Finite Group Databasebeta

← #19142879

Subgroups — #19142879

Order 5504.

1068 conjugacy classes of subgroups; only the 201 normal subgroups are shown.

normal subgroups only

This group has 1068 conjugacy classes of subgroups — too many to draw. Only its 201 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
1C23×C4×D43C211yesmaximal, characteristic
2C2×#13063301C212yesmaximal, complemented
3#14582771C212yesmaximal, complemented
4#14582770C214yesmaximal, complemented
5#14582759C211yesmaximal, characteristic, complemented
6#14582756C212yesmaximal, complemented
7#14582754C212yesmaximal
8C2×C22⋊Q8×C43C211yesmaximal, characteristic, complemented
9C24×D43C2211yescharacteristic
10C23×Dic43C2211yescharacteristic
11C22×C4×D43—24no—
12C22×C4×D43—28no—
13C22×C4×D43C2212yes—
14C22×C4×D43C2212yes—
15#13063301—28no—
16C2×C4⋊Dic43C2212yescomplemented
17C2×Dic43⋊C4C2214yescomplemented
18C2×C22⋊Dic43C2212yes—
19C2×(C2×D43)⋊C4C2214yes—
20C2×(C2×D43)⋊C4C2214yes—
21C43⋊#95C2214yescomplemented
22C43⋊#95C2212yescomplemented
23C22×Q8×C43C2211yescharacteristic, complemented
24C23×C4×C43C2211yescharacteristic
25C22⋊Q8×C43—24no—
26C2×C4⋊C4×C43C2211yescharacteristic, complemented
27C2×C4⋊C4×C43C2212yescomplemented
28C2×C22⋊C4×C43C2212yes—
29C24×C43C2311yescharacteristic
30C23×D43—22no—
31C23×D43—22no—
32C23×D43—24no—
33C23×D43—24no—
34C23×D43Q814yescomplemented
35C23×D43C2312yes—
36C2×Q8×C43—22no—
37C2×Q8×C43—24no—
38C22×C4×C43—24no—
39C22×C4×C43D414yes—
40C22×C4×C43C2312yes—
41C22×C4×C43C2312yes—
42C22×C4×C43C2311yescharacteristic
43C22×C4×C43C2311yescharacteristic
44C22×Dic43—24no—
45C22×Dic43—22no—
46C22×Dic43C2314yes—
47C22×Dic43C2312yes—
48C2×C4×D43—232no—
49C2×C4×D43—24no—
50C2×C4×D43—24no—
51C2×C4×D43—24no—
52C2×C4×D43—24no—
53C2×C4×D43—24no—
54C2×C4×D43D418yes—
55C4⋊C4×C43—22no—
56C4⋊C4×C43—24no—
57C22⋊C4×C43—24no—
58C22⋊Dic43—24no—
59(C2×D43)⋊C4—28no—
60(C2×D43)⋊C4—28no—
61C4⋊Dic43—24no—
62Dic43⋊C4—28no—
63C23×C43—22no—
64C23×C43—22no—
65C23×C43C2×D412yes—
66C23×C43C2×Q812yes—
67C23×C43#4112yes—
68C23×C43C2411yescharacteristic
69C22×D43—28no—
70C22×D43—216no—
71C22×D43—28no—
72C22×D43—28no—
73C22×D43—22no—
74C22×D43—22no—
75C22×D43—22no—
76C22×D43—22no—
77C22×D43—216no—
78C22×D43C2×D414yes—
79C22×D43C2×Q814yes—
80C22×D43#4114yes—
81Q8×C43—44no—
82C2×C4×C43—22no—
83C2×C4×C43—28no—
84C2×C4×C43—24no—
85C2×C4×C43—22no—
86C2×C4×C43—22no—
87C2×C4×C43—24no—
88C2×C4×C43C2×D414yes—
89C2×C4×C43C2×D412yes—
90C2×C4×C43C2×D412yes—
91C2×Dic43—28no—
92C2×Dic43—22no—
93C2×Dic43—22no—
94C2×Dic43—24no—
95C2×Dic43—24no—
96C2×Dic43—24no—
97C2×Dic43C2×D414yes—
98C4×D43—216no—
99C4×D43—216no—
100C22×C43—22no—
101C22×C43—24no—
102C22×C43—22no—
103C22×C43—24no—
104C22×C43C22⋊Q814yescomplemented
105C22×C43C22×D412yes—
106C22×C43C22×D411yescharacteristic
107C22×C43C22×Q811yescharacteristic
108C22×C43C2×#4112yes—
109C22×C43C2×#4111yescharacteristic
110C2×D43—232no—
111C2×D43—24no—
112C2×D43—24no—
113C2×D43—24no—
114C2×D43—24no—
115C2×D43—24no—
116C2×D43—24no—
117C2×D43C22⋊Q818yescomplemented
118C4×C43—22no—
119C4×C43—42no—
120C4×C43—42no—
121C4×C43#12014yes—
122Dic43—24no—
123Dic43—44no—
124#2848—431nomaximal, Sylow
125C2×C43—24no—
126C2×C43C2×#12011yescharacteristic
127C2×C43C2×C22⋊Q812yes—
128C2×C43C2×C22⋊Q811yescharacteristic
129C2×C43#52511yescharacteristic
130C2×C43#52512yes—
131D43—28no—
132C24×C4—431no—
133C2×C22⋊Q8—432no—
134C2×C22⋊Q8D4311yescharacteristic, complemented
135#385—434no—
136#385—432no—
137#380—431no—
138#380—432no—
139#379—432no—
140C43#284811yescharacteristic, complemented, Sylow
141C25—431no—
142C22×Q8C2×D4311yescharacteristic, complemented
143C23×C4—432no—
144C23×C4—431no—
145C23×C4—432no—
146C23×C4—868no—
147C23×C4—864no—
148C23×C4C2×D4311yescharacteristic
149C22⋊Q8—24no—
150C22⋊Q8—868no—
151C2×C4⋊C4—434no—
152C2×C4⋊C4—432no—
153C2×C4⋊C4C2×D4311yescharacteristic, complemented
154C2×C4⋊C4C2×D4312yescomplemented
155C2×C22⋊C4—432no—
156C2×C22⋊C4—434no—
157C2×C22⋊C4—434no—
158C2×C22⋊C4C2×D4312yes—
159#95—434no—
160#95—432no—
161C24—432no—
162C24—434no—
163C24—864no—
164C24—864no—
165C24—862no—
166C24—862no—
167C24C22×D4311yescharacteristic
168C2×Q8—24no—
169C2×Q8—22no—
170C22×C4—24no—
171C22×C4—432no—
172C22×C4—438no—
173C22×C4—434no—
174C22×C4—864no—
175C22×C4—864no—
176C22×C4—8632no—
177C22×C4—864no—
178C22×C4—862no—
179C22×C4—864no—
180C22×C4—864no—
181C22×C4—864no—
182C22×C4C22⋊D4314yes—
183C22×C4C22×D4312yes—
184C22×C4C22×D4312yes—
185C22×C4C22×D4311yescharacteristic
186C22×C4C22×D4311yescharacteristic
187C4⋊C4—22no—
188C4⋊C4—24no—
189C4⋊C4—868no—
190C4⋊C4—864no—
191C22⋊C4—24no—
192C22⋊C4—868no—
193C22⋊C4—868no—
194C22⋊C4—864no—
195C23—22no—
196C23—22no—
197C23—434no—
198C23—434no—
199C23—434no—
200C23—8616no—
201C23—868no—
202C23—868no—
203C23—8616no—
204C23—868no—
205C23—862no—
206C23—862no—
207C23—862no—
208C23—862no—
209C23Q8×D4312yes—
210C23#1062190212yes—
211C23C2×C22⋊D4312yes—
212C23C23×D4311yescharacteristic
213Q8—44no—
214C2×C4—22no—
215C2×C4—28no—
216C2×C4—24no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—24no—
220C2×C4—434no—
221C2×C4—8616no—
222C2×C4—868no—
223C2×C4—862no—
224C2×C4—862no—
225C2×C4—8616no—
226C2×C4—864no—
227C2×C4—864no—
228C2×C4—864no—
229C2×C4D4×D4314yes—
230C2×C4C2×C22⋊D4312yes—
231C2×C4C2×C22⋊D4312yes—
232C22—22no—
233C22—24no—
234C22—22no—
235C22—24no—
236C22—438no—
237C22—8632no—
238C22—864no—
239C22—864no—
240C22—864no—
241C22—864no—
242C22—864no—
243C22—864no—
244C22#1306330114yescomplemented
245C22C2×D4×D4312yes—
246C22C22×C22⋊D4311yescharacteristic
247C22C2×Q8×D4311yescharacteristic
248C22#41×D4312yes—
249C22C2×#1062190211yescharacteristic
250C4—22no—
251C4—42no—
252C4—42no—
253C4—864no—
254C4—1724no—
255C4#1306329314yes—
256C2—24no—
257C2—868no—
258C2C2×#1306329311yescharacteristic
259C2C22⋊Q8×D4312yes—
260C2C2×#1306330111yescharacteristic
261C2#1458340611yescharacteristic
262C2#1458341212yes—
subgroupG/H
center, FrattiniC23C23×D43
commutator, socleC23×C43C24
FittingC2×C22⋊Q8×C43C2
hypercenterC2×C22⋊Q8D43
hypocenterC43#2848
LCSG ▷ C23×C43 ▷ C43
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×#13063301C2semidirect
3. #14582771C2semidirect
4. #14582770C2semidirect
5. #14582759C2semidirect
6. #14582756C2semidirect
8. C2×C22⋊Q8×C43C2semidirect
16. C2×C4⋊Dic43C22semidirect
17. C2×Dic43⋊C4C22semidirect
21. C43⋊#95C22semidirect
22. C43⋊#95C22semidirect
23. C22×Q8×C43C22semidirect
26. C2×C4⋊C4×C43C22semidirect
27. C2×C4⋊C4×C43C22semidirect
34. C23×D43Q8semidirect
104. C22×C43C22⋊Q8semidirect
117. C2×D43C22⋊Q8semidirect
134. C2×C22⋊Q8D43semidirect
140. C43#2848semidirect
142. C22×Q8C2×D43semidirect
153. C2×C4⋊C4C2×D43semidirect
154. C2×C4⋊C4C2×D43semidirect
244. C22#13063301semidirect