Finite Group Databasebeta

← #24908504

Subgroups — #24908504

Order 10624.

406 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 73 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×#14581818C212yesmaximal, complemented
2#19128282C211yesmaximal, characteristic, complemented
3C2×#14581720C212yesmaximal, complemented
4#19127470C211yesmaximal, characteristic, complemented
5#390×C83C211yesmaximal, characteristic, complemented
6C2×#13050590C2212yes—
7#14581818—24no—
8#14581818—24no—
9C2×#13050578—22no—
10C2×#13050575—22no—
11C2×#13050574C2212yes—
12#14581772—22no—
13#14581767—22no—
14#14581766—22no—
15#14581720—24no—
16C22×D4×C83C2211yescharacteristic
17C2×C4⋊C4×C83C2211yescharacteristic
18C2×C22⋊C4×C83—22no—
19C2×C42×C83C2211yescharacteristic
20#13050590—22no—
21#13050590—22no—
22#13050590—22no—
23#13050590—22no—
24C2×Dic166D412yes—
25C2×D332—22no—
26C2×D332D412yes—
27C2×C22⋊D83—22no—
28C2×C4×D83—22no—
29#13050578—28no—
30#13050575—28no—
31#13050574—22no—
32#13050574D414yescomplemented
33C2×C83⋊C8—22no—
34C4⋊Dic83—22no—
35(C2×D83)⋊C4—24no—
36C24×C83—21no—
37C2×D4×C83—22no—
38C2×D4×C83—22no—
39C2×D4×C83—21no—
40C2×D4×C83D412yescomplemented
41C22×C4×C83—21no—
42C22×C4×C83D412yes—
43C22×C4×C83C2311yescharacteristic
44C4⋊C4×C83—22no—
45C22⋊C4×C83—44no—
46C42×C83—22no—
47C23×C83—22no—
48C23×C83—42no—
49C23×C83—42no—
50C23×C83—41no—
51C23×C83—41no—
52C23×C83C2×D411yescharacteristic
53C22×D83—22no—
54D4×C83—24no—
55D4×C83—42no—
56C2×C4×C83—22no—
57C2×C4×C83—22no—
58C2×C4×C83—41no—
59C2×C4×C83—41no—
60C2×C4×C83—42no—
61C2×C4×C83C2×D412yes—
62C2×C4×C83C2×D412yes—
63C2×C4×C83C2×D411yescharacteristic
64C2×C4×C83#4111yescharacteristic
65C22⋊D83—42no—
66C22⋊D83—42no—
67C2×Dic83—22no—
68D332—22no—
69D332—22no—
70D332—82no—
71C4×D83—42no—
72C4×D83—42no—
73Dic166—22no—
74Dic166—22no—
75C83⋊C8—24no—
76C22×C83—21no—
77C22×C83—21no—
78C22×C83—44no—
79C22×C83—42no—
80C22×C83—82no—
81C22×C83—81no—
82C22×C83—81no—
83C22×C83#12011yescharacteristic
84C22×C83#12111yescharacteristic
85C22×C83#12711yescharacteristic
86C2×D83—42no—
87C2×D83—42no—
88C2×D83—42no—
89C4×C83—42no—
90C4×C83—42no—
91C4×C83#12012yes—
92C4×C83#12112yes—
93Dic83—42no—
94C2×C83—21no—
95C2×C83—21no—
96C2×C83—82no—
97C2×C83#39211yescharacteristic
98C2×C83#45311yescharacteristic
99C2×C83#45411yescharacteristic
100D83—82no—
101#2006—831nomaximal, Sylow
102C83#200611yescharacteristic, complemented, Sylow
103C2×AGL(1,ℤ/8)—832no—
104#537—831no—
105C2×#104—832no—
106#390D8311yescharacteristic, complemented
107#343—831no—
108C2×#41—832no—
109C22×D4C2×D8311yescharacteristic
110AGL(1,ℤ/8)—1664no—
111AGL(1,ℤ/8)—1664no—
112C2×SD16—1662no—
113C2×D8—1662no—
114C2×M16—832no—
115C22⋊Q8—1662no—
116#121—1662no—
117#121—1662no—
118C2×C4⋊C4C2×D8311yescharacteristic
119C2×C22⋊C4—22no—
120C2×C42C2×D8311yescharacteristic
121#104—1664no—
122C24—21no—
123#41—1662no—
124#41—1662no—
125#41—1662no—
126#41—1662no—
127C2×Q8—832no—
128C2×D4—22no—
129C2×D4—22no—
130C2×D4—21no—
131C2×D4—832no—
132C2×D4—1662no—
133C2×D4—1662no—
134C2×D4D33212yescomplemented
135C22×C4—21no—
136C22×C4—1662no—
137C22×C4C22⋊D8312yes—
138C22×C4C22×D8311yescharacteristic
139SD16—1668no—
140D8—1668no—
141M16—834no—
142M16—1662no—
143C2×C8—1662no—
144C4⋊C4—22no—
145C4⋊C4—1662no—
146C22⋊C4—44no—
147C22⋊C4—1664no—
148C42—22no—
149C23—22no—
150C23—42no—
151C23—42no—
152C23—41no—
153C23—41no—
154C23—1662no—
155C23C2×C22⋊D8311yescharacteristic
156Q8—1662no—
157Q8—1662no—
158D4—24no—
159D4—42no—
160D4—3322no—
161D4—3322no—
162D4—6642no—
163D4—1662no—
164D4—1662no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—42no—
170C2×C4—3322no—
171C2×C4—3322no—
172C2×C4—1662no—
173C2×C4D4×D8312yes—
174C2×C4D4×D8312yes—
175C2×C4C2×D33211yescharacteristic
176C2×C4#1305059011yescharacteristic
177C8—1664no—
178C22—21no—
179C22—21no—
180C22—44no—
181C22—42no—
182C22—82no—
183C22—81no—
184C22—81no—
185C22—3322no—
186C22—3322no—
187C22—3322no—
188C22#1458176011yescharacteristic
189C22#1458176611yescharacteristic
190C22#1458178811yescharacteristic
191C4—42no—
192C4—42no—
193C4—3322no—
194C4#1458176212yes—
195C4#1458176512yes—
196C2—21no—
197C2—21no—
198C2—82no—
199C2—6642no—
200C2#1912761011yescharacteristic
201C2#1912786811yescharacteristic
202C2#1912787311yescharacteristic
subgroupG/H
centerC22#14581760
commutator, cosocleC22×C4×C83C23
FrattiniC22×C4C22×D83
Fitting#390×C83C2
socleC22×C83#120
hypercenter#390D83
hypocenterC83#2006
LCSG ▷ C22×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ #390

Complements

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

NH ≅ G/Ntype
1. C2×#14581818C2semidirect
2. #19128282C2semidirect
3. C2×#14581720C2semidirect
4. #19127470C2semidirect
5. #390×C83C2semidirect
32. #13050574D4semidirect
40. C2×D4×C83D4semidirect
102. C83#2006semidirect
106. #390D83semidirect
134. C2×D4D332semidirect

Maximal subgroups

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

subgroupindex
C2×#145818182
#191282822
C2×#145817202
#191274702
#390×C832
#200683