Finite Group Databasebeta

← #28401052

Subgroups — #28401052

Order 14464.

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

normal subgroups only

This group has 564 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×#16813653C211yesmaximal, characteristic, complemented
2C2×#16813647C211yesmaximal, characteristic, complemented
3C2×#16813642C211yesmaximal, characteristic, complemented
4C22×#14268634C211yesmaximal, characteristic, complemented
5#20299167C211yesmaximal, characteristic, complemented
6#20298990C212yesmaximal, complemented
7#20298988C212yesmaximal, complemented
8#20298980C212yesmaximal, complemented
9#20298972C212yesmaximal, complemented
10#20298554C211yesmaximal, characteristic, complemented
11#536×C113C211yesmaximal, characteristic, complemented
12C2×#14268647C2211yescharacteristic
13C22×D452C2211yescharacteristic
14#16813653—24no—
15#16813647—24no—
16#16813642—24no—
17C2×#14268637C2211yescharacteristic, complemented
18C2×#14268637C2212yescomplemented
19C2×#14268635C2211yescharacteristic, complemented
20C2×#14268634—24no—
21C2×#14268634C2212yes—
22C2×#14268634C2212yes—
23C2×#14268634C2212yes—
24C2×#14268634C2211yescharacteristic, complemented
25C2×#14268634C2212yescomplemented
26C2×#14268632C2212yes—
27C2×#14268632C2211yescharacteristic, complemented
28C2×#14268631C2211yescharacteristic
29C22×#11744751C2211yescharacteristic
30#16813619C2212yes—
31#16813610C2212yes—
32#16813597—22no—
33#16813592—22no—
34#16813533C2212yescomplemented
35C4×#11744751C2212yescomplemented
36C2×#41×C113C2211yescharacteristic
37C22×Q8×C113C2211yescharacteristic
38C4⋊Q8×C113C2212yescomplemented
39#124×C113C2212yescomplemented
40#123×C113—22no—
41C22⋊Q8×C113—22no—
42#117×C113C2211yescharacteristic
43C23×D113—21no—
44#14268647—22no—
45#14268647—22no—
46C2×Dic226C2311yescharacteristic
47C2×D452—22no—
48C2×D452—22no—
49C2×D452—21no—
50C2×D452—24no—
51C2×D452C2312yes—
52C2×D452C2311yescharacteristic
53C2×C22⋊D113—21no—
54C2×C4×D113—21no—
55#14268637—22no—
56#14268637—24no—
57#14268635—22no—
58#14268634—24no—
59#14268634—24no—
60#14268634—24no—
61#14268634—22no—
62#14268634—24no—
63#14268632—24no—
64#14268632—22no—
65#14268631D414yescomplemented
66C2×#11744751D414yescomplemented
67C2×#11744751C2311yescharacteristic
68C2×#11744751C2311yescharacteristic
69C2×#11744751C2312yes—
70#14268624—24no—
71#41×C113—22no—
72#41×C113—22no—
73C2×Q8×C113—21no—
74C2×Q8×C113—22no—
75C2×Q8×C113—22no—
76C2×Q8×C113C2312yes—
77C2×Q8×C113C2311yescharacteristic
78C2×D4×C113—21no—
79C2×D4×C113C2311yescharacteristic
80C22×C4×C113—21no—
81C22×C4×C113—21no—
82C22×C4×C113C2311yescharacteristic
83C4⋊C4×C113—24no—
84C4⋊C4×C113D412yes—
85C22⋊C4×C113—24no—
86C22⋊C4×C113D412yes—
87C42×C113C2312yes—
88C23×C113—21no—
89C23×C113C2×D411yescharacteristic
90C22×D113—22no—
91C22×D113—21no—
92C22×D113—42no—
93C22×D113—82no—
94Q8×C113—22no—
95Q8×C113—22no—
96Q8×C113—21no—
97Q8×C113—21no—
98Q8×C113—42no—
99D4×C113—21no—
100D4×C113—21no—
101D4×C113—42no—
102C2×C4×C113—22no—
103C2×C4×C113—21no—
104C2×C4×C113—42no—
105C2×C4×C113—42no—
106C2×C4×C113C2×D412yes—
107C2×C4×C113C2×D412yes—
108C2×C4×C113C2×D414yes—
109C2×C4×C113C2×D411yescharacteristic
110C2×C4×C113C2411yescharacteristic
111C22⋊D113—42no—
112C2×Dic113—21no—
113D452—22no—
114D452—21no—
115D452—22no—
116D452—21no—
117D452—42no—
118D452—84no—
119C4×D113—42no—
120Dic226—21no—
121Dic226—21no—
122#11744751C2×D412yes—
123#11744751C2×D412yes—
124#11744751C2×D414yes—
125C22×C113—22no—
126C22×C113—41no—
127C22×C113—42no—
128C22×C113#12712yes—
129C22×C113C22×D411yescharacteristic
130C2×D113—41no—
131C2×D113—42no—
132C2×D113—42no—
133C2×D113—44no—
134C2×D113—84no—
135C4×C113—24no—
136C4×C113—41no—
137C4×C113—42no—
138C4×C113#12712yes—
139C4×C113C22×D411yescharacteristic
140C4×C113C22×D411yescharacteristic
141Dic113—41no—
142C2×C113—22no—
143C2×C113—81no—
144C2×C113C2×#12711yescharacteristic
145C2×C113#57712yes—
146D113—81no—
147D113—82no—
148#3154—1131nomaximal, Sylow
149C113#315411yescharacteristic, complemented, Sylow
150C2×#137—1131no—
151C2×AGL(1,ℤ/8)—1131no—
152C2×#135—1131no—
153C22×SD16—1131no—
154#536D11311yescharacteristic, complemented
155#535—1131no—
156#497—1132no—
157#496—1132no—
158#495—1132no—
159#492—1132no—
160#405—1131no—
161C2×#41—1131no—
162C2×#41C2×D11311yescharacteristic
163C22×Q8C2×D11311yescharacteristic
164C22×D4—1131no—
165#137—2264no—
166AGL(1,ℤ/8)—2264no—
167#135—2264no—
168C2×Q16—1131no—
169C2×Q16—1132no—
170C2×SD16—1132no—
171C2×SD16—1132no—
172C2×SD16—1131no—
173C2×SD16—1132no—
174C2×SD16—1132no—
175C2×SD16—1131no—
176C2×SD16—2264no—
177C2×D8—1131no—
178C2×D8—1132no—
179C2×M16—1131no—
180C22×C8—1131no—
181C4⋊Q8C2×D11312yescomplemented
182#127—1132no—
183#124—1132no—
184#124C2×D11312yescomplemented
185#123—22no—
186C22⋊Q8—22no—
187#121—2262no—
188#120—2262no—
189#117C2×D11311yescharacteristic
190#97—1132no—
191C4×C8—1132no—
192C24—2261no—
193#41—22no—
194#41—22no—
195#41—2262no—
196#41—2262no—
197C2×Q8—21no—
198C2×Q8—22no—
199C2×Q8—22no—
200C2×Q8—1131no—
201C2×Q8C22×D11312yes—
202C2×Q8C22×D11311yescharacteristic
203C2×D4—21no—
204C2×D4—1132no—
205C2×D4—1131no—
206C2×D4—2261no—
207C2×D4—2262no—
208C2×D4—2262no—
209C2×D4—2264no—
210C2×D4—2261no—
211C2×D4C22×D11311yescharacteristic
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—2261no—
215C22×C4C22×D11311yescharacteristic
216Q16—2262no—
217Q16—2264no—
218SD16—2262no—
219SD16—2264no—
220SD16—2264no—
221SD16—2262no—
222SD16—2264no—
223SD16—2264no—
224D8—2262no—
225D8—2264no—
226M16—1134no—
227C2×C8—1134no—
228C2×C8—1131no—
229C2×C8—1132no—
230C2×C8—1131no—
231C4⋊C4—24no—
232C4⋊C4C22⋊D11312yes—
233C22⋊C4—24no—
234C22⋊C4—2264no—
235C22⋊C4C22⋊D11312yes—
236C42C22×D11312yes—
237C23—21no—
238C23—9042no—
239C23—4522no—
240C23—2262no—
241C23—2261no—
242C23C2×C22⋊D11311yescharacteristic
243Q8—22no—
244Q8—21no—
245Q8—22no—
246Q8—21no—
247Q8—42no—
248Q8—2261no—
249Q8—2261no—
250D4—21no—
251D4—21no—
252D4—42no—
253D4—9044no—
254D4—4522no—
255D4—4522no—
256D4—2262no—
257D4—2261no—
258D4—2262no—
259D4—2261no—
260C2×C4—22no—
261C2×C4—21no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—4522no—
265C2×C4—2261no—
266C2×C4D4×D11312yes—
267C2×C4D4×D11312yes—
268C2×C4C2×C22⋊D11314yes—
269C2×C4C2×C22⋊D11311yescharacteristic
270C2×C4C23×D11311yescharacteristic
271C8—1132no—
272C8—1132no—
273C8—1134no—
274C22—22no—
275C22—41no—
276C22—42no—
277C22—9044no—
278C22—4521no—
279C22—4522no—
280C22—4522no—
281C22—4524no—
282C22#1681361812yes—
283C22C22×C22⋊D11311yescharacteristic
284C4—24no—
285C4—41no—
286C4—42no—
287C4—4521no—
288C4#1681361812yes—
289C4C2×D4×D11311yescharacteristic
290C4C2×D4×D11311yescharacteristic
291C2—22no—
292C2—81no—
293C2—9042no—
294C2—9041no—
295C2C2×#1681361811yescharacteristic
296C2#2029940612yes—
subgroupG/H
centerC22C22×C22⋊D113
commutator, cosocleC2×C4×C113C24
FrattiniC2×C4C23×D113
Fitting#536×C113C2
socleC22×C113C22×D4
hypercenter#536D113
hypocenterC113#3154
LCSG ▷ C2×C4×C113 ▷ C2×C113 ▷ C113
UCSC1 ◁ C22 ◁ #117 ◁ #536

Complements

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

NH ≅ G/Ntype
1. C2×#16813653C2semidirect
2. C2×#16813647C2semidirect
3. C2×#16813642C2semidirect
4. C22×#14268634C2semidirect
5. #20299167C2semidirect
6. #20298990C2semidirect
7. #20298988C2semidirect
8. #20298980C2semidirect
9. #20298972C2semidirect
10. #20298554C2semidirect
11. #536×C113C2semidirect
17. C2×#14268637C22semidirect
18. C2×#14268637C22semidirect
19. C2×#14268635C22semidirect
24. C2×#14268634C22semidirect
25. C2×#14268634C22semidirect
27. C2×#14268632C22semidirect
34. #16813533C22semidirect
35. C4×#11744751C22semidirect
38. C4⋊Q8×C113C22semidirect
39. #124×C113C22semidirect
65. #14268631D4semidirect
66. C2×#11744751D4semidirect
149. C113#3154semidirect
154. #536D113semidirect
181. C4⋊Q8C2×D113semidirect
184. #124C2×D113semidirect

Maximal subgroups

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

subgroupindex
C2×#168136532
C2×#168136472
C2×#168136422
C22×#142686342
#202991672
#202989902
#202989882
#202989802
#202989722
#202985542
#536×C1132
#3154113