Finite Group Databasebeta

← #24917940

Subgroups — #24917940

Order 10624.

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×#14581822C211yesmaximal, characteristic, complemented
2C2×#14581817C211yesmaximal, characteristic, complemented
3C22×#13050579C211yesmaximal, characteristic, complemented
4#19128272C211yesmaximal, characteristic, complemented
5#19128265C211yesmaximal, characteristic, complemented
6#19128094C212yesmaximal, complemented
7#19128089C212yesmaximal, complemented
8#19128082C212yesmaximal, complemented
9#19128072C212yesmaximal, complemented
10#19127656C211yesmaximal, characteristic, complemented
11C2×#135×C83C211yesmaximal, characteristic, complemented
12C22×Dic166C2211yescharacteristic
13C22×D332C2211yescharacteristic
14#14581822—24no—
15#14581817—24no—
16C2×#13050580C2212yescomplemented
17C2×#13050579—24no—
18C2×#13050579C2212yes—
19C2×#13050579C2212yes—
20C2×#13050578C2212yescomplemented
21C2×#13050577C2212yes—
22C2×#13050575C2212yes—
23C2×#13050574C2211yescharacteristic
24#14581790C2212yescomplemented
25#14581788C2212yes—
26#14581781C2212yes—
27#14581779C2212yescomplemented
28#14581773—22no—
29#14581771—22no—
30#14581766—22no—
31#14581760—22no—
32#14581751C2211yescharacteristic
33#14581702C2212yescomplemented
34C8×Dic83C2212yescomplemented
35C2×#41×C83C2211yescharacteristic
36C2×#41×C83C2211yescharacteristic
37#135×C83—24no—
38C2×Q16×C83C2211yescharacteristic, complemented
39C2×SD16×C83C2211yescharacteristic, complemented
40C2×SD16×C83C2211yescharacteristic, complemented
41C2×D8×C83C2211yescharacteristic, complemented
42C22×C8×C83C2211yescharacteristic
43C23×D83—21no—
44C2×Dic166—22no—
45C2×Dic166—22no—
46C2×Dic166—21no—
47C2×Dic166C2312yes—
48C2×D332—22no—
49C2×D332—22no—
50C2×D332—21no—
51C2×D332C2312yes—
52C2×C22⋊D83—24no—
53C22×Dic83—21no—
54#13050580—24no—
55#13050579—24no—
56#13050579—24no—
57#13050578—24no—
58#13050577—24no—
59#13050575—24no—
60#13050574D414yescomplemented
61C2×C83⋊C8C2312yes—
62C4⋊Dic83D412yes—
63Dic83⋊C4—24no—
64C22⋊Dic83—24no—
65C22⋊Dic83D412yes—
66(C2×D83)⋊C4—24no—
67C4×Dic83C2312yes—
68#41×C83—22no—
69#41×C83—22no—
70#41×C83—22no—
71#41×C83—22no—
72C2×Q8×C83C2311yescharacteristic
73C2×Q8×C83C2311yescharacteristic
74C2×D4×C83—21no—
75C2×D4×C83—21no—
76C2×D4×C83C2311yescharacteristic
77C2×D4×C83C2311yescharacteristic
78C22×C4×C83—21no—
79C22×C4×C83—21no—
80C22×C4×C83C2311yescharacteristic
81Q16×C83—22no—
82SD16×C83—22no—
83SD16×C83—22no—
84D8×C83—22no—
85C2×C8×C83D414yescomplemented
86C2×C8×C83C2311yescharacteristic
87C2×C8×C83C2311yescharacteristic
88C23×C83—21no—
89C23×C83—21no—
90C23×C83C2×D411yescharacteristic
91C22×D83—22no—
92C22×D83—42no—
93C22×D83—82no—
94Q8×C83—21no—
95Q8×C83—21no—
96Q8×C83—21no—
97Q8×C83—21no—
98D4×C83—21no—
99D4×C83—21no—
100D4×C83—21no—
101D4×C83—21no—
102D4×C83—42no—
103D4×C83—42no—
104C2×C4×C83—21no—
105C2×C4×C83—21no—
106C2×C4×C83—42no—
107C2×C4×C83—42no—
108C2×C4×C83C2×D411yescharacteristic
109C2×C4×C83C2×D412yes—
110C2×C4×C83C2×D412yes—
111C2×C4×C83C2411yescharacteristic
112C22⋊D83—84no—
113C2×Dic83—22no—
114C2×Dic83—42no—
115C2×Dic83C2×D414yes—
116D332—22no—
117D332—22no—
118D332—42no—
119Dic166—22no—
120Dic166—22no—
121Dic166—42no—
122C8×C83C2×D412yes—
123C8×C83C2×D412yes—
124C83⋊C8C2×D414yes—
125C22×C83—22no—
126C22×C83—41no—
127C22×C83—41no—
128C22×C83—42no—
129C22×C83—42no—
130C22×C83#12712yes—
131C22×C83C22×D411yescharacteristic
132C2×D83—42no—
133C2×D83—44no—
134C2×D83—84no—
135C4×C83—41no—
136C4×C83—41no—
137C4×C83#12712yes—
138C4×C83C22×D411yescharacteristic
139C4×C83C22×D411yescharacteristic
140Dic83—24no—
141Dic83—42no—
142C2×C83—22no—
143C2×C83—81no—
144C2×C83—81no—
145C2×C83C2×#12711yescharacteristic
146C2×C83#57712yes—
147D83—82no—
148#3154—831nomaximal, Sylow
149C83#315411yescharacteristic, complemented, Sylow
150C2×#137—831no—
151C2×AGL(1,ℤ/8)—831no—
152C2×#135D8311yescharacteristic, complemented
153C22×SD16—831no—
154#536—831no—
155#535—831no—
156#497—832no—
157#496—832no—
158#495—832no—
159#492—832no—
160#405—831no—
161C2×#41C2×D8311yescharacteristic
162C2×#41C2×D8311yescharacteristic
163C22×Q8—831no—
164C22×D4—831no—
165#137—1664no—
166AGL(1,ℤ/8)—1664no—
167#135—24no—
168C2×Q16—832no—
169C2×Q16C2×D8311yescharacteristic, complemented
170C2×SD16—832no—
171C2×SD16—832no—
172C2×SD16—832no—
173C2×SD16—832no—
174C2×SD16—1664no—
175C2×SD16C2×D8311yescharacteristic, complemented
176C2×SD16C2×D8311yescharacteristic, complemented
177C2×D8—832no—
178C2×D8C2×D8311yescharacteristic, complemented
179C2×M16—831no—
180C22×C8C2×D8311yescharacteristic
181C4⋊Q8—832no—
182#127—832no—
183#124—832no—
184#124—832no—
185#123—1662no—
186C22⋊Q8—1662no—
187#121—1662no—
188#120—1662no—
189#117—831no—
190#97—832no—
191C4×C8—832no—
192C24—1661no—
193#41—22no—
194#41—22no—
195#41—22no—
196#41—22no—
197C2×Q8—832no—
198C2×Q8—1662no—
199C2×Q8—1662no—
200C2×Q8—1661no—
201C2×Q8C22×D8311yescharacteristic
202C2×Q8C22×D8311yescharacteristic
203C2×D4—21no—
204C2×D4—21no—
205C2×D4—832no—
206C2×D4—1664no—
207C2×D4—1661no—
208C2×D4—1662no—
209C2×D4—1662no—
210C2×D4C22×D8311yescharacteristic
211C2×D4C22×D8311yescharacteristic
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—1661no—
215C22×C4C22×D8311yescharacteristic
216Q16—22no—
217Q16—1664no—
218SD16—22no—
219SD16—22no—
220SD16—1664no—
221SD16—1664no—
222SD16—1664no—
223SD16—1664no—
224D8—22no—
225D8—1664no—
226M16—834no—
227C2×C8—832no—
228C2×C8C22⋊D8314yescomplemented
229C2×C8C22×D8311yescharacteristic
230C2×C8C22×D8311yescharacteristic
231C4⋊C4—832no—
232C4⋊C4—1664no—
233C22⋊C4—832no—
234C22⋊C4—1664no—
235C22⋊C4—1664no—
236C42—832no—
237C23—21no—
238C23—21no—
239C23—3322no—
240C23—6642no—
241C23—1662no—
242C23D4×D8311yescharacteristic
243Q8—21no—
244Q8—21no—
245Q8—21no—
246Q8—21no—
247Q8—3322no—
248Q8—1662no—
249Q8—1662no—
250D4—21no—
251D4—21no—
252D4—21no—
253D4—21no—
254D4—42no—
255D4—42no—
256D4—3322no—
257D4—6644no—
258D4—1662no—
259D4—1662no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—3322no—
265C2×C4—834no—
266C2×C4—1662no—
267C2×C4D4×D8311yescharacteristic
268C2×C4D4×D8312yes—
269C2×C4C2×C22⋊D8312yes—
270C2×C4C23×D8311yescharacteristic
271C8—834no—
272C8C2×C22⋊D8312yes—
273C8C2×C22⋊D8312yes—
274C22—22no—
275C22—41no—
276C22—41no—
277C22—42no—
278C22—42no—
279C22—3322no—
280C22—3324no—
281C22—6644no—
282C22#1458178812yes—
283C22C2×D4×D8311yescharacteristic
284C4—41no—
285C4—41no—
286C4—3322no—
287C4—1664no—
288C4#1458178812yes—
289C4C2×D4×D8311yescharacteristic
290C4C22×C22⋊D8311yescharacteristic
291C2—22no—
292C2—81no—
293C2—81no—
294C2—6642no—
295C2C2×#1458178811yescharacteristic
296C2#1912850812yes—
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
FittingC2×#135×C83C2
socleC22×C83C22×D4
hypercenterC2×#135D83
hypocenterC83#3154
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#135

Complements

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

NH ≅ G/Ntype
1. C2×#14581822C2semidirect
2. C2×#14581817C2semidirect
3. C22×#13050579C2semidirect
4. #19128272C2semidirect
5. #19128265C2semidirect
6. #19128094C2semidirect
7. #19128089C2semidirect
8. #19128082C2semidirect
9. #19128072C2semidirect
10. #19127656C2semidirect
11. C2×#135×C83C2semidirect
16. C2×#13050580C22semidirect
20. C2×#13050578C22semidirect
24. #14581790C22semidirect
27. #14581779C22semidirect
33. #14581702C22semidirect
34. C8×Dic83C22semidirect
38. C2×Q16×C83C22semidirect
39. C2×SD16×C83C22semidirect
40. C2×SD16×C83C22semidirect
41. C2×D8×C83C22semidirect
60. #13050574D4semidirect
85. C2×C8×C83D4semidirect
149. C83#3154semidirect
152. C2×#135D83semidirect
169. C2×Q16C2×D83semidirect
175. C2×SD16C2×D83semidirect
176. C2×SD16C2×D83semidirect
178. C2×D8C2×D83semidirect
228. C2×C8C22⋊D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145818222
C2×#145818172
C22×#130505792
#191282722
#191282652
#191280942
#191280892
#191280822
#191280722
#191276562
C2×#135×C832
#315483