Finite Group Databasebeta

← #28402220

Subgroups — #28402220

Order 14464.

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

normal subgroups only

This group has 472 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#20299253C211yesmaximal, characteristic
3#20298977C211yesmaximal, characteristic, complemented
4#20298854C212yesmaximal, complemented
5#20298808C211yesmaximal, characteristic, complemented
6#20298800C211yesmaximal, characteristic, complemented
7#20298758C212yesmaximal, complemented
8#20298754C212yesmaximal, complemented
9C4×#14268634C211yesmaximal, characteristic, complemented
10#20298676C211yesmaximal, characteristic, complemented
11D4×Q8×C113C211yesmaximal, characteristic, complemented
12C2×#14268647C2212yes—
13#16813653—28no—
14C2×#14268637C2212yes—
15C2×#14268637C2212yes—
16C2×#14268637C2212yes—
17C2×#14268634C2212yes—
18C2×#14268634C2211yescharacteristic
19C2×#14268631C2212yes—
20#16813622C2211yescharacteristic
21#16813604—22no—
22#16813601C2212yes—
23#16813593—21no—
24C4×D452C2211yescharacteristic
25C4⋊C4×D113—21no—
26#16813553C2211yescharacteristic
27C4⋊#11744751C2211yescharacteristic
28#16813546C2211yescharacteristic
29#16813546C2212yes—
30#16813545C2212yes—
31#16813542C2211yescharacteristic
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×Dic226C2312yes—
45C2×D452C2311yescharacteristic
46C2×C22⋊D113—22no—
47C2×C4×D113—21no—
48C2×C4×D113—22no—
49#14268637—24no—
50#14268637—24no—
51#14268637—42no—
52#14268634—24no—
53#14268634D414yescomplemented
54#14268631—24no—
55C2×#11744751C2312yes—
56C2×#11744751C2311yescharacteristic
57C2×#11744751C2311yescharacteristic
58C4⋊Dic113—21no—
59C4⋊Dic113—21no—
60C4⋊Dic113C2311yescharacteristic
61#14268626—22no—
62C22⋊Dic113—22no—
63#14268624—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—21no—
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—
101C2×Dic113—21no—
102C2×Dic113—22no—
103D452—21no—
104D452C2×D412yes—
105C4×D113—44no—
106C4×D113—42no—
107Dic226—22no—
108Dic226—22no—
109#11744751—22no—
110#11744751—21no—
111#11744751C2×D412yes—
112C22×C113—22no—
113C22×C113—22no—
114C22×C113—22no—
115C22×C113C22×D411yescharacteristic
116C2×D113—22no—
117C2×D113—41no—
118C2×D113—41no—
119C4×C113—22no—
120C4×C113—22no—
121C4×C113—21no—
122C4×C113—42no—
123C4×C113—41no—
124C4×C113#13712yescomplemented
125C4×C113C22×D411yescharacteristic
126C4×C113H(5,𝔽2)11yescharacteristic
127Dic113—42no—
128Dic113—41no—
129C2×C113—42no—
130C2×C113D4211yescharacteristic
131C2×C113C2×#13711yescharacteristic
132C2×C113#57811yescharacteristic
133D113—42no—
134#3276—1131nomaximal, Sylow
135C113#327611yescharacteristic, complemented, Sylow
136C2×#137—1132no—
137D4×Q8D11311yescharacteristic, complemented
138#547—1131no—
139#494—1131no—
140#470—1132no—
141#462—1131no—
142#461—1131no—
143#452—1132no—
144#451—1132no—
145C4×SD16—1131no—
146#436—1131no—
147C2×#41—1132no—
148C22×Q8C2×D11312yes—
149#137—2268no—
150C2×Q16—1132no—
151C2×Q16—1132no—
152C2×Q16—1132no—
153C2×SD16—1131no—
154C2×SD16—1132no—
155C2×M16—1132no—
156C4⋊Q8—21no—
157C4⋊Q8—1131no—
158C4⋊Q8C2×D11311yescharacteristic
159#123—2262no—
160C22⋊Q8—22no—
161C22⋊Q8—1132no—
162C22⋊Q8C2×D11312yes—
163#121—2261no—
164C4×Q8C2×D11311yescharacteristic
165C4×D4—21no—
166C4×D4—1131no—
167C4×D4C2×D11311yescharacteristic
168C2×C4⋊C4—2261no—
169#106—1131no—
170C4⋊C8—1131no—
171#103—1131no—
172#103—1132no—
173#103—1132no—
174#102—1131no—
175C22⋊C8—1132no—
176C4×C8—1131no—
177#41—4522no—
178#41—2264no—
179C2×Q8—22no—
180C2×Q8—24no—
181C2×Q8—42no—
182C2×Q8—1132no—
183C2×Q8C22×D11312yes—
184C2×Q8C22×D11311yescharacteristic
185C2×D4—1131no—
186C2×D4—2262no—
187C2×D4C22⋊D11311yescharacteristic
188C22×C4—22no—
189C22×C4—2261no—
190C22×C4—2262no—
191C22×C4C22×D11312yes—
192Q16—4522no—
193Q16—2264no—
194Q16—2264no—
195SD16—1134no—
196SD16—2264no—
197M16—2264no—
198C2×C8—1131no—
199C2×C8—1131no—
200C2×C8—1132no—
201C4⋊C4—21no—
202C4⋊C4—21no—
203C4⋊C4—21no—
204C4⋊C4—22no—
205C4⋊C4—1131no—
206C4⋊C4—2261no—
207C4⋊C4—2262no—
208C4⋊C4—2261no—
209C4⋊C4C22⋊D11311yescharacteristic
210C4⋊C4C22×D11311yescharacteristic
211C22⋊C4—22no—
212C22⋊C4—2262no—
213C22⋊C4—2261no—
214C22⋊C4C22⋊D11312yes—
215C42—21no—
216C42C22×D11311yescharacteristic
217C23—2261no—
218C23C2×C22⋊D11312yes—
219Q8—21no—
220Q8—22no—
221Q8—22no—
222Q8—44no—
223Q8—2262no—
224Q8—2262no—
225Q8D4×D11312yes—
226D4—22no—
227D4—1132no—
228D4—4524no—
229D4—2261no—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—44no—
236C2×C4—4524no—
237C2×C4—4522no—
238C2×C4—2261no—
239C2×C4—2262no—
240C2×C4C2×C22⋊D11311yescharacteristic
241C2×C4C2×C22⋊D11312yes—
242C2×C4C2×C22⋊D11311yescharacteristic
243C2×C4C23×D11311yescharacteristic
244C8—1132no—
245C8—2262no—
246C8—2261no—
247C22—22no—
248C22—22no—
249C22—22no—
250C22—4521no—
251C22—4521no—
252C22—2262no—
253C22C22×C22⋊D11311yescharacteristic
254C4—21no—
255C4—22no—
256C4—22no—
257C4—41no—
258C4—42no—
259C4—4521no—
260C4—4522no—
261C4#1681365312yescomplemented
262C4C2×D4×D11311yescharacteristic
263C4#1681366711yescharacteristic
264C2—42no—
265C2—4522no—
266C2D4×C22⋊D11311yescharacteristic
267C2C2×#1681365311yescharacteristic
268C2#2029941311yescharacteristic
subgroupG/H
centerC22C22×C22⋊D113
commutator, cosocleC2×C4×C113C24
FrattiniC2×C4C23×D113
FittingD4×Q8×C113C2
socleC22×C113C22×D4
prime radical#20299253C2
hypercenterD4×Q8D113
hypocenterC113#3276
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. #20298977C2semidirect
4. #20298854C2semidirect
5. #20298808C2semidirect
6. #20298800C2semidirect
7. #20298758C2semidirect
8. #20298754C2semidirect
9. C4×#14268634C2semidirect
10. #20298676C2semidirect
11. D4×Q8×C113C2semidirect
53. #14268634D4semidirect
124. C4×C113#137semidirect
135. C113#3276semidirect
137. D4×Q8D113semidirect
261. C4#16813653semidirect

Maximal subgroups

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

subgroupindex
C2×#168136532
#202992532
#202989772
#202988542
#202988082
#202988002
#202987582
#202987542
C4×#142686342
#202986762
D4×Q8×C1132
#3276113