Finite Group Databasebeta

← #28399833

Subgroups — #28399833

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×#14268648C211yesmaximal, characteristic
2C2×#16813654C212yesmaximal, complemented
3C22⋊Q8×D113C211yesmaximal, characteristic, complemented
4#20298763C212yesmaximal, complemented
5#20298759C212yesmaximal, complemented
6#20298754C211yesmaximal, characteristic, complemented
7#20298751C211yesmaximal, characteristic, complemented
8#20298748C211yesmaximal, characteristic, complemented
9#20298598C212yesmaximal, complemented
10#20298560C211yesmaximal, characteristic, complemented
11#451×C113C211yesmaximal, characteristic, complemented
12C2×#14268648—28no—
13C2×#14268648—21no—
14C2×#14268648—21no—
15C2×#14268648C2212yes—
16C2×#14268648C2212yes—
17C2×Q8×D113C2211yescharacteristic, complemented
18C22×D452—21no—
19C22×D452C2211yescharacteristic
20C22×C4×D113—21no—
21C22×C4×D113C2211yescharacteristic
22#16813654—28no—
23C2×#14268637C2212yescomplemented
24C2×#14268636C2212yescomplemented
25C2×#14268634C2212yescomplemented
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#16813546C2212yescomplemented
36#16813544C2212yescomplemented
37#16813542C2212yescomplemented
38C22⋊#11744751C2211yescharacteristic
39#16813534C2212yescomplemented
40C22×Q8×C113C2211yescharacteristic
41C2×Q16×C113C2212yescomplemented
42C22⋊Q8×C113C2211yescharacteristic
43#103×C113C2212yescomplemented
44C22⋊C8×C113C2211yescharacteristic
45C23×D113—21no—
46C23×D113D411yescharacteristic
47#14268648—28no—
48#14268648—28no—
49#14268648—24no—
50#14268648—24no—
51#14268648—24no—
52#14268648D418yes—
53C2×Dic226—21no—
54C2×Dic226C2311yescharacteristic
55Q8×D113—24no—
56C2×D452—28no—
57C2×D452—22no—
58C2×D452—22no—
59C2×D452—21no—
60C2×D452—24no—
61C2×D452C2312yes—
62C22×Dic113D411yescharacteristic
63C2×C4×D113—28no—
64C2×C4×D113—22no—
65C2×C4×D113—22no—
66C2×C4×D113—24no—
67C2×C4×D113—21no—
68C2×C4×D113—21no—
69C2×C4×D113D411yescharacteristic
70C2×C4×D113D411yescharacteristic
71C2×C4×D113C2311yescharacteristic
72C2×C4×D113C2311yescharacteristic
73#14268637—24no—
74#14268636—24no—
75#14268634—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—
86C2×Q8×C113—24no—
87C2×Q8×C113—21no—
88C2×Q8×C113C2311yescharacteristic
89C2×Q8×C113C2312yes—
90C22×C4×C113—21no—
91C22×C4×C113C2311yescharacteristic
92Q16×C113—24no—
93C2×C8×C113C2312yes—
94C4⋊C4×C113—21no—
95C4⋊C4×C113C2311yescharacteristic
96C22⋊C4×C113—21no—
97C23×C113C2×D411yescharacteristic
98C22×D113—24no—
99C22×D113—22no—
100C22×D113—22no—
101C22×D113—22no—
102C22×D113—41no—
103C22×D113—42no—
104C22×D113—42no—
105C22×D113C2×D411yescharacteristic
106C22×D113C2×D411yescharacteristic
107Q8×C113—24no—
108Q8×C113—22no—
109Q8×C113—22no—
110Q8×C113C2×D414yes—
111C2×C4×C113—24no—
112C2×C4×C113—22no—
113C2×C4×C113—22no—
114C2×C4×C113—21no—
115C2×C4×C113—21no—
116C2×C4×C113C2×D411yescharacteristic
117C2×C4×C113C2411yescharacteristic
118C2×Dic113—22no—
119C2×Dic113—21no—
120C2×Dic113—21no—
121C2×Dic113C2×D411yescharacteristic
122C2×Dic113C2×D411yescharacteristic
123D452—28no—
124D452—24no—
125D452—28no—
126D452—22no—
127D452C2×D414yes—
128C4×D113—28no—
129C4×D113—22no—
130C4×D113—22no—
131C4×D113—28no—
132C4×D113—22no—
133C4×D113—42no—
134C4×D113—42no—
135C4×D113C2×D414yes—
136Dic226—22no—
137Dic226—42no—
138C8×C113—22no—
139#11744751—22no—
140C22×C113—21no—
141C22×C113—21no—
142C22×C113#13712yescomplemented
143C22×C113C22×D411yescharacteristic
144C2×D113—21no—
145C2×D113—21no—
146C2×D113—21no—
147C2×D113—24no—
148C2×D113—21no—
149C2×D113—21no—
150C2×D113—44no—
151C2×D113—48no—
152C2×D113—42no—
153C2×D113—42no—
154C2×D113#12012yes—
155C4×C113—24no—
156C4×C113—21no—
157C4×C113—41no—
158C4×C113—41no—
159C4×C113C22×D412yes—
160Dic113—21no—
161Dic113—41no—
162Dic113—41no—
163Dic113#12012yes—
164C2×C113—22no—
165C2×C113C2×#12011yescharacteristic
166C2×C113C2×#13711yescharacteristic
167C2×C113#57511yescharacteristic
168D113—22no—
169D113—41no—
170D113—44no—
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#451D11311yescharacteristic, complemented
180#450—1131no—
181#450—1131no—
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×Q8C2×D11311yescharacteristic
191C22×D4—1131no—
192C22×D4—2261no—
193C23×C4—1131no—
194C23×C4—2261no—
195#137—2268no—
196C2×Q16—1132no—
197C2×Q16C2×D11312yescomplemented
198C2×SD16—1132no—
199C2×SD16—1132no—
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#103C2×D11312yescomplemented
212#102—1132no—
213#102—1132no—
214C22⋊C8—1131no—
215C22⋊C8—1132no—
216C22⋊C8C2×D11311yescharacteristic
217C24—1131no—
218C24—2261no—
219#41—1138no—
220#41—2264no—
221#41—2264no—
222#41—2268no—
223#41—2268no—
224#41—2264no—
225C2×Q8—24no—
226C2×Q8—21no—
227C2×Q8—1131no—
228C2×Q8—2261no—
229C2×Q8—2264no—
230C2×Q8C22×D11311yescharacteristic
231C2×Q8C22×D11312yes—
232C2×D4—1132no—
233C2×D4—2264no—
234C2×D4—2262no—
235C2×D4—2262no—
236C2×D4—2268no—
237C2×D4—2261no—
238C22×C4—21no—
239C22×C4—1131no—
240C22×C4—1131no—
241C22×C4—1131no—
242C22×C4—1131no—
243C22×C4—1131no—
244C22×C4—2264no—
245C22×C4—2262no—
246C22×C4—2262no—
247C22×C4—2261no—
248C22×C4—2261no—
249C22×C4—2268no—
250C22×C4C22×D11311yescharacteristic
251Q16—24no—
252Q16—2264no—
253SD16—2264no—
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—1131no—
270C23—1131no—
271C23—4522no—
272C23—4521no—
273C23—4522no—
274C23—2262no—
275C23—2264no—
276C23—2262no—
277C23—2262no—
278C23D4×D11311yescharacteristic
279Q8—24no—
280Q8—22no—
281Q8—22no—
282Q8—4522no—
283Q8—2262no—
284Q8D4×D11314yes—
285D4—1134no—
286D4—2268no—
287D4—2268no—
288D4—2264no—
289D4—2262no—
290C2×C4—24no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—1131no—
296C2×C4—1134no—
297C2×C4—1131no—
298C2×C4—4522no—
299C2×C4—4522no—
300C2×C4—2262no—
301C2×C4—2268no—
302C2×C4—2262no—
303C2×C4—2268no—
304C2×C4—2262no—
305C2×C4—2261no—
306C2×C4—2261no—
307C2×C4—2262no—
308C2×C4D4×D11311yescharacteristic
309C2×C4C23×D11311yescharacteristic
310C8—22no—
311C8—2262no—
312C22—21no—
313C22—21no—
314C22—1132no—
315C22—4528no—
316C22—4524no—
317C22—4522no—
318C22—4522no—
319C22—2264no—
320C22—2261no—
321C22—2261no—
322C22—2261no—
323C22—2261no—
324C22—2261no—
325C22#1681365412yescomplemented
326C22C2×D4×D11311yescharacteristic
327C4—24no—
328C4—21no—
329C4—41no—
330C4—41no—
331C4—1132no—
332C4—4521no—
333C4—4521no—
334C4—2261no—
335C4C2×D4×D11312yes—
336C2—22no—
337C2—4524no—
338C2—4521no—
339C2—2262no—
340C2#120×D11311yescharacteristic
341C2C2×#1681365411yescharacteristic
342C2#2029939711yescharacteristic
subgroupG/H
centerC22C2×D4×D113
commutator, cosocleC2×C4×C113C24
FrattiniC2×C4C23×D113
Fitting#451×C113C2
socleC22×C113C22×D4
prime radicalC22×#14268648C2
hypercenter#451D113
hypocenterC113#3007
LCSG ▷ C2×C4×C113 ▷ C2×C113 ▷ C113
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
2. C2×#16813654C2semidirect
3. C22⋊Q8×D113C2semidirect
4. #20298763C2semidirect
5. #20298759C2semidirect
6. #20298754C2semidirect
7. #20298751C2semidirect
8. #20298748C2semidirect
9. #20298598C2semidirect
10. #20298560C2semidirect
11. #451×C113C2semidirect
17. C2×Q8×D113C22semidirect
23. C2×#14268637C22semidirect
24. C2×#14268636C22semidirect
25. C2×#14268634C22semidirect
26. C2×#14268630C22semidirect
27. #16813602C22semidirect
31. #16813598C22semidirect
33. C4⋊C4×D113C22semidirect
35. #16813546C22semidirect
36. #16813544C22semidirect
37. #16813542C22semidirect
39. #16813534C22semidirect
41. C2×Q16×C113C22semidirect
43. #103×C113C22semidirect
142. C22×C113#137semidirect
172. C113#3007semidirect
179. #451D113semidirect
197. C2×Q16C2×D113semidirect
211. #103C2×D113semidirect
325. C22#16813654semidirect

Maximal subgroups

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

subgroupindex
C22×#142686482
C2×#168136542
C22⋊Q8×D1132
#202987632
#202987592
#202987542
#202987512
#202987482
#202985982
#202985602
#451×C1132
#3007113