Finite Group Databasebeta

← #28399880

Subgroups — #28399880

Order 14464.

764 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 764 conjugacy classes of subgroups — too many to draw. Only its 149 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×#16813668C211yesmaximal, characteristic, complemented
2C2×#16813650C211yesmaximal, characteristic, complemented
3C2×#16813642C211yesmaximal, characteristic, complemented
4#20299179C211yesmaximal, characteristic, complemented
5#20298763C212yesmaximal, complemented
6#20298750C212yesmaximal, complemented
7#20298746C212yesmaximal, complemented
8#20298733C212yesmaximal, complemented
9C2×#16813542C211yesmaximal, characteristic, complemented
10#20298566C211yesmaximal, characteristic, complemented
11#417×C113C211yesmaximal, characteristic, complemented
12#16813668—24no—
13#16813668—24no—
14C2×#14268648C2212yes—
15C2×#14268647—21no—
16C2×#14268647C2211yescharacteristic
17C2×#14268647C2211yescharacteristic, complemented
18C22×D452—21no—
19C22×D452C2211yescharacteristic
20C2×D4×D113—22no—
21C2×D4×D113C2212yes—
22#16813650—24no—
23#16813642—24no—
24C2×#14268637C2211yescharacteristic, complemented
25C2×#14268636C2212yescomplemented
26C2×#14268635C2211yescharacteristic, complemented
27C2×#14268634C2211yescharacteristic, complemented
28C2×D904C2212yescomplemented
29C2×#14268632C2211yescharacteristic, complemented
30C22×#11744751C2211yescharacteristic, complemented
31#16813602—21no—
32#16813602C2212yescomplemented
33#16813597—21no—
34#16813597C2212yescomplemented
35#16813596—22no—
36#16813542C2212yescomplemented
37#16813542C2212yescomplemented
38#16813534C2212yescomplemented
39#16813534C2212yescomplemented
40C2×#41×C113C2211yescharacteristic
41C2×M16×C113C2211yescharacteristic, complemented
42C2×C4⋊C4×C113C2211yescharacteristic, complemented
43#103×C113C2212yescomplemented
44#102×C113C2212yescomplemented
45C23×D113—21no—
46C23×D113—22no—
47#14268648—24no—
48#14268648—24no—
49#14268647—28no—
50#14268647—22no—
51#14268647—22no—
52#14268647—22no—
53#14268647—22no—
54C2×Dic226D411yescharacteristic
55C2×Dic226C2311yescharacteristic
56C2×D452—28no—
57C2×D452—22no—
58C2×D452—21no—
59C2×D452—22no—
60C2×D452—22no—
61C2×D452D411yescharacteristic
62C2×D452D414yescomplemented
63C2×D452C2311yescharacteristic
64C2×D452C2312yes—
65C2×C22⋊D113—21no—
66C2×C22⋊D113—22no—
67C2×C22⋊D113D412yes—
68D4×D113—216no—
69D4×D113—24no—
70D4×D113—24no—
71C2×C4×D113—22no—
72C2×C4×D113—21no—
73C2×C4×D113C2312yes—
74#14268637—22no—
75#14268636—24no—
76#14268635—22no—
77#14268634—22no—
78D904—24no—
79#14268632—22no—
80C2×#11744751—22no—
81C2×#11744751C2311yescharacteristic
82C2×#11744751C2311yescharacteristic
83C4⋊Dic113—22no—
84#14268624—22no—
85#14268624—22no—
86#41×C113—22no—
87#41×C113D414yescomplemented
88C2×Q8×C113C2311yescharacteristic
89C2×D4×C113—21no—
90C2×D4×C113C2311yescharacteristic
91C22×C4×C113—21no—
92C22×C4×C113—21no—
93C22×C4×C113C2311yescharacteristic
94M16×C113—22no—
95C2×C8×C113C2312yes—
96C4⋊C4×C113—21no—
97C4⋊C4×C113C2312yes—
98C23×C113—21no—
99C23×C113C2×D411yescharacteristic
100C22×D113—22no—
101C22×D113—28no—
102C22×D113—24no—
103C22×D113—21no—
104C22×D113—82no—
105C22×D113—82no—
106C22×D113—81no—
107C22×D113—81no—
108C22×D113C2×D412yes—
109Q8×C113—21no—
110Q8×C113C2×D412yes—
111D4×C113—24no—
112D4×C113—21no—
113D4×C113C2×D412yes—
114C2×C4×C113—21no—
115C2×C4×C113—24no—
116C2×C4×C113—22no—
117C2×C4×C113—81no—
118C2×C4×C113C2×D411yescharacteristic
119C2×C4×C113C2×D412yes—
120C2×C4×C113C2×D412yes—
121C2×C4×C113C2411yescharacteristic
122C22⋊D113—28no—
123C22⋊D113—24no—
124C22⋊D113—42no—
125C2×Dic113—21no—
126C2×Dic113C2×D412yes—
127D452—28no—
128D452—22no—
129D452—24no—
130D452—22no—
131D452—21no—
132D452—21no—
133D452—82no—
134D452C2×D414yes—
135C4×D113—28no—
136C4×D113—22no—
137C4×D113—22no—
138C4×D113—42no—
139Dic226—22no—
140Dic226—21no—
141Dic226—21no—
142C8×C113—22no—
143#11744751—21no—
144#11744751—21no—
145C22×C113—21no—
146C22×C113—21no—
147C22×C113—22no—
148C22×C113—41no—
149C22×C113—41no—
150C22×C113#12012yes—
151C22×C113C22×D411yescharacteristic
152C2×D113—22no—
153C2×D113—24no—
154C2×D113—22no—
155C2×D113—22no—
156C2×D113—42no—
157C2×D113—42no—
158C2×D113—41no—
159C2×D113—41no—
160C2×D113—41no—
161C2×D113—84no—
162C2×D113—88no—
163C4×C113—22no—
164C4×C113—42no—
165C4×C113#12012yes—
166C4×C113C22×D411yescharacteristic
167C4×C113C22×D411yescharacteristic
168Dic113—22no—
169Dic113—41no—
170C2×C113—22no—
171C2×C113—42no—
172C2×C113C2×#12011yescharacteristic
173C2×C113#57611yescharacteristic
174C2×C113#57711yescharacteristic
175D113—42no—
176D113—44no—
177D113—81no—
178#3012—1131nomaximal, Sylow
179C113#301211yescharacteristic, complemented, Sylow
180C2×H(5,𝔽2)—1131no—
181C2×AGL(1,ℤ/8)—1131no—
182C2×#135—1131no—
183#537—1131no—
184#452—1132no—
185#450—1132no—
186#449—1132no—
187#447—1132no—
188#417D11311yescharacteristic, complemented
189C2×#102—1131no—
190#408—1131no—
191H(5,𝔽2)—2264no—
192H(5,𝔽2)—2264no—
193C2×#41—1131no—
194C2×#41—1132no—
195C2×#41—1131no—
196C2×#41—2261no—
197C2×#41C2×D11311yescharacteristic
198C22×D4—1132no—
199C22×D4—1131no—
200C22×D4—2261no—
201C22×D4—2262no—
202AGL(1,ℤ/8)—2264no—
203#135—2264no—
204C2×Q16—1131no—
205C2×SD16—1131no—
206C2×SD16—1131no—
207C2×SD16—1132no—
208C2×D8—1131no—
209C2×D8—1132no—
210C2×M16C2×D11311yescharacteristic, complemented
211C22×C8—1131no—
212C22⋊Q8—1132no—
213C22⋊Q8—2261no—
214#121—1132no—
215#121—2261no—
216#121—2262no—
217C2×C4⋊C4C2×D11311yescharacteristic, complemented
218#103C2×D11312yescomplemented
219#102—1132no—
220#102—1132no—
221#102C2×D11312yescomplemented
222C22⋊C8—1132no—
223C22⋊C8—1132no—
224C24—2262no—
225C24—2261no—
226#41—22no—
227#41—2268no—
228#41—2262no—
229#41—2262no—
230#41—2264no—
231#41—2264no—
232#41—2262no—
233#41—2262no—
234#41D45214yescomplemented
235C2×Q8—1131no—
236C2×Q8—1131no—
237C2×Q8C22×D11311yescharacteristic
238C2×D4—21no—
239C2×D4—1132no—
240C2×D4—1134no—
241C2×D4—1131no—
242C2×D4—1132no—
243C2×D4—1131no—
244C2×D4—2262no—
245C2×D4—2262no—
246C2×D4—2268no—
247C2×D4—2262no—
248C2×D4—2264no—
249C2×D4—22616no—
250C2×D4—2261no—
251C2×D4—2264no—
252C2×D4—2261no—
253C2×D4—2262no—
254C2×D4C22×D11311yescharacteristic
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—1132no—
258C22×C4—2262no—
259C22×C4—2261no—
260C22×C4C22×D11311yescharacteristic
261Q16—2262no—
262SD16—2264no—
263SD16—2262no—
264SD16—2262no—
265D8—2264no—
266D8—2262no—
267M16—22no—
268C2×C8—1131no—
269C2×C8—1131no—
270C2×C8—2262no—
271C2×C8C22×D11312yes—
272C4⋊C4—21no—
273C4⋊C4—2262no—
274C4⋊C4C22×D11312yes—
275C22⋊C4—2262no—
276C22⋊C4—2262no—
277C23—21no—
278C23—1132no—
279C23—9041no—
280C23—9041no—
281C23—9042no—
282C23—9042no—
283C23—2268no—
284C23—2264no—
285C23—2261no—
286C23—2262no—
287C23D4×D11311yescharacteristic
288Q8—21no—
289Q8—2261no—
290Q8—2261no—
291Q8—2262no—
292Q8C2×D45212yes—
293D4—24no—
294D4—21no—
295D4—1134no—
296D4—9042no—
297D4—4522no—
298D4—2268no—
299D4—2268no—
300D4—2264no—
301D4—2264no—
302D4—2262no—
303D4—2261no—
304D4—2261no—
305D4—2262no—
306D4C2×D45212yes—
307C2×C4—21no—
308C2×C4—24no—
309C2×C4—22no—
310C2×C4—81no—
311C2×C4—1132no—
312C2×C4—4522no—
313C2×C4—2268no—
314C2×C4—2261no—
315C2×C4—2262no—
316C2×C4—2262no—
317C2×C4D4×D11311yescharacteristic
318C2×C4D4×D11312yes—
319C2×C4C2×D45212yes—
320C2×C4C23×D11311yescharacteristic
321C8—22no—
322C8—2261no—
323C8—2261no—
324C22—21no—
325C22—21no—
326C22—22no—
327C22—41no—
328C22—41no—
329C22—9044no—
330C22—9048no—
331C22—4521no—
332C22—4521no—
333C22—4521no—
334C22—4522no—
335C22—4522no—
336C22—2264no—
337C22—2262no—
338C22—2262no—
339C22—2262no—
340C22#1681359212yes—
341C22C2×D4×D11311yescharacteristic
342C4—22no—
343C4—42no—
344C4—4521no—
345C4—2262no—
346C4#1681359212yes—
347C4C2×D4×D11311yescharacteristic
348C4C22×D45211yescharacteristic
349C2—22no—
350C2—42no—
351C2—9041no—
352C2—4524no—
353C2—4522no—
354C2C2×#1681359211yescharacteristic
355C2#2029940211yescharacteristic
356C2#2029940711yescharacteristic
subgroupG/H
centerC22C2×D4×D113
commutator, cosocleC2×C4×C113C24
FrattiniC2×C4C23×D113
Fitting#417×C113C2
socleC22×C113C22×D4
hypercenter#417D113
hypocenterC113#3012
LCSG ▷ C2×C4×C113 ▷ C2×C113 ▷ C113
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
1. C2×#16813668C2semidirect
2. C2×#16813650C2semidirect
3. C2×#16813642C2semidirect
4. #20299179C2semidirect
5. #20298763C2semidirect
6. #20298750C2semidirect
7. #20298746C2semidirect
8. #20298733C2semidirect
9. C2×#16813542C2semidirect
10. #20298566C2semidirect
11. #417×C113C2semidirect
17. C2×#14268647C22semidirect
24. C2×#14268637C22semidirect
25. C2×#14268636C22semidirect
26. C2×#14268635C22semidirect
27. C2×#14268634C22semidirect
28. C2×D904C22semidirect
29. C2×#14268632C22semidirect
30. C22×#11744751C22semidirect
32. #16813602C22semidirect
34. #16813597C22semidirect
36. #16813542C22semidirect
37. #16813542C22semidirect
38. #16813534C22semidirect
39. #16813534C22semidirect
41. C2×M16×C113C22semidirect
42. C2×C4⋊C4×C113C22semidirect
43. #103×C113C22semidirect
44. #102×C113C22semidirect
62. C2×D452D4semidirect
87. #41×C113D4semidirect
179. C113#3012semidirect
188. #417D113semidirect
210. C2×M16C2×D113semidirect
217. C2×C4⋊C4C2×D113semidirect
218. #103C2×D113semidirect
221. #102C2×D113semidirect
234. #41D452semidirect

Maximal subgroups

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

subgroupindex
C2×#168136682
C2×#168136502
C2×#168136422
#202991792
#202987632
#202987502
#202987462
#202987332
C2×#168135422
#202985662
#417×C1132
#3012113