Finite Group Databasebeta

← #28401378

Subgroups — #28401378

Order 14464.

462 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 115 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×#16813598C211yesmaximal, characteristic, complemented
2#121×D113C211yesmaximal, characteristic
3#20298932C212yesmaximal, complemented
4#20298912C211yesmaximal, characteristic, complemented
5#20298909C211yesmaximal, characteristic, complemented
6#20298847C212yesmaximal, complemented
7#20298841C212yesmaximal, complemented
8#20298747C212yesmaximal, complemented
9#20298737C211yesmaximal, characteristic, complemented
10#20298560C211yesmaximal, characteristic, complemented
11#448×C113C211yesmaximal, characteristic, complemented
12C2×D4×D113—21no—
13C2×D4×D113C2211yescharacteristic
14C22×C4×D113C2211yescharacteristic
15C2×#14268634C2212yescomplemented
16C2×#14268630C2212yescomplemented
17#16813598—21no—
18#16813598—21no—
19#16813598C2212yes—
20#16813598C2212yescomplemented
21#16813597C2211yescharacteristic
22#16813596C2211yescharacteristic
23#16813595—21no—
24#16813594—21no—
25#16813593C2211yescharacteristic
26C2×C4⋊Dic113C2211yescharacteristic, complemented
27C2×#14268626—21no—
28C4⋊C4×D113C2211yescharacteristic
29C2×#14268624—21no—
30C22⋊C4×D113—21no—
31#16813556C2212yescomplemented
32#16813555C2212yescomplemented
33#16813546C2212yescomplemented
34#16813544C2212yescomplemented
35#16813543C2212yescomplemented
36C22⋊#11744751C2211yescharacteristic, complemented
37#16813534C2212yes—
38C22×Q8×C113C2211yescharacteristic, complemented
39C2×SD16×C113C2212yescomplemented
40#121×C113C2211yescharacteristic
41#103×C113C2212yescomplemented
42C22⋊C8×C113C2211yescharacteristic, complemented
43C23×D113—21no—
44C23×D113D411yescharacteristic
45C2×D452—21no—
46C2×D452C2311yescharacteristic
47C2×C22⋊D113—21no—
48C2×C22⋊D113—21no—
49C2×C22⋊D113—21no—
50D4×D113—42no—
51D4×D113—44no—
52C22×Dic113—21no—
53C22×Dic113D411yescharacteristic
54C2×C4×D113—21no—
55C2×C4×D113—42no—
56C2×C4×D113D411yescharacteristic
57C2×C4×D113D411yescharacteristic
58C2×C4×D113C2311yescharacteristic
59C2×C4×D113C2311yescharacteristic
60#14268634—42no—
61#14268630—42no—
62C2×#11744751C2312yes—
63C4⋊Dic113—21no—
64C4⋊Dic113C2311yescharacteristic
65C4⋊Dic113C2312yes—
66#14268626—21no—
67#14268626—22no—
68#14268626—22no—
69C22⋊Dic113—21no—
70#14268624—22no—
71#14268624—22no—
72#14268624—21no—
73#14268624—21no—
74C2×Q8×C113—21no—
75C2×Q8×C113—42no—
76C2×Q8×C113C2312yes—
77C2×D4×C113—21no—
78C2×D4×C113C2311yescharacteristic
79C22×C4×C113—21no—
80C22×C4×C113C2311yescharacteristic
81SD16×C113—42no—
82C2×C8×C113C2312yes—
83C4⋊C4×C113C2311yescharacteristic
84C22⋊C4×C113—21no—
85C23×C113—21no—
86C23×C113C2×D411yescharacteristic
87C22×D113—22no—
88C22×D113—22no—
89C22×D113—21no—
90C22×D113—41no—
91C22×D113—42no—
92C22×D113—42no—
93C22×D113—42no—
94C22×D113C2×D411yescharacteristic
95C22×D113C2×D411yescharacteristic
96Q8×C113—22no—
97Q8×C113—22no—
98Q8×C113—42no—
99D4×C113—22no—
100D4×C113—42no—
101C2×C4×C113—22no—
102C2×C4×C113—21no—
103C2×C4×C113—22no—
104C2×C4×C113—42no—
105C2×C4×C113C2×D411yescharacteristic
106C2×C4×C113C2411yescharacteristic
107C22⋊D113—42no—
108C22⋊D113—42no—
109C22⋊D113—42no—
110C2×Dic113—22no—
111C2×Dic113—21no—
112C2×Dic113—41no—
113C2×Dic113—42no—
114C2×Dic113C2×D411yescharacteristic
115C2×Dic113C2×D411yescharacteristic
116D452—22no—
117D452—42no—
118C4×D113—22no—
119C4×D113—22no—
120C4×D113—41no—
121C4×D113—41no—
122C4×D113—42no—
123C8×C113—22no—
124#11744751—22no—
125C22×C113—21no—
126C22×C113—21no—
127C22×C113—41no—
128C22×C113—41no—
129C22×C113—41no—
130C22×C113AGL(1,ℤ/8)12yes—
131C22×C113C22×D411yescharacteristic
132C2×D113—21no—
133C2×D113—21no—
134C2×D113—21no—
135C2×D113—21no—
136C2×D113—21no—
137C2×D113—44no—
138C2×D113—41no—
139C2×D113—41no—
140C2×D113—41no—
141C2×D113—84no—
142C2×D113#13712yes—
143C4×C113—21no—
144C4×C113—42no—
145C4×C113—41no—
146C4×C113H(5,𝔽2)12yes—
147Dic113—21no—
148Dic113—21no—
149Dic113—41no—
150Dic113—42no—
151C2×C113—22no—
152C2×C113—81no—
153C2×C113#53411yescharacteristic
154C2×C113C2×AGL(1,ℤ/8)11yescharacteristic
155C2×C113C2×#13711yescharacteristic
156D113—22no—
157D113—41no—
158D113—81no—
159#3193—1131nomaximal, Sylow
160C113#319311yescharacteristic, complemented, Sylow
161C2×C22⋊Q8—1131no—
162C2×#121—1131no—
163#484—1132no—
164#480—1131no—
165#480—1131no—
166#468—1132no—
167#468—1132no—
168#450—1132no—
169#448—1131no—
170#448D11311yescharacteristic, complemented
171#407—1131no—
172C22×Q8C2×D11311yescharacteristic, complemented
173C22×D4—1131no—
174C22×D4—2261no—
175C23×C4—1131no—
176C2×SD16—1132no—
177C2×SD16C2×D11312yescomplemented
178C2×M16—1132no—
179C22⋊Q8—1132no—
180C22⋊Q8—1132no—
181C22⋊Q8—2261no—
182C22⋊Q8—2261no—
183#121—1131no—
184#121—1131no—
185#121—1131no—
186#121—2261no—
187#121—2261no—
188#121C2×D11311yescharacteristic
189C2×C4⋊C4—1131no—
190C2×C4⋊C4—1131no—
191C2×C4⋊C4—2261no—
192C2×C22⋊C4—2261no—
193C2×C22⋊C4—2261no—
194#107—1132no—
195#107—1132no—
196#103—1132no—
197#103C2×D11312yescomplemented
198#102—1132no—
199#102—1132no—
200C22⋊C8—1131no—
201C22⋊C8—1132no—
202C22⋊C8C2×D11311yescharacteristic, complemented
203C24—1131no—
204C24—2261no—
205C2×Q8—21no—
206C2×Q8—42no—
207C2×Q8C22×D11312yes—
208C2×D4—21no—
209C2×D4—1131no—
210C2×D4—4524no—
211C2×D4—4522no—
212C2×D4—2261no—
213C2×D4—2261no—
214C2×D4—2261no—
215C2×D4—2261no—
216C2×D4C22×D11311yescharacteristic
217C22×C4—21no—
218C22×C4—1131no—
219C22×C4—1131no—
220C22×C4—1131no—
221C22×C4—1131no—
222C22×C4—1131no—
223C22×C4—4522no—
224C22×C4—2261no—
225C22×C4—2261no—
226C22×C4C22×D11311yescharacteristic
227SD16—42no—
228SD16—4522no—
229M16—4522no—
230C2×C8—1132no—
231C2×C8C22×D11312yes—
232C4⋊C4—1132no—
233C4⋊C4—1131no—
234C4⋊C4—2261no—
235C4⋊C4—2262no—
236C4⋊C4—2262no—
237C4⋊C4—2261no—
238C4⋊C4C22×D11311yescharacteristic
239C22⋊C4—21no—
240C22⋊C4—2261no—
241C22⋊C4—2262no—
242C22⋊C4—2262no—
243C22⋊C4—2261no—
244C22⋊C4—2261no—
245C23—21no—
246C23—1131no—
247C23—1131no—
248C23—4521no—
249C23—4522no—
250C23—4522no—
251C23—4522no—
252C23—2262no—
253C23—2261no—
254C23—2262no—
255C23D4×D11311yescharacteristic
256Q8—22no—
257Q8—22no—
258Q8—42no—
259D4—22no—
260D4—42no—
261D4—4522no—
262D4—4522no—
263D4—4522no—
264D4—4522no—
265D4—2262no—
266C2×C4—22no—
267C2×C4—21no—
268C2×C4—22no—
269C2×C4—42no—
270C2×C4—1131no—
271C2×C4—1131no—
272C2×C4—4521no—
273C2×C4—4521no—
274C2×C4—4522no—
275C2×C4—4522no—
276C2×C4—4521no—
277C2×C4—2262no—
278C2×C4—2261no—
279C2×C4—2262no—
280C2×C4—2262no—
281C2×C4D4×D11311yescharacteristic
282C2×C4C23×D11311yescharacteristic
283C8—22no—
284C8—2262no—
285C22—21no—
286C22—21no—
287C22—41no—
288C22—41no—
289C22—41no—
290C22—1132no—
291C22—9044no—
292C22—4524no—
293C22—4521no—
294C22—4521no—
295C22—4521no—
296C22—2261no—
297C22—2261no—
298C22—2261no—
299C22—2261no—
300C22—2261no—
301C22#1681364912yes—
302C22C2×D4×D11311yescharacteristic
303C4—21no—
304C4—41no—
305C4—42no—
306C4—4521no—
307C4—4522no—
308C4—2261no—
309C4—2261no—
310C4#1681366712yes—
311C2—22no—
312C2—81no—
313C2—9041no—
314C2—4521no—
315C2—2262no—
316C2#2029915811yescharacteristic
317C2C2×#1681364911yescharacteristic
318C2#137×D11311yescharacteristic
subgroupG/H
centerC22C2×D4×D113
commutator, cosocleC2×C4×C113C24
FrattiniC2×C4C23×D113
Fitting#448×C113C2
socleC22×C113C22×D4
prime radical#121×D113C2
hypercenter#448D113
hypocenterC113#3193
LCSG ▷ C2×C4×C113 ▷ C2×C113 ▷ C113
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
1. C2×#16813598C2semidirect
3. #20298932C2semidirect
4. #20298912C2semidirect
5. #20298909C2semidirect
6. #20298847C2semidirect
7. #20298841C2semidirect
8. #20298747C2semidirect
9. #20298737C2semidirect
10. #20298560C2semidirect
11. #448×C113C2semidirect
15. C2×#14268634C22semidirect
16. C2×#14268630C22semidirect
20. #16813598C22semidirect
26. C2×C4⋊Dic113C22semidirect
31. #16813556C22semidirect
32. #16813555C22semidirect
33. #16813546C22semidirect
34. #16813544C22semidirect
35. #16813543C22semidirect
36. C22⋊#11744751C22semidirect
38. C22×Q8×C113C22semidirect
39. C2×SD16×C113C22semidirect
41. #103×C113C22semidirect
42. C22⋊C8×C113C22semidirect
160. C113#3193semidirect
170. #448D113semidirect
172. C22×Q8C2×D113semidirect
177. C2×SD16C2×D113semidirect
197. #103C2×D113semidirect
202. C22⋊C8C2×D113semidirect

Maximal subgroups

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

subgroupindex
C2×#168135982
#121×D1132
#202989322
#202989122
#202989092
#202988472
#202988412
#202987472
#202987372
#202985602
#448×C1132
#3193113