Finite Group Databasebeta

← #24911945

Subgroups — #24911945

Order 10624.

512 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 149 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×#14581772C211yesmaximal, characteristic
2C2×C4⋊C4×D83C211yesmaximal, characteristic
3#19127643C214yesmaximal, complemented
4#19127607C211yesmaximal, characteristic, complemented
5#19127606C211yesmaximal, characteristic, complemented
6#19127566C211yesmaximal, characteristic, complemented
7#19127563C214yesmaximal
8C4×(C2×D83)⋊C4C211yesmaximal, characteristic
9#382×C83C211yesmaximal, characteristic, complemented
10C22×Dic166C2211yescharacteristic
11C22×C4×D83C2212yes—
12C22×C4×D83C2211yescharacteristic
13#14581772—24no—
14C2×C4⋊Dic83C2211yescharacteristic
15C2×C4⋊Dic83C2212yes—
16C2×Dic83⋊C4C2212yes—
17C4⋊C4×D83—24no—
18C2×(C2×D83)⋊C4C2212yes—
19C2×(C2×D83)⋊C4C2212yes—
20C2×C4×Dic83C2211yescharacteristic
21C83⋊#95C2214yes—
22C83⋊#95C2214yes—
23C83⋊#95C2214yes—
24C83⋊#95C2212yes—
25C2×C4⋊C4×C83C2211yescharacteristic
26C2×C4⋊C4×C83C2211yescharacteristic
27C2×C42×C83C2211yescharacteristic
28#95×C83C2214yes—
29C23×D83C2311yescharacteristic
30C2×Dic166—22no—
31C2×Dic166—22no—
32C2×Dic166—22no—
33C22×Dic83C2312yes—
34C22×Dic83C2312yes—
35C22×Dic83C2312yes—
36C22×Dic83C2311yescharacteristic
37C2×C4×D83—22no—
38C2×C4×D83—22no—
39C2×C4×D83—24no—
40C2×C4×D83—28no—
41C2×C4×D83D414yes—
42C4⋊Dic83—22no—
43C4⋊Dic83—24no—
44Dic83⋊C4—24no—
45(C2×D83)⋊C4—24no—
46(C2×D83)⋊C4—24no—
47C4×Dic83—22no—
48C22×C4×C83C2312yes—
49C22×C4×C83C2312yes—
50C22×C4×C83C2312yes—
51C22×C4×C83C2311yescharacteristic
52C4⋊C4×C83—22no—
53C4⋊C4×C83—22no—
54C42×C83—22no—
55C23×C83C2411yescharacteristic
56C22×D83—21no—
57C22×D83—21no—
58C22×D83—22no—
59C22×D83C2×D412yes—
60C22×D83#4114yes—
61C2×C4×C83—22no—
62C2×C4×C83—22no—
63C2×C4×C83—22no—
64C2×C4×C83—22no—
65C2×C4×C83—24no—
66C2×C4×C83—24no—
67C2×C4×C83—22no—
68C2×C4×C83C2×D412yes—
69C2×C4×C83#4112yes—
70C2×C4×C83#4112yes—
71C2×Dic83—21no—
72C2×Dic83—21no—
73C2×Dic83—22no—
74C2×Dic83—22no—
75C2×Dic83—22no—
76C2×Dic83—22no—
77C2×Dic83—24no—
78C2×Dic83—22no—
79C2×Dic83—22no—
80C2×Dic83—22no—
81C2×Dic83C2×D412yes—
82C4×D83—28no—
83C4×D83—48no—
84Dic166—44no—
85C22×C83C22×D411yescharacteristic
86C22×C83C2×#4112yes—
87C22×C83C2×#4111yescharacteristic
88C22×C83C2×#4111yescharacteristic
89C22×C832−1+412yes—
90C2×D83—22no—
91C2×D83—22no—
92C2×D83—24no—
93C2×D83—24no—
94C2×D83#12414yes—
95C4×C83—42no—
96C4×C83—42no—
97C4×C83—42no—
98C4×C83#12314yes—
99Dic83—22no—
100Dic83—42no—
101Dic83—42no—
102Dic83—42no—
103C2×C83C2×#12311yescharacteristic
104C2×C83C2×#12411yescharacteristic
105C2×C83#53611yescharacteristic
106C2×C83#55212yes—
107C2×C83#55212yes—
108D83—24no—
109#2503—831nomaximal, Sylow
110C83#250311yescharacteristic, complemented, Sylow
111C2×C22⋊Q8—831no—
112C22×C4⋊C4—831no—
113#400—834no—
114#391—831no—
115#391—831no—
116#382—831no—
117#382D8311yescharacteristic, complemented
118#381—834no—
119C4×C22⋊C4—831no—
120C22×Q8—831no—
121C23×C4—831no—
122C23×C4—832no—
123C22⋊Q8—1664no—
124C2×C4⋊C4—831no—
125C2×C4⋊C4—832no—
126C2×C4⋊C4—832no—
127C2×C4⋊C4—1664no—
128C2×C4⋊C4C2×D8311yescharacteristic
129C2×C4⋊C4C2×D8311yescharacteristic
130C2×C22⋊C4—832no—
131C2×C22⋊C4—832no—
132C2×C42—831no—
133C2×C42C2×D8311yescharacteristic
134#95—832no—
135#95—834no—
136#95—834no—
137#95—834no—
138#95C2×D8314yes—
139C24—831no—
140C2×Q8—1662no—
141C2×Q8—1662no—
142C2×Q8—1662no—
143C22×C4—831no—
144C22×C4—834no—
145C22×C4—832no—
146C22×C4—832no—
147C22×C4—832no—
148C22×C4—1664no—
149C22×C4—1668no—
150C22×C4—1662no—
151C22×C4—1662no—
152C22×C4C22×D8312yes—
153C22×C4C22×D8312yes—
154C22×C4C22×D8312yes—
155C22×C4C22×D8311yescharacteristic
156C4⋊C4—22no—
157C4⋊C4—22no—
158C4⋊C4—1664no—
159C4⋊C4—1664no—
160C4⋊C4—1662no—
161C22⋊C4—1664no—
162C22⋊C4—1664no—
163C42—22no—
164C42—1662no—
165C23—832no—
166C23—834no—
167C23—1661no—
168C23—1661no—
169C23—1662no—
170C23C23×D8311yescharacteristic
171Q8—3324no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—24no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—24no—
179C2×C4—3328no—
180C2×C4—832no—
181C2×C4—1662no—
182C2×C4—1662no—
183C2×C4—1662no—
184C2×C4—1664no—
185C2×C4—1662no—
186C2×C4—1662no—
187C2×C4—1662no—
188C2×C4—1662no—
189C2×C4—1661no—
190C2×C4—1661no—
191C2×C4—1668no—
192C2×C4D4×D8312yes—
193C2×C4#1305059012yes—
194C2×C4#1305059112yes—
195C22—834no—
196C22—1664no—
197C22—1664no—
198C22—1662no—
199C22—1662no—
200C22C2×D4×D8311yescharacteristic
201C22#41×D8312yes—
202C22C2×#1305059011yescharacteristic
203C22C2×#1305059111yescharacteristic
204C22#1458183912yes—
205C4—42no—
206C4—42no—
207C4—42no—
208C4—3322no—
209C4—3322no—
210C4—3322no—
211C4—1662no—
212C4#1458177714yes—
213C2—1664no—
214C2C2×#1458177711yescharacteristic
215C2#124×D8311yescharacteristic
216C2#1912827111yescharacteristic
217C2#1912838912yes—
218C2#1912839012yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#382×C83C2
prime radicalC23×D83C23
hypercenter#382D83
hypocenterC83#2503
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #382

Complements

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

NH ≅ G/Ntype
3. #19127643C2semidirect
4. #19127607C2semidirect
5. #19127606C2semidirect
6. #19127566C2semidirect
9. #382×C83C2semidirect
110. C83#2503semidirect
117. #382D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145817722
C2×C4⋊C4×D832
#191276432
#191276072
#191276062
#191275662
#191275632
C4×(C2×D83)⋊C42
#382×C832
#250383