Finite Group Databasebeta

← #100894

Subgroups — #100894

Order 512.

493 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 493 conjugacy classes of subgroups — too many to draw. Only its 80 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#27919C211yesmaximal, characteristic
2#14494C211yesmaximal, characteristic
3#9889C212yesmaximal
4#8718C211yesmaximal, characteristic
5#7036C212yesmaximal
6C2×#484—21no—
7C2×#483—21no—
8#2857C2211yescharacteristic
9C2×#382C2211yescharacteristic
10#2097—22no—
11#1960C412yescomplemented
12#1847C412yescomplemented
13#1749C2212yes—
14#1749C2211yescharacteristic
15#1443—22no—
16#1441—21no—
17#1436—22no—
18#1436—21no—
19#1434C2212yes—
20#1295—22no—
21#1294—22no—
22#1294—22no—
23C2×C4⋊Q8C2×C411yescharacteristic
24C2×C22⋊Q8—21no—
25C22×C4⋊C4—21no—
26C22×C4⋊C4D411yescharacteristic
27C22×C4⋊C4D411yescharacteristic
28C22×C42—21no—
29C22×C42C2311yescharacteristic
30#484—42no—
31#484—42no—
32#483—42no—
33#483—42no—
34C2×#107—22no—
35C2×#106—22no—
36C2×#103—22no—
37C2×#103—22no—
38C2×C22⋊C8—21no—
39C2×C22⋊C8—22no—
40C2×#97—22no—
41C2×#97C2×C412yes—
42#391—21no—
43#389—22no—
44#384C2×C411yescharacteristic
45#382—22no—
46#382C2×C412yes—
47#382D412yes—
48#380—22no—
49C2×#95—22no—
50C2×#95—22no—
51C2×#95—22no—
52C2×#95—22no—
53C2×#95—22no—
54C2×#95—21no—
55C2×#95—21no—
56C2×#95D412yes—
57C2×#95D412yes—
58#336—21no—
59#336—22no—
60C22×Q8—21no—
61C23×C4—21no—
62C23×C4—22no—
63C23×C4—21no—
64C23×C4—22no—
65C23×C4C2×D411yescharacteristic
66C23×C4C2×D412yes—
67C22×C8—22no—
68C22×C8—42no—
69C4⋊Q8—42no—
70C22⋊Q8—41no—
71C22⋊Q8—41no—
72C22⋊Q8—41no—
73C22⋊Q8—41no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—22no—
78C2×C4⋊C4—22no—
79C2×C4⋊C4—41no—
80C2×C4⋊C4—41no—
81C2×C4⋊C4—41no—
82C2×C4⋊C4—41no—
83C2×C4⋊C4—44no—
84C2×C4⋊C4—41no—
85C2×C4⋊C4—81no—
86C2×C4⋊C4C22⋊C412yes—
87C2×C4⋊C4C22⋊C412yes—
88C2×C22⋊C4—41no—
89C2×C42—42no—
90C2×C42—81no—
91C2×C42—82no—
92C2×C42C22×C411yescharacteristic
93C2×C42C2×D411yescharacteristic
94C2×C42#4112yes—
95#107—44no—
96#106—44no—
97#103—44no—
98#103—44no—
99C22⋊C8—22no—
100C22⋊C8—22no—
101C22⋊C8—24no—
102C22⋊C8—42no—
103#97—42no—
104#97—44no—
105#95—24no—
106#95—24no—
107#95—42no—
108#95—42no—
109#95—44no—
110#95—44no—
111#95—44no—
112#95—42no—
113#95—81no—
114#95—81no—
115#95—82no—
116C24#12011yescharacteristic
117C2×Q8—41no—
118C2×Q8—41no—
119C2×Q8—42no—
120C2×Q8—42no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—22no—
124C22×C4—22no—
125C22×C4—22no—
126C22×C4—24no—
127C22×C4—22no—
128C22×C4—22no—
129C22×C4—41no—
130C22×C4—42no—
131C22×C4—42no—
132C22×C4—42no—
133C22×C4—42no—
134C22×C4—42no—
135C22×C4—42no—
136C22×C4—42no—
137C22×C4—41no—
138C22×C4—41no—
139C22×C4—41no—
140C22×C4—42no—
141C22×C4—41no—
142C22×C4—42no—
143C22×C4—42no—
144C22×C4—42no—
145C22×C4—42no—
146C22×C4—42no—
147C22×C4—42no—
148C22×C4—42no—
149C22×C4—42no—
150C22×C4—41no—
151C22×C4—41no—
152C22×C4—42no—
153C22×C4—42no—
154C22×C4—42no—
155C22×C4C2×C22⋊C411yescharacteristic
156C22×C4C4×D412yes—
157C22×C4#12112yes—
158C22×C4#12311yescharacteristic
159C2×C8—24no—
160C2×C8—42no—
161C2×C8—42no—
162C2×C8—44no—
163C2×C8—82no—
164C2×C8—82no—
165C4⋊C4—42no—
166C4⋊C4—42no—
167C4⋊C4—41no—
168C4⋊C4—41no—
169C4⋊C4—81no—
170C4⋊C4—81no—
171C4⋊C4—82no—
172C4⋊C4—82no—
173C4⋊C4—81no—
174C4⋊C4—81no—
175C4⋊C4—84no—
176C22⋊C4—81no—
177C22⋊C4—81no—
178C42—22no—
179C42—22no—
180C42—42no—
181C42—84no—
182C23—21no—
183C23—22no—
184C23—21no—
185C23#38611yescharacteristic
186C23#45311yescharacteristic
187C23#45311yescharacteristic
188C23#45512yes—
189C23#45612yes—
190Q8—82no—
191Q8—82no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—21no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—21no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—21no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—44no—
206C2×C4—44no—
207C2×C4—44no—
208C2×C4—44no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—44no—
212C2×C4—82no—
213C2×C4—82no—
214C2×C4—81no—
215C2×C4—82no—
216C2×C4—82no—
217C2×C4—82no—
218C2×C4—81no—
219C2×C4—82no—
220C2×C4—82no—
221C2×C4—81no—
222C2×C4—81no—
223C2×C4—82no—
224C2×C4—82no—
225C2×C4—82no—
226C2×C4—82no—
227C2×C4—82no—
228C2×C4—82no—
229C2×C4—82no—
230C2×C4—81no—
231C2×C4—81no—
232C2×C4—81no—
233C2×C4—82no—
234C2×C4—81no—
235C2×C4—82no—
236C2×C4#41012yes—
237C8—44no—
238C8—84no—
239C22—22no—
240C22—22no—
241C22—22no—
242C22—22no—
243C22—21no—
244C22—21no—
245C22—21no—
246C22—21no—
247C22#165911yescharacteristic
248C22#166011yescharacteristic
249C22#166112yes—
250C22#188212yes—
251C22#188711yescharacteristic
252C22#189911yescharacteristic
253C22#190612yes—
254C22#191211yescharacteristic
255C4—22no—
256C4—42no—
257C4—42no—
258C4—41no—
259C4—84no—
260C4—82no—
261C4—82no—
262C4—162no—
263C4—161no—
264C4—161no—
265C2—21no—
266C2—22no—
267C2—21no—
268C2#900311yescharacteristic
269C2#936212yes—
270C2#938012yes—
271C2#944111yescharacteristic
272C2#946411yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#120
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
11. #1960C4semidirect
12. #1847C4semidirect

Maximal subgroups

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

subgroupindex
#279192
#144942
#98892
#87182
#70362