Finite Group Databasebeta

← #24919882

Subgroups — #24919882

Order 10624.

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

normal subgroups only

This group has 364 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#19128415C212yesmaximal, complemented
2#19128105C211yesmaximal, characteristic, complemented
3#19128024C212yesmaximal, complemented
4#19128022C212yesmaximal, complemented
5#19127989C212yesmaximal, complemented
6#19127934C211yesmaximal, characteristic, complemented
7D8×Dic83C211yesmaximal, characteristic, complemented
8C8×C22⋊D83C211yesmaximal, characteristic, complemented
9#19127739C211yesmaximal, characteristic, complemented
10#19127738C211yesmaximal, characteristic, complemented
11#466×C83C211yesmaximal, characteristic, complemented
12C2×C8×D83C2211yescharacteristic
13#14581791C2212yes—
14#14581779—22no—
15#14581778—22no—
16#14581764—22no—
17#14581763C2212yes—
18D4×Dic83C2212yes—
19C4×C22⋊D83C2211yescharacteristic
20#14581751C2212yes—
21#14581750C2212yes—
22#14581748—22no—
23#14581726C2211yescharacteristic
24#14581726C2211yescharacteristic
25#14581725C2212yes—
26#14581724C2212yes—
27#14581723C2212yes—
28Dic83⋊C8C2211yescharacteristic
29#14581713C2212yes—
30#14581713C2212yes—
31C22⋊(C83⋊C8)C2211yescharacteristic
32(C2×D83)⋊C8C2211yescharacteristic
33C8×Dic83C2211yescharacteristic
34C2×D8×C83C2211yescharacteristic
35C22×C8×C83C2211yescharacteristic
36#121×C83C2212yes—
37#107×C83C2211yescharacteristic
38#102×C83C2212yes—
39C2×Dic166—22no—
40C2×C22⋊D83—22no—
41C2×C22⋊D83D411yescharacteristic
42C22×Dic83—22no—
43C2×C4×D83C2311yescharacteristic
44C2×C83⋊C8C2311yescharacteristic
45C2×C83⋊C8C2311yescharacteristic
46C8×D83—22no—
47C4⋊Dic83C2312yes—
48C4⋊Dic83C2312yes—
49Dic83⋊C4—22no—
50Dic83⋊C4D411yescharacteristic
51C22⋊Dic83—22no—
52C22⋊Dic83—22no—
53C22⋊Dic83—22no—
54C22⋊Dic83D411yescharacteristic
55(C2×D83)⋊C4—22no—
56(C2×D83)⋊C4D411yescharacteristic
57C4×Dic83—22no—
58C4×Dic83—22no—
59C4×Dic83C2311yescharacteristic
60C2×D4×C83—22no—
61C2×D4×C83C2312yes—
62C22×C4×C83C2311yescharacteristic
63D8×C83—22no—
64C2×C8×C83—22no—
65C2×C8×C83C2311yescharacteristic
66C2×C8×C83C2311yescharacteristic
67C4⋊C4×C83C2312yes—
68C22⋊C4×C83—22no—
69C23×C83—22no—
70C23×C83C2×D411yescharacteristic
71C22×D83C2×D411yescharacteristic
72D4×C83—22no—
73D4×C83—22no—
74D4×C83—82no—
75C2×C4×C83—21no—
76C2×C4×C83—21no—
77C2×C4×C83—22no—
78C2×C4×C83C2×D411yescharacteristic
79C2×C4×C83C2411yescharacteristic
80C22⋊D83—22no—
81C22⋊D83—82no—
82C2×Dic83—22no—
83C2×Dic83—22no—
84C2×Dic83—44no—
85C2×Dic83C2×D411yescharacteristic
86C2×Dic83C2×D411yescharacteristic
87C2×Dic83C2×D411yescharacteristic
88C4×D83—21no—
89C4×D83—21no—
90Dic166—82no—
91C8×C83—21no—
92C8×C83#4112yes—
93C83⋊C8—21no—
94C83⋊C8—21no—
95C22×C83—21no—
96C22×C83—21no—
97C22×C83—21no—
98C22×C83—42no—
99C22×C83—42no—
100C22×C83—42no—
101C22×C83C22×D411yescharacteristic
102C2×D83—21no—
103C2×D83—21no—
104C2×D83—21no—
105C4×C83—21no—
106C4×C83—42no—
107C4×C83C2×#4111yescharacteristic
108C4×C832−1+411yescharacteristic
109Dic83—21no—
110Dic83—21no—
111Dic83—42no—
112Dic83—42no—
113Dic83#13512yes—
114C2×C83—41no—
115C2×C83—82no—
116C2×C83#54711yescharacteristic
117C2×C83C2×#13511yescharacteristic
118C2×C83#57611yescharacteristic
119D83—41no—
120#3341—831nomaximal, Sylow
121C83#334111yescharacteristic, complemented, Sylow
122#555—832no—
123#500—831no—
124#482—832no—
125#482—832no—
126#476—832no—
127#466—831no—
128#466D8311yescharacteristic, complemented
129C4×D8—831no—
130C8×D4—831no—
131#427—831no—
132#427—831no—
133C2×D8C2×D8311yescharacteristic
134C22×C8—831no—
135C22×C8C2×D8311yescharacteristic
136C4⋊Q8—832no—
137#124—1662no—
138#124—1662no—
139#121—832no—
140#121—1662no—
141#121C2×D8312yes—
142C4×D4—831no—
143C4×D4—832no—
144#117—832no—
145#117—832no—
146#117—1662no—
147#107—831no—
148#107—831no—
149#107—832no—
150#107C2×D8311yescharacteristic
151#106—832no—
152#106—832no—
153C4⋊C8—831no—
154#102—832no—
155#102—832no—
156#102C2×D8312yes—
157C22⋊C8—831no—
158C22⋊C8—831no—
159C4×C8—831no—
160C2×Q8—1662no—
161C2×D4—22no—
162C2×D4—831no—
163C2×D4—1662no—
164C2×D4C22×D8312yes—
165C22×C4—831no—
166C22×C4—1662no—
167C22×C4C22×D8311yescharacteristic
168D8—22no—
169C2×C8—22no—
170C2×C8—831no—
171C2×C8—831no—
172C2×C8—1662no—
173C2×C8C22×D8311yescharacteristic
174C2×C8C22×D8311yescharacteristic
175C4⋊C4—831no—
176C4⋊C4—832no—
177C4⋊C4—832no—
178C4⋊C4—1662no—
179C4⋊C4C22×D8312yes—
180C22⋊C4—22no—
181C22⋊C4—831no—
182C22⋊C4—831no—
183C22⋊C4—1662no—
184C22⋊C4—1662no—
185C22⋊C4—1662no—
186C22⋊C4—1662no—
187C42—831no—
188C42—1662no—
189C42—1662no—
190C23—22no—
191C23—831no—
192C23D4×D8311yescharacteristic
193Q8—6642no—
194D4—22no—
195D4—22no—
196D4—82no—
197D4—6642no—
198D4—1662no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—22no—
202C2×C4—3324no—
203C2×C4—831no—
204C2×C4—831no—
205C2×C4—831no—
206C2×C4—1662no—
207C2×C4—1662no—
208C2×C4—1661no—
209C2×C4—1661no—
210C2×C4D4×D8311yescharacteristic
211C2×C4C23×D8311yescharacteristic
212C8—21no—
213C8—1661no—
214C8—1661no—
215C8#1305058912yes—
216C22—21no—
217C22—21no—
218C22—21no—
219C22—42no—
220C22—42no—
221C22—42no—
222C22—1661no—
223C22—1661no—
224C22—1661no—
225C22C2×D4×D8311yescharacteristic
226C4—21no—
227C4—42no—
228C4—3322no—
229C4—3322no—
230C4—832no—
231C4—1661no—
232C4—1661no—
233C4C2×#1305058911yescharacteristic
234C4#1458183911yescharacteristic
235C2—41no—
236C2—82no—
237C2—3321no—
238C2#1912835111yescharacteristic
239C2#135×D8311yescharacteristic
240C2#1912850211yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#466×C83C2
socleC22×C83C22×D4
hypercenter#466D83
hypocenterC83#3341
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ #466

Complements

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

NH ≅ G/Ntype
1. #19128415C2semidirect
2. #19128105C2semidirect
3. #19128024C2semidirect
4. #19128022C2semidirect
5. #19127989C2semidirect
6. #19127934C2semidirect
7. D8×Dic83C2semidirect
8. C8×C22⋊D83C2semidirect
9. #19127739C2semidirect
10. #19127738C2semidirect
11. #466×C83C2semidirect
121. C83#3341semidirect
128. #466D83semidirect

Maximal subgroups

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

subgroupindex
#191284152
#191281052
#191280242
#191280222
#191279892
#191279342
D8×Dic832
C8×C22⋊D832
#191277392
#191277382
#466×C832
#334183