Finite Group Databasebeta

← #124142

Subgroups — #124142

Order 512.

634 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 634 conjugacy classes of subgroups — too many to draw. Only its 98 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#14288C211yesmaximal, characteristic
2C4×#324C212yesmaximal, complemented
3#7131C212yesmaximal, complemented
4#7103C212yesmaximal
5C22×#124C411yescharacteristic
6C2×#383C411yescharacteristic
7C2×C4×C4⋊C4C2211yescharacteristic
8C4×C22⋊C8—22no—
9C2×#324—22no—
10#1446C411yescharacteristic
11#1446C2211yescharacteristic
12#1439—22no—
13#1438C411yescharacteristic
14#1438C2211yescharacteristic
15#1437—22no—
16#1280C2214yes—
17C23×Q8C2×C411yescharacteristic
18C23×D4C2×C411yescharacteristic
19C2×#124—22no—
20C2×#124—42no—
21C2×#124—42no—
22C2×#124C814yescomplemented
23C22×C4⋊C4D412yes—
24C22×C22⋊C4—21no—
25C22×C22⋊C4—21no—
26C22×C22⋊C4—22no—
27C22×C22⋊C4—22no—
28C22×C42C2×C411yescharacteristic
29C22×C42D412yes—
30C2×C22⋊C8—24no—
31C2×C4×C8—42no—
32#383—42no—
33C4×C4⋊C4—42no—
34C2×#95—21no—
35C2×#95—21no—
36C2×#95—22no—
37C2×#95—22no—
38C2×#95C2×C411yescharacteristic
39C2×#95C2×C411yescharacteristic
40C2×#95C2×C411yescharacteristic
41C2×#95C2311yescharacteristic
42#336—42no—
43#324—28no—
44C25—21no—
45C22×Q8—21no—
46C22×Q8—41no—
47C22×Q8—41no—
48C22×Q8—82no—
49C22×Q8C2×C812yes—
50C22×D4—21no—
51C22×D4—41no—
52C22×D4—41no—
53C22×D4—82no—
54C22×D4C2×C812yes—
55C23×C4—21no—
56C23×C4—21no—
57C23×C4—21no—
58C23×C4C22⋊C411yescharacteristic
59C23×C4C22⋊C412yes—
60C23×C4C22⋊C411yescharacteristic
61C23×C4C22×C411yescharacteristic
62C23×C4C2×D412yes—
63C22×C8—44no—
64#124—48no—
65#124—48no—
66C2×C4⋊C4—24no—
67C2×C4⋊C4—24no—
68C2×C4⋊C4—42no—
69C2×C22⋊C4—24no—
70C2×C22⋊C4—24no—
71C2×C22⋊C4—42no—
72C2×C22⋊C4—42no—
73C2×C22⋊C4—42no—
74C2×C22⋊C4—42no—
75C2×C22⋊C4—44no—
76C2×C22⋊C4—44no—
77C2×C22⋊C4—82no—
78C2×C22⋊C4—82no—
79C2×C22⋊C4—82no—
80C2×C22⋊C4—82no—
81C2×C42—22no—
82C2×C42—22no—
83C2×C42—22no—
84C2×C42—22no—
85C2×C42—82no—
86C2×C42C2×C812yes—
87C2×C42M1612yes—
88C22⋊C8—28no—
89C22⋊C8—44no—
90C4×C8—84no—
91#95—41no—
92#95—41no—
93#95—41no—
94#95—41no—
95#95—42no—
96#95—42no—
97#95—82no—
98#95—82no—
99C24—22no—
100C24—42no—
101C24—42no—
102C24—42no—
103C24—42no—
104C24—41no—
105C24—41no—
106C24—42no—
107C24—42no—
108C24C2×C22⋊C411yescharacteristic
109C2×Q8—42no—
110C2×Q8—42no—
111C2×Q8—82no—
112C2×Q8—82no—
113C2×Q8—84no—
114C2×Q8—84no—
115C2×D4—42no—
116C2×D4—42no—
117C2×D4—82no—
118C2×D4—84no—
119C2×D4—84no—
120C2×D4—82no—
121C22×C4—21no—
122C22×C4—22no—
123C22×C4—22no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—22no—
128C22×C4—22no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—22no—
132C22×C4—41no—
133C22×C4—41no—
134C22×C4—42no—
135C22×C4—42no—
136C22×C4—42no—
137C22×C4—42no—
138C22×C4—42no—
139C22×C4—42no—
140C22×C4—42no—
141C22×C4—41no—
142C22×C4—41no—
143C22×C4—42no—
144C22×C4—42no—
145C22×C4—41no—
146C22×C4—41no—
147C22×C4—41no—
148C22×C4—41no—
149C22×C4—41no—
150C22×C4—41no—
151C22×C4—42no—
152C22×C4—42no—
153C22×C4C22⋊C812yes—
154C22×C4C22⋊C812yes—
155C22×C4#9912yes—
156C22×C4#10012yes—
157C22×C4#10414yes—
158C22×C4C22×C811yescharacteristic
159C22×C4C2×M1611yescharacteristic
160C22×C4#13514yes—
161C2×C8—48no—
162C2×C8—84no—
163C2×C8—84no—
164C4⋊C4—48no—
165C22⋊C4—48no—
166C22⋊C4—48no—
167C22⋊C4—88no—
168C22⋊C4—88no—
169C42—24no—
170C42—24no—
171C42—44no—
172C42—82no—
173C23—22no—
174C23—21no—
175C23—21no—
176C23—42no—
177C23—42no—
178C23—44no—
179C23—42no—
180C23—44no—
181C23—82no—
182C23—82no—
183C23—82no—
184C23—82no—
185C23—82no—
186C23—82no—
187C23—82no—
188C23—82no—
189C23—82no—
190C23—82no—
191C23—81no—
192C23—82no—
193C23—81no—
194C23—81no—
195C23—82no—
196C23—81no—
197C23C2×C22⋊C811yescharacteristic
198C23C2×#9911yescharacteristic
199C23C2×#10011yescharacteristic
200C23#41612yes—
201C23C2×#10412yes—
202Q8—84no—
203Q8—84no—
204D4—84no—
205D4—84no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—24no—
212C2×C4—24no—
213C2×C4—24no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—41no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—41no—
225C2×C4—44no—
226C2×C4—44no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—42no—
232C2×C4—82no—
233C2×C4—82no—
234C2×C4—82no—
235C2×C4—82no—
236C2×C4—82no—
237C2×C4—82no—
238C2×C4—81no—
239C2×C4—81no—
240C2×C4—81no—
241C2×C4—81no—
242C2×C4—82no—
243C2×C4—82no—
244C2×C4—81no—
245C2×C4—81no—
246C2×C4#32314yes—
247C8—88no—
248C22—21no—
249C22—21no—
250C22—21no—
251C22—21no—
252C22—22no—
253C22—22no—
254C22—22no—
255C22—22no—
256C22—84no—
257C22—84no—
258C22—82no—
259C22—82no—
260C22—82no—
261C22—162no—
262C22—162no—
263C22—162no—
264C22—162no—
265C22C2×#32311yescharacteristic
266C22#147612yes—
267C22#150112yes—
268C22#152511yescharacteristic
269C22#154311yescharacteristic
270C22#212212yes—
271C22#212711yescharacteristic
272C22#213311yescharacteristic
273C4—24no—
274C4—42no—
275C4—41no—
276C4—41no—
277C4—44no—
278C4—44no—
279C4—82no—
280C4—161no—
281C4—161no—
282C2—21no—
283C2—21no—
284C2—22no—
285C2—162no—
286C2#756311yescharacteristic
287C2#1142711yescharacteristic
288C2#1143211yescharacteristic
289C2#1173012yes—
290C2#1173812yes—
subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C22×C8
Frattini, cosocleC2×#95C23
prime radicalC23×D4C2×C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C2×#95 ▷ C24 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C4×#324C2semidirect
3. #7131C2semidirect
22. C2×#124C8semidirect

Maximal subgroups

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

subgroupindex
#142882
C4×#3242
#71312
#71032