Finite Group Databasebeta

← #28401022

Subgroups — #28401022

Order 14464.

656 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 153 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×#16813650C211yesmaximal, characteristic, complemented
2C2×#16813648C211yesmaximal, characteristic, complemented
3C2×#16813647C211yesmaximal, characteristic, complemented
4#20299144C211yesmaximal, characteristic, complemented
5C2×#16813618C211yesmaximal, characteristic, complemented
6#20298986C212yesmaximal, complemented
7#20298984C212yesmaximal, complemented
8#20298973C212yesmaximal, complemented
9#20298969C212yesmaximal, complemented
10M16×Dic113C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C113C211yesmaximal, characteristic, complemented
12C2×#14268647C2211yescharacteristic
13C22×D452C2211yescharacteristic
14C22×C22⋊D113—22no—
15#16813650—24no—
16#16813648—24no—
17#16813647—24no—
18C2×#14268636C2212yescomplemented
19C2×#14268635C2212yescomplemented
20C2×#14268634C2212yescomplemented
21C2×D904C2212yescomplemented
22C2×#14268632C2212yescomplemented
23C2×#14268632C2212yescomplemented
24C2×#14268631C2211yescharacteristic
25#16813620C2211yescharacteristic, complemented
26#16813618—22no—
27#16813618C2212yes—
28#16813618C2212yes—
29#16813618C2211yescharacteristic, complemented
30#16813611C2211yescharacteristic, complemented
31#16813609C2211yescharacteristic, complemented
32#16813594—22no—
33C4×C22⋊D113—22no—
34C2×C4×Dic113C2211yescharacteristic
35#16813532C2212yescomplemented
36C8×Dic113C2212yescomplemented
37C2×#41×C113C2211yescharacteristic
38C22×D4×C113C2211yescharacteristic
39AGL(1,ℤ/8)×C113—24no—
40C2×SD16×C113C2212yescomplemented
41C2×D8×C113C2212yescomplemented
42C2×M16×C113C2211yescharacteristic
43C23×D113—21no—
44#14268647—22no—
45#14268647—22no—
46C2×Dic226C2311yescharacteristic
47C2×D452—22no—
48C2×D452—22no—
49C2×D452—21no—
50C2×D452C2311yescharacteristic
51C2×D452C2312yes—
52C2×C22⋊D113—21no—
53C2×C22⋊D113—22no—
54C2×C22⋊D113—24no—
55C2×C22⋊D113—24no—
56C2×C22⋊D113—48no—
57C22×Dic113D412yes—
58C2×C4×D113—21no—
59#14268636—24no—
60#14268635—24no—
61#14268634—24no—
62D904—24no—
63#14268632—24no—
64#14268632—24no—
65#14268631D414yescomplemented
66C2×#11744751C2312yes—
67#14268626—22no—
68C22⋊Dic113—22no—
69#14268624—22no—
70C4×Dic113D412yes—
71C4×Dic113C2311yescharacteristic
72C4×Dic113C2311yescharacteristic
73C24×C113—21no—
74#41×C113—22no—
75#41×C113—22no—
76C2×Q8×C113C2311yescharacteristic
77C2×D4×C113—21no—
78C2×D4×C113—22no—
79C2×D4×C113—22no—
80C2×D4×C113—21no—
81C2×D4×C113C2311yescharacteristic
82C2×D4×C113C2312yes—
83C22×C4×C113—21no—
84C22×C4×C113C2311yescharacteristic
85SD16×C113—24no—
86D8×C113—24no—
87M16×C113D414yescomplemented
88C2×C8×C113C2312yes—
89C23×C113—21no—
90C23×C113—22no—
91C23×C113—44no—
92C23×C113—42no—
93C23×C113C2×D411yescharacteristic
94C22×D113—21no—
95C22×D113—22no—
96C22×D113—44no—
97C22×D113—42no—
98Q8×C113—21no—
99Q8×C113—21no—
100D4×C113—21no—
101D4×C113—22no—
102D4×C113—22no—
103D4×C113—21no—
104D4×C113—42no—
105D4×C113—42no—
106C2×C4×C113—21no—
107C2×C4×C113—42no—
108C2×C4×C113C2×D411yescharacteristic
109C2×C4×C113C2×D412yes—
110C2×C4×C113C2×D412yes—
111C2×C4×C113C2411yescharacteristic
112C22⋊D113—48no—
113C22⋊D113—42no—
114C22⋊D113—44no—
115C22⋊D113—48no—
116C2×Dic113—24no—
117C2×Dic113—21no—
118C2×Dic113C2×D412yes—
119C2×Dic113C2×D412yes—
120D452—21no—
121D452—22no—
122D452—21no—
123D452—22no—
124D452—42no—
125C4×D113—42no—
126Dic226—21no—
127Dic226—21no—
128C8×C113C2×D414yescomplemented
129#11744751C2×D414yescomplemented
130C22×C113—22no—
131C22×C113—41no—
132C22×C113—42no—
133C22×C113—44no—
134C22×C113—42no—
135C22×C113—84no—
136C22×C113#12712yes—
137C22×C113C22×D411yescharacteristic
138C2×D113—41no—
139C2×D113—42no—
140C2×D113—44no—
141C2×D113—42no—
142C2×D113—84no—
143C4×C113—41no—
144C4×C113#12712yes—
145C4×C113C22×D411yescharacteristic
146C4×C113C22×D411yescharacteristic
147Dic113—22no—
148Dic113—41no—
149Dic113AGL(1,ℤ/8)14yescomplemented
150C2×C113—22no—
151C2×C113—81no—
152C2×C113—82no—
153C2×C113C2×#12711yescharacteristic
154C2×C113C2×AGL(1,ℤ/8)12yes—
155D113—81no—
156D113—82no—
157#3151—1131nomaximal, Sylow
158C113#315111yescharacteristic, complemented, Sylow
159C2×AGL(1,ℤ/8)—1131no—
160C2×AGL(1,ℤ/8)—1131no—
161C2×AGL(1,ℤ/8)—1131no—
162C2×AGL(1,ℤ/8)D11311yescharacteristic, complemented
163#532—1131no—
164C2×#127—1131no—
165#496—1132no—
166#496—1132no—
167#493—1132no—
168#492—1132no—
169C4×M16—1131no—
170C2×#41—1131no—
171C2×#41C2×D11311yescharacteristic
172C22×D4—1131no—
173C22×D4—2262no—
174C22×D4C2×D11311yescharacteristic
175AGL(1,ℤ/8)—24no—
176AGL(1,ℤ/8)—2264no—
177AGL(1,ℤ/8)—2264no—
178AGL(1,ℤ/8)—2264no—
179C2×SD16—1132no—
180C2×SD16—1132no—
181C2×SD16—1132no—
182C2×SD16C2×D11312yescomplemented
183C2×D8—1132no—
184C2×D8—1132no—
185C2×D8—1132no—
186C2×D8C2×D11312yescomplemented
187C2×M16—1131no—
188C2×M16C2×D11311yescharacteristic
189C4⋊Q8—1131no—
190#127—1132no—
191#127—1132no—
192#127—1131no—
193#127—2262no—
194#124—1131no—
195#124—1131no—
196#121—2262no—
197C4×D4—2262no—
198C2×C42—1131no—
199#97—1132no—
200C4×C8—1132no—
201C24—21no—
202C24—2261no—
203#41—22no—
204#41—22no—
205#41—2262no—
206#41—2262no—
207C2×Q8—1131no—
208C2×Q8C22×D11311yescharacteristic
209C2×D4—21no—
210C2×D4—22no—
211C2×D4—21no—
212C2×D4—22no—
213C2×D4—1131no—
214C2×D4—1132no—
215C2×D4—4528no—
216C2×D4—2262no—
217C2×D4—2264no—
218C2×D4—2264no—
219C2×D4—2261no—
220C2×D4—2261no—
221C2×D4—2262no—
222C2×D4—2262no—
223C2×D4C22×D11311yescharacteristic
224C2×D4C22×D11312yes—
225C22×C4—21no—
226C22×C4—1132no—
227C22×C4—2261no—
228C22×C4C22×D11311yescharacteristic
229SD16—24no—
230SD16—2264no—
231SD16—2264no—
232SD16—2264no—
233D8—24no—
234D8—2264no—
235D8—2264no—
236D8—2264no—
237M16—1134no—
238M16C22⋊D11314yescomplemented
239C2×C8—1132no—
240C2×C8C22×D11312yes—
241C4⋊C4—2262no—
242C22⋊C4—2262no—
243C22⋊C4—2262no—
244C42—1132no—
245C42—1131no—
246C42—1131no—
247C23—21no—
248C23—22no—
249C23—44no—
250C23—42no—
251C23—4524no—
252C23—4522no—
253C23—2261no—
254C23—2262no—
255C23D4×D11311yescharacteristic
256Q8—21no—
257Q8—21no—
258Q8—2261no—
259Q8—2261no—
260D4—21no—
261D4—22no—
262D4—21no—
263D4—22no—
264D4—42no—
265D4—42no—
266D4—4528no—
267D4—4524no—
268D4—4522no—
269D4—4528no—
270D4—4522no—
271D4—2261no—
272D4—2262no—
273D4—2261no—
274D4—2262no—
275C2×C4—21no—
276C2×C4—42no—
277C2×C4—1132no—
278C2×C4—1132no—
279C2×C4—4522no—
280C2×C4—2264no—
281C2×C4—2261no—
282C2×C4D4×D11311yescharacteristic
283C2×C4D4×D11312yes—
284C2×C4C2×C22⋊D11312yes—
285C2×C4C23×D11311yescharacteristic
286C8—1134no—
287C8C2×C22⋊D11314yescomplemented
288C22—22no—
289C22—44no—
290C22—42no—
291C22—41no—
292C22—42no—
293C22—84no—
294C22—9044no—
295C22—4524no—
296C22—4522no—
297C22—4521no—
298C22—4522no—
299C22#1681361812yes—
300C22C2×D4×D11311yescharacteristic
301C4—41no—
302C4—1134no—
303C4—4521no—
304C4—2262no—
305C4#1681361812yes—
306C4C2×D4×D11311yescharacteristic
307C4C22×C22⋊D11311yescharacteristic
308C2—22no—
309C2—82no—
310C2—81no—
311C2—9042no—
312C2—9041no—
313C2C2×#1681361811yescharacteristic
314C2AGL(1,ℤ/8)×D11312yes—
subgroupG/H
centerC22C2×D4×D113
commutator, cosocleC2×C4×C113C24
FrattiniC2×C4C23×D113
FittingC2×AGL(1,ℤ/8)×C113C2
socleC22×C113C22×D4
hypercenterC2×AGL(1,ℤ/8)D113
hypocenterC113#3151
LCSG ▷ C2×C4×C113 ▷ C2×C113 ▷ C113
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#16813650C2semidirect
2. C2×#16813648C2semidirect
3. C2×#16813647C2semidirect
4. #20299144C2semidirect
5. C2×#16813618C2semidirect
6. #20298986C2semidirect
7. #20298984C2semidirect
8. #20298973C2semidirect
9. #20298969C2semidirect
10. M16×Dic113C2semidirect
11. C2×AGL(1,ℤ/8)×C113C2semidirect
18. C2×#14268636C22semidirect
19. C2×#14268635C22semidirect
20. C2×#14268634C22semidirect
21. C2×D904C22semidirect
22. C2×#14268632C22semidirect
23. C2×#14268632C22semidirect
25. #16813620C22semidirect
29. #16813618C22semidirect
30. #16813611C22semidirect
31. #16813609C22semidirect
35. #16813532C22semidirect
36. C8×Dic113C22semidirect
40. C2×SD16×C113C22semidirect
41. C2×D8×C113C22semidirect
65. #14268631D4semidirect
87. M16×C113D4semidirect
128. C8×C113C2×D4semidirect
129. #11744751C2×D4semidirect
149. Dic113AGL(1,ℤ/8)semidirect
158. C113#3151semidirect
162. C2×AGL(1,ℤ/8)D113semidirect
182. C2×SD16C2×D113semidirect
186. C2×D8C2×D113semidirect
238. M16C22⋊D113semidirect
287. C8C2×C22⋊D113semidirect

Maximal subgroups

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

subgroupindex
C2×#168136502
C2×#168136482
C2×#168136472
#202991442
C2×#168136182
#202989862
#202989842
#202989732
#202989692
M16×Dic1132
C2×AGL(1,ℤ/8)×C1132
#3151113