Finite Group Databasebeta

← #19137953

Subgroups — #19137953

Order 5504.

840 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 840 conjugacy classes of subgroups — too many to draw. Only its 171 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
1C22×Q8×D43C211yesmaximal, characteristic, complemented
2C2×#13063302C212yesmaximal, complemented
3C2×(C22⋊D43)⋊C4C211yesmaximal, characteristic
4#14582820C211yesmaximal, characteristic, complemented
5#14582818C211yesmaximal, characteristic, complemented
6#14582811C211yesmaximal, characteristic, complemented
7#14582803C212yesmaximal, complemented
8#14582775C212yesmaximal
9#14582773C212yesmaximal, complemented
10#14582758C211yesmaximal, characteristic, complemented
11C43×#393C211yesmaximal, characteristic, complemented
12C22×Dic86C2211yescharacteristic
13C22×Dic86C2212yes—
14C2×Q8×D43—24no—
15C2×Q8×D43—28no—
16C22×C22⋊D43C2211yescharacteristic
17C23×Dic43C2211yescharacteristic
18C22×C4×D43C2212yes—
19C22×C4×D43C2211yescharacteristic
20#13063302—28no—
21(C22⋊D43)⋊C4—24no—
22C2×C4⋊Dic43C2212yes—
23C2×Dic43⋊C4C2212yes—
24C2×Dic43⋊C4C2211yescharacteristic
25C2×Dic43⋊C4C2212yes—
26C2×C22⋊Dic43C2212yes—
27C2×(C2×D43)⋊C4C2211yescharacteristic
28C2×(C2×D43)⋊C4C2212yes—
29C2×C4×Dic43C2211yescharacteristic
30C43⋊#95C2212yes—
31C43⋊#95C2212yes—
32C43⋊#95C2212yes—
33C43⋊#95C2211yescharacteristic
34C22×Q8×C43C2211yescharacteristic
35C2×C22⋊C4×C43C2211yescharacteristic
36C2×C22⋊C4×C43C2212yes—
37#95×C43C2212yes—
38#95×C43C2211yescharacteristic
39C24×C43C2311yescharacteristic
40C23×D43C2311yescharacteristic
41C2×Q8×C43—22no—
42C2×Q8×C43—24no—
43C22×C4×C43C2311yescharacteristic
44C22×C4×C43C2312yes—
45C22×C4×C43C2311yescharacteristic
46C22×C4×C43C2312yes—
47C2×C22⋊D43—22no—
48C2×C22⋊D43—22no—
49C2×C22⋊D43Q814yescomplemented
50C22×Dic43—22no—
51C22×Dic43—24no—
52C22×Dic43C2312yes—
53C22×Dic43C2311yescharacteristic
54C22×Dic43C2312yes—
55C22×Dic43C2311yescharacteristic
56C22×Dic43C2311yescharacteristic
57Q8×D43—232no—
58C2×C4×D43—24no—
59C2×C4×D43—24no—
60C2×C4×D43—24no—
61C2×C4×D43—24no—
62C2×C4×D43D414yes—
63C2×Dic86—22no—
64C2×Dic86—24no—
65C2×Dic86—24no—
66C2×Dic86—24no—
67C2×Dic86D418yescomplemented
68C22⋊C4×C43—22no—
69C22⋊C4×C43—24no—
70C22⋊Dic43—24no—
71(C2×D43)⋊C4—22no—
72(C2×D43)⋊C4—24no—
73C4⋊Dic43—24no—
74Dic43⋊C4—22no—
75Dic43⋊C4—24no—
76Dic43⋊C4—24no—
77C4×Dic43—22no—
78C23×C43—22no—
79C23×C43—21no—
80C23×C43—21no—
81C23×C43—22no—
82C23×C43C2×Q812yescomplemented
83C23×C43C2411yescharacteristic
84C22×D43—22no—
85C22×D43—22no—
86C22×D43C2×D412yes—
87C22×D43C2×Q812yes—
88C22×D43#4112yes—
89Q8×C43—28no—
90C2×C4×C43—22no—
91C2×C4×C43—22no—
92C2×C4×C43—22no—
93C2×C4×C43—21no—
94C2×C4×C43—22no—
95C2×C4×C43—22no—
96C2×C4×C43—22no—
97C2×C4×C43—22no—
98C2×C4×C43C2×D412yes—
99C2×C4×C43C2×D414yes—
100C22⋊D43—28no—
101C2×Dic43—21no—
102C2×Dic43—22no—
103C2×Dic43—22no—
104C2×Dic43—21no—
105C2×Dic43—22no—
106C2×Dic43—22no—
107C2×Dic43—21no—
108C2×Dic43—22no—
109C2×Dic43—22no—
110C2×Dic43—44no—
111C2×Dic43C2×D414yes—
112C2×Dic43C2×D412yes—
113C2×Dic43C2×D414yes—
114C2×Dic43C2×Q812yescomplemented
115C4×D43—216no—
116C4×D43—28no—
117Dic86—28no—
118Dic86—216no—
119C22×C43—22no—
120C22×C43—24no—
121C22×C43—41no—
122C22×C43—41no—
123C22×C43—42no—
124C22×C43C22×D411yescharacteristic
125C22×C43C22×D412yes—
126C22×C43C22×Q811yescharacteristic
127C22×C43C2×#4111yescharacteristic
128C22×C432−1+412yes—
129C2×D43—22no—
130C2×D43—22no—
131C2×D43—24no—
132C2×D43—24no—
133C2×D43C22⋊Q814yescomplemented
134C4×C43—22no—
135C4×C43—24no—
136C4×C43—41no—
137C4×C43—42no—
138Dic43—24no—
139Dic43—22no—
140Dic43—41no—
141Dic43—42no—
142Dic43#12014yes—
143#2409—431nomaximal, Sylow
144C2×C43—42no—
145C2×C43C2×#12011yescharacteristic
146C2×C43C2×C22⋊Q811yescharacteristic
147C2×C43#53611yescharacteristic
148C2×C43#54812yes—
149C2×C43D4×Q812yes—
150D43—24no—
151C23×Q8—431no—
152C2×C22⋊Q8—432no—
153C2×C4×D4—431no—
154#395—431no—
155#395—431no—
156#393—431no—
157#393D4311yescharacteristic, complemented
158#391—432no—
159#386—432no—
160#385—432no—
161#380—431no—
162C43#240911yescharacteristic, complemented, Sylow
163C22×Q8—432no—
164C22×Q8—431no—
165C22×Q8—864no—
166C22×Q8—868no—
167C22×Q8C2×D4311yescharacteristic
168C22×D4—431no—
169C23×C4—431no—
170C23×C4—431no—
171C23×C4—432no—
172C22⋊Q8—868no—
173C4×D4—864no—
174C2×C4⋊C4—431no—
175C2×C4⋊C4—432no—
176C2×C4⋊C4—432no—
177C2×C4⋊C4—432no—
178C2×C22⋊C4—432no—
179C2×C22⋊C4—431no—
180C2×C22⋊C4—432no—
181C2×C22⋊C4C2×D4311yescharacteristic
182C2×C22⋊C4C2×D4312yes—
183C2×C42—431no—
184#95—431no—
185#95—432no—
186#95—432no—
187#95—432no—
188#95C2×D4311yescharacteristic
189#95C2×D4312yes—
190C24—431no—
191C24C22×D4311yescharacteristic
192C2×Q8—22no—
193C2×Q8—24no—
194C2×Q8—438no—
195C2×Q8—864no—
196C2×Q8—864no—
197C2×Q8—864no—
198C2×Q8—8632no—
199C2×Q8—862no—
200C2×D4—434no—
201C2×D4—862no—
202C2×D4—862no—
203C22×C4—431no—
204C22×C4—431no—
205C22×C4—432no—
206C22×C4—434no—
207C22×C4—432no—
208C22×C4—431no—
209C22×C4—864no—
210C22×C4—864no—
211C22×C4—864no—
212C22×C4—862no—
213C22×C4—864no—
214C22×C4—864no—
215C22×C4C22×D4311yescharacteristic
216C22×C4C22×D4312yes—
217C22×C4C22×D4311yescharacteristic
218C22×C4C22×D4312yes—
219C4⋊C4—864no—
220C4⋊C4—864no—
221C4⋊C4—862no—
222C4⋊C4—864no—
223C22⋊C4—22no—
224C22⋊C4—24no—
225C22⋊C4—862no—
226C22⋊C4—864no—
227C22⋊C4—864no—
228C42—862no—
229C23—22no—
230C23—21no—
231C23—21no—
232C23—22no—
233C23—432no—
234C23—432no—
235C23—432no—
236C23—862no—
237C23—862no—
238C23Q8×D4312yescomplemented
239C23C23×D4311yescharacteristic
240Q8—28no—
241Q8—8616no—
242Q8—868no—
243D4—868no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—21no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—432no—
253C2×C4—434no—
254C2×C4—434no—
255C2×C4—432no—
256C2×C4—868no—
257C2×C4—8616no—
258C2×C4—861no—
259C2×C4—862no—
260C2×C4—862no—
261C2×C4—861no—
262C2×C4—862no—
263C2×C4—861no—
264C2×C4—862no—
265C2×C4—862no—
266C2×C4—862no—
267C2×C4—1724no—
268C2×C4D4×D4312yes—
269C2×C4D4×D4314yes—
270C22—24no—
271C22—22no—
272C22—42no—
273C22—41no—
274C22—41no—
275C22—434no—
276C22—864no—
277C22—864no—
278C22—862no—
279C22—862no—
280C22C2×D4×D4312yes—
281C22C2×D4×D4311yescharacteristic
282C22C2×Q8×D4311yescharacteristic
283C22#41×D4311yescharacteristic
284C22#1306337212yes—
285C4—22no—
286C4—24no—
287C4—42no—
288C4—41no—
289C4—434no—
290C4—864no—
291C4—862no—
292C4—1722no—
293C4—1721no—
294C2—42no—
295C2—864no—
296C2#120×D4311yescharacteristic
297C2C22⋊Q8×D4311yescharacteristic
298C2#1458346911yescharacteristic
299C2#1458355712yes—
300C2#1458356212yes—
subgroupG/H
center, FrattiniC23C23×D43
commutator, socleC23×C43C24
FittingC43×#393C2
hypercenter#393D43
hypocenterC43#2409
LCSG ▷ C23×C43 ▷ C43
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. C22×Q8×D43C2semidirect
2. C2×#13063302C2semidirect
4. #14582820C2semidirect
5. #14582818C2semidirect
6. #14582811C2semidirect
7. #14582803C2semidirect
9. #14582773C2semidirect
10. #14582758C2semidirect
11. C43×#393C2semidirect
49. C2×C22⋊D43Q8semidirect
67. C2×Dic86D4semidirect
82. C23×C43C2×Q8semidirect
114. C2×Dic43C2×Q8semidirect
133. C2×D43C22⋊Q8semidirect
157. #393D43semidirect
162. C43#2409semidirect
238. C23Q8×D43semidirect