Finite Group Databasebeta

← #24920116

Subgroups — #24920116

Order 10624.

350 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 117 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#19128425C211yesmaximal, characteristic, complemented
2#19128408C211yesmaximal, characteristic, complemented
3#19128057C212yesmaximal, complemented
4#19128051C211yesmaximal, characteristic, complemented
5#19128006C212yesmaximal, complemented
6#19127992C211yesmaximal, characteristic, complemented
7#19127918C212yesmaximal
8#19127825C211yesmaximal, characteristic, complemented
9SD16×Dic83C212yesmaximal, complemented
10C4×#13050575C211yesmaximal, characteristic
11#461×C83C211yesmaximal, characteristic, complemented
12C2×#13050575C2211yescharacteristic
13#14581792C2211yescharacteristic
14#14581787—22no—
15#14581783C2212yes—
16#14581781C2212yes—
17#14581772—21no—
18#14581771—21no—
19#14581769—22no—
20Q8×Dic83C2212yes—
21C4×Dic166C2211yescharacteristic
22C4×D332C2211yescharacteristic
23D4×Dic83C2212yes—
24#14581750—21no—
25C2×C4⋊Dic83—21no—
26#14581725C2211yescharacteristic
27#14581724C2212yes—
28Dic83⋊C8C2212yes—
29#14581716C2212yes—
30#14581716C2212yes—
31#14581714C2212yes—
32#14581713C2211yescharacteristic
33#14581712C2211yescharacteristic
34C4×C83⋊C8C2211yescharacteristic
35C8×Dic83C2212yes—
36C2×SD16×C83C2212yes—
37C4⋊Q8×C83C2211yescharacteristic
38C4×D4×C83C2211yescharacteristic
39C4⋊C8×C83C2211yescharacteristic
40#103×C83C2212yes—
41C2×Dic166—21no—
42C2×Dic166D411yescharacteristic
43C2×D332C2311yescharacteristic
44C22×Dic83—22no—
45C2×C4×D83—21no—
46#13050575—22no—
47C2×C83⋊C8C2311yescharacteristic
48C2×C83⋊C8C2311yescharacteristic
49C4⋊Dic83—21no—
50C4⋊Dic83—22no—
51C4⋊Dic83—22no—
52C4⋊Dic83D411yescharacteristic
53C4⋊Dic83C2312yes—
54C4⋊Dic83C2311yescharacteristic
55Dic83⋊C4—22no—
56Dic83⋊C4D412yes—
57C22⋊Dic83—22no—
58(C2×D83)⋊C4—21no—
59(C2×D83)⋊C4—22no—
60(C2×D83)⋊C4—22no—
61C4×Dic83—22no—
62C4×Dic83C2312yes—
63C2×Q8×C83C2312yes—
64C2×D4×C83C2311yescharacteristic
65C22×C4×C83—21no—
66SD16×C83—24no—
67C2×C8×C83C2312yes—
68C4⋊C4×C83—21no—
69C4⋊C4×C83—21no—
70C4⋊C4×C83C2311yescharacteristic
71C22⋊C4×C83—21no—
72C42×C83C2311yescharacteristic
73C23×C83—21no—
74C22×D83—21no—
75Q8×C83—22no—
76Q8×C83—22no—
77D4×C83—21no—
78D4×C83#4112yes—
79C2×C4×C83—21no—
80C2×C4×C83—22no—
81C2×C4×C83—42no—
82C2×C4×C83C2×D411yescharacteristic
83C2×C4×C83C2×D411yescharacteristic
84C2×C4×C83C2411yescharacteristic
85C2×Dic83—21no—
86C2×Dic83—22no—
87C2×Dic83—82no—
88C2×Dic83C2×D412yes—
89C2×Dic83C2×D412yes—
90D332—21no—
91D332—21no—
92C4×D83—42no—
93Dic166—22no—
94Dic166—81no—
95C8×C83—22no—
96C83⋊C8—21no—
97C83⋊C8—21no—
98C22×C83—22no—
99C22×C83—41no—
100C22×C83—41no—
101C22×C83C22×D411yescharacteristic
102C2×D83—41no—
103C2×D83—41no—
104C2×D83—41no—
105C4×C83—21no—
106C4×C83—42no—
107C4×C83—41no—
108C4×C83#13512yes—
109C4×C83C2×#4111yescharacteristic
110C4×C832−1+411yescharacteristic
111Dic83—22no—
112Dic83—22no—
113Dic83—41no—
114Dic83—42no—
115C2×C83—42no—
116C2×C83#54811yescharacteristic
117C2×C83C2×#13511yescharacteristic
118C2×C83#57711yescharacteristic
119D83—81no—
120#3360—831nomaximal, Sylow
121C83#336011yescharacteristic, complemented, Sylow
122#556—831no—
123#554—831no—
124#488—832no—
125#487—831no—
126#479—832no—
127#476—831no—
128#463—832no—
129#461D8311yescharacteristic, complemented
130C8×Q8—831no—
131C4×SD16—832no—
132C4×D8—831no—
133C2×SD16C2×D8312yes—
134C2×D8—831no—
135C4⋊Q8—831no—
136C4⋊Q8C2×D8311yescharacteristic
137#126—1662no—
138#125—832no—
139#124—832no—
140C22⋊Q8—1662no—
141C22⋊Q8—1661no—
142C22⋊Q8—1661no—
143C4×Q8—831no—
144C4×Q8—832no—
145C4×D4—832no—
146C4×D4—831no—
147C4×D4C2×D8311yescharacteristic
148#117—1661no—
149C2×C4⋊C4—1661no—
150#107—831no—
151#106—832no—
152C4⋊C8—832no—
153C4⋊C8C2×D8311yescharacteristic
154#103—832no—
155#103—832no—
156#103C2×D8312yes—
157#102—831no—
158#102—832no—
159#102—831no—
160C4×C8—832no—
161C4×C8—831no—
162C2×Q8—831no—
163C2×Q8—1661no—
164C2×Q8C22×D8312yes—
165C2×D4—831no—
166C2×D4C22×D8311yescharacteristic
167C22×C4—21no—
168C22×C4—1661no—
169C22×C4—1662no—
170SD16—24no—
171D8—1662no—
172C2×C8—831no—
173C2×C8—831no—
174C2×C8C22×D8312yes—
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—831no—
178C4⋊C4—832no—
179C4⋊C4—832no—
180C4⋊C4—831no—
181C4⋊C4—1662no—
182C4⋊C4—1662no—
183C4⋊C4—1662no—
184C4⋊C4—1661no—
185C4⋊C4C22×D8311yescharacteristic
186C22⋊C4—21no—
187C22⋊C4—1662no—
188C22⋊C4—1662no—
189C22⋊C4—1662no—
190C22⋊C4—1661no—
191C42—832no—
192C42—1662no—
193C42C22×D8311yescharacteristic
194C23—21no—
195C23—1661no—
196Q8—22no—
197Q8—22no—
198Q8—6641no—
199Q8—1662no—
200D4—21no—
201D4—1661no—
202D4—1661no—
203D4#1305058912yes—
204C2×C4—22no—
205C2×C4—21no—
206C2×C4—42no—
207C2×C4—3322no—
208C2×C4—832no—
209C2×C4—832no—
210C2×C4—6642no—
211C2×C4—1661no—
212C2×C4—1662no—
213C2×C4D4×D8311yescharacteristic
214C2×C4D4×D8311yescharacteristic
215C2×C4C23×D8311yescharacteristic
216C8—22no—
217C8—1661no—
218C8—1661no—
219C22—22no—
220C22—41no—
221C22—41no—
222C22—3321no—
223C22—3321no—
224C22—3321no—
225C22C2×D4×D8311yescharacteristic
226C4—21no—
227C4—42no—
228C4—41no—
229C4—3321no—
230C4—3322no—
231C4—1662no—
232C4—1662no—
233C4#1458181312yes—
234C4C2×#1305058911yescharacteristic
235C4#1458183911yescharacteristic
236C2—42no—
237C2—6641no—
238C2#1912835711yescharacteristic
239C2C2×#1458181311yescharacteristic
240C2#1912851011yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#461×C83C2
socleC22×C83C22×D4
prime radicalC2×#13050575C22
hypercenter#461D83
hypocenterC83#3360
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. #19128425C2semidirect
2. #19128408C2semidirect
3. #19128057C2semidirect
4. #19128051C2semidirect
5. #19128006C2semidirect
6. #19127992C2semidirect
8. #19127825C2semidirect
9. SD16×Dic83C2semidirect
11. #461×C83C2semidirect
121. C83#3360semidirect
129. #461D83semidirect

Maximal subgroups

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

subgroupindex
#191284252
#191284082
#191280572
#191280512
#191280062
#191279922
#191279182
#191278252
SD16×Dic832
C4×#130505752
#461×C832
#336083