Finite Group Databasebeta

← #24918815

Subgroups — #24918815

Order 10624.

360 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 121 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#19128235C212yesmaximal, complemented
2#19128009C212yesmaximal, complemented
3#19128002C212yesmaximal, complemented
4#19127942C211yesmaximal, characteristic, complemented
5#19127937C211yesmaximal, characteristic
6#19127930C212yesmaximal, complemented
7#19127924C212yesmaximal, complemented
8#19127917C212yesmaximal, complemented
9C2×C4⋊C8×C83C211yesmaximal, characteristic, complemented
10C2×Dic332C2211yescharacteristic
11C2×#13050579C2212yes—
12C2×D664C2211yescharacteristic
13#14581786—22no—
14#14581784C2212yes—
15#14581780C2212yes—
16#14581775—22no—
17#14581771C2212yes—
18#14581766C2212yes—
19C4×Dic166C2212yes—
20C4×D332C2212yes—
21C4×C22⋊D83—22no—
22#14581749—22no—
23#14581726C2211yescharacteristic
24#14581726C2211yescharacteristic
25#14581724C2212yes—
26#14581717C2212yes—
27#14581717C2212yes—
28#14581714C2212yes—
29#14581714C2212yes—
30C22×C8×C83C2211yescharacteristic
31C22×C8×C83C2211yescharacteristic
32C2×C42×C83C2211yescharacteristic
33C4⋊C8×C83C2212yes—
34C4⋊C8×C83C2212yes—
35C2×Dic166C2312yes—
36C2×D332C2312yes—
37C2×C22⋊D83—22no—
38C2×C4×D83—22no—
39Dic332—41no—
40#13050579—42no—
41D664—41no—
42C4⋊Dic83C2312yes—
43C4⋊Dic83C2312yes—
44Dic83⋊C4—22no—
45Dic83⋊C4—22no—
46Dic83⋊C4—22no—
47Dic83⋊C4—22no—
48C22⋊Dic83—22no—
49(C2×D83)⋊C4—22no—
50(C2×D83)⋊C4—22no—
51(C2×D83)⋊C4—22no—
52(C2×D83)⋊C4—22no—
53C4×Dic83—22no—
54C22×C4×C83D411yescharacteristic
55C22×C4×C83D411yescharacteristic
56C22×C4×C83C2311yescharacteristic
57C2×C8×C83—41no—
58C2×C8×C83—41no—
59C2×C8×C83C2311yescharacteristic
60C2×C8×C83C2312yes—
61C2×C8×C83C2311yescharacteristic
62C42×C83D412yes—
63C42×C83C2312yes—
64C23×C83C2×D411yescharacteristic
65C22×D83—22no—
66C2×C4×C83—22no—
67C2×C4×C83—21no—
68C2×C4×C83—21no—
69C2×C4×C83—41no—
70C2×C4×C83C2×D411yescharacteristic
71C2×C4×C83C2×D412yes—
72C2×C4×C83C2×D412yes—
73C2×C4×C83C2411yescharacteristic
74C22⋊D83—44no—
75C2×Dic83—22no—
76C2×Dic83—22no—
77C2×Dic83—22no—
78D332—22no—
79D332—22no—
80C4×D83—44no—
81Dic166—22no—
82Dic166—22no—
83C8×C83—21no—
84C8×C83—22no—
85C8×C83—21no—
86C22×C83—21no—
87C22×C83—21no—
88C22×C83#13512yes—
89C22×C83C22×D411yescharacteristic
90C2×D83—42no—
91C2×D83—42no—
92C2×D83—42no—
93C4×C83—21no—
94C4×C83—22no—
95C4×C83—22no—
96C4×C83H(5,𝔽2)11yescharacteristic
97C4×C832−1+411yescharacteristic
98Dic83—42no—
99Dic83—42no—
100Dic83—42no—
101C2×C83—22no—
102C2×C83#53711yescharacteristic
103C2×C83C2×#13511yescharacteristic
104C2×C83#57511yescharacteristic
105D83—82no—
106#3250—831nomaximal, Sylow
107C83#325011yescharacteristic, complemented, Sylow
108#529—832no—
109#479—832no—
110#478—832no—
111#467—831no—
112#466—831no—
113#465—832no—
114#464—832no—
115#463—832no—
116C2×C4⋊C8D8311yescharacteristic, complemented
117C2×Q16—831no—
118C2×SD16—832no—
119C2×D8—831no—
120C22×C8C2×D8311yescharacteristic
121C22×C8C2×D8311yescharacteristic
122#126—1662no—
123#125—832no—
124#124—832no—
125#123—1662no—
126C22⋊Q8—832no—
127#121—832no—
128C4×Q8—832no—
129C4×D4—832no—
130C4×D4—1662no—
131#117—1662no—
132C2×C42C2×D8311yescharacteristic
133#107—831no—
134#107—831no—
135#106—832no—
136C4⋊C8C2×D8312yes—
137C4⋊C8C2×D8312yes—
138#103—832no—
139#103—832no—
140#102—832no—
141#102—832no—
142C2×Q8—832no—
143C2×D4—832no—
144C2×D4—1662no—
145C22×C4—1662no—
146C22×C4D33211yescharacteristic
147C22×C4D33211yescharacteristic
148C22×C4C22×D8311yescharacteristic
149Q16—3321no—
150SD16—3322no—
151D8—3321no—
152C2×C8—41no—
153C2×C8—41no—
154C2×C8C22×D8311yescharacteristic
155C2×C8C22×D8312yes—
156C2×C8C22×D8311yescharacteristic
157C4⋊C4—832no—
158C4⋊C4—832no—
159C4⋊C4—1662no—
160C4⋊C4—1662no—
161C4⋊C4—1662no—
162C4⋊C4—1662no—
163C22⋊C4—1662no—
164C22⋊C4—1662no—
165C22⋊C4—1662no—
166C22⋊C4—1662no—
167C22⋊C4—1662no—
168C42—1662no—
169C42D33212yes—
170C42C22×D8312yes—
171C23—1662no—
172C23C2×D33211yescharacteristic
173Q8—1662no—
174Q8—1662no—
175D4—3324no—
176D4—1662no—
177D4—1662no—
178C2×C4—22no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—41no—
182C2×C4—3324no—
183C2×C4—1662no—
184C2×C4—1662no—
185C2×C4—1662no—
186C2×C4C2×D33212yes—
187C2×C4C2×D33211yescharacteristic
188C2×C4C2×D33212yes—
189C2×C4C23×D8311yescharacteristic
190C8—21no—
191C8—22no—
192C8—21no—
193C22—21no—
194C22—21no—
195C22—3322no—
196C22—3322no—
197C22—3322no—
198C22#1458181412yes—
199C22C22×D33211yescharacteristic
200C4—22no—
201C4—22no—
202C4—21no—
203C4—3322no—
204C4—3322no—
205C4—3322no—
206C4#1458183711yescharacteristic
207C4#1458184011yescharacteristic
208C2—22no—
209C2—6642no—
210C2#1912828211yescharacteristic
211C2C2×#1458181411yescharacteristic
212C2#1912850111yescharacteristic
subgroupG/H
centerC22C22×D332
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
FittingC2×C4⋊C8×C83C2
socleC22×C83C22×D4
prime radical#19127937C2
hypercenterC2×C4⋊C8D83
hypocenterC83#3250
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4⋊C8

Complements

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

NH ≅ G/Ntype
1. #19128235C2semidirect
2. #19128009C2semidirect
3. #19128002C2semidirect
4. #19127942C2semidirect
6. #19127930C2semidirect
7. #19127924C2semidirect
8. #19127917C2semidirect
9. C2×C4⋊C8×C83C2semidirect
107. C83#3250semidirect
116. C2×C4⋊C8D83semidirect

Maximal subgroups

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

subgroupindex
#191282352
#191280092
#191280022
#191279422
#191279372
#191279302
#191279242
#191279172
C2×C4⋊C8×C832
#325083