Finite Group Databasebeta

← #24920108

Subgroups — #24920108

Order 10624.

350 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 123 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#19128411C211yesmaximal, characteristic, complemented
2#19128055C212yesmaximal, complemented
3#19128051C211yesmaximal, characteristic, complemented
4#19128008C212yesmaximal, complemented
5#19127991C211yesmaximal, characteristic, complemented
6#19127913C212yesmaximal
7#19127903C211yesmaximal, characteristic, complemented
8Q8×C83⋊C8C211yesmaximal, characteristic, complemented
9C4×#13050577C212yesmaximal, complemented
10C4×#13050575C211yesmaximal, characteristic
11#556×C83C211yesmaximal, characteristic, complemented
12C2×#13050577C2212yes—
13C2×#13050575C2211yescharacteristic
14#14581792C2211yescharacteristic
15#14581784C2212yes—
16#14581770—22no—
17#14581768—21no—
18C4×D332C2211yescharacteristic
19C4×D332C2212yes—
20C4⋊C4×D83—21no—
21#14581725C2211yescharacteristic
22#14581723C2212yes—
23C4⋊(C83⋊C8)C2211yescharacteristic
24C4⋊(C83⋊C8)C2212yes—
25#14581716C2212yes—
26#14581716C2212yes—
27#14581716C2212yes—
28#14581713C2212yes—
29#14581713C2211yescharacteristic
30#14581712C2211yescharacteristic
31C4×C83⋊C8C2211yescharacteristic
32C4×C83⋊C8C2212yes—
33C4⋊Q8×C83C2211yescharacteristic
34#126×C83—22no—
35#124×C83C2212yes—
36C22⋊Q8×C83—21no—
37C4×Q8×C83C2212yes—
38C4×Q8×C83C2211yescharacteristic
39C4×D4×C83C2211yescharacteristic
40#117×C83—21no—
41C2×Dic166—21no—
42C2×D332C2311yescharacteristic
43C2×C4×D83—21no—
44C2×C4×D83—22no—
45#13050577—24no—
46#13050575—22no—
47C2×C83⋊C8C2311yescharacteristic
48C2×C83⋊C8C2312yes—
49C2×C83⋊C8C2311yescharacteristic
50C4⋊Dic83—21no—
51C4⋊Dic83C2311yescharacteristic
52C4⋊Dic83C2312yes—
53Dic83⋊C4—22no—
54Dic83⋊C4—22no—
55(C2×D83)⋊C4—22no—
56(C2×D83)⋊C4—21no—
57C2×Q8×C83D411yescharacteristic
58C2×Q8×C83C2312yes—
59C2×D4×C83C2311yescharacteristic
60C22×C4×C83—21no—
61C4⋊C4×C83—21no—
62C4⋊C4×C83—22no—
63C4⋊C4×C83—21no—
64C4⋊C4×C83D412yes—
65C4⋊C4×C83D411yescharacteristic
66C4⋊C4×C83C2311yescharacteristic
67C22⋊C4×C83—22no—
68C22⋊C4×C83—21no—
69C22⋊C4×C83—22no—
70C42×C83—22no—
71C42×C83C2312yes—
72C42×C83C2311yescharacteristic
73C23×C83—21no—
74C22×D83—21no—
75Q8×C83—22no—
76Q8×C83—22no—
77Q8×C83—22no—
78D4×C83—21no—
79D4×C83—21no—
80C2×C4×C83—21no—
81C2×C4×C83—22no—
82C2×C4×C83—42no—
83C2×C4×C83C2×D412yes—
84C2×C4×C83C2×D411yescharacteristic
85C2×C4×C83C2×D412yes—
86C2×C4×C83C2×D411yescharacteristic
87C2×C4×C83C2411yescharacteristic
88C2×Dic83—21no—
89C2×Dic83—22no—
90D332—21no—
91D332#4112yes—
92C4×D83—42no—
93C4×D83—82no—
94Dic166—81no—
95C83⋊C8—21no—
96C83⋊C8—22no—
97C83⋊C8—21no—
98C22×C83—41no—
99C22×C83—41no—
100C22×C83—41no—
101C22×C83C22×D411yescharacteristic
102C2×D83—22no—
103C2×D83—41no—
104C2×D83—41no—
105C4×C83—22no—
106C4×C83—21no—
107C4×C83—22no—
108C4×C83—42no—
109C4×C83—41no—
110C4×C83#13512yes—
111C4×C83C2×#4111yescharacteristic
112C4×C832−1+411yescharacteristic
113Dic83—41no—
114Dic83—42no—
115C2×C83—81no—
116C2×C83#54811yescharacteristic
117C2×C83C2×#13511yescharacteristic
118C2×C83#57711yescharacteristic
119D83—42no—
120#3360—831nomaximal, Sylow
121C83#336011yescharacteristic, complemented, Sylow
122#556D8311yescharacteristic, complemented
123#554—831no—
124#488—832no—
125#487—831no—
126#479—832no—
127#476—831no—
128#463—832no—
129#461—831no—
130C8×Q8—831no—
131C4×SD16—832no—
132C4×D8—831no—
133C2×SD16—832no—
134C2×D8—831no—
135C4⋊Q8—831no—
136C4⋊Q8C2×D8311yescharacteristic
137#126—22no—
138#125—832no—
139#124C2×D8312yes—
140C22⋊Q8—21no—
141C22⋊Q8—1661no—
142C22⋊Q8—1662no—
143C4×Q8C2×D8312yes—
144C4×Q8C2×D8311yescharacteristic
145C4×D4—831no—
146C4×D4—832no—
147C4×D4C2×D8311yescharacteristic
148#117—21no—
149C2×C4⋊C4—1661no—
150#107—831no—
151#106—832no—
152C4⋊C8—831no—
153C4⋊C8—832no—
154#103—832no—
155#103—832no—
156#103—832no—
157#102—831no—
158#102—832no—
159#102—831no—
160C4×C8—831no—
161C4×C8—832no—
162C2×Q8—1661no—
163C2×Q8C22⋊D8311yescharacteristic
164C2×Q8C22×D8312yes—
165C2×D4—831no—
166C2×D4C22×D8311yescharacteristic
167C22×C4—21no—
168C22×C4—1661no—
169C22×C4—1662no—
170SD16—1664no—
171D8—1662no—
172C2×C8—831no—
173C2×C8—832no—
174C2×C8—831no—
175C4⋊C4—21no—
176C4⋊C4—22no—
177C4⋊C4—21no—
178C4⋊C4—832no—
179C4⋊C4—831no—
180C4⋊C4—1662no—
181C4⋊C4—1661no—
182C4⋊C4—1662no—
183C4⋊C4C22⋊D8311yescharacteristic
184C4⋊C4C22⋊D8312yes—
185C4⋊C4C22×D8311yescharacteristic
186C22⋊C4—22no—
187C22⋊C4—21no—
188C22⋊C4—22no—
189C22⋊C4—1661no—
190C22⋊C4—1662no—
191C42—22no—
192C42C22×D8311yescharacteristic
193C42C22×D8312yes—
194C23—21no—
195C23—1661no—
196Q8—22no—
197Q8—22no—
198Q8—22no—
199Q8—6641no—
200D4—21no—
201D4—21no—
202D4—832no—
203D4—1661no—
204C2×C4—21no—
205C2×C4—22no—
206C2×C4—42no—
207C2×C4—3322no—
208C2×C4—6642no—
209C2×C4—1661no—
210C2×C4—1662no—
211C2×C4C2×C22⋊D8312yes—
212C2×C4C2×C22⋊D8312yes—
213C2×C4C2×C22⋊D8311yescharacteristic
214C2×C4C2×C22⋊D8311yescharacteristic
215C2×C4C23×D8311yescharacteristic
216C8—1661no—
217C8—1662no—
218C8—1661no—
219C22—41no—
220C22—41no—
221C22—41no—
222C22—3321no—
223C22—3321no—
224C22—1662no—
225C22C22×C22⋊D8311yescharacteristic
226C4—22no—
227C4—22no—
228C4—21no—
229C4—42no—
230C4—41no—
231C4—3321no—
232C4—3322no—
233C4#1458181212yes—
234C4#41×D8311yescharacteristic
235C4#1458183911yescharacteristic
236C2—81no—
237C2—3322no—
238C2#1912835811yescharacteristic
239C2C2×#1458181211yescharacteristic
240C2#1912850911yescharacteristic
subgroupG/H
centerC22C22×C22⋊D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#556×C83C2
socleC22×C83C22×D4
prime radicalC2×#13050575C22
hypercenter#556D83
hypocenterC83#3360
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #556

Complements

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

NH ≅ G/Ntype
1. #19128411C2semidirect
2. #19128055C2semidirect
3. #19128051C2semidirect
4. #19128008C2semidirect
5. #19127991C2semidirect
7. #19127903C2semidirect
8. Q8×C83⋊C8C2semidirect
9. C4×#13050577C2semidirect
11. #556×C83C2semidirect
121. C83#3360semidirect
122. #556D83semidirect

Maximal subgroups

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

subgroupindex
#191284112
#191280552
#191280512
#191280082
#191279912
#191279132
#191279032
Q8×C83⋊C82
C4×#130505772
C4×#130505752
#556×C832
#336083