Finite Group Databasebeta

← #28399829

Subgroups — #28399829

Order 14464.

754 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 754 conjugacy classes of subgroups — too many to draw. Only its 141 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
1C22×#14268646C211yesmaximal, characteristic
2C2×#16813651C212yesmaximal, complemented
3C22⋊Q8×D113C211yesmaximal, characteristic, complemented
4#20298761C212yesmaximal, complemented
5#20298757C212yesmaximal, complemented
6#20298756C211yesmaximal, characteristic, complemented
7#20298753C211yesmaximal, characteristic, complemented
8#20298749C211yesmaximal, characteristic, complemented
9#20298596C212yesmaximal, complemented
10#20298560C211yesmaximal, characteristic, complemented
11#450×C113C211yesmaximal, characteristic, complemented
12C2×#14268646—28no—
13C2×#14268646—21no—
14C2×#14268646—21no—
15C2×#14268646C2212yes—
16C2×#14268646C2212yes—
17C22×Dic226C2211yescharacteristic
18C2×Q8×D113C2211yescharacteristic, complemented
19C22×C22⋊D113—21no—
20C23×Dic113—21no—
21C22×C4×D113C2211yescharacteristic
22#16813651—28no—
23C2×Dic452C2212yescomplemented
24C2×#14268637C2212yescomplemented
25C2×#14268635C2212yescomplemented
26C2×#14268630C2212yescomplemented
27#16813602C2211yescharacteristic, complemented
28#16813601C2211yescharacteristic
29#16813600—21no—
30#16813599—21no—
31#16813598C2211yescharacteristic, complemented
32C4⋊C4×D113—21no—
33C4⋊C4×D113C2211yescharacteristic, complemented
34C22⋊C4×D113—21no—
35#16813547C2212yescomplemented
36#16813545C2212yescomplemented
37#16813543C2212yescomplemented
38C22⋊#11744751C2211yescharacteristic
39#16813534C2212yescomplemented
40C22×D4×C113C2211yescharacteristic
41C2×SD16×C113C2212yescomplemented
42C22⋊Q8×C113C2211yescharacteristic
43#102×C113C2212yescomplemented
44C22⋊C8×C113C2211yescharacteristic
45C23×D113D411yescharacteristic
46#14268646—28no—
47#14268646—28no—
48#14268646—24no—
49#14268646—24no—
50#14268646—24no—
51#14268646D418yes—
52C2×Dic226—21no—
53C2×Dic226—24no—
54C2×Dic226—21no—
55C2×Dic226C2311yescharacteristic
56C2×Dic226C2312yes—
57Q8×D113—24no—
58C2×C22⋊D113—28no—
59C2×C22⋊D113—22no—
60C2×C22⋊D113—22no—
61C22×Dic113—28no—
62C22×Dic113—21no—
63C22×Dic113—22no—
64C22×Dic113—22no—
65C22×Dic113D411yescharacteristic
66C2×C4×D113—24no—
67C2×C4×D113—21no—
68C2×C4×D113—21no—
69C2×C4×D113D411yescharacteristic
70C2×C4×D113D411yescharacteristic
71C2×C4×D113C2311yescharacteristic
72C2×C4×D113C2311yescharacteristic
73Dic452—24no—
74#14268637—24no—
75#14268635—24no—
76#14268630—24no—
77C2×#11744751C2312yes—
78C4⋊Dic113—21no—
79C4⋊Dic113C2311yescharacteristic
80#14268626—21no—
81#14268626—21no—
82#14268626—21no—
83C22⋊Dic113—21no—
84#14268624—21no—
85#14268624—21no—
86C24×C113—21no—
87C2×Q8×C113C2311yescharacteristic
88C2×D4×C113—24no—
89C2×D4×C113—21no—
90C2×D4×C113C2312yes—
91C22×C4×C113C2311yescharacteristic
92SD16×C113—24no—
93C2×C8×C113C2312yes—
94C4⋊C4×C113—21no—
95C4⋊C4×C113C2311yescharacteristic
96C22⋊C4×C113—21no—
97C23×C113—24no—
98C23×C113—22no—
99C23×C113—42no—
100C23×C113—42no—
101C23×C113C2×D411yescharacteristic
102C22×D113—22no—
103C22×D113—22no—
104C22×D113—41no—
105C22×D113C2×D411yescharacteristic
106C22×D113C2×D411yescharacteristic
107Q8×C113—22no—
108D4×C113—24no—
109D4×C113—22no—
110D4×C113C2×D414yes—
111C2×C4×C113—22no—
112C2×C4×C113—21no—
113C2×C4×C113—21no—
114C2×C4×C113C2×D411yescharacteristic
115C2×C4×C113C2411yescharacteristic
116C22⋊D113—28no—
117C22⋊D113—28no—
118C2×Dic113—24no—
119C2×Dic113—28no—
120C2×Dic113—22no—
121C2×Dic113—22no—
122C2×Dic113—28no—
123C2×Dic113—21no—
124C2×Dic113—21no—
125C2×Dic113C2×D411yescharacteristic
126C2×Dic113C2×D411yescharacteristic
127C4×D113—22no—
128C4×D113—22no—
129C4×D113—22no—
130C4×D113—42no—
131C4×D113—42no—
132C4×D113C2×D414yes—
133Dic226—24no—
134Dic226—22no—
135Dic226—22no—
136Dic226—42no—
137Dic226C2×D414yes—
138C8×C113—22no—
139#11744751—22no—
140C22×C113—21no—
141C22×C113—21no—
142C22×C113—24no—
143C22×C113—48no—
144C22×C113—42no—
145C22×C113—42no—
146C22×C113#13712yescomplemented
147C22×C113C22×D411yescharacteristic
148C2×D113—21no—
149C2×D113—21no—
150C2×D113—21no—
151C2×D113—21no—
152C2×D113—21no—
153C2×D113—44no—
154C2×D113#12012yes—
155C4×C113—21no—
156C4×C113—41no—
157C4×C113—41no—
158C4×C113C22×D412yes—
159Dic113—24no—
160Dic113—21no—
161Dic113—41no—
162Dic113—41no—
163Dic113#12012yes—
164C2×C113—22no—
165C2×C113—44no—
166C2×C113C2×#12011yescharacteristic
167C2×C113C2×#13711yescharacteristic
168C2×C113#57511yescharacteristic
169D113—22no—
170D113—41no—
171#3007—1131nomaximal, Sylow
172C113#300711yescharacteristic, complemented, Sylow
173C22×#41—1131no—
174C2×#137—1132no—
175C2×C22⋊Q8—1131no—
176#452—1132no—
177#452—1132no—
178#451—1131no—
179#451—1131no—
180#450—1131no—
181#450D11311yescharacteristic, complemented
182#417—1132no—
183#407—1131no—
184C2×#41—1132no—
185C2×#41—1132no—
186C2×#41—2261no—
187C2×#41—2261no—
188C2×#41—2268no—
189C22×Q8—1131no—
190C22×Q8—1131no—
191C22×D4—2261no—
192C22×D4C2×D11311yescharacteristic
193C23×C4—1131no—
194C23×C4—2261no—
195#137—2268no—
196C2×Q16—1132no—
197C2×Q16—1132no—
198C2×SD16—1132no—
199C2×SD16C2×D11312yescomplemented
200C2×M16—1132no—
201C22⋊Q8—1131no—
202C22⋊Q8—1131no—
203C22⋊Q8—1131no—
204C22⋊Q8—2261no—
205C22⋊Q8—2261no—
206C22⋊Q8C2×D11311yescharacteristic
207C2×C4⋊C4—1131no—
208C2×C4⋊C4—2261no—
209C2×C22⋊C4—2261no—
210#103—1132no—
211#103—1132no—
212#102—1132no—
213#102C2×D11312yescomplemented
214C22⋊C8—1131no—
215C22⋊C8—1132no—
216C22⋊C8C2×D11311yescharacteristic
217C24—21no—
218C24—1131no—
219#41—1138no—
220#41—2264no—
221#41—2264no—
222#41—2268no—
223#41—2268no—
224#41—2264no—
225C2×Q8—1132no—
226C2×Q8—1131no—
227C2×Q8—2261no—
228C2×Q8—2264no—
229C2×Q8—2264no—
230C2×Q8—2261no—
231C2×Q8C22×D11311yescharacteristic
232C2×D4—24no—
233C2×D4—21no—
234C2×D4—2262no—
235C2×D4—2262no—
236C2×D4—2268no—
237C2×D4C22×D11312yes—
238C22×C4—1131no—
239C22×C4—1131no—
240C22×C4—1131no—
241C22×C4—1131no—
242C22×C4—1131no—
243C22×C4—2264no—
244C22×C4—2261no—
245C22×C4—2261no—
246C22×C4—2262no—
247C22×C4—2262no—
248C22×C4—2268no—
249C22×C4—2261no—
250C22×C4C22×D11311yescharacteristic
251Q16—2264no—
252Q16—2264no—
253SD16—24no—
254SD16—2264no—
255M16—2264no—
256C2×C8—1132no—
257C2×C8C22×D11312yes—
258C4⋊C4—21no—
259C4⋊C4—1131no—
260C4⋊C4—2261no—
261C4⋊C4—2261no—
262C4⋊C4—2261no—
263C4⋊C4—2261no—
264C4⋊C4C22×D11311yescharacteristic
265C22⋊C4—21no—
266C22⋊C4—2261no—
267C22⋊C4—2261no—
268C22⋊C4—2261no—
269C23—24no—
270C23—22no—
271C23—42no—
272C23—42no—
273C23—1131no—
274C23—1131no—
275C23—4521no—
276C23—2262no—
277C23—2262no—
278C23D4×D11311yescharacteristic
279Q8—22no—
280Q8—1134no—
281Q8—4522no—
282Q8—2264no—
283Q8—2262no—
284Q8—2262no—
285D4—24no—
286D4—22no—
287D4—2268no—
288D4—2268no—
289D4D4×D11314yes—
290C2×C4—22no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—1131no—
294C2×C4—1134no—
295C2×C4—1131no—
296C2×C4—4522no—
297C2×C4—4522no—
298C2×C4—2262no—
299C2×C4—2262no—
300C2×C4—2262no—
301C2×C4—2268no—
302C2×C4—2261no—
303C2×C4—2261no—
304C2×C4—2264no—
305C2×C4—2268no—
306C2×C4—2262no—
307C2×C4—2262no—
308C2×C4D4×D11311yescharacteristic
309C2×C4C23×D11311yescharacteristic
310C8—22no—
311C8—2262no—
312C22—21no—
313C22—21no—
314C22—24no—
315C22—48no—
316C22—42no—
317C22—42no—
318C22—1132no—
319C22—4524no—
320C22—2261no—
321C22—2261no—
322C22—2261no—
323C22—2261no—
324C22—2261no—
325C22#1681365112yescomplemented
326C22C2×D4×D11311yescharacteristic
327C4—21no—
328C4—41no—
329C4—41no—
330C4—1132no—
331C4—4521no—
332C4—4521no—
333C4—2261no—
334C4—2264no—
335C4C2×D4×D11312yes—
336C2—22no—
337C2—44no—
338C2—4521no—
339C2—2262no—
340C2#120×D11311yescharacteristic
341C2C2×#1681365111yescharacteristic
342C2#2029939411yescharacteristic
subgroupG/H
centerC22C2×D4×D113
commutator, cosocleC2×C4×C113C24
FrattiniC2×C4C23×D113
Fitting#450×C113C2
socleC22×C113C22×D4
prime radicalC22×#14268646C2
hypercenter#450D113
hypocenterC113#3007
LCSG ▷ C2×C4×C113 ▷ C2×C113 ▷ C113
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
2. C2×#16813651C2semidirect
3. C22⋊Q8×D113C2semidirect
4. #20298761C2semidirect
5. #20298757C2semidirect
6. #20298756C2semidirect
7. #20298753C2semidirect
8. #20298749C2semidirect
9. #20298596C2semidirect
10. #20298560C2semidirect
11. #450×C113C2semidirect
18. C2×Q8×D113C22semidirect
23. C2×Dic452C22semidirect
24. C2×#14268637C22semidirect
25. C2×#14268635C22semidirect
26. C2×#14268630C22semidirect
27. #16813602C22semidirect
31. #16813598C22semidirect
33. C4⋊C4×D113C22semidirect
35. #16813547C22semidirect
36. #16813545C22semidirect
37. #16813543C22semidirect
39. #16813534C22semidirect
41. C2×SD16×C113C22semidirect
43. #102×C113C22semidirect
146. C22×C113#137semidirect
172. C113#3007semidirect
181. #450D113semidirect
199. C2×SD16C2×D113semidirect
213. #102C2×D113semidirect
325. C22#16813651semidirect

Maximal subgroups

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

subgroupindex
C22×#142686462
C2×#168136512
C22⋊Q8×D1132
#202987612
#202987572
#202987562
#202987532
#202987492
#202985962
#202985602
#450×C1132
#3007113