Finite Group Databasebeta

← #28402799

Subgroups — #28402799

Order 14464.

448 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 123 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
1D4×D452C211yesmaximal, characteristic
2#20298946C211yesmaximal, characteristic, complemented
3#20298922C212yesmaximal, complemented
4#20298875C211yesmaximal, characteristic, complemented
5#20298846C212yesmaximal, complemented
6#20298798C211yesmaximal, characteristic, complemented
7#20298748C212yesmaximal, complemented
8C4×#14268634C211yesmaximal, characteristic, complemented
9#20298676C211yesmaximal, characteristic, complemented
10#20298602C212yesmaximal, complemented
11#556×C113C211yesmaximal, characteristic, complemented
12C22×D452C2212yes—
13C2×D4×D113—21no—
14C2×#14268634C2211yescharacteristic
15C2×#14268634C2212yescomplemented
16C2×#14268631C2212yescomplemented
17#16813619C2211yescharacteristic
18#16813597—21no—
19#16813596C2212yes—
20#16813592—22no—
21C4×D452C2211yescharacteristic
22#16813555C2212yescomplemented
23#16813553C2211yescharacteristic, complemented
24C4⋊#11744751C2211yescharacteristic, complemented
25#16813546C2211yescharacteristic, complemented
26#16813542C2211yescharacteristic
27#16813542C2212yes—
28#16813542C2212yescomplemented
29#16813542C2212yescomplemented
30#16813542C2212yescomplemented
31C22⋊#11744751C2212yescomplemented
32C4×#11744751C2211yescharacteristic, complemented
33C4⋊Q8×C113C2211yescharacteristic, complemented
34#126×C113—22no—
35#124×C113—21no—
36C22⋊Q8×C113C2212yescomplemented
37C4×Q8×C113—21no—
38C4×Q8×C113C2211yescharacteristic, complemented
39C4×D4×C113C2211yescharacteristic
40#117×C113C2212yescomplemented
41C23×D113—22no—
42C2×D452—24no—
43C2×D452—22no—
44C2×D452—21no—
45C2×D452—21no—
46C2×D452—42no—
47C2×D452C2312yes—
48C2×D452C2311yescharacteristic
49C2×C22⋊D113—22no—
50D4×D113—44no—
51C2×C4×D113—21no—
52#14268634—22no—
53#14268634—42no—
54#14268631—42no—
55C2×#11744751C2311yescharacteristic
56C2×#11744751C2312yes—
57C2×#11744751C2311yescharacteristic
58C4⋊Dic113C2311yescharacteristic
59#14268624—21no—
60#14268624—22no—
61C2×Q8×C113—21no—
62C2×Q8×C113C2311yescharacteristic
63C2×D4×C113D411yescharacteristic
64C22×C4×C113C2312yes—
65C4⋊C4×C113—22no—
66C4⋊C4×C113—21no—
67C4⋊C4×C113—21no—
68C4⋊C4×C113D411yescharacteristic
69C4⋊C4×C113C2311yescharacteristic
70C4⋊C4×C113C2312yes—
71C22⋊C4×C113—22no—
72C22⋊C4×C113—22no—
73C22⋊C4×C113D412yes—
74C42×C113—22no—
75C42×C113—21no—
76C42×C113C2311yescharacteristic
77C23×C113C2×D412yes—
78C22×D113—21no—
79C22×D113—22no—
80C22×D113—44no—
81C22×D113—44no—
82C22×D113—82no—
83Q8×C113—21no—
84Q8×C113—21no—
85Q8×C113—42no—
86D4×C113—22no—
87C2×C4×C113—22no—
88C2×C4×C113—22no—
89C2×C4×C113—21no—
90C2×C4×C113—21no—
91C2×C4×C113—22no—
92C2×C4×C113C2×D412yes—
93C2×C4×C113C2×D411yescharacteristic
94C2×C4×C113C2×D411yescharacteristic
95C2×C4×C113C2411yescharacteristic
96C22⋊D113—44no—
97C2×Dic113—21no—
98D452—22no—
99D452—21no—
100D452—22no—
101D452—44no—
102D452—42no—
103D452—81no—
104D452#4112yes—
105C4×D113—42no—
106#11744751—21no—
107#11744751—21no—
108#11744751—22no—
109C22×C113—22no—
110C22×C113—22no—
111C22×C113—22no—
112C22×C113C22×D411yescharacteristic
113C2×D113—22no—
114C2×D113—41no—
115C2×D113—41no—
116C2×D113—42no—
117C2×D113—42no—
118C2×D113—42no—
119C2×D113—88no—
120C4×C113—22no—
121C4×C113—21no—
122C4×C113—41no—
123C4×C113—42no—
124C4×C113—41no—
125C4×C113AGL(1,ℤ/8)12yes—
126C4×C113C2×#4111yescharacteristic
127C4×C113H(5,𝔽2)11yescharacteristic
128Dic113—41no—
129C2×C113—42no—
130C2×C113#54611yescharacteristic
131C2×C113C2×AGL(1,ℤ/8)11yescharacteristic
132C2×C113#57711yescharacteristic
133D113—42no—
134D113—82no—
135#3324—1131nomaximal, Sylow
136C113#332411yescharacteristic, complemented, Sylow
137#556D11311yescharacteristic, complemented
138D42—1131no—
139#486—1131no—
140#482—1132no—
141#474—1131no—
142#468—1132no—
143#460—1131no—
144#450—1132no—
145C4×SD16—1131no—
146#436—1131no—
147#418—1132no—
148C22×D4—1132no—
149C22×D4—2261no—
150C2×SD16—1132no—
151C2×SD16—1131no—
152C2×M16—1132no—
153C4⋊Q8C2×D11311yescharacteristic, complemented
154#127—1131no—
155#126—22no—
156#124—21no—
157C22⋊Q8C2×D11312yescomplemented
158#121—1132no—
159#121—2261no—
160#120—2262no—
161C4×Q8—21no—
162C4×Q8C2×D11311yescharacteristic, complemented
163C4×D4—1131no—
164C4×D4C2×D11311yescharacteristic
165#117C2×D11312yescomplemented
166#107—1132no—
167#106—1131no—
168C4⋊C8—1131no—
169#103—1131no—
170#102—1131no—
171#102—1132no—
172#102—1132no—
173#102—1132no—
174#102—1132no—
175C22⋊C8—1132no—
176C4×C8—1131no—
177C24—2262no—
178C2×Q8—21no—
179C2×Q8C22×D11311yescharacteristic
180C2×D4—1131no—
181C2×D4—1132no—
182C2×D4—4522no—
183C2×D4—4524no—
184C2×D4—2264no—
185C2×D4—2262no—
186C2×D4—2261no—
187C2×D4—2262no—
188C2×D4—2261no—
189C2×D4C22⋊D11311yescharacteristic
190C22×C4—2261no—
191C22×C4C22×D11312yes—
192SD16—4522no—
193SD16—2262no—
194M16—4522no—
195C2×C8—1131no—
196C2×C8—1131no—
197C2×C8—1132no—
198C4⋊C4—22no—
199C4⋊C4—21no—
200C4⋊C4—21no—
201C4⋊C4—1131no—
202C4⋊C4C22⋊D11311yescharacteristic
203C4⋊C4C22×D11311yescharacteristic
204C4⋊C4C22×D11312yes—
205C22⋊C4—22no—
206C22⋊C4—22no—
207C22⋊C4—2261no—
208C22⋊C4—2262no—
209C22⋊C4C22⋊D11312yes—
210C42—22no—
211C42—21no—
212C42C22×D11311yescharacteristic
213C23—9042no—
214C23—4524no—
215C23—4524no—
216C23—2262no—
217C23—2261no—
218C23C2×C22⋊D11312yes—
219Q8—21no—
220Q8—21no—
221Q8—42no—
222D4—22no—
223D4—1132no—
224D4—9041no—
225D4—4524no—
226D4—4524no—
227D4—4522no—
228D4—2262no—
229D4—2262no—
230D4—2261no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—22no—
236C2×C4—4522no—
237C2×C4—2261no—
238C2×C4C2×C22⋊D11312yes—
239C2×C4C2×C22⋊D11311yescharacteristic
240C2×C4C2×C22⋊D11311yescharacteristic
241C2×C4C23×D11311yescharacteristic
242C8—2261no—
243C8—2261no—
244C8—2262no—
245C22—22no—
246C22—22no—
247C22—22no—
248C22—9048no—
249C22—4522no—
250C22—4522no—
251C22—4522no—
252C22—4521no—
253C22—4521no—
254C22—2262no—
255C22C22×C22⋊D11311yescharacteristic
256C4—21no—
257C4—22no—
258C4—42no—
259C4—41no—
260C4—41no—
261C4—4521no—
262C4#1681364712yes—
263C4#41×D11311yescharacteristic
264C4#1681366811yescharacteristic
265C2—42no—
266C2—9042no—
267C2—4522no—
268C2#2029923611yescharacteristic
269C2C2×#1681364711yescharacteristic
270C2#2029940611yescharacteristic
subgroupG/H
centerC22C22×C22⋊D113
commutator, cosocleC2×C4×C113C24
FrattiniC2×C4C23×D113
Fitting#556×C113C2
socleC22×C113C22×D4
prime radicalD4×D452C2
hypercenter#556D113
hypocenterC113#3324
LCSG ▷ C2×C4×C113 ▷ C2×C113 ▷ C113
UCSC1 ◁ C22 ◁ C4×D4 ◁ #556

Complements

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

NH ≅ G/Ntype
2. #20298946C2semidirect
3. #20298922C2semidirect
4. #20298875C2semidirect
5. #20298846C2semidirect
6. #20298798C2semidirect
7. #20298748C2semidirect
8. C4×#14268634C2semidirect
9. #20298676C2semidirect
10. #20298602C2semidirect
11. #556×C113C2semidirect
15. C2×#14268634C22semidirect
16. C2×#14268631C22semidirect
22. #16813555C22semidirect
23. #16813553C22semidirect
24. C4⋊#11744751C22semidirect
25. #16813546C22semidirect
28. #16813542C22semidirect
29. #16813542C22semidirect
30. #16813542C22semidirect
31. C22⋊#11744751C22semidirect
32. C4×#11744751C22semidirect
33. C4⋊Q8×C113C22semidirect
36. C22⋊Q8×C113C22semidirect
38. C4×Q8×C113C22semidirect
40. #117×C113C22semidirect
136. C113#3324semidirect
137. #556D113semidirect
153. C4⋊Q8C2×D113semidirect
157. C22⋊Q8C2×D113semidirect
162. C4×Q8C2×D113semidirect
165. #117C2×D113semidirect

Maximal subgroups

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

subgroupindex
D4×D4522
#202989462
#202989222
#202988752
#202988462
#202987982
#202987482
C4×#142686342
#202986762
#202986022
#556×C1132
#3324113