Finite Group Databasebeta

← #24920110

Subgroups — #24920110

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#19128421C211yesmaximal, characteristic, complemented
2#19128408C211yesmaximal, characteristic, complemented
3#19128054C212yesmaximal, complemented
4#19128052C211yesmaximal, characteristic, complemented
5#19128007C212yesmaximal, complemented
6#19127991C211yesmaximal, characteristic, complemented
7#19127914C212yesmaximal
8#19127906C211yesmaximal, characteristic, complemented
9C8×Dic166C211yesmaximal, characteristic, complemented
10#19127792C212yesmaximal, complemented
11C4×D8×C83C211yesmaximal, characteristic
12C2×#13050578C2212yes—
13#14581792C2211yescharacteristic
14#14581792C2211yescharacteristic
15#14581785—22no—
16#14581783C2212yes—
17#14581779C2212yes—
18#14581771—21no—
19#14581771—21no—
20#14581769—22no—
21C4×Dic166C2211yescharacteristic
22#14581757C2212yes—
23D4×Dic83C2212yes—
24#14581751—21no—
25C2×C4⋊Dic83—21no—
26#14581723C2212yes—
27C4⋊(C83⋊C8)C2211yescharacteristic
28Dic83⋊C8C2212yes—
29#14581717C2212yes—
30#14581717C2212yes—
31#14581715C2212yes—
32#14581713C2212yes—
33C8×Dic83C2212yes—
34C2×D8×C83C2211yescharacteristic
35C4×D4×C83C2211yescharacteristic
36C4×D4×C83C2211yescharacteristic
37#107×C83C2211yescharacteristic
38#102×C83C2211yescharacteristic
39#102×C83C2211yescharacteristic
40C4×C8×C83C2211yescharacteristic
41C2×Dic166—21no—
42C2×Dic166D411yescharacteristic
43C2×Dic166C2312yes—
44C22×Dic83—22no—
45#13050578—24no—
46C2×C83⋊C8C2312yes—
47C4⋊Dic83—21no—
48C4⋊Dic83—21no—
49C4⋊Dic83—21no—
50C4⋊Dic83—22no—
51C4⋊Dic83D411yescharacteristic
52C4⋊Dic83C2312yes—
53Dic83⋊C4—22no—
54Dic83⋊C4—22no—
55Dic83⋊C4D412yes—
56C22⋊Dic83—22no—
57C22⋊Dic83—22no—
58C22⋊Dic83—22no—
59C4×Dic83—22no—
60C4×Dic83C2312yes—
61C2×D4×C83C2311yescharacteristic
62C2×D4×C83C2311yescharacteristic
63C22×C4×C83—21no—
64C22×C4×C83—21no—
65D8×C83—22no—
66C2×C8×C83C2311yescharacteristic
67C2×C8×C83C2311yescharacteristic
68C4⋊C4×C83C2311yescharacteristic
69C4⋊C4×C83C2311yescharacteristic
70C22⋊C4×C83—21no—
71C22⋊C4×C83—21no—
72C42×C83C2311yescharacteristic
73C23×C83—21no—
74C23×C83—21no—
75D4×C83—21no—
76D4×C83—21no—
77D4×C83—21no—
78D4×C83#4112yes—
79C2×C4×C83—21no—
80C2×C4×C83—21no—
81C2×C4×C83—42no—
82C2×C4×C83—42no—
83C2×C4×C83C2×D411yescharacteristic
84C2×C4×C83C2×D411yescharacteristic
85C2×C4×C83C2411yescharacteristic
86C2×Dic83—22no—
87C2×Dic83—22no—
88C2×Dic83—82no—
89C2×Dic83C2×D412yes—
90C2×Dic83C2×D412yes—
91Dic166—22no—
92Dic166—22no—
93Dic166—22no—
94Dic166—81no—
95C8×C83—21no—
96C8×C83—21no—
97C83⋊C8—22no—
98C22×C83—22no—
99C22×C83—41no—
100C22×C83—41no—
101C22×C83—41no—
102C22×C83—41no—
103C22×C83—41no—
104C22×C83C22×D411yescharacteristic
105C4×C83—21no—
106C4×C83—41no—
107C4×C83—41no—
108C4×C83#13512yes—
109C4×C83C2×#4111yescharacteristic
110C4×C832−1+411yescharacteristic
111Dic83—22no—
112Dic83—22no—
113Dic83—42no—
114Dic83—42no—
115C2×C83—42no—
116C2×C83—81no—
117C2×C83#54811yescharacteristic
118C2×C83C2×#13511yescharacteristic
119C2×C83#57711yescharacteristic
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#461—831no—
130C8×Q8—831no—
131C4×SD16—832no—
132C4×D8D8311yescharacteristic
133C2×SD16—832no—
134C2×D8C2×D8311yescharacteristic
135C4⋊Q8—831no—
136C4⋊Q8—831no—
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×D4C2×D8311yescharacteristic
147C4×D4C2×D8311yescharacteristic
148#117—1661no—
149C2×C4⋊C4—1661no—
150#107C2×D8311yescharacteristic
151#106—832no—
152C4⋊C8—832no—
153C4⋊C8—831no—
154#103—832no—
155#103—832no—
156#103—832no—
157#102—832no—
158#102C2×D8311yescharacteristic
159#102C2×D8311yescharacteristic
160C4×C8—832no—
161C4×C8C2×D8311yescharacteristic
162C2×Q8—831no—
163C2×Q8—832no—
164C2×Q8—1661no—
165C2×D4C22×D8311yescharacteristic
166C2×D4C22×D8311yescharacteristic
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—1662no—
170SD16—1664no—
171D8—22no—
172C2×C8—832no—
173C2×C8C22×D8311yescharacteristic
174C2×C8C22×D8311yescharacteristic
175C4⋊C4—831no—
176C4⋊C4—832no—
177C4⋊C4—832no—
178C4⋊C4—1662no—
179C4⋊C4—1662no—
180C4⋊C4—1662no—
181C4⋊C4—1661no—
182C4⋊C4—1661no—
183C4⋊C4—1661no—
184C4⋊C4C22×D8311yescharacteristic
185C4⋊C4C22×D8311yescharacteristic
186C22⋊C4—21no—
187C22⋊C4—21no—
188C22⋊C4—1662no—
189C22⋊C4—1662no—
190C22⋊C4—1662no—
191C42—832no—
192C42—1662no—
193C42C22×D8311yescharacteristic
194C23—21no—
195C23—21no—
196Q8—6641no—
197Q8—1662no—
198Q8—1662no—
199Q8—1662no—
200D4—21no—
201D4—21no—
202D4—21no—
203D4#1305059012yes—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—832no—
209C2×C4—832no—
210C2×C4—6642no—
211C2×C4—1662no—
212C2×C4—1662no—
213C2×C4D4×D8311yescharacteristic
214C2×C4D4×D8311yescharacteristic
215C2×C4C23×D8311yescharacteristic
216C8—21no—
217C8—21no—
218C8—1662no—
219C22—22no—
220C22—41no—
221C22—41no—
222C22—41no—
223C22—41no—
224C22—41no—
225C22C2×D4×D8311yescharacteristic
226C4—21no—
227C4—41no—
228C4—41no—
229C4—3322no—
230C4—3322no—
231C4—1662no—
232C4—1662no—
233C4#1458181112yes—
234C4C2×#1305059011yescharacteristic
235C4#1458183911yescharacteristic
236C2—42no—
237C2—81no—
238C2#1912836011yescharacteristic
239C2C2×#1458181111yescharacteristic
240C2#1912850811yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
FittingC4×D8×C83C2
socleC22×C83C22×D4
prime radicalC2×D8×C83C22
hypercenterC4×D8D83
hypocenterC83#3360
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ C4×D8

Complements

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

NH ≅ G/Ntype
1. #19128421C2semidirect
2. #19128408C2semidirect
3. #19128054C2semidirect
4. #19128052C2semidirect
5. #19128007C2semidirect
6. #19127991C2semidirect
8. #19127906C2semidirect
9. C8×Dic166C2semidirect
10. #19127792C2semidirect
121. C83#3360semidirect

Maximal subgroups

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

subgroupindex
#191284212
#191284082
#191280542
#191280522
#191280072
#191279912
#191279142
#191279062
C8×Dic1662
#191277922
C4×D8×C832
#336083