Finite Group Databasebeta

← #28401036

Subgroups — #28401036

Order 14464.

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

normal subgroups only

This group has 572 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×#16813655C211yesmaximal, characteristic, complemented
2C2×#16813652C211yesmaximal, characteristic, complemented
3C2×#16813648C211yesmaximal, characteristic, complemented
4#20299144C211yesmaximal, characteristic, complemented
5C2×#16813609C211yesmaximal, characteristic, complemented
6#20298989C212yesmaximal, complemented
7#20298986C212yesmaximal, complemented
8#20298981C212yesmaximal, complemented
9#20298979C212yesmaximal, complemented
10M16×Dic113C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C113C211yesmaximal, characteristic, complemented
12C2×#14268647C2211yescharacteristic
13C22×Dic226C2211yescharacteristic
14#16813655—24no—
15#16813652—24no—
16#16813648—24no—
17C2×Dic452C2212yescomplemented
18C2×#14268637C2212yescomplemented
19C2×#14268636C2212yescomplemented
20C2×#14268635C2212yes—
21C2×#14268635C2212yescomplemented
22C2×#14268632C2212yes—
23C2×#14268631C2211yescharacteristic
24#16813620C2211yescharacteristic, complemented
25#16813618C2211yescharacteristic, complemented
26#16813611C2211yescharacteristic, complemented
27#16813609—22no—
28#16813609C2212yes—
29#16813609C2212yes—
30#16813609C2211yescharacteristic, complemented
31#16813594—22no—
32C4×C22⋊D113—22no—
33C2×C22⋊Dic113—22no—
34C2×C4×Dic113C2211yescharacteristic
35#16813532C2212yescomplemented
36C8×Dic113C2212yescomplemented
37C2×#41×C113C2211yescharacteristic
38C22×D4×C113C2211yescharacteristic
39AGL(1,ℤ/8)×C113—24no—
40C2×SD16×C113C2212yes—
41C2×D8×C113C2212yes—
42C2×M16×C113C2211yescharacteristic
43#14268647—22no—
44#14268647—22no—
45C2×Dic226—22no—
46C2×Dic226—22no—
47C2×Dic226—21no—
48C2×Dic226C2312yes—
49C2×Dic226C2311yescharacteristic
50C2×D452C2311yescharacteristic
51C2×C22⋊D113—21no—
52C2×C22⋊D113—22no—
53C22×Dic113—21no—
54C22×Dic113D412yes—
55C2×C4×D113—21no—
56Dic452—24no—
57#14268637—24no—
58#14268636—24no—
59#14268635—24no—
60#14268635—24no—
61#14268632—24no—
62#14268631D414yescomplemented
63C2×#11744751C2312yes—
64#14268626—22no—
65C22⋊Dic113—24no—
66C22⋊Dic113—24no—
67C22⋊Dic113—22no—
68#14268624—22no—
69C4×Dic113D412yes—
70C4×Dic113C2311yescharacteristic
71C4×Dic113C2311yescharacteristic
72C24×C113—21no—
73#41×C113—22no—
74#41×C113—22no—
75C2×Q8×C113C2311yescharacteristic
76C2×D4×C113—21no—
77C2×D4×C113—22no—
78C2×D4×C113—22no—
79C2×D4×C113—21no—
80C2×D4×C113C2312yes—
81C2×D4×C113C2311yescharacteristic
82C22×C4×C113—21no—
83C22×C4×C113C2311yescharacteristic
84SD16×C113—24no—
85D8×C113—24no—
86M16×C113D414yescomplemented
87C2×C8×C113C2312yes—
88C23×C113—21no—
89C23×C113—22no—
90C23×C113—44no—
91C23×C113—42no—
92C23×C113C2×D411yescharacteristic
93C22×D113—21no—
94Q8×C113—21no—
95Q8×C113—21no—
96D4×C113—21no—
97D4×C113—22no—
98D4×C113—22no—
99D4×C113—21no—
100D4×C113—42no—
101D4×C113—42no—
102C2×C4×C113—21no—
103C2×C4×C113—42no—
104C2×C4×C113C2×D411yescharacteristic
105C2×C4×C113C2×D412yes—
106C2×C4×C113C2×D412yes—
107C2×C4×C113C2411yescharacteristic
108C22⋊D113—44no—
109C22⋊D113—42no—
110C2×Dic113—22no—
111C2×Dic113—21no—
112C2×Dic113—42no—
113C2×Dic113—42no—
114C2×Dic113C2×D412yes—
115C2×Dic113C2×D412yes—
116D452—21no—
117D452—21no—
118C4×D113—42no—
119Dic226—21no—
120Dic226—22no—
121Dic226—22no—
122Dic226—21no—
123Dic226—42no—
124C8×C113C2×D414yes—
125#11744751C2×D414yes—
126C22×C113—22no—
127C22×C113—41no—
128C22×C113—42no—
129C22×C113—42no—
130C22×C113—44no—
131C22×C113—84no—
132C22×C113#12712yes—
133C22×C113C22×D411yescharacteristic
134C2×D113—41no—
135C2×D113—42no—
136C4×C113—41no—
137C4×C113#12712yes—
138C4×C113C22×D411yescharacteristic
139C4×C113C22×D411yescharacteristic
140Dic113—22no—
141Dic113—22no—
142Dic113—42no—
143Dic113—41no—
144C2×C113—22no—
145C2×C113—81no—
146C2×C113—82no—
147C2×C113C2×#12711yescharacteristic
148C2×C113#57512yes—
149D113—81no—
150#3153—1131nomaximal, Sylow
151C113#315311yescharacteristic, complemented, Sylow
152C2×#137—1131no—
153C2×#137—1131no—
154C2×AGL(1,ℤ/8)—1131no—
155C2×AGL(1,ℤ/8)D11311yescharacteristic, complemented
156#532—1131no—
157C2×#124—1131no—
158#497—1132no—
159#496—1132no—
160#495—1132no—
161#495—1132no—
162C4×M16—1131no—
163C2×#41—1131no—
164C2×#41C2×D11311yescharacteristic
165C22×Q8—1131no—
166C22×D4C2×D11311yescharacteristic
167#137—2264no—
168#137—2264no—
169AGL(1,ℤ/8)—24no—
170AGL(1,ℤ/8)—2264no—
171C2×Q16—1132no—
172C2×Q16—1132no—
173C2×SD16—1132no—
174C2×SD16—1132no—
175C2×SD16—1132no—
176C2×SD16C2×D11312yes—
177C2×D8—1132no—
178C2×D8C2×D11312yes—
179C2×M16—1131no—
180C2×M16C2×D11311yescharacteristic
181C4⋊Q8—1131no—
182#127—1131no—
183#124—1131no—
184#124—1132no—
185#124—1132no—
186#124—1131no—
187#124—2262no—
188#121—2262no—
189C4×D4—2262no—
190C2×C22⋊C4—2262no—
191C2×C42—1131no—
192#97—1132no—
193C4×C8—1132no—
194C24—21no—
195#41—22no—
196#41—22no—
197#41—2262no—
198#41—2262no—
199C2×Q8—1132no—
200C2×Q8—1131no—
201C2×Q8—2262no—
202C2×Q8—2262no—
203C2×Q8—2261no—
204C2×Q8C22×D11311yescharacteristic
205C2×D4—21no—
206C2×D4—22no—
207C2×D4—21no—
208C2×D4—22no—
209C2×D4—1131no—
210C2×D4—2262no—
211C2×D4—2261no—
212C2×D4C22×D11312yes—
213C2×D4C22×D11311yescharacteristic
214C22×C4—21no—
215C22×C4—1132no—
216C22×C4—2261no—
217C22×C4—2261no—
218C22×C4C22×D11311yescharacteristic
219Q16—2264no—
220Q16—2264no—
221SD16—24no—
222SD16—2264no—
223SD16—2264no—
224SD16—2264no—
225D8—24no—
226D8—2264no—
227M16—1134no—
228M16C22⋊D11314yescomplemented
229C2×C8—1132no—
230C2×C8C22×D11312yes—
231C4⋊C4—2262no—
232C22⋊C4—2262no—
233C22⋊C4—2264no—
234C22⋊C4—2264no—
235C22⋊C4—2262no—
236C42—1132no—
237C42—1131no—
238C42—1131no—
239C23—22no—
240C23—21no—
241C23—44no—
242C23—42no—
243C23—2261no—
244C23D4×D11311yescharacteristic
245Q8—21no—
246Q8—21no—
247Q8—4522no—
248Q8—2261no—
249Q8—2262no—
250Q8—2262no—
251Q8—2261no—
252D4—22no—
253D4—21no—
254D4—21no—
255D4—22no—
256D4—42no—
257D4—42no—
258D4—4522no—
259D4—4524no—
260D4—2261no—
261D4—2261no—
262C2×C4—21no—
263C2×C4—42no—
264C2×C4—1132no—
265C2×C4—1132no—
266C2×C4—4522no—
267C2×C4—4522no—
268C2×C4—4522no—
269C2×C4—2262no—
270C2×C4—2261no—
271C2×C4D4×D11311yescharacteristic
272C2×C4D4×D11312yes—
273C2×C4C2×C22⋊D11312yes—
274C2×C4C23×D11311yescharacteristic
275C8—1134no—
276C8C2×C22⋊D11314yes—
277C22—22no—
278C22—41no—
279C22—42no—
280C22—42no—
281C22—44no—
282C22—84no—
283C22—4521no—
284C22—4522no—
285C22#1681361812yes—
286C22C2×D4×D11311yescharacteristic
287C4—41no—
288C4—4521no—
289C4—4522no—
290C4—2262no—
291C4—2262no—
292C4#1681361812yes—
293C4C2×D4×D11311yescharacteristic
294C4C22×C22⋊D11311yescharacteristic
295C2—22no—
296C2—82no—
297C2—81no—
298C2—9041no—
299C2C2×#1681361811yescharacteristic
300C2#2029939412yes—
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#3153
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×#16813655C2semidirect
2. C2×#16813652C2semidirect
3. C2×#16813648C2semidirect
4. #20299144C2semidirect
5. C2×#16813609C2semidirect
6. #20298989C2semidirect
7. #20298986C2semidirect
8. #20298981C2semidirect
9. #20298979C2semidirect
10. M16×Dic113C2semidirect
11. C2×AGL(1,ℤ/8)×C113C2semidirect
17. C2×Dic452C22semidirect
18. C2×#14268637C22semidirect
19. C2×#14268636C22semidirect
21. C2×#14268635C22semidirect
24. #16813620C22semidirect
25. #16813618C22semidirect
26. #16813611C22semidirect
30. #16813609C22semidirect
35. #16813532C22semidirect
36. C8×Dic113C22semidirect
62. #14268631D4semidirect
86. M16×C113D4semidirect
151. C113#3153semidirect
155. C2×AGL(1,ℤ/8)D113semidirect
228. M16C22⋊D113semidirect

Maximal subgroups

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

subgroupindex
C2×#168136552
C2×#168136522
C2×#168136482
#202991442
C2×#168136092
#202989892
#202989862
#202989812
#202989792
M16×Dic1132
C2×AGL(1,ℤ/8)×C1132
#3153113