Finite Group Databasebeta

← #102899

Subgroups — #102899

Order 512.

548 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 112 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#28280C212yesmaximal
3#9638C211yesmaximal, characteristic
4#9637C212yesmaximal, complemented
5#7050C211yesmaximal, characteristic, complemented
6C2×#476—22no—
7C4×C4⋊Q8C2212yes—
8C4×#127C2211yescharacteristic
9#2087—22no—
10#2065—24no—
11#1987—22no—
12#1982—22no—
13#1948—22no—
14#1947—22no—
15#1842C2211yescharacteristic
16#1842C2212yes—
17#1802—22no—
18C42×C8C2211yescharacteristic
19C2×Q8⋊C8—24no—
20C2×D4⋊C8—22no—
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⋊C8—22no—
33C2×C4⋊C8D414yes—
34C2×C4⋊C8D412yes—
35C2×#102—24no—
36C2×#102—24no—
37C2×C4×C8—21no—
38C2×C4×C8—22no—
39#391—22no—
40#390—22no—
41#390—21no—
42#389—22no—
43#389—22no—
44#384D411yescharacteristic
45#384D412yes—
46C4×C4⋊C4—22no—
47C4×C4⋊C4—22no—
48C43C2311yescharacteristic
49#340—24no—
50#340—24no—
51#340—22no—
52Q8⋊C8—48no—
53D4⋊C8—44no—
54C22×Q8—22no—
55C22×D4—22no—
56C22×D4—21no—
57C23×C4—41no—
58C22×C8—22no—
59C22×C8—24no—
60C22×C8—41no—
61C4⋊Q8—22no—
62C4⋊Q8—22no—
63C4⋊Q8—22no—
64C4⋊Q8SD1614yes—
65#127—21no—
66#127—22no—
67#127Q1612yes—
68C4×Q8—44no—
69C4×Q8—44no—
70C4×D4—42no—
71C4×D4—42no—
72C4×D4—42no—
73C4×D4—42no—
74C4×D4—42no—
75C4×D4—42no—
76C2×C4⋊C4—22no—
77C2×C4⋊C4—22no—
78C2×C4⋊C4—22no—
79C2×C4⋊C4—22no—
80C2×C4⋊C4—22no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—22no—
83C2×C4⋊C4—42no—
84C2×C22⋊C4—42no—
85C2×C22⋊C4—41no—
86C2×C42—42no—
87C2×C42C2×D411yescharacteristic
88C2×C42C2×D412yes—
89C2×C42C2×D412yes—
90C2×C42C2×D411yescharacteristic
91C2×C42#4111yescharacteristic
92#107—82no—
93#106—48no—
94C4⋊C8—42no—
95C4⋊C8—44no—
96C4⋊C8—42no—
97C4⋊C8—42no—
98C4⋊C8—42no—
99C4⋊C8—42no—
100C4⋊C8—42no—
101C4⋊C8—42no—
102#102—48no—
103#102—84no—
104C4×C8—22no—
105C4×C8—24no—
106C4×C8—41no—
107C4×C8—42no—
108#95—42no—
109#95—42no—
110C24—41no—
111C2×Q8—24no—
112C2×Q8—44no—
113C2×Q8—44no—
114C2×Q8—42no—
115C2×D4—24no—
116C2×D4—22no—
117C2×D4—42no—
118C2×D4—42no—
119C2×D4—42no—
120C2×D4—42no—
121C2×D4—42no—
122C2×D4—42no—
123C2×D4—42no—
124C2×D4—41no—
125C22×C4—42no—
126C22×C4—42no—
127C22×C4—41no—
128C22×C4—82no—
129C22×C4—82no—
130C22×C4—81no—
131C22×C4—81no—
132C22×C4#12011yescharacteristic
133C22×C4#12012yes—
134C22×C4#12112yes—
135C22×C4#12111yescharacteristic
136C22×C4#12711yescharacteristic
137C2×C8—42no—
138C2×C8—44no—
139C2×C8—44no—
140C2×C8—44no—
141C2×C8—44no—
142C2×C8—44no—
143C2×C8—42no—
144C4⋊C4—24no—
145C4⋊C4—24no—
146C4⋊C4—42no—
147C4⋊C4—44no—
148C4⋊C4—42no—
149C4⋊C4—42no—
150C4⋊C4—42no—
151C4⋊C4—42no—
152C4⋊C4—42no—
153C4⋊C4—42no—
154C4⋊C4—42no—
155C4⋊C4—42no—
156C4⋊C4—84no—
157C22⋊C4—82no—
158C22⋊C4—82no—
159C22⋊C4—82no—
160C42—21no—
161C42—22no—
162C42—21no—
163C42—22no—
164C42—41no—
165C42—82no—
166C42—82no—
167C42C2×SD1612yes—
168C42C2×Q1611yescharacteristic
169C42#13512yes—
170C42#13511yescharacteristic
171C42AGL(1,ℤ/8)12yes—
172C42#13712yes—
173C42#13712yes—
174C23—42no—
175C23—82no—
176C23—81no—
177C23—82no—
178C23—81no—
179C23#39211yescharacteristic
180Q8—48no—
181Q8—84no—
182D4—44no—
183D4—44no—
184D4—44no—
185D4—82no—
186D4—82no—
187D4—82no—
188C2×C4—21no—
189C2×C4—22no—
190C2×C4—21no—
191C2×C4—22no—
192C2×C4—21no—
193C2×C4—22no—
194C2×C4—44no—
195C2×C4—82no—
196C2×C4—82no—
197C2×C4—82no—
198C2×C4—82no—
199C2×C4—81no—
200C2×C4—82no—
201C2×C4—82no—
202C2×C4—161no—
203C2×C4—161no—
204C2×C4—162no—
205C2×C4#44812yes—
206C2×C4#44912yes—
207C2×C4#45012yes—
208C2×C4#45112yes—
209C2×C4#45212yes—
210C2×C4#45212yes—
211C2×C4#45311yescharacteristic
212C2×C4#45312yes—
213C2×C4#45411yescharacteristic
214C2×C4#45412yes—
215C2×C4#46512yes—
216C2×C4#46711yescharacteristic
217C2×C4#46812yes—
218C2×C4#47011yescharacteristic
219C8—44no—
220C8—82no—
221C8—84no—
222C22—82no—
223C22—84no—
224C22—162no—
225C22—161no—
226C22—161no—
227C22#200012yes—
228C22#200111yescharacteristic
229C22#200612yes—
230C22#200611yescharacteristic
231C22#201711yescharacteristic
232C4—22no—
233C4—22no—
234C4—21no—
235C4—21no—
236C4—41no—
237C4—84no—
238C4—162no—
239C4—161no—
240C4#162011yescharacteristic
241C4#162311yescharacteristic
242C4#162612yes—
243C4#163112yes—
244C4#163212yes—
245C4#163712yes—
246C4#163812yes—
247C2—162no—
248C2#1023011yescharacteristic
249C2#1023612yes—
250C2#1025612yes—
251C2#1025811yescharacteristic
252C2#1030011yescharacteristic
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. #9637C2semidirect
5. #7050C2semidirect

Maximal subgroups

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

subgroupindex
#292902
#282802
#96382
#96372
#70502