Finite Group Databasebeta

← #28402236

Subgroups — #28402236

Order 14464.

482 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 482 conjugacy classes of subgroups — too many to draw. Only its 133 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
1C2×#16813653C212yesmaximal, complemented
2#20299269C211yesmaximal, characteristic
3#20298972C211yesmaximal, characteristic, complemented
4#20298846C212yesmaximal, complemented
5#20298805C211yesmaximal, characteristic, complemented
6#20298798C211yesmaximal, characteristic, complemented
7#20298758C212yesmaximal, complemented
8#20298737C212yesmaximal, complemented
9C4×#14268637C211yesmaximal, characteristic, complemented
10#20298676C211yesmaximal, characteristic, complemented
11D4×Q8×C113C211yesmaximal, characteristic, complemented
12C2×#14268647C2212yes—
13#16813653—28no—
14C2×#14268637C2212yes—
15C2×#14268637C2211yescharacteristic
16C2×#14268634C2212yes—
17C2×#14268634C2212yes—
18C2×#14268634C2212yes—
19C2×#14268631C2212yes—
20#16813619C2211yescharacteristic
21#16813618—21no—
22#16813596C2212yes—
23#16813595—22no—
24C4×Dic226C2211yescharacteristic
25#16813584—21no—
26#16813555C2211yescharacteristic
27C4⋊#11744751C2211yescharacteristic
28#16813546C2212yes—
29#16813546C2211yescharacteristic
30#16813545C2211yescharacteristic
31#16813542C2212yes—
32C22⋊#11744751C2212yes—
33C4×#11744751C2211yescharacteristic
34C22×Q8×C113C2212yes—
35C4⋊Q8×C113—21no—
36C4⋊Q8×C113C2211yescharacteristic
37C22⋊Q8×C113—22no—
38C22⋊Q8×C113C2212yes—
39C4×Q8×C113C2211yescharacteristic
40C4×D4×C113—21no—
41C4×D4×C113C2211yescharacteristic
42#14268647—24no—
43#14268647—42no—
44C2×Dic226C2311yescharacteristic
45C2×D452—21no—
46C2×D452—21no—
47C2×D452C2312yes—
48C2×C22⋊D113—22no—
49C2×C22⋊D113—22no—
50C2×C4×D113—22no—
51#14268637—24no—
52#14268637D414yescomplemented
53#14268634—24no—
54#14268634—24no—
55#14268634—42no—
56#14268631—24no—
57C2×#11744751C2312yes—
58C2×#11744751C2311yescharacteristic
59C2×#11744751C2311yescharacteristic
60C4⋊Dic113C2311yescharacteristic
61#14268626—21no—
62#14268624—22no—
63C4×Dic113—21no—
64C2×Q8×C113—22no—
65C2×Q8×C113—24no—
66C2×Q8×C113—42no—
67C2×Q8×C113C2312yes—
68C2×Q8×C113C2311yescharacteristic
69C2×D4×C113D411yescharacteristic
70C22×C4×C113—22no—
71C22×C4×C113C2312yes—
72C4⋊C4×C113—21no—
73C4⋊C4×C113—21no—
74C4⋊C4×C113—21no—
75C4⋊C4×C113—22no—
76C4⋊C4×C113D411yescharacteristic
77C4⋊C4×C113C2311yescharacteristic
78C22⋊C4×C113—22no—
79C22⋊C4×C113D412yes—
80C42×C113—21no—
81C42×C113C2311yescharacteristic
82C23×C113C2×D412yes—
83C22×D113—22no—
84Q8×C113—21no—
85Q8×C113—22no—
86Q8×C113—22no—
87Q8×C113—44no—
88Q8×C113C2×D412yes—
89D4×C113—22no—
90C2×C4×C113—21no—
91C2×C4×C113—22no—
92C2×C4×C113—22no—
93C2×C4×C113—21no—
94C2×C4×C113—22no—
95C2×C4×C113—44no—
96C2×C4×C113C2×D411yescharacteristic
97C2×C4×C113C2×D412yes—
98C2×C4×C113C2×D411yescharacteristic
99C2×C4×C113C2411yescharacteristic
100C22⋊D113—44no—
101C22⋊D113—82no—
102C2×Dic113—21no—
103C2×Dic113—21no—
104D452—22no—
105D452—22no—
106D452—42no—
107D452—81no—
108C4×D113—44no—
109Dic226—21no—
110Dic226C2×D412yes—
111#11744751—22no—
112#11744751—21no—
113#11744751C2×D412yes—
114C22×C113—22no—
115C22×C113—22no—
116C22×C113—22no—
117C22×C113C22×D411yescharacteristic
118C2×D113—42no—
119C2×D113—42no—
120C2×D113—42no—
121C4×C113—22no—
122C4×C113—22no—
123C4×C113—21no—
124C4×C113—42no—
125C4×C113—41no—
126C4×C113#13712yescomplemented
127C4×C113C22×D411yescharacteristic
128C4×C113H(5,𝔽2)11yescharacteristic
129Dic113—22no—
130Dic113—41no—
131C2×C113—42no—
132C2×C113D4211yescharacteristic
133C2×C113C2×#13711yescharacteristic
134C2×C113#57711yescharacteristic
135D113—82no—
136#3278—1131nomaximal, Sylow
137C113#327811yescharacteristic, complemented, Sylow
138C2×#137—1132no—
139#550—1131no—
140D4×Q8D11311yescharacteristic, complemented
141#492—1131no—
142#468—1132no—
143#462—1131no—
144#460—1131no—
145#452—1132no—
146#448—1132no—
147C4×Q16—1131no—
148#436—1131no—
149C2×#41—1132no—
150C22×Q8C2×D11312yes—
151#137—2268no—
152C2×Q16—1131no—
153C2×Q16—1132no—
154C2×SD16—1132no—
155C2×SD16—1132no—
156C2×SD16—1132no—
157C2×M16—1132no—
158C4⋊Q8—21no—
159C4⋊Q8C2×D11311yescharacteristic
160#127—1131no—
161#127—2261no—
162C22⋊Q8—22no—
163C22⋊Q8C2×D11312yes—
164#121—1132no—
165#121—2262no—
166C4×Q8—1131no—
167C4×Q8C2×D11311yescharacteristic
168C4×D4—21no—
169C4×D4—2261no—
170C4×D4C2×D11311yescharacteristic
171#107—1131no—
172C4⋊C8—1131no—
173#103—1131no—
174#103—1132no—
175#103—1131no—
176#102—1132no—
177C22⋊C8—1132no—
178C4×C8—1131no—
179#41—4522no—
180#41—2264no—
181C2×Q8—22no—
182C2×Q8—24no—
183C2×Q8—42no—
184C2×Q8—1131no—
185C2×Q8C22×D11312yes—
186C2×Q8C22×D11311yescharacteristic
187C2×D4—1132no—
188C2×D4—2262no—
189C2×D4—2262no—
190C2×D4—2261no—
191C2×D4—2261no—
192C2×D4C22⋊D11311yescharacteristic
193C22×C4—22no—
194C22×C4—2262no—
195C22×C4C22×D11312yes—
196Q16—1134no—
197Q16—2264no—
198SD16—4522no—
199SD16—2264no—
200SD16—2264no—
201M16—2264no—
202C2×C8—1131no—
203C2×C8—1131no—
204C2×C8—1132no—
205C4⋊C4—21no—
206C4⋊C4—21no—
207C4⋊C4—21no—
208C4⋊C4—22no—
209C4⋊C4—1131no—
210C4⋊C4—2261no—
211C4⋊C4C22⋊D11311yescharacteristic
212C4⋊C4C22×D11311yescharacteristic
213C22⋊C4—22no—
214C22⋊C4—2262no—
215C22⋊C4C22⋊D11312yes—
216C42—21no—
217C42—2261no—
218C42C22×D11311yescharacteristic
219C23—2262no—
220C23C2×C22⋊D11312yes—
221Q8—21no—
222Q8—22no—
223Q8—22no—
224Q8—44no—
225Q8—1132no—
226Q8—2261no—
227Q8D4×D11312yes—
228D4—22no—
229D4—9041no—
230D4—9042no—
231D4—4524no—
232D4—4522no—
233D4—2262no—
234D4—2262no—
235C2×C4—21no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—21no—
239C2×C4—22no—
240C2×C4—44no—
241C2×C4—4524no—
242C2×C4—2261no—
243C2×C4—2261no—
244C2×C4C2×C22⋊D11311yescharacteristic
245C2×C4C2×C22⋊D11312yes—
246C2×C4C2×C22⋊D11311yescharacteristic
247C2×C4C23×D11311yescharacteristic
248C8—1132no—
249C8—2262no—
250C8—2261no—
251C22—22no—
252C22—22no—
253C22—22no—
254C22—4522no—
255C22—4522no—
256C22—4522no—
257C22C22×C22⋊D11311yescharacteristic
258C4—21no—
259C4—22no—
260C4—22no—
261C4—41no—
262C4—42no—
263C4—4521no—
264C4—2262no—
265C4#1681365312yescomplemented
266C4C2×D4×D11311yescharacteristic
267C4#1681366711yescharacteristic
268C2—42no—
269C2—9042no—
270C2D4×C22⋊D11311yescharacteristic
271C2C2×#1681365311yescharacteristic
272C2#2029940611yescharacteristic
subgroupG/H
centerC22C22×C22⋊D113
commutator, cosocleC2×C4×C113C24
FrattiniC2×C4C23×D113
FittingD4×Q8×C113C2
socleC22×C113C22×D4
prime radical#20299269C2
hypercenterD4×Q8D113
hypocenterC113#3278
LCSG ▷ C2×C4×C113 ▷ C2×C113 ▷ C113
UCSC1 ◁ C22 ◁ C4×D4 ◁ D4×Q8

Complements

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

NH ≅ G/Ntype
1. C2×#16813653C2semidirect
3. #20298972C2semidirect
4. #20298846C2semidirect
5. #20298805C2semidirect
6. #20298798C2semidirect
7. #20298758C2semidirect
8. #20298737C2semidirect
9. C4×#14268637C2semidirect
10. #20298676C2semidirect
11. D4×Q8×C113C2semidirect
52. #14268637D4semidirect
126. C4×C113#137semidirect
137. C113#3278semidirect
140. D4×Q8D113semidirect
265. C4#16813653semidirect

Maximal subgroups

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

subgroupindex
C2×#168136532
#202992692
#202989722
#202988462
#202988052
#202987982
#202987582
#202987372
C4×#142686372
#202986762
D4×Q8×C1132
#3278113