Finite Group Databasebeta

← #24918164

Subgroups — #24918164

Order 10624.

410 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 115 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
1#19128252C211yesmaximal, characteristic, complemented
2#19128230C211yesmaximal, characteristic, complemented
3#19128107C211yesmaximal, characteristic, complemented
4#19128091C211yesmaximal, characteristic, complemented
5#19128066C211yesmaximal, characteristic, complemented
6#19127936C212yesmaximal, complemented
7#19127933C212yesmaximal, complemented
8#19127810C212yesmaximal, complemented
9#19127804C212yesmaximal, complemented
10#97×D83C211yesmaximal, characteristic, complemented
11#489×C83C211yesmaximal, characteristic, complemented
12C2×#13050591—21no—
13C2×#13050577C2212yes—
14C2×#13050575C2212yes—
15C2×C8×D83C2212yes—
16#14581789C2211yescharacteristic
17#14581785—21no—
18#14581784C2211yescharacteristic
19#14581781—21no—
20#14581774—21no—
21#14581768C2211yescharacteristic
22#14581767—21no—
23#14581767C2212yes—
24#14581763C2211yescharacteristic
25Q8×Dic83C2211yescharacteristic
26D4×Dic83C2211yescharacteristic
27#14581754—21no—
28#14581754C2212yes—
29(C22⋊D83)⋊C4—21no—
30#14581748—21no—
31C42×D83C2211yescharacteristic
32#14581725C2212yes—
33#14581723C2212yes—
34#14581714C2212yes—
35#14581714C2212yes—
36#14581703C2211yescharacteristic
37#14581702C2212yes—
38C4⋊Q8×C83C2211yescharacteristic
39#124×C83C2211yescharacteristic
40#103×C83C2212yes—
41#102×C83C2212yes—
42#97×C83C2211yescharacteristic
43#13050591—44no—
44C2×D332—22no—
45C2×D332—21no—
46C2×D332—21no—
47C2×D332C2312yes—
48C2×C22⋊D83—21no—
49C22×Dic83—21no—
50C2×C4×D83—22no—
51C2×C4×D83D411yescharacteristic
52C2×C4×D83D411yescharacteristic
53C2×C4×D83C2311yescharacteristic
54#13050577—24no—
55#13050575—24no—
56C2×C83⋊C8C2312yes—
57C8×D83—24no—
58C4⋊Dic83—21no—
59C4⋊Dic83C2311yescharacteristic
60C4⋊Dic83C2311yescharacteristic
61Dic83⋊C4—21no—
62Dic83⋊C4—21no—
63Dic83⋊C4—21no—
64C22⋊Dic83—21no—
65C22⋊Dic83—21no—
66(C2×D83)⋊C4—21no—
67(C2×D83)⋊C4—22no—
68(C2×D83)⋊C4—22no—
69C4×Dic83—21no—
70C4×Dic83D411yescharacteristic
71C4×Dic83D411yescharacteristic
72C4×Dic83C2311yescharacteristic
73C2×Q8×C83—21no—
74C2×Q8×C83C2311yescharacteristic
75C2×D4×C83C2311yescharacteristic
76C2×C8×C83C2312yes—
77C4⋊C4×C83—21no—
78C4⋊C4×C83C2312yes—
79C22⋊C4×C83—21no—
80C22⋊C4×C83—21no—
81C42×C83C2311yescharacteristic
82C23×C83—21no—
83C22×D83—22no—
84C22×D83C2×D411yescharacteristic
85Q8×C83—22no—
86Q8×C83—42no—
87D4×C83—22no—
88C2×C4×C83—21no—
89C2×C4×C83—22no—
90C2×C4×C83C2×D411yescharacteristic
91C2×C4×C83C2×D411yescharacteristic
92C2×C4×C83C2411yescharacteristic
93C22⋊D83—42no—
94C2×Dic83—21no—
95C2×Dic83—21no—
96C2×Dic83—42no—
97C2×Dic83C2×D411yescharacteristic
98C2×Dic83C2×D411yescharacteristic
99C2×Dic83C2×D411yescharacteristic
100D332—22no—
101D332—22no—
102D332—42no—
103D332—42no—
104D332—44no—
105C4×D83—22no—
106C4×D83—22no—
107C4×D83—41no—
108C4×D83—44no—
109C8×C83—22no—
110C83⋊C8#4114yescomplemented
111C22×C83—41no—
112C22×C83—41no—
113C22×C83—41no—
114C22×C83C22×D411yescharacteristic
115C2×D83—21no—
116C2×D83—21no—
117C2×D83—42no—
118C2×D83—42no—
119C2×D83—42no—
120C2×D83AGL(1,ℤ/8)12yes—
121C4×C83—21no—
122C4×C83—21no—
123C4×C83—41no—
124C4×C83—42no—
125C4×C83C2×#4112yes—
126Dic83—21no—
127Dic83—21no—
128Dic83—21no—
129Dic83—41no—
130Dic83—41no—
131C2×C83—81no—
132C2×C83#53211yescharacteristic
133C2×C83C2×AGL(1,ℤ/8)11yescharacteristic
134C2×C83#57511yescharacteristic
135D83—22no—
136D83—82no—
137#3180—831nomaximal, Sylow
138C83#318011yescharacteristic, complemented, Sylow
139#532—831no—
140#529—831no—
141#501—831no—
142#496—831no—
143#490—831no—
144#489D8311yescharacteristic, complemented
145#466—832no—
146#465—832no—
147#442—832no—
148#440—832no—
149C2×#97—831no—
150C2×#41—1661no—
151C2×SD16—832no—
152C2×D8—832no—
153C22×C8—832no—
154C4⋊Q8C2×D8311yescharacteristic
155#127—831no—
156#126—1661no—
157#125—831no—
158#124—1661no—
159#124C2×D8311yescharacteristic
160#123—1661no—
161C22⋊Q8—831no—
162#121—831no—
163#121—832no—
164#121—1661no—
165C4×Q8—831no—
166C4×D4—831no—
167C4×D4—832no—
168C4×D4—1661no—
169C4×D4—1661no—
170#117—1661no—
171C2×C42—831no—
172#107—832no—
173#106—832no—
174#103C2×D8312yes—
175#102—832no—
176#102—832no—
177#102C2×D8312yes—
178#97—832no—
179#97—831no—
180#97C2×D8311yescharacteristic
181#41—3324no—
182C2×Q8—21no—
183C2×Q8C22×D8311yescharacteristic
184C2×D4—832no—
185C2×D4—1662no—
186C2×D4—1661no—
187C2×D4—1661no—
188C2×D4—1661no—
189C2×D4C22×D8311yescharacteristic
190C22×C4—831no—
191C22×C4—831no—
192C22×C4—831no—
193C22×C4—1662no—
194C22×C4—1661no—
195SD16—1664no—
196D8—1664no—
197C2×C8—832no—
198C2×C8—1664no—
199C2×C8C22×D8312yes—
200C4⋊C4—21no—
201C4⋊C4—831no—
202C4⋊C4—831no—
203C4⋊C4—1661no—
204C4⋊C4—1661no—
205C4⋊C4—1661no—
206C4⋊C4—1661no—
207C4⋊C4C22×D8312yes—
208C22⋊C4—21no—
209C22⋊C4—21no—
210C22⋊C4—1662no—
211C22⋊C4—1662no—
212C22⋊C4—1661no—
213C22⋊C4—1661no—
214C22⋊C4—1661no—
215C42—831no—
216C42—831no—
217C42—831no—
218C42—1661no—
219C42C22×D8311yescharacteristic
220C23—21no—
221C23—831no—
222C23—1662no—
223Q8—22no—
224Q8—42no—
225D4—22no—
226D4—3322no—
227D4—3322no—
228D4—3324no—
229D4—3322no—
230D4—1662no—
231D4—1662no—
232C2×C4—21no—
233C2×C4—22no—
234C2×C4—3321no—
235C2×C4—3324no—
236C2×C4—3322no—
237C2×C4—831no—
238C2×C4—831no—
239C2×C4—831no—
240C2×C4—1662no—
241C2×C4—1661no—
242C2×C4—1661no—
243C2×C4—1662no—
244C2×C4D4×D8311yescharacteristic
245C2×C4D4×D8311yescharacteristic
246C2×C4C23×D8311yescharacteristic
247C8—22no—
248C8—834no—
249C22—41no—
250C22—41no—
251C22—41no—
252C22—3322no—
253C22—3322no—
254C22—3322no—
255C22—832no—
256C22—1661no—
257C22—1661no—
258C22C2×D4×D8311yescharacteristic
259C4—21no—
260C4—21no—
261C4—42no—
262C4—41no—
263C4—3321no—
264C4—3321no—
265C4—1661no—
266C4—1661no—
267C4—1661no—
268C4#41×D8312yes—
269C2—81no—
270C2—6642no—
271C2—1662no—
272C2#1912825011yescharacteristic
273C2AGL(1,ℤ/8)×D8311yescharacteristic
274C2#1912850011yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#489×C83C2
socleC22×C83C22×D4
hypercenter#489D83
hypocenterC83#3180
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #489

Complements

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

NH ≅ G/Ntype
1. #19128252C2semidirect
2. #19128230C2semidirect
3. #19128107C2semidirect
4. #19128091C2semidirect
5. #19128066C2semidirect
6. #19127936C2semidirect
7. #19127933C2semidirect
8. #19127810C2semidirect
9. #19127804C2semidirect
10. #97×D83C2semidirect
11. #489×C83C2semidirect
110. C83⋊C8#41semidirect
138. C83#3180semidirect
144. #489D83semidirect

Maximal subgroups

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

subgroupindex
#191282522
#191282302
#191281072
#191280912
#191280662
#191279362
#191279332
#191278102
#191278042
#97×D832
#489×C832
#318083