Finite Group Databasebeta

← #100800

Subgroups — #100800

Order 512.

528 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 88 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#28445C211yesmaximal, characteristic
2#23355C211yesmaximal, characteristic, complemented
3#8592C211yesmaximal, characteristic, complemented
4#7719C212yesmaximal, complemented
5#7041C212yesmaximal, complemented
6C2×#496—21no—
7C2×#463—22no—
8#2858C2211yescharacteristic
9C4×#127C2211yescharacteristic
10#2013C412yes—
11#2011C412yes—
12#1959—21no—
13#1869—24no—
14#1869—22no—
15#1839C2212yes—
16#1831C2212yes—
17#1767C2211yescharacteristic
18C2×#331—21no—
19#1296—21no—
20#1270—24no—
21C22×SD16—22no—
22C22×D8—22no—
23C23×C8—22no—
24C2×C4⋊Q8C2×C411yescharacteristic
25C2×#127C2×C411yescharacteristic
26C2×#127D411yescharacteristic
27C2×#124—21no—
28C2×#124—21no—
29C2×C4×D4—22no—
30C2×C4×D4D412yes—
31#496—42no—
32#496—42no—
33#463—44no—
34#463—44no—
35C2×C4⋊C8—21no—
36C2×C4⋊C8—22no—
37C2×#103—22no—
38C2×#102—22no—
39C2×C22⋊C8—24no—
40C2×C22⋊C8—22no—
41C2×#97—21no—
42#390—22no—
43#390—22no—
44#390—22no—
45#390C2×C412yes—
46#384D411yescharacteristic
47C43C2311yescharacteristic
48#336—22no—
49#336—22no—
50#336—22no—
51#336—22no—
52#336C2×C412yes—
53#336D414yes—
54#331—42no—
55C22×Q8—21no—
56C22×D4—21no—
57C22×D4—22no—
58C22×D4—22no—
59C22×D4C22⋊C412yes—
60C23×C4—22no—
61C2×SD16—44no—
62C2×SD16—44no—
63C2×SD16—82no—
64C2×D8—44no—
65C2×D8—44no—
66C2×D8—82no—
67C22×C8—22no—
68C22×C8—24no—
69C22×C8—42no—
70C22×C8—84no—
71C22×C8—82no—
72C4⋊Q8—81no—
73#127—22no—
74#127—81no—
75#127D812yes—
76#127SD1612yes—
77#124—42no—
78#124—42no—
79#124—41no—
80#124—41no—
81#124—42no—
82C4×D4—42no—
83C4×D4—42no—
84C4×D4—82no—
85C4×D4—82no—
86C2×C4⋊C4—22no—
87C2×C4⋊C4—22no—
88C2×C4⋊C4C22⋊C412yes—
89C2×C22⋊C4—24no—
90C2×C22⋊C4—22no—
91C2×C22⋊C4—41no—
92C2×C22⋊C4—42no—
93C2×C22⋊C4—41no—
94C2×C42C22×C411yescharacteristic
95C2×C42C2×D411yescharacteristic
96C2×C42C2×D412yes—
97C2×C42C2×D411yescharacteristic
98C2×C42#4112yes—
99C4⋊C8—24no—
100C4⋊C8—42no—
101C4⋊C8—42no—
102#103—44no—
103#102—44no—
104C22⋊C8—84no—
105C22⋊C8—82no—
106#97—41no—
107#97—41no—
108C24—22no—
109C24—41no—
110C2×Q8—42no—
111C2×Q8—42no—
112C2×Q8—41no—
113C2×Q8—41no—
114C2×D4—22no—
115C2×D4—24no—
116C2×D4—24no—
117C2×D4—42no—
118C2×D4—42no—
119C2×D4—42no—
120C2×D4—42no—
121C2×D4—44no—
122C2×D4—41no—
123C2×D4—41no—
124C2×D4—82no—
125C2×D4—82no—
126C2×D4—82no—
127C2×D4—82no—
128C2×D4#10214yes—
129C22×C4—22no—
130C22×C4—44no—
131C22×C4—44no—
132C22×C4—42no—
133C22×C4—42no—
134C22×C4—41no—
135C22×C4C2×C22⋊C411yescharacteristic
136C22×C4C4×D412yes—
137C22×C4#12011yescharacteristic
138C22×C4#12112yes—
139C22×C4#12311yescharacteristic
140SD16—88no—
141D8—88no—
142C2×C8—24no—
143C2×C8—42no—
144C2×C8—42no—
145C2×C8—48no—
146C2×C8—44no—
147C2×C8—88no—
148C2×C8—82no—
149C2×C8—82no—
150C2×C8—82no—
151C4⋊C4—42no—
152C4⋊C4—82no—
153C4⋊C4—82no—
154C22⋊C4—82no—
155C22⋊C4—84no—
156C22⋊C4—81no—
157C22⋊C4—81no—
158C22⋊C4—82no—
159C22⋊C4—82no—
160C22⋊C4—82no—
161C42—22no—
162C42—41no—
163C42—42no—
164C42—42no—
165C42C2×D811yescharacteristic
166C42C2×SD1611yescharacteristic
167C42#13512yes—
168C23—24no—
169C23—44no—
170C23—42no—
171C23—44no—
172C23—42no—
173C23—82no—
174C23—81no—
175C23—81no—
176C23—82no—
177C23—81no—
178C23#38611yescharacteristic
179Q8—82no—
180Q8—82no—
181D4—44no—
182D4—82no—
183D4—84no—
184D4—84no—
185D4—82no—
186D4—82no—
187D4—164no—
188C2×C4—22no—
189C2×C4—21no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—21no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—21no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—42no—
199C2×C4—44no—
200C2×C4—82no—
201C2×C4—82no—
202C2×C4—84no—
203C2×C4—81no—
204C2×C4—81no—
205C2×C4—81no—
206C2×C4C2×#10211yescharacteristic
207C2×C4#41611yescharacteristic
208C2×C4#45312yes—
209C2×C4#46512yes—
210C2×C4#46612yes—
211C8—44no—
212C8—84no—
213C8—162no—
214C22—48no—
215C22—44no—
216C22—82no—
217C22—84no—
218C22—82no—
219C22—161no—
220C22—162no—
221C22—162no—
222C22—161no—
223C22—161no—
224C22#188112yes—
225C22#188711yescharacteristic
226C22#189311yescharacteristic
227C22#190412yes—
228C22#191411yescharacteristic
229C4—22no—
230C4—22no—
231C4—21no—
232C4—21no—
233C4—22no—
234C4—41no—
235C4—42no—
236C4—82no—
237C4—161no—
238C2—84no—
239C2—321no—
240C2#900711yescharacteristic
241C2#930512yes—
242C2#933312yes—
243C2#939711yescharacteristic
244C2#942811yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC2×C42C22×C4
Frattini, cosocleC43C23
prime radical#28445C2
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
2. #23355C2semidirect
3. #8592C2semidirect
4. #7719C2semidirect
5. #7041C2semidirect

Maximal subgroups

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

subgroupindex
#284452
#233552
#85922
#77192
#70412