Finite Group Databasebeta

← #28399797

Subgroups — #28399797

Order 14464.

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

normal subgroups only

This group has 764 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, complemented
2C2×#16813645C212yesmaximal, complemented
3#20299111C211yesmaximal, characteristic, complemented
4#20298759C212yesmaximal, complemented
5#20298748C211yesmaximal, characteristic, complemented
6#20298746C212yesmaximal, complemented
7#20298737C211yesmaximal, characteristic, complemented
8#20298734C211yesmaximal, characteristic, complemented
9#20298593C212yesmaximal, complemented
10C22⋊C8×D113C211yesmaximal, characteristic, complemented
11#451×C113C211yesmaximal, characteristic, complemented
12C2×#14268648—28no—
13C2×#14268648—21no—
14C2×#14268648—21no—
15C2×#14268648C2212yes—
16C2×#14268648C2212yes—
17C2×#14268648C2211yescharacteristic, complemented
18C22×D452—21no—
19C22×D452C2211yescharacteristic
20C22×C4×D113—21no—
21C22×C4×D113C2211yescharacteristic
22#16813645—28no—
23C2×#14268634C2212yescomplemented
24C2×#14268634C2212yescomplemented
25C2×D904C2212yescomplemented
26C2×C8×D113C2212yescomplemented
27#16813607—21no—
28#16813598C2211yescharacteristic, complemented
29#16813597C2211yescharacteristic, complemented
30#16813596C2211yescharacteristic
31#16813592—21no—
32#16813584—21no—
33#16813582—21no—
34#16813580C2211yescharacteristic, complemented
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#14268648—24no—
53#14268648D418yescomplemented
54C2×D452—28no—
55C2×D452—22no—
56C2×D452—22no—
57C2×D452—21no—
58C2×D452—24no—
59C2×D452—21no—
60C2×D452—21no—
61C2×D452C2311yescharacteristic
62C2×D452C2312yes—
63C2×C22⋊D113—21no—
64C22×Dic113D411yescharacteristic
65C2×C4×D113—28no—
66C2×C4×D113—22no—
67C2×C4×D113—22no—
68C2×C4×D113—24no—
69C2×C4×D113—21no—
70C2×C4×D113D411yescharacteristic
71C2×C4×D113D411yescharacteristic
72C2×C4×D113C2311yescharacteristic
73C2×C4×D113C2311yescharacteristic
74#14268634—24no—
75#14268634—24no—
76D904—24no—
77C2×#11744751C2312yes—
78C8×D113—24no—
79C4⋊Dic113C2311yescharacteristic
80#14268626—21no—
81#14268624—21no—
82#14268624—21no—
83#14268624—21no—
84#14268624—21no—
85C4×Dic113—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—21no—
103C22×D113—41no—
104C22×D113—42no—
105C22×D113—42no—
106C22×D113C2×D411yescharacteristic
107C22×D113C2×D411yescharacteristic
108Q8×C113—24no—
109Q8×C113—22no—
110Q8×C113—22no—
111Q8×C113C2×D414yes—
112C2×C4×C113—24no—
113C2×C4×C113—22no—
114C2×C4×C113—22no—
115C2×C4×C113—21no—
116C2×C4×C113—21no—
117C2×C4×C113C2×D411yescharacteristic
118C2×C4×C113C2411yescharacteristic
119C22⋊D113—42no—
120C2×Dic113—22no—
121C2×Dic113—21no—
122C2×Dic113C2×D411yescharacteristic
123C2×Dic113C2×D411yescharacteristic
124D452—28no—
125D452—24no—
126D452—28no—
127D452—22no—
128D452—22no—
129D452—42no—
130D452—42no—
131D452C2×D414yes—
132C4×D113—28no—
133C4×D113—22no—
134C4×D113—22no—
135C4×D113—28no—
136C4×D113—22no—
137C4×D113—42no—
138C4×D113C2×D414yes—
139C8×C113—22no—
140#11744751—22no—
141C22×C113—21no—
142C22×C113—21no—
143C22×C113#13512yescomplemented
144C22×C113C22×D411yescharacteristic
145C2×D113—21no—
146C2×D113—21no—
147C2×D113—21no—
148C2×D113—24no—
149C2×D113—21no—
150C2×D113—21no—
151C2×D113—44no—
152C2×D113—48no—
153C2×D113—42no—
154C2×D113—42no—
155C2×D113—41no—
156C2×D113—41no—
157C2×D113—41no—
158C2×D113#12012yes—
159C4×C113—24no—
160C4×C113—21no—
161C4×C113—41no—
162C4×C113—41no—
163C4×C113C22×D412yes—
164Dic113—21no—
165Dic113—41no—
166Dic113#12012yes—
167C2×C113—22no—
168C2×C113C2×#12011yescharacteristic
169C2×C113C2×#13511yescharacteristic
170C2×C113#57511yescharacteristic
171D113—22no—
172D113—41no—
173D113—44no—
174D113—81no—
175#3002—1131nomaximal, Sylow
176C113#300211yescharacteristic, complemented, Sylow
177C22×#41—1131no—
178C2×#135—1132no—
179#525—1131no—
180#452—1132no—
181#451D11311yescharacteristic, complemented
182#450—1131no—
183#449—1132no—
184#448—1131no—
185#447—1131no—
186#416—1132no—
187C2×C22⋊C8—1131no—
188C2×#41—1132no—
189C2×#41—1131no—
190C2×#41—1132no—
191C2×#41—2261no—
192C2×#41—2261no—
193C2×#41—2268no—
194C22×Q8C2×D11311yescharacteristic
195C22×D4—1131no—
196C22×D4—2261no—
197C23×C4—1131no—
198C23×C4—2261no—
199#135—2268no—
200C2×Q16C2×D11312yescomplemented
201C2×SD16—1132no—
202C2×SD16—1132no—
203C2×D8—1132no—
204C22×C8—1132no—
205#123—2261no—
206C22⋊Q8—1131no—
207C22⋊Q8C2×D11311yescharacteristic
208#121—1131no—
209#121—1131no—
210#120—2261no—
211C4×D4—2261no—
212C4×D4—2261no—
213#117—1131no—
214#103—1132no—
215#103C2×D11312yescomplemented
216#102—1132no—
217#102—1132no—
218C22⋊C8—1131no—
219C22⋊C8—1132no—
220C22⋊C8C2×D11311yescharacteristic
221C24—1131no—
222C24—2261no—
223#41—1138no—
224#41—2264no—
225#41—2264no—
226#41—2264no—
227#41—2268no—
228#41—2268no—
229#41—2264no—
230C2×Q8—24no—
231C2×Q8—21no—
232C2×Q8C22×D11311yescharacteristic
233C2×Q8C22×D11312yes—
234C2×D4—1132no—
235C2×D4—1131no—
236C2×D4—2264no—
237C2×D4—2261no—
238C2×D4—2261no—
239C2×D4—2262no—
240C2×D4—2262no—
241C2×D4—2268no—
242C2×D4—2261no—
243C2×D4—2261no—
244C22×C4—21no—
245C22×C4—1131no—
246C22×C4—1131no—
247C22×C4—1131no—
248C22×C4—1131no—
249C22×C4—1131no—
250C22×C4—2261no—
251C22×C4—2264no—
252C22×C4—2262no—
253C22×C4—2262no—
254C22×C4—2268no—
255C22×C4C22×D11311yescharacteristic
256Q16—24no—
257SD16—2264no—
258SD16—2264no—
259D8—2264no—
260C2×C8—1132no—
261C2×C8—2264no—
262C2×C8C22×D11312yes—
263C4⋊C4—21no—
264C4⋊C4—1131no—
265C4⋊C4—2261no—
266C4⋊C4C22×D11311yescharacteristic
267C22⋊C4—21no—
268C22⋊C4—2261no—
269C22⋊C4—2261no—
270C22⋊C4—2261no—
271C22⋊C4—2261no—
272C42—2261no—
273C23—1131no—
274C23—1131no—
275C23—4522no—
276C23—4521no—
277C23—4522no—
278C23—2262no—
279C23—2264no—
280C23—2261no—
281C23—2262no—
282C23—2262no—
283C23D4×D11311yescharacteristic
284Q8—24no—
285Q8—22no—
286Q8—22no—
287Q8D4×D11314yes—
288D4—1134no—
289D4—4522no—
290D4—4522no—
291D4—4522no—
292D4—2268no—
293D4—2268no—
294D4—2264no—
295D4—2262no—
296D4—2262no—
297C2×C4—24no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—1131no—
303C2×C4—1134no—
304C2×C4—1131no—
305C2×C4—4522no—
306C2×C4—2262no—
307C2×C4—2268no—
308C2×C4—2262no—
309C2×C4—2268no—
310C2×C4—2262no—
311C2×C4—2261no—
312C2×C4—2262no—
313C2×C4D4×D11311yescharacteristic
314C2×C4C23×D11311yescharacteristic
315C8—22no—
316C8—2262no—
317C22—21no—
318C22—21no—
319C22—1132no—
320C22—4528no—
321C22—4524no—
322C22—4521no—
323C22—4521no—
324C22—4521no—
325C22—4522no—
326C22—4522no—
327C22—2264no—
328C22—2261no—
329C22—2261no—
330C22—2261no—
331C22—2261no—
332C22—2261no—
333C22#1681364512yescomplemented
334C22C2×D4×D11311yescharacteristic
335C4—24no—
336C4—21no—
337C4—41no—
338C4—41no—
339C4—1132no—
340C4—4521no—
341C4—2261no—
342C4C2×D4×D11312yes—
343C2—22no—
344C2—9041no—
345C2—4524no—
346C2—4521no—
347C2—2262no—
348C2#120×D11311yescharacteristic
349C2C2×#1681364511yescharacteristic
350C2#2029939711yescharacteristic
subgroupG/H
centerC22C2×D4×D113
commutator, cosocleC2×C4×C113C24
FrattiniC2×C4C23×D113
Fitting#451×C113C2
socleC22×C113C22×D4
hypercenter#451D113
hypocenterC113#3002
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
1. C22×#14268648C2semidirect
2. C2×#16813645C2semidirect
3. #20299111C2semidirect
4. #20298759C2semidirect
5. #20298748C2semidirect
6. #20298746C2semidirect
7. #20298737C2semidirect
8. #20298734C2semidirect
9. #20298593C2semidirect
10. C22⋊C8×D113C2semidirect
11. #451×C113C2semidirect
17. C2×#14268648C22semidirect
23. C2×#14268634C22semidirect
24. C2×#14268634C22semidirect
25. C2×D904C22semidirect
26. C2×C8×D113C22semidirect
28. #16813598C22semidirect
29. #16813597C22semidirect
34. #16813580C22semidirect
35. #16813546C22semidirect
36. #16813544C22semidirect
37. #16813542C22semidirect
39. #16813534C22semidirect
41. C2×Q16×C113C22semidirect
43. #103×C113C22semidirect
53. #14268648D4semidirect
143. C22×C113#135semidirect
176. C113#3002semidirect
181. #451D113semidirect
200. C2×Q16C2×D113semidirect
215. #103C2×D113semidirect
333. C22#16813645semidirect

Maximal subgroups

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

subgroupindex
C22×#142686482
C2×#168136452
#202991112
#202987592
#202987482
#202987462
#202987372
#202987342
#202985932
C22⋊C8×D1132
#451×C1132
#3002113