Finite Group Databasebeta

← #28400205

Subgroups — #28400205

Order 14464.

496 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 143 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×#16813655C211yesmaximal, characteristic, complemented
2C2×M16×D113C211yesmaximal, characteristic, complemented
3#20299110C211yesmaximal, characteristic
4C2×#16813602C211yesmaximal, characteristic, complemented
5#20298857C212yesmaximal, complemented
6#20298845C212yesmaximal, complemented
7#20298840C212yesmaximal, complemented
8#20298828C212yesmaximal, complemented
9#20298606C211yesmaximal, characteristic, complemented
10#20298597C211yesmaximal, characteristic, complemented
11#428×C113C211yesmaximal, characteristic, complemented
12C2×#14268647C2211yescharacteristic
13C22×Dic226C2211yescharacteristic
14C22×C4×D113C2211yescharacteristic
15#16813655—24no—
16C2×Dic452C2212yescomplemented
17C2×#14268636C2212yescomplemented
18C2×#14268631C2211yescharacteristic
19C2×#14268630C2212yes—
20M16×D113—24no—
21C2×C8×D113C2212yes—
22#16813606—21no—
23#16813602—21no—
24#16813602—21no—
25#16813602C2212yes—
26#16813602C2211yescharacteristic
27#16813602C2212yes—
28#16813602C2211yescharacteristic
29#16813597C2211yescharacteristic
30#16813597C2211yescharacteristic
31#16813592—21no—
32C4×D452—21no—
33C4×D452—21no—
34C2×C4⋊Dic113—21no—
35C2×#14268624—21no—
36#16813545C2212yescomplemented
37#16813545C2212yescomplemented
38#16813545C2212yescomplemented
39#16813542C2212yescomplemented
40C2×M16×C113C2211yescharacteristic
41#117×C113C2211yescharacteristic
42C2×C4⋊C4×C113C2211yescharacteristic
43#107×C113C2212yescomplemented
44#106×C113C2212yescomplemented
45C23×D113D411yescharacteristic
46#14268647—22no—
47#14268647—22no—
48C2×Dic226—22no—
49C2×Dic226—22no—
50C2×Dic226—21no—
51C2×Dic226C2312yes—
52C2×Dic226C2311yescharacteristic
53C2×D452—21no—
54C2×D452—21no—
55C2×D452C2311yescharacteristic
56C2×C22⋊D113—21no—
57C22×Dic113—21no—
58C22×Dic113D411yescharacteristic
59C2×C4×D113—22no—
60C2×C4×D113—22no—
61C2×C4×D113—21no—
62C2×C4×D113D411yescharacteristic
63C2×C4×D113D411yescharacteristic
64C2×C4×D113C2311yescharacteristic
65C2×C4×D113C2311yescharacteristic
66Dic452—24no—
67#14268636—24no—
68#14268631—22no—
69#14268630—24no—
70C2×#11744751C2312yes—
71C8×D113—24no—
72C4⋊Dic113—22no—
73C4⋊Dic113—22no—
74C4⋊Dic113—21no—
75C4⋊Dic113—21no—
76#14268624—22no—
77#14268624—22no—
78#14268624—21no—
79#14268624—21no—
80#14268624—21no—
81#14268624—21no—
82C22×C4×C113—21no—
83C22×C4×C113C2311yescharacteristic
84M16×C113D414yescomplemented
85C2×C8×C113C2312yes—
86C4⋊C4×C113—21no—
87C4⋊C4×C113C2311yescharacteristic
88C4⋊C4×C113C2311yescharacteristic
89C4⋊C4×C113C2312yes—
90C22⋊C4×C113—21no—
91C42×C113—21no—
92C23×C113C2×D411yescharacteristic
93C22×D113—22no—
94C22×D113—22no—
95C22×D113—21no—
96C22×D113—41no—
97C22×D113C2×D411yescharacteristic
98C22×D113C2×D411yescharacteristic
99C2×C4×C113—21no—
100C2×C4×C113—21no—
101C2×C4×C113—22no—
102C2×C4×C113—42no—
103C2×C4×C113C2×D411yescharacteristic
104C2×C4×C113C2×D412yes—
105C2×C4×C113#4112yes—
106C2×C4×C113C2411yescharacteristic
107C22⋊D113—42no—
108C2×Dic113—22no—
109C2×Dic113—21no—
110C2×Dic113—22no—
111C2×Dic113—42no—
112C2×Dic113C2×D411yescharacteristic
113C2×Dic113C2×D411yescharacteristic
114D452—21no—
115D452—21no—
116D452—42no—
117D452—42no—
118C4×D113—22no—
119C4×D113—22no—
120C4×D113—21no—
121C4×D113—21no—
122C4×D113—21no—
123C4×D113—21no—
124C4×D113—42no—
125Dic226—22no—
126Dic226—21no—
127Dic226—22no—
128Dic226—21no—
129Dic226—42no—
130C8×C113C2×D414yes—
131#11744751—22no—
132C22×C113—21no—
133C22×C113—21no—
134C22×C113#12112yes—
135C22×C113C22×D411yescharacteristic
136C2×D113—21no—
137C2×D113—21no—
138C2×D113—21no—
139C2×D113—21no—
140C2×D113—21no—
141C2×D113—44no—
142C2×D113—41no—
143C2×D113—41no—
144C2×D113—41no—
145C2×D113#13712yescomplemented
146C4×C113—41no—
147C4×C113—41no—
148C4×C113—42no—
149C4×C113#12112yes—
150C4×C113C22×D411yescharacteristic
151C4×C113C2×#4111yescharacteristic
152Dic113—21no—
153Dic113—21no—
154Dic113—42no—
155Dic113—41no—
156C2×C113—22no—
157C2×C113C2×#12111yescharacteristic
158C2×C113C2×#13711yescharacteristic
159C2×C113#57511yescharacteristic
160D113—22no—
161D113—41no—
162D113—81no—
163#3056—1131nomaximal, Sylow
164C113#305611yescharacteristic, complemented, Sylow
165C2×#137—1131no—
166C22×M16—1131no—
167#525—1131no—
168C2×C22⋊Q8—1131no—
169#470—1132no—
170#468—1132no—
171#467—1132no—
172#465—1132no—
173#428D11311yescharacteristic, complemented
174#419—1131no—
175#417—1131no—
176C2×#41—1131no—
177C22×Q8—1131no—
178C23×C4—1131no—
179#137—2264no—
180C2×Q16—1132no—
181C2×SD16—1132no—
182C2×M16—1131no—
183C2×M16—1132no—
184C2×M16—2264no—
185C2×M16C2×D11311yescharacteristic
186C22×C8—1132no—
187#123—2261no—
188C22⋊Q8—1132no—
189C22⋊Q8—1132no—
190C22⋊Q8—1131no—
191C22⋊Q8—1131no—
192C22⋊Q8—2261no—
193C22⋊Q8—2261no—
194#121—1131no—
195#121—1131no—
196#120—2261no—
197C4×D4—2261no—
198C4×D4—2261no—
199#117C2×D11311yescharacteristic
200C2×C4⋊C4—2261no—
201C2×C4⋊C4C2×D11311yescharacteristic
202C2×C22⋊C4—2261no—
203#107C2×D11312yescomplemented
204#106C2×D11312yescomplemented
205#103—1132no—
206#103—1132no—
207#103—1132no—
208#102—1132no—
209C24—1131no—
210#41—2262no—
211#41—2262no—
212C2×Q8—1132no—
213C2×Q8—1131no—
214C2×Q8—2262no—
215C2×Q8—2262no—
216C2×Q8—2261no—
217C2×D4—1131no—
218C2×D4—2261no—
219C2×D4—2261no—
220C2×D4—2261no—
221C22×C4—21no—
222C22×C4—1131no—
223C22×C4—1131no—
224C22×C4—1131no—
225C22×C4—1131no—
226C22×C4—1131no—
227C22×C4—2261no—
228C22×C4—2261no—
229C22×C4—2262no—
230C22×C4—2262no—
231C22×C4C22×D11311yescharacteristic
232Q16—2264no—
233SD16—2264no—
234M16—2262no—
235M16—2264no—
236M16D45214yescomplemented
237C2×C8—1132no—
238C2×C8—2264no—
239C2×C8C22×D11312yes—
240C4⋊C4—21no—
241C4⋊C4—2262no—
242C4⋊C4—2262no—
243C4⋊C4—2261no—
244C4⋊C4—2261no—
245C4⋊C4C22×D11312yes—
246C4⋊C4C22×D11311yescharacteristic
247C4⋊C4C22×D11311yescharacteristic
248C22⋊C4—21no—
249C22⋊C4—2261no—
250C22⋊C4—2261no—
251C22⋊C4—2262no—
252C22⋊C4—2262no—
253C22⋊C4—2261no—
254C22⋊C4—2261no—
255C42—21no—
256C23—1131no—
257C23—1131no—
258C23—4521no—
259C23—2262no—
260C23—2261no—
261C23—2262no—
262C23D4×D11311yescharacteristic
263Q8—4522no—
264Q8—2262no—
265Q8—2261no—
266Q8—2262no—
267Q8—2261no—
268D4—4522no—
269D4—4522no—
270D4—4522no—
271D4—2261no—
272D4—2261no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—22no—
276C2×C4—42no—
277C2×C4—1131no—
278C2×C4—1131no—
279C2×C4—4522no—
280C2×C4—4522no—
281C2×C4—2261no—
282C2×C4—2262no—
283C2×C4—2262no—
284C2×C4—2262no—
285C2×C4—2262no—
286C2×C4—2261no—
287C2×C4—2261no—
288C2×C4—2261no—
289C2×C4—2261no—
290C2×C4D4×D11311yescharacteristic
291C2×C4C2×D45212yes—
292C2×C4#1426864812yes—
293C2×C4C23×D11311yescharacteristic
294C8—2262no—
295C8C2×D45214yes—
296C22—21no—
297C22—21no—
298C22—1132no—
299C22—4524no—
300C22—4521no—
301C22—4521no—
302C22—4521no—
303C22—2261no—
304C22—2261no—
305C22—2261no—
306C22—2261no—
307C22—2261no—
308C22#1681359712yes—
309C22C2×D4×D11311yescharacteristic
310C4—41no—
311C4—42no—
312C4—41no—
313C4—4521no—
314C4—4522no—
315C4—2261no—
316C4—2261no—
317C4#1681359712yes—
318C4C22×D45211yescharacteristic
319C4C2×#1426864811yescharacteristic
320C2—22no—
321C2—9041no—
322C2—4521no—
323C2—2262no—
324C2C2×#1681359711yescharacteristic
325C2#137×D11311yescharacteristic
326C2#2029939411yescharacteristic
subgroupG/H
centerC22C2×D4×D113
commutator, cosocleC2×C4×C113C24
FrattiniC2×C4C23×D113
Fitting#428×C113C2
socleC22×C113C22×D4
prime radical#20299110C2
hypercenter#428D113
hypocenterC113#3056
LCSG ▷ C2×C4×C113 ▷ C2×C113 ▷ C113
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
1. C2×#16813655C2semidirect
2. C2×M16×D113C2semidirect
4. C2×#16813602C2semidirect
5. #20298857C2semidirect
6. #20298845C2semidirect
7. #20298840C2semidirect
8. #20298828C2semidirect
9. #20298606C2semidirect
10. #20298597C2semidirect
11. #428×C113C2semidirect
16. C2×Dic452C22semidirect
17. C2×#14268636C22semidirect
36. #16813545C22semidirect
37. #16813545C22semidirect
38. #16813545C22semidirect
39. #16813542C22semidirect
43. #107×C113C22semidirect
44. #106×C113C22semidirect
84. M16×C113D4semidirect
145. C2×D113#137semidirect
164. C113#3056semidirect
173. #428D113semidirect
203. #107C2×D113semidirect
204. #106C2×D113semidirect
236. M16D452semidirect

Maximal subgroups

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

subgroupindex
C2×#168136552
C2×M16×D1132
#202991102
C2×#168136022
#202988572
#202988452
#202988402
#202988282
#202986062
#202985972
#428×C1132
#3056113