Finite Group Databasebeta

← #24912414

Subgroups — #24912414

Order 10624.

676 conjugacy classes of subgroups; only the 185 normal subgroups are shown.

normal subgroups only

This group has 676 conjugacy classes of subgroups — too many to draw. Only its 185 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
1C2×#14581791C211yesmaximal, characteristic, complemented
2C2×#14581769C214yesmaximal, complemented
3C2×D4×Dic83C211yesmaximal, characteristic
4#19127637C214yesmaximal, complemented
5#19127604C211yesmaximal, characteristic
6#19127571C211yesmaximal, characteristic, complemented
7#19127570C211yesmaximal, characteristic, complemented
8C4×Dic83⋊C4C211yesmaximal, characteristic, complemented
9#390×C83C211yesmaximal, characteristic
10C22×Dic166C2212yes—
11C23×Dic83C2212yes—
12#14581791—24no—
13#14581769—216no—
14D4×Dic83—24no—
15C2×C4⋊Dic83C2211yescharacteristic
16C2×C4⋊Dic83C2211yescharacteristic
17C2×C4⋊Dic83C2214yes—
18C2×Dic83⋊C4C2214yes—
19C2×Dic83⋊C4C2212yes—
20C2×Dic83⋊C4C2212yes—
21C2×C22⋊Dic83C2214yes—
22C2×C22⋊Dic83C2212yes—
23C2×C4×Dic83C2211yescharacteristic
24C2×C4×Dic83C2211yescharacteristic
25C83⋊#95C2212yes—
26C22×D4×C83C2211yescharacteristic
27C2×C4⋊C4×C83C2211yescharacteristic
28C2×C22⋊C4×C83C2214yes—
29C2×C42×C83C2211yescharacteristic
30C2×Dic166—28no—
31C2×Dic166—24no—
32C22×Dic83—28no—
33C22×Dic83—24no—
34C22×Dic83C2312yes—
35C22×Dic83C2312yes—
36C22×Dic83C2311yescharacteristic
37C22×Dic83C2312yes—
38C22×Dic83C2311yescharacteristic
39C4⋊Dic83—28no—
40C4⋊Dic83—22no—
41C4⋊Dic83—22no—
42Dic83⋊C4—28no—
43Dic83⋊C4—24no—
44Dic83⋊C4D418yescomplemented
45C22⋊Dic83—28no—
46C22⋊Dic83—24no—
47C4×Dic83—22no—
48C4×Dic83D414yes—
49C24×C83C2312yes—
50C2×D4×C83—22no—
51C2×D4×C83—22no—
52C2×D4×C83Q814yescomplemented
53C22×C4×C83C2312yes—
54C22×C4×C83C2312yes—
55C22×C4×C83C2311yescharacteristic
56C4⋊C4×C83—22no—
57C22⋊C4×C83—28no—
58C42×C83—22no—
59C23×C83—24no—
60C23×C83—22no—
61C23×C83—22no—
62C23×C83—24no—
63C23×C83C2×Q814yes—
64C23×C83C2411yescharacteristic
65D4×C83—28no—
66C2×C4×C83—22no—
67C2×C4×C83—22no—
68C2×C4×C83—24no—
69C2×C4×C83—22no—
70C2×C4×C83C2×D414yes—
71C2×C4×C83C2×D412yes—
72C2×C4×C83C2×Q812yes—
73C2×C4×C83#4112yes—
74C2×Dic83—22no—
75C2×Dic83—22no—
76C2×Dic83—22no—
77C2×Dic83—22no—
78C2×Dic83—24no—
79C2×Dic83—24no—
80C2×Dic83—22no—
81C2×Dic83—48no—
82C2×Dic83C2×D412yes—
83C2×Dic83C2×D414yes—
84C2×Dic83C2×D412yes—
85C2×Dic83C2×D414yes—
86Dic166—48no—
87C22×C83—24no—
88C22×C83—28no—
89C22×C83—42no—
90C22×C83—42no—
91C22×C83—44no—
92C22×C83C22×D411yescharacteristic
93C22×C83C22×D412yes—
94C22×C83C22×Q811yescharacteristic
95C22×C83C2×#4111yescharacteristic
96C22×C832−1+412yes—
97C4×C83—24no—
98C4×C83—42no—
99C4×C83C22⋊Q814yescomplemented
100Dic83—22no—
101Dic83—24no—
102Dic83—42no—
103Dic83—42no—
104Dic83#12714yes—
105C2×C83—44no—
106C2×C83C2×C22⋊Q811yescharacteristic
107C2×C83C2×#12711yescharacteristic
108C2×C83#53611yescharacteristic
109C2×C83#54712yes—
110C2×C83D4×Q812yes—
111#2542—831nomaximal, Sylow
112C83#254211yescharacteristic, complemented, Sylow
113C2×C4⋊Q8—831no—
114C2×C22⋊Q8—834no—
115C2×C4×D4—831no—
116#399—834no—
117#390—831no—
118#390D8311yescharacteristic
119#384—831no—
120#384—831no—
121C4×C4⋊C4—831no—
122C22×Q8—832no—
123C22×D4C2×D8311yescharacteristic
124C23×C4—832no—
125C4⋊Q8—1664no—
126C22⋊Q8—16616no—
127C4×D4—1664no—
128C2×C4⋊C4—832no—
129C2×C4⋊C4—831no—
130C2×C4⋊C4—834no—
131C2×C4⋊C4—834no—
132C2×C4⋊C4—831no—
133C2×C4⋊C4—832no—
134C2×C4⋊C4C2×D8311yescharacteristic
135C2×C22⋊C4—832no—
136C2×C22⋊C4—834no—
137C2×C22⋊C4C2×D8314yes—
138C2×C42—831no—
139C2×C42—831no—
140C2×C42C2×D8311yescharacteristic
141#95—832no—
142C24C22×D8312yes—
143C2×Q8—1668no—
144C2×Q8—1664no—
145C2×D4—22no—
146C2×D4—22no—
147C2×D4Dic16614yescomplemented
148C22×C4—831no—
149C22×C4—831no—
150C22×C4—832no—
151C22×C4—832no—
152C22×C4—832no—
153C22×C4—1664no—
154C22×C4—1668no—
155C22×C4C22×D8312yes—
156C22×C4C22×D8312yes—
157C22×C4C22×D8311yescharacteristic
158C4⋊C4—22no—
159C4⋊C4—838no—
160C4⋊C4—1668no—
161C4⋊C4—1664no—
162C4⋊C4—1662no—
163C4⋊C4—1668no—
164C4⋊C4—1662no—
165C22⋊C4—28no—
166C22⋊C4—1664no—
167C22⋊C4—1668no—
168C42—22no—
169C42—834no—
170C42—1662no—
171C23—24no—
172C23—22no—
173C23—22no—
174C23—24no—
175C23C2×Dic16614yes—
176C23C23×D8311yescharacteristic
177Q8—3328no—
178D4—28no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—24no—
182C2×C4—22no—
183C2×C4—3328no—
184C2×C4—832no—
185C2×C4—834no—
186C2×C4—832no—
187C2×C4—834no—
188C2×C4—1662no—
189C2×C4—1664no—
190C2×C4—1664no—
191C2×C4—1662no—
192C2×C4—1662no—
193C2×C4—1662no—
194C2×C4—1662no—
195C2×C4D4×D8314yes—
196C2×C4D4×D8312yes—
197C2×C4C2×Dic16612yes—
198C2×C4#1305058912yes—
199C22—28no—
200C22—24no—
201C22—44no—
202C22—42no—
203C22—42no—
204C22C2×D4×D8312yes—
205C22C2×D4×D8311yescharacteristic
206C22C22×Dic16611yescharacteristic
207C22C2×#1305058911yescharacteristic
208C22#1458183912yes—
209C4—24no—
210C4—42no—
211C4—3322no—
212C4—3322no—
213C4—834no—
214C4—1664no—
215C4—1662no—
216C4#1458176914yescomplemented
217C2—44no—
218C2C2×#1458176911yescharacteristic
219C2#127×D8311yescharacteristic
220C2#1912827111yescharacteristic
221C2#1912835112yes—
222C2D4×Dic16612yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#390×C83C2
prime radicalC22×D4×C83C22
hypercenter#390D83
hypocenterC83#2542
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
1. C2×#14581791C2semidirect
2. C2×#14581769C2semidirect
4. #19127637C2semidirect
6. #19127571C2semidirect
7. #19127570C2semidirect
8. C4×Dic83⋊C4C2semidirect
44. Dic83⋊C4D4semidirect
52. C2×D4×C83Q8semidirect
99. C4×C83C22⋊Q8semidirect
112. C83#2542semidirect
147. C2×D4Dic166semidirect
216. C4#14581769semidirect

Maximal subgroups

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

subgroupindex
C2×#145817912
C2×#145817692
C2×D4×Dic832
#191276372
#191276042
#191275712
#191275702
C4×Dic83⋊C42
#390×C832
#254283