Finite Group Databasebeta

← #101773

Subgroups — #101773

Order 512.

413 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 413 conjugacy classes of subgroups — too many to draw. Only its 92 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#28720C211yesmaximal, characteristic
2#23269C211yesmaximal, characteristic
3#8583C211yesmaximal, characteristic
4#7680C212yesmaximal, complemented
5#7027C212yesmaximal, complemented
6C2×#488—21no—
7#2859C2211yescharacteristic
8C4×#124C2211yescharacteristic
9#2072C412yes—
10#2071C412yes—
11#1991—21no—
12#1984—21no—
13#1923—22no—
14#1869—21no—
15#1805—21no—
16#1804—21no—
17#1767C2212yes—
18C4×C4⋊C8C2211yescharacteristic
19C8×C22⋊C4—22no—
20#1744C2212yes—
21C2×#329—21no—
22#1295—21no—
23#1271—22no—
24#1271—22no—
25C23×C8—21no—
26C2×#125—21no—
27C2×#124D411yescharacteristic
28C2×C4×Q8D411yescharacteristic
29C2×C4×D4D411yescharacteristic
30#488—42no—
31#488—42no—
32C2×C4⋊C8—22no—
33C2×C4⋊C8—21no—
34C2×C4⋊C8—21no—
35C2×#103—22no—
36C2×#102—22no—
37C2×C22⋊C8—22no—
38C2×C22⋊C8—21no—
39C2×#97—22no—
40C2×C4×C8—21no—
41C2×C4×C8—22no—
42C2×C4×C8—21no—
43#391C2×C411yescharacteristic
44#390C2×C411yescharacteristic
45#389—22no—
46#389—22no—
47#388—22no—
48#384C2×C412yes—
49#382—21no—
50#382D411yescharacteristic
51C4×C22⋊C4—22no—
52C43C2311yescharacteristic
53#340—22no—
54#340—22no—
55#336—22no—
56#336—22no—
57#336—22no—
58#336C2×C412yes—
59#336D412yes—
60#336D412yes—
61#329—42no—
62C22×Q8C22⋊C411yescharacteristic
63C22×D4C22⋊C411yescharacteristic
64C23×C4—21no—
65C22×C8—22no—
66C22×C8—22no—
67C22×C8—22no—
68C22×C8—44no—
69C22×C8—42no—
70C22×C8—81no—
71#125—41no—
72#125—41no—
73#125—42no—
74#124—22no—
75#124D812yes—
76#124SD1612yes—
77C4×Q8—41no—
78C4×Q8—41no—
79C4×D4—41no—
80C4×D4—41no—
81C2×C4⋊C4—22no—
82C2×C4⋊C4—22no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—22no—
85C2×C4⋊C4—42no—
86C2×C4⋊C4C22⋊C411yescharacteristic
87C2×C4⋊C4C22⋊C411yescharacteristic
88C2×C22⋊C4—22no—
89C2×C22⋊C4—22no—
90C2×C22⋊C4—21no—
91C2×C42—21no—
92C2×C42C22×C411yescharacteristic
93C2×C42C2×D411yescharacteristic
94C2×C42C2×D412yes—
95C2×C42C2×D411yescharacteristic
96C2×C42#4112yes—
97C4⋊C8—24no—
98C4⋊C8—42no—
99C4⋊C8—41no—
100C4⋊C8—41no—
101C4⋊C8—81no—
102#103—44no—
103#102—44no—
104C22⋊C8—44no—
105C22⋊C8—81no—
106#97—42no—
107#97—42no—
108C4×C8—44no—
109C4×C8—41no—
110C4×C8—41no—
111C4×C8—81no—
112#95—22no—
113#95—22no—
114#95—41no—
115#95—41no—
116C24—21no—
117C2×Q8—21no—
118C2×Q8—41no—
119C2×Q8—41no—
120C2×Q8#10212yes—
121C2×D4—21no—
122C2×D4—41no—
123C2×D4—41no—
124C2×D4#10212yes—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—42no—
129C22×C4—42no—
130C22×C4—41no—
131C22×C4—41no—
132C22×C4—42no—
133C22×C4C2×C22⋊C411yescharacteristic
134C22×C4C4×D412yes—
135C22×C4#12011yescharacteristic
136C22×C4#12112yes—
137C22×C4#12311yescharacteristic
138C2×C8—24no—
139C2×C8—42no—
140C2×C8—42no—
141C2×C8—42no—
142C2×C8—42no—
143C2×C8—44no—
144C2×C8—42no—
145C2×C8—42no—
146C2×C8—42no—
147C2×C8—42no—
148C2×C8—84no—
149C2×C8—82no—
150C2×C8—82no—
151C4⋊C4—41no—
152C4⋊C4—41no—
153C4⋊C4—42no—
154C4⋊C4—42no—
155C4⋊C4—42no—
156C4⋊C4—42no—
157C4⋊C4—81no—
158C4⋊C4—82no—
159C4⋊C4—82no—
160C4⋊C4—82no—
161C22⋊C4—24no—
162C22⋊C4—24no—
163C22⋊C4—42no—
164C22⋊C4—42no—
165C22⋊C4—81no—
166C42—22no—
167C42—22no—
168C42—21no—
169C42—21no—
170C42—42no—
171C42—41no—
172C42—41no—
173C42—41no—
174C42C2×D811yescharacteristic
175C42C2×SD1611yescharacteristic
176C42#13512yes—
177C23—22no—
178C23—42no—
179C23—41no—
180C23—42no—
181C23—41no—
182C23#38611yescharacteristic
183Q8—42no—
184Q8—81no—
185D4—42no—
186D4—81no—
187C2×C4—22no—
188C2×C4—21no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—21no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—21no—
196C2×C4—22no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—42no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—42no—
203C2×C4—81no—
204C2×C4—81no—
205C2×C4—82no—
206C2×C4—82no—
207C2×C4—82no—
208C2×C4—82no—
209C2×C4C2×#10211yescharacteristic
210C2×C4#41611yescharacteristic
211C2×C4#44312yes—
212C2×C4#44412yes—
213C2×C4#45511yescharacteristic
214C2×C4#45511yescharacteristic
215C2×C4#46512yes—
216C2×C4#46612yes—
217C8—44no—
218C8—82no—
219C8—82no—
220C8—84no—
221C22—42no—
222C22—44no—
223C22—82no—
224C22—81no—
225C22—81no—
226C22#187612yes—
227C22#188511yescharacteristic
228C22#189311yescharacteristic
229C22#190612yes—
230C22#191511yescharacteristic
231C4—22no—
232C4—22no—
233C4—21no—
234C4—21no—
235C4—21no—
236C4—21no—
237C4—22no—
238C4—22no—
239C4—41no—
240C4—42no—
241C4—81no—
242C4—81no—
243C4—162no—
244C2—82no—
245C2#906711yescharacteristic
246C2#920812yes—
247C2#926612yes—
248C2#939311yescharacteristic
249C2#943911yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC2×C42C22×C4
Frattini, cosocleC43C23
prime radicalC22×D4C22⋊C4
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C43 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7680C2semidirect
5. #7027C2semidirect

Maximal subgroups

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

subgroupindex
#287202
#232692
#85832
#76802
#70272