Finite Group Databasebeta

← #28399843

Subgroups — #28399843

Order 14464.

784 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 784 conjugacy classes of subgroups — too many to draw. Only its 149 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×#16813667C211yesmaximal, characteristic, complemented
2C2×#16813650C211yesmaximal, characteristic, complemented
3C22×#14268632C211yesmaximal, characteristic, complemented
4#20299167C211yesmaximal, characteristic, complemented
5#20298744C214yesmaximal, complemented
6#20298733C214yesmaximal, complemented
7#20298592C211yesmaximal, characteristic, complemented
8#20298566C211yesmaximal, characteristic, complemented
9#418×C113C211yesmaximal, characteristic, complemented
10#16813667—24no—
11#16813667—24no—
12C2×#14268647C2211yescharacteristic
13C2×#14268647C2211yescharacteristic
14C2×#14268646—21no—
15C2×#14268646C2212yescomplemented
16C22×D452C2211yescharacteristic, complemented
17C22×C22⋊D113—22no—
18C2×D4×D113—21no—
19C2×D4×D113C2212yescomplemented
20#16813650—24no—
21C2×#14268636C2212yescomplemented
22C2×D904C2212yescomplemented
23C2×#14268632—24no—
24C2×#14268632C2214yescomplemented
25C22×#11744751C2211yescharacteristic, complemented
26#16813619—21no—
27#16813610—21no—
28#16813597C2214yescomplemented
29#16813592—22no—
30#16813545C2212yescomplemented
31#16813542C2212yescomplemented
32#16813534C2214yescomplemented
33C22×D4×C113C2211yescharacteristic
34C2×M16×C113C2211yescharacteristic, complemented
35#117×C113C2211yescharacteristic, complemented
36#102×C113C2214yescomplemented
37C23×D113—22no—
38C23×D113—21no—
39#14268647—22no—
40#14268647—22no—
41#14268647—22no—
42#14268647D414yescomplemented
43#14268646—28no—
44#14268646—24no—
45#14268646—24no—
46C2×Dic226D411yescharacteristic
47C2×Dic226C2311yescharacteristic
48C2×D452—22no—
49C2×D452—22no—
50C2×D452—21no—
51C2×D452—24no—
52C2×D452D411yescharacteristic
53C2×D452C2312yes—
54C2×D452C2311yescharacteristic
55C2×C22⋊D113—216no—
56C2×C22⋊D113—21no—
57C2×C22⋊D113—22no—
58C2×C22⋊D113—22no—
59C2×C22⋊D113D412yes—
60D4×D113—28no—
61D4×D113—24no—
62D4×D113—24no—
63C22×Dic113—22no—
64C2×C4×D113—21no—
65C2×C4×D113C2312yes—
66#14268636—24no—
67D904—24no—
68#14268632—28no—
69C2×#11744751—22no—
70C2×#11744751C2312yes—
71#14268624—24no—
72C24×C113—21no—
73C2×D4×C113—21no—
74C2×D4×C113—22no—
75C2×D4×C113D414yescomplemented
76C2×D4×C113C2312yes—
77C22×C4×C113C2311yescharacteristic
78M16×C113—22no—
79C2×C8×C113C2312yes—
80C4⋊C4×C113C2312yes—
81C22⋊C4×C113—21no—
82C42×C113—21no—
83C23×C113—22no—
84C23×C113—24no—
85C23×C113—82no—
86C23×C113C2×D411yescharacteristic
87C22×D113—21no—
88C22×D113—28no—
89C22×D113—22no—
90C22×D113—42no—
91C22×D113—84no—
92C22×D113—82no—
93C22×D113C2×D412yes—
94D4×C113—24no—
95D4×C113—22no—
96D4×C113C2×D414yes—
97C2×C4×C113—22no—
98C2×C4×C113C2×D411yescharacteristic
99C2×C4×C113C2×D412yes—
100C2×C4×C113C2×D412yes—
101C2×C4×C113C2411yescharacteristic
102C22⋊D113—28no—
103C22⋊D113—28no—
104C22⋊D113—24no—
105C22⋊D113—24no—
106C2×Dic113—28no—
107C2×Dic113—21no—
108C2×Dic113C2×D412yes—
109D452—22no—
110D452—21no—
111D452—22no—
112D452—22no—
113D452—42no—
114D452—84no—
115D452C2×D412yes—
116C4×D113—24no—
117C4×D113—22no—
118C4×D113—22no—
119Dic226—22no—
120Dic226—21no—
121Dic226C2×D412yes—
122C8×C113—22no—
123#11744751—22no—
124C22×C113—22no—
125C22×C113—24no—
126C22×C113—44no—
127C22×C113—84no—
128C22×C113#12012yes—
129C22×C113C22×D411yescharacteristic
130C2×D113—22no—
131C2×D113—22no—
132C2×D113—24no—
133C2×D113—42no—
134C2×D113—42no—
135C2×D113—44no—
136C2×D113—88no—
137C2×D113—84no—
138C4×C113—42no—
139C4×C113#12012yes—
140C4×C113C22×D411yescharacteristic
141C4×C113C22×D411yescharacteristic
142Dic113—22no—
143Dic113—22no—
144C2×C113—22no—
145C2×C113—44no—
146C2×C113C2×#12011yescharacteristic
147C2×C113#57612yes—
148D113—42no—
149D113—42no—
150D113—82no—
151#3008—1131nomaximal, Sylow
152C113#300811yescharacteristic, complemented, Sylow
153C2×H(5,𝔽2)—1131no—
154C2×AGL(1,ℤ/8)—1131no—
155C22×D8—1131no—
156#535—1131no—
157#449—1134no—
158#447—1134no—
159#418D11311yescharacteristic, complemented
160#416—1131no—
161#408—1131no—
162H(5,𝔽2)—2264no—
163H(5,𝔽2)—2264no—
164C2×#41—1131no—
165C2×#41—1132no—
166C2×#41—1131no—
167C2×#41—2261no—
168C22×D4—1132no—
169C22×D4—1131no—
170C22×D4—2261no—
171C22×D4—2262no—
172C22×D4C2×D11311yescharacteristic
173AGL(1,ℤ/8)—2264no—
174C2×SD16—1132no—
175C2×D8—1134no—
176C2×D8—1132no—
177C2×D8—2264no—
178C2×M16C2×D11311yescharacteristic, complemented
179C22×C8—1131no—
180#127—2261no—
181#124—2261no—
182#121—1134no—
183#120—2262no—
184#117C2×D11311yescharacteristic, complemented
185#103—1132no—
186#102—1132no—
187#102C2×D11314yescomplemented
188C22⋊C8—1134no—
189C24—21no—
190C24—2262no—
191C24—2261no—
192#41—1134no—
193#41—2264no—
194#41—2268no—
195#41—2264no—
196#41—2262no—
197#41—2262no—
198#41—2262no—
199C2×Q8—1131no—
200C2×Q8—1131no—
201C2×D4—21no—
202C2×D4—22no—
203C2×D4—1132no—
204C2×D4—1131no—
205C2×D4—1131no—
206C2×D4—1132no—
207C2×D4—2264no—
208C2×D4—2262no—
209C2×D4—2264no—
210C2×D4—2268no—
211C2×D4—2262no—
212C2×D4—22616no—
213C2×D4—2261no—
214C2×D4—2264no—
215C2×D4—2261no—
216C2×D4—2262no—
217C2×D4—2262no—
218C2×D4D45214yescomplemented
219C2×D4C22×D11312yes—
220C22×C4—1132no—
221C22×C4—2262no—
222C22×C4—2261no—
223C22×C4C22×D11311yescharacteristic
224SD16—2264no—
225D8—2264no—
226D8—2268no—
227M16—22no—
228C2×C8—1132no—
229C2×C8—2262no—
230C2×C8C22×D11312yes—
231C4⋊C4C22×D11312yes—
232C22⋊C4—21no—
233C22⋊C4—2264no—
234C42—21no—
235C23—22no—
236C23—24no—
237C23—82no—
238C23—1132no—
239C23—9042no—
240C23—9044no—
241C23—4522no—
242C23—2268no—
243C23—2262no—
244C23—2261no—
245C23D4×D11311yescharacteristic
246Q8—1132no—
247Q8—2262no—
248Q8—2261no—
249D4—24no—
250D4—22no—
251D4—1132no—
252D4—9044no—
253D4—4522no—
254D4—2268no—
255D4—2268no—
256D4—2264no—
257D4—2264no—
258D4—2262no—
259D4—2262no—
260D4—2262no—
261D4—2261no—
262D4C2×D45214yes—
263C2×C4—22no—
264C2×C4—1132no—
265C2×C4—2268no—
266C2×C4—2261no—
267C2×C4—2264no—
268C2×C4—2262no—
269C2×C4—2262no—
270C2×C4D4×D11311yescharacteristic
271C2×C4D4×D11312yes—
272C2×C4C2×D45212yes—
273C2×C4C23×D11311yescharacteristic
274C8—22no—
275C8—2262no—
276C22—22no—
277C22—24no—
278C22—44no—
279C22—84no—
280C22—9044no—
281C22—9048no—
282C22—4522no—
283C22—4524no—
284C22—4522no—
285C22—2262no—
286C22—2262no—
287C22—2264no—
288C22#1681359212yes—
289C22C2×D4×D11311yescharacteristic
290C4—42no—
291C4—2262no—
292C4—2262no—
293C4#1681359212yes—
294C4C2×D4×D11311yescharacteristic
295C4C22×D45211yescharacteristic
296C2—22no—
297C2—44no—
298C2—9042no—
299C2—4522no—
300C2—4522no—
301C2C2×#1681359211yescharacteristic
302C2#2029940212yes—
subgroupG/H
centerC22C2×D4×D113
commutator, cosocleC2×C4×C113C24
FrattiniC2×C4C23×D113
Fitting#418×C113C2
socleC22×C113C22×D4
hypercenter#418D113
hypocenterC113#3008
LCSG ▷ C2×C4×C113 ▷ C2×C113 ▷ C113
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
1. C2×#16813667C2semidirect
2. C2×#16813650C2semidirect
3. C22×#14268632C2semidirect
4. #20299167C2semidirect
5. #20298744C2semidirect
6. #20298733C2semidirect
7. #20298592C2semidirect
8. #20298566C2semidirect
9. #418×C113C2semidirect
15. C2×#14268646C22semidirect
16. C22×D452C22semidirect
19. C2×D4×D113C22semidirect
21. C2×#14268636C22semidirect
22. C2×D904C22semidirect
24. C2×#14268632C22semidirect
25. C22×#11744751C22semidirect
28. #16813597C22semidirect
30. #16813545C22semidirect
31. #16813542C22semidirect
32. #16813534C22semidirect
34. C2×M16×C113C22semidirect
35. #117×C113C22semidirect
36. #102×C113C22semidirect
42. #14268647D4semidirect
75. C2×D4×C113D4semidirect
152. C113#3008semidirect
159. #418D113semidirect
178. C2×M16C2×D113semidirect
184. #117C2×D113semidirect
187. #102C2×D113semidirect
218. C2×D4D452semidirect

Maximal subgroups

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

subgroupindex
C2×#168136672
C2×#168136502
C22×#142686322
#202991672
#202987442
#202987332
#202985922
#202985662
#418×C1132
#3008113