subgroups only

This group has 497 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#29286C211yesmaximal, characteristic, complemented
2#28202C212yesmaximal
3#9644C211yesmaximal, characteristic
4#7590C212yesmaximal, complemented
5#7040C211yesmaximal, characteristic, complemented
6C2×#476—22no—
7C4×C4⋊Q8C2212yes—
8C4×#127C2211yescharacteristic
9#2086—22no—
10#2084—22no—
11#2041—22no—
12#1988—22no—
13#1947—22no—
14#1926—24no—
15#1842C2211yescharacteristic
16#1831C2212yes—
17#1802—22no—
18#1744C2211yescharacteristic
19C2×D4⋊C8—22no—
20#1272—24no—
21C2×C4⋊Q8D412yes—
22C2×#127D411yescharacteristic
23C2×C4×Q8—22no—
24C2×C4×D4—22no—
25C2×C4×D4—21no—
26#476—44no—
27#476—44no—
28C2×#107—22no—
29C2×#106—24no—
30C2×C4⋊C8—22no—
31C2×C4⋊C8—22no—
32C2×C4⋊C8D412yes—
33C2×#102—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—
57C22×C8—24no—
58C22×C8—22no—
59C22×C8—41no—
60C4⋊Q8—22no—
61C4⋊Q8—22no—
62C4⋊Q8—22no—
63C4⋊Q8SD1614yes—
64#127—42no—
65C4×Q8—84no—
66C4×D4—42no—
67C4×D4—42no—
68C4×D4—42no—
69C4×D4—42no—
70C4×D4—42no—
71C4×D4—42no—
72C2×C4⋊C4—22no—
73C2×C4⋊C4—22no—
74C2×C4⋊C4—22no—
75C2×C4⋊C4—22no—
76C2×C4⋊C4—22no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—22no—
79C2×C4⋊C4—42no—
80C2×C22⋊C4—42no—
81C2×C22⋊C4—41no—
82C2×C42—42no—
83C2×C42C2×D411yescharacteristic
84C2×C42C2×D411yescharacteristic
85C2×C42C2×D412yes—
86C2×C42C2×D412yes—
87C2×C42#4111yescharacteristic
88#107—82no—
89#106—48no—
90C4⋊C8—42no—
91C4⋊C8—42no—
92C4⋊C8—42no—
93C4⋊C8—42no—
94C4⋊C8—42no—
95#102—48no—
96#102—48no—
97C4×C8—22no—
98C4×C8—41no—
99#95—42no—
100#95—42no—
101C24—41no—
102C2×Q8—24no—
103C2×Q8—42no—
104C2×Q8—82no—
105C2×Q8—82no—
106C2×D4—24no—
107C2×D4—22no—
108C2×D4—42no—
109C2×D4—42no—
110C2×D4—42no—
111C2×D4—42no—
112C2×D4—41no—
113C2×D4—42no—
114C2×D4—42no—
115C2×D4—42no—
116C22×C4—42no—
117C22×C4—42no—
118C22×C4—41no—
119C22×C4—82no—
120C22×C4—82no—
121C22×C4—81no—
122C22×C4—81no—
123C22×C4#12012yes—
124C22×C4#12011yescharacteristic
125C22×C4#12112yes—
126C22×C4#12111yescharacteristic
127C22×C4#12711yescharacteristic
128C2×C8—42no—
129C2×C8—44no—
130C2×C8—44no—
131C2×C8—44no—
132C2×C8—44no—
133C2×C8—42no—
134C2×C8—82no—
135C4⋊C4—24no—
136C4⋊C4—24no—
137C4⋊C4—44no—
138C4⋊C4—42no—
139C4⋊C4—42no—
140C4⋊C4—42no—
141C4⋊C4—42no—
142C4⋊C4—42no—
143C4⋊C4—42no—
144C4⋊C4—42no—
145C4⋊C4—82no—
146C4⋊C4—84no—
147C22⋊C4—82no—
148C22⋊C4—82no—
149C22⋊C4—82no—
150C42—21no—
151C42—21no—
152C42—22no—
153C42—42no—
154C42—41no—
155C42—84no—
156C42C2×SD1612yes—
157C42#13512yes—
158C42AGL(1,ℤ/8)12yes—
159C42#13712yes—
160C23—42no—
161C23—82no—
162C23—81no—
163C23—82no—
164C23—81no—
165C23#39211yescharacteristic
166Q8—84no—
167Q8—84no—
168D4—44no—
169D4—44no—
170D4—44no—
171D4—82no—
172D4—82no—
173D4—82no—
174C2×C4—21no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—21no—
179C2×C4—22no—
180C2×C4—21no—
181C2×C4—22no—
182C2×C4—44no—
183C2×C4—82no—
184C2×C4—82no—
185C2×C4—82no—
186C2×C4—81no—
187C2×C4—82no—
188C2×C4—82no—
189C2×C4—82no—
190C2×C4—161no—
191C2×C4—161no—
192C2×C4—162no—
193C2×C4#44812yes—
194C2×C4#44912yes—
195C2×C4#45012yes—
196C2×C4#45212yes—
197C2×C4#45311yescharacteristic
198C2×C4#45411yescharacteristic
199C2×C4#46512yes—
200C2×C4#46812yes—
201C2×C4#47212yes—
202C8—84no—
203C8—82no—
204C8—82no—
205C22—84no—
206C22—82no—
207C22—162no—
208C22—161no—
209C22—161no—
210C22#200012yes—
211C22#200611yescharacteristic
212C22#200912yes—
213C22#201011yescharacteristic
214C22#201811yescharacteristic
215C4—22no—
216C4—22no—
217C4—22no—
218C4—22no—
219C4—22no—
220C4—41no—
221C4—84no—
222C4—162no—
223C4—161no—
224C4#162011yescharacteristic
225C4#162311yescharacteristic
226C4#162612yes—
227C2—162no—
228C2#1000112yes—
229C2#1002611yescharacteristic
230C2#1018812yes—
231C2#1021611yescharacteristic
232C2#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. #29286C2semidirect
4. #7590C2semidirect
5. #7040C2semidirect

Maximal subgroups

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

subgroupindex
#292862
#282022
#96442
#75902
#70402