Finite Group Databasebeta

← #24907770

Subgroups — #24907770

Order 10624.

404 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 125 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×#14581771C211yesmaximal, characteristic
2C2×#14581715C212yesmaximal, complemented
3C2×C22⋊(C83⋊C8)C211yesmaximal, characteristic
4#19127463C212yesmaximal, complemented
5C22×C4⋊C4×C83C211yesmaximal, characteristic
6C22×Dic166C2211yescharacteristic
7C22×C83⋊C8C2212yes—
8#14581771—22no—
9#14581771C414yescomplemented
10C2×C4⋊Dic83C2211yescharacteristic
11C2×Dic83⋊C4—21no—
12C2×C22⋊Dic83—21no—
13#14581715—24no—
14C22⋊(C83⋊C8)—22no—
15C23×C4×C83—21no—
16C23×C4×C83C2211yescharacteristic
17C2×C4⋊C4×C83—21no—
18C2×C4⋊C4×C83—24no—
19C2×C4⋊C4×C83C2212yes—
20C2×Dic166—22no—
21C2×Dic166—22no—
22C2×Dic166—21no—
23C2×Dic166C2×C412yescomplemented
24C22×Dic83—21no—
25C22×Dic83—21no—
26C2×C83⋊C8—22no—
27C2×C83⋊C8—22no—
28C2×C83⋊C8—22no—
29C4⋊Dic83—21no—
30C4⋊Dic83C2×C412yescomplemented
31Dic83⋊C4—24no—
32C22⋊Dic83—24no—
33C24×C83D411yescharacteristic
34C22×C4×C83—22no—
35C22×C4×C83—22no—
36C22×C4×C83—24no—
37C22×C4×C83—22no—
38C22×C4×C83—42no—
39C22×C4×C83—42no—
40C22×C4×C83C2×C412yes—
41C22×C4×C83D412yes—
42C22×C4×C83D411yescharacteristic
43C22×C4×C83C2311yescharacteristic
44C4⋊C4×C83—24no—
45C4⋊C4×C83—22no—
46C4⋊C4×C83D414yes—
47C23×C83—21no—
48C23×C83—21no—
49C23×C83—21no—
50C23×C83—21no—
51C23×C83C22⋊C412yes—
52C23×C83SD1612yes—
53C23×C83Q1612yes—
54C23×C83C2×D411yescharacteristic
55C2×C4×C83—21no—
56C2×C4×C83—21no—
57C2×C4×C83—24no—
58C2×C4×C83—24no—
59C2×C4×C83—24no—
60C2×C4×C83—48no—
61C2×C4×C83—42no—
62C2×C4×C83—42no—
63C2×C4×C83C22⋊C412yes—
64C2×C4×C83C22×C411yescharacteristic
65C2×C4×C83C2×D412yes—
66C2×C4×C83C2×D411yescharacteristic
67C2×C4×C83#4112yes—
68C2×Dic83—22no—
69C2×Dic83—22no—
70C2×Dic83—41no—
71C2×Dic83—41no—
72C2×Dic83—41no—
73C2×Dic83—41no—
74Dic166—24no—
75Dic166—42no—
76C83⋊C8—42no—
77C22×C83—22no—
78C22×C83—22no—
79C22×C83—22no—
80C22×C83—22no—
81C22×C83—22no—
82C22×C83—22no—
83C22×C83#10314yescomplemented
84C22×C83C2×C22⋊C411yescharacteristic
85C22×C83#12011yescharacteristic
86C22×C83#12311yescharacteristic
87C22×C83C2×SD1611yescharacteristic
88C22×C83C2×Q1611yescharacteristic
89C22×C83AGL(1,ℤ/8)11yescharacteristic
90C22×C83#13711yescharacteristic
91C4×C83—22no—
92C4×C83—44no—
93C4×C83C4×D412yes—
94C4×C83#12112yes—
95Dic83—42no—
96Dic83—42no—
97C2×C83—24no—
98C2×C83#38611yescharacteristic
99C2×C83C2×#10311yescharacteristic
100C2×C83#41711yescharacteristic
101C2×C83#45011yescharacteristic
102C2×C83#45111yescharacteristic
103C2×C83#48311yescharacteristic
104C2×C83#48411yescharacteristic
105#1872—831nomaximal, Sylow
106C83#187211yescharacteristic, complemented, Sylow
107C2×C22⋊Q8—831no—
108C22×C4⋊C4D8311yescharacteristic
109C2×#103—832no—
110C2×C22⋊C8—831no—
111#340—832no—
112C22×Q8—831no—
113C23×C4—21no—
114C23×C4C2×D8311yescharacteristic
115C22×C8—832no—
116C22⋊Q8—834no—
117C22⋊Q8—1662no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—24no—
120C2×C4⋊C4—831no—
121C2×C4⋊C4—1661no—
122C2×C4⋊C4C2×D8312yes—
123C2×C22⋊C4—1661no—
124#103—1664no—
125C22⋊C8—1662no—
126C24C22⋊D8311yescharacteristic
127C2×Q8—832no—
128C2×Q8—1662no—
129C2×Q8—1662no—
130C2×Q8—1661no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—24no—
134C22×C4—22no—
135C22×C4—42no—
136C22×C4—42no—
137C22×C4—1661no—
138C22×C4—1661no—
139C22×C4C4×D8312yes—
140C22×C4D33212yes—
141C22×C4C22⋊D8311yescharacteristic
142C22×C4C22×D8311yescharacteristic
143C2×C8—1662no—
144C2×C8—1662no—
145C2×C8—1662no—
146C4⋊C4—24no—
147C4⋊C4—22no—
148C4⋊C4—832no—
149C4⋊C4—1664no—
150C4⋊C4—1661no—
151C4⋊C4C22⋊D8314yes—
152C22⋊C4—1664no—
153C23—21no—
154C23—21no—
155C23—21no—
156C23—21no—
157C23(C2×D83)⋊C412yes—
158C23#1305057812yes—
159C23#1305058012yes—
160C23C2×C22⋊D8311yescharacteristic
161Q8—3322no—
162Q8—1664no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—24no—
166C2×C4—24no—
167C2×C4—24no—
168C2×C4—48no—
169C2×C4—42no—
170C2×C4—42no—
171C2×C4—3321no—
172C2×C4—3321no—
173C2×C4—3321no—
174C2×C4—3321no—
175C2×C4—1662no—
176C2×C4—1662no—
177C2×C4(C2×D83)⋊C412yes—
178C2×C4C2×C4×D8311yescharacteristic
179C2×C4C2×C22⋊D8312yes—
180C2×C4C2×D33211yescharacteristic
181C2×C4#1305059012yes—
182C8—3322no—
183C22—22no—
184C22—22no—
185C22—22no—
186C22—22no—
187C22—22no—
188C22—22no—
189C22#1458171514yescomplemented
190C22C2×(C2×D83)⋊C411yescharacteristic
191C22#1458176111yescharacteristic
192C22#1458177511yescharacteristic
193C22C2×#1305057811yescharacteristic
194C22C2×#1305058011yescharacteristic
195C22#1458181811yescharacteristic
196C22#1458182311yescharacteristic
197C4—22no—
198C4—44no—
199C4—3322no—
200C4—3322no—
201C4C4×C22⋊D8312yes—
202C4#1458176612yes—
203C2—24no—
204C2#1912758011yescharacteristic
205C2C2×#1458171511yescharacteristic
206C2#1912770011yescharacteristic
207C2#1912785211yescharacteristic
208C2#1912785711yescharacteristic
209C2#1912803111yescharacteristic
210C2#1912803611yescharacteristic
subgroupG/H
centerC23C2×C22⋊D83
commutatorC2×C4×C83C22×C4
FrattiniC22×C4C22×D83
FittingC22×C4⋊C4×C83C2
socleC23×C83C2×D4
prime radicalC24×C83D4
cosocleC22×C4×C83C23
hypercenterC22×C4⋊C4D83
hypocenterC83#1872
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#14581715C2semidirect
4. #19127463C2semidirect
9. #14581771C4semidirect
23. C2×Dic166C2×C4semidirect
30. C4⋊Dic83C2×C4semidirect
83. C22×C83#103semidirect
106. C83#1872semidirect
189. C22#14581715semidirect

Maximal subgroups

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

subgroupindex
C2×#145817712
C2×#145817152
C2×C22⋊(C83⋊C8)2
#191274632
C22×C4⋊C4×C832
#187283