Finite Group Databasebeta

← #24918198

Subgroups — #24918198

Order 10624.

446 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 119 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#19128248C212yesmaximal, complemented
2#19128098C211yesmaximal, characteristic, complemented
3#19128076C211yesmaximal, characteristic, complemented
4#19128046C212yesmaximal, complemented
5#19127955C211yesmaximal, characteristic, complemented
6#19127953C212yesmaximal, complemented
7#19127951C211yesmaximal, characteristic, complemented
8#19127814C211yesmaximal, characteristic, complemented
9#19127811C212yesmaximal, complemented
10#19127653C211yesmaximal, characteristic, complemented
11#442×C83C211yesmaximal, characteristic, complemented
12C2×#13050590—22no—
13C2×D664C2211yescharacteristic
14C2×#13050575C2212yes—
15C2×#13050573C2211yescharacteristic
16C2×#13050573C2211yescharacteristic
17#14581791C2212yes—
18#14581788C2212yes—
19#14581778—22no—
20#14581767C2212yes—
21#14581765—22no—
22#14581763C2212yes—
23C4×D332C2212yes—
24C4×C22⋊D83—22no—
25C42×D83C2211yescharacteristic
26#14581724C2211yescharacteristic
27#14581723C2212yes—
28#14581714C2212yes—
29#14581713C2212yes—
30#14581712C2212yes—
31#14581703C2211yescharacteristic
32C8×Dic83C2211yescharacteristic
33C8×Dic83C2211yescharacteristic
34C2×D8×C83C2211yescharacteristic
35C4×D4×C83C2212yes—
36#106×C83C2211yescharacteristic
37#102×C83C2212yes—
38#97×C83C2211yescharacteristic
39#13050590—48no—
40C2×Dic166—22no—
41C2×D332—22no—
42C2×D332C2312yes—
43C2×C22⋊D83—22no—
44C2×C22⋊D83—22no—
45C2×C22⋊D83—22no—
46C2×C4×D83—22no—
47C2×C4×D83D412yes—
48C2×C4×D83C2311yescharacteristic
49D664—22no—
50#13050575—24no—
51#13050573—22no—
52#13050573—22no—
53C2×C83⋊C8C2311yescharacteristic
54C2×C83⋊C8C2311yescharacteristic
55C4⋊Dic83C2312yes—
56Dic83⋊C4—22no—
57C22⋊Dic83—22no—
58(C2×D83)⋊C4—22no—
59(C2×D83)⋊C4—22no—
60C4×Dic83D412yes—
61C4×Dic83C2311yescharacteristic
62C2×D4×C83C2312yes—
63C22×C4×C83—22no—
64D8×C83—22no—
65C2×C8×C83C2311yescharacteristic
66C2×C8×C83C2311yescharacteristic
67C4⋊C4×C83C2312yes—
68C22⋊C4×C83—22no—
69C42×C83C2311yescharacteristic
70C23×C83—22no—
71C22×D83—22no—
72C22×D83C2×D411yescharacteristic
73D4×C83—22no—
74D4×C83—22no—
75C2×C4×C83—22no—
76C2×C4×C83—44no—
77C2×C4×C83C2×D412yes—
78C2×C4×C83C2411yescharacteristic
79C22⋊D83—44no—
80C22⋊D83—44no—
81C22⋊D83—44no—
82C2×Dic83—22no—
83C2×Dic83C2×D411yescharacteristic
84C2×Dic83C2×D412yes—
85D332—22no—
86D332—22no—
87D332—44no—
88C4×D83—21no—
89C4×D83—21no—
90C4×D83—24no—
91C4×D83—44no—
92Dic166—44no—
93C8×C83—21no—
94C8×C83#4112yescomplemented
95C83⋊C8—21no—
96C83⋊C8#4112yescomplemented
97C22×C83—42no—
98C22×C83—42no—
99C22×C83—42no—
100C22×C83C22×D411yescharacteristic
101C2×D83—21no—
102C2×D83—22no—
103C2×D83—42no—
104C2×D83—42no—
105C2×D83—42no—
106C4×C83—22no—
107C4×C83—42no—
108C4×C83C2×#4111yescharacteristic
109C4×C83C2×#4111yescharacteristic
110Dic83—21no—
111Dic83—42no—
112Dic83AGL(1,ℤ/8)14yes—
113C2×C83—82no—
114C2×C83#53211yescharacteristic
115C2×C83C2×AGL(1,ℤ/8)12yes—
116D83—41no—
117D83—82no—
118#3184—831nomaximal, Sylow
119C83#318411yescharacteristic, complemented, Sylow
120#532—832no—
121#498—831no—
122#493—831no—
123#486—832no—
124#469—831no—
125#469—831no—
126#469—832no—
127#442—832no—
128#442—831no—
129#442D8311yescharacteristic, complemented
130C4×M16—831no—
131C2×#41—1662no—
132C2×D8—832no—
133C2×D8—831no—
134C2×D8C2×D8311yescharacteristic
135C2×M16—831no—
136C2×M16—831no—
137C4⋊Q8—832no—
138#127—832no—
139#124—1662no—
140#121—832no—
141#121—832no—
142#121—1662no—
143C4×D4—832no—
144C4×D4—1662no—
145C4×D4C2×D8312yes—
146C2×C42—831no—
147#106—831no—
148#106—832no—
149#106C2×D8311yescharacteristic
150#102—832no—
151#102—832no—
152#102—832no—
153#102C2×D8312yes—
154#97—831no—
155#97C2×D8311yescharacteristic
156C4×C8—831no—
157C4×C8—831no—
158#41—3328no—
159C2×Q8—1662no—
160C2×D4—832no—
161C2×D4—1662no—
162C2×D4—1662no—
163C2×D4—1662no—
164C2×D4—1662no—
165C2×D4C22×D8312yes—
166C22×C4—22no—
167C22×C4—832no—
168C22×C4—831no—
169C22×C4—1662no—
170D8—22no—
171D8—1664no—
172D8—1662no—
173M16—1662no—
174M16—1662no—
175C2×C8—831no—
176C2×C8—831no—
177C2×C8C22×D8311yescharacteristic
178C2×C8C22×D8311yescharacteristic
179C4⋊C4—832no—
180C4⋊C4—1662no—
181C4⋊C4C22×D8312yes—
182C22⋊C4—22no—
183C22⋊C4—1662no—
184C22⋊C4—1662no—
185C22⋊C4—1662no—
186C42—832no—
187C42—831no—
188C42C22×D8311yescharacteristic
189C23—22no—
190C23—831no—
191C23—1662no—
192Q8—3324no—
193D4—22no—
194D4—22no—
195D4—3324no—
196D4—3324no—
197D4—3324no—
198D4—3324no—
199D4—1662no—
200D4—1662no—
201C2×C4—22no—
202C2×C4—44no—
203C2×C4—3324no—
204C2×C4—832no—
205C2×C4—831no—
206C2×C4—1664no—
207C2×C4—1662no—
208C2×C4—1661no—
209C2×C4—1661no—
210C2×C4D4×D8312yes—
211C2×C4C23×D8311yescharacteristic
212C8—21no—
213C8—832no—
214C8—1661no—
215C8#1305059012yescomplemented
216C22—42no—
217C22—42no—
218C22—42no—
219C22—3322no—
220C22—3322no—
221C22—3322no—
222C22—1662no—
223C22—1661no—
224C22C2×D4×D8311yescharacteristic
225C4—22no—
226C4—42no—
227C4—3322no—
228C4—834no—
229C4—1661no—
230C4#41×D8311yescharacteristic
231C4C2×#1305059011yescharacteristic
232C2—82no—
233C2—3321no—
234C2—6642no—
235C2#1912824811yescharacteristic
236C2AGL(1,ℤ/8)×D8312yes—
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#442×C83C2
socleC22×C83C22×D4
hypercenter#442D83
hypocenterC83#3184
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #442

Complements

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

NH ≅ G/Ntype
1. #19128248C2semidirect
2. #19128098C2semidirect
3. #19128076C2semidirect
4. #19128046C2semidirect
5. #19127955C2semidirect
6. #19127953C2semidirect
7. #19127951C2semidirect
8. #19127814C2semidirect
9. #19127811C2semidirect
10. #19127653C2semidirect
11. #442×C83C2semidirect
94. C8×C83#41semidirect
96. C83⋊C8#41semidirect
119. C83#3184semidirect
129. #442D83semidirect
215. C8#13050590semidirect

Maximal subgroups

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

subgroupindex
#191282482
#191280982
#191280762
#191280462
#191279552
#191279532
#191279512
#191278142
#191278112
#191276532
#442×C832
#318483