Finite Group Databasebeta

← #24920545

Subgroups — #24920545

Order 10624.

356 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 113 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#19128420C212yesmaximal, complemented
2#19128092C212yesmaximal, complemented
3#19127920C212yesmaximal, complemented
4#19127914C212yesmaximal
5#19127909C212yesmaximal, complemented
6#19127831C211yesmaximal, characteristic, complemented
7#19127802C212yesmaximal, complemented
8C4×Dic332C211yesmaximal, characteristic, complemented
9#495×C83C211yesmaximal, characteristic
10C2×Dic332C2211yescharacteristic
11C2×#13050580C2212yes—
12C2×#13050578C2212yes—
13#14581790C2212yes—
14#14581785—22no—
15#14581785—22no—
16#14581779C2212yes—
17#14581769—22no—
18Q8×Dic83C2212yes—
19C4×Dic166C2212yes—
20C4×Dic166C2211yescharacteristic
21D4×Dic83C2212yes—
22#14581748—22no—
23#14581726C2211yescharacteristic
24C4⋊(C83⋊C8)C2211yescharacteristic
25Dic83⋊C8C2212yes—
26#14581717C2212yes—
27#14581716C2212yes—
28#14581713C2212yes—
29#14581702C2212yes—
30C2×Q16×C83C2211yescharacteristic
31C2×SD16×C83C2212yes—
32C2×D8×C83C2211yescharacteristic
33#124×C83C2212yes—
34C4×C8×C83C2211yescharacteristic
35C2×Dic166D411yescharacteristic
36C2×Dic166C2312yes—
37C22×Dic83—22no—
38Dic332—22no—
39#13050580—42no—
40#13050578—42no—
41C2×C83⋊C8C2312yes—
42C4⋊Dic83—22no—
43C4⋊Dic83D411yescharacteristic
44C4⋊Dic83C2312yes—
45Dic83⋊C4—22no—
46Dic83⋊C4—22no—
47Dic83⋊C4—22no—
48Dic83⋊C4D412yes—
49C22⋊Dic83—22no—
50C22⋊Dic83—22no—
51C22⋊Dic83—22no—
52C4×Dic83—22no—
53C4×Dic83—22no—
54C4×Dic83C2312yes—
55C2×Q8×C83C2312yes—
56C2×D4×C83C2312yes—
57Q16×C83—41no—
58SD16×C83—24no—
59D8×C83—41no—
60C2×C8×C83C2311yescharacteristic
61C2×C8×C83C2311yescharacteristic
62C22⋊C4×C83—22no—
63C22⋊C4×C83—22no—
64C42×C83C2311yescharacteristic
65C23×C83—22no—
66Q8×C83—22no—
67Q8×C83—22no—
68D4×C83—22no—
69D4×C83—22no—
70C2×C4×C83—22no—
71C2×C4×C83C2×D411yescharacteristic
72C2×C4×C83C2×D411yescharacteristic
73C2×C4×C83C2411yescharacteristic
74C2×Dic83—22no—
75C2×Dic83—22no—
76C2×Dic83—44no—
77C2×Dic83C2×D412yes—
78C2×Dic83C2×D412yes—
79Dic166—22no—
80Dic166—22no—
81Dic166—41no—
82C8×C83—21no—
83C8×C83#4112yes—
84C83⋊C8—22no—
85C22×C83—42no—
86C22×C83—42no—
87C22×C83—42no—
88C22×C83C22×D411yescharacteristic
89C4×C83—21no—
90C4×C83—21no—
91C4×C83—42no—
92C4×C83C2×#4111yescharacteristic
93C4×C83H(5,𝔽2)11yescharacteristic
94Dic83—22no—
95Dic83—22no—
96Dic83—42no—
97Dic83—42no—
98C2×C83—82no—
99C2×C83#55011yescharacteristic
100C2×C83#57511yescharacteristic
101C2×C83#57811yescharacteristic
102#3402—831nomaximal, Sylow
103C83#340211yescharacteristic, complemented, Sylow
104#556—832no—
105#497—832no—
106#495D8311yescharacteristic
107#464—832no—
108#463—832no—
109#462—832no—
110#446—831no—
111#440—832no—
112C4×Q16—831no—
113C2×Q16—832no—
114C2×Q16—831no—
115C2×Q16C2×D8311yescharacteristic
116C2×SD16—832no—
117C2×SD16C2×D8312yes—
118C2×D8C2×D8311yescharacteristic
119C4⋊Q8—832no—
120#126—1662no—
121#126—1662no—
122#124—832no—
123#124C2×D8312yes—
124C22⋊Q8—1662no—
125C4×Q8—831no—
126C4×Q8—832no—
127C4×Q8—832no—
128C4×D4—832no—
129#117—1662no—
130#107—831no—
131C4⋊C8—832no—
132C4⋊C8—831no—
133#103—832no—
134#103—832no—
135#102—832no—
136#97—832no—
137C4×C8C2×D8311yescharacteristic
138C2×Q8—831no—
139C2×Q8—832no—
140C2×Q8C22×D8312yes—
141C2×D4C22×D8312yes—
142C22×C4—1662no—
143Q16—41no—
144Q16—3322no—
145Q16—1662no—
146SD16—24no—
147SD16—3322no—
148D8—41no—
149C2×C8—832no—
150C2×C8C22×D8311yescharacteristic
151C2×C8C22×D8311yescharacteristic
152C4⋊C4—832no—
153C4⋊C4—831no—
154C4⋊C4—832no—
155C4⋊C4—1662no—
156C4⋊C4—1662no—
157C4⋊C4—1662no—
158C4⋊C4—1662no—
159C22⋊C4—22no—
160C22⋊C4—22no—
161C22⋊C4—1662no—
162C22⋊C4—1662no—
163C22⋊C4—1662no—
164C42—832no—
165C42—1662no—
166C42—1662no—
167C42C22×D8311yescharacteristic
168C23—22no—
169Q8—22no—
170Q8—22no—
171Q8—3321no—
172Q8—1662no—
173Q8—1662no—
174D4—22no—
175D4—22no—
176C2×C4—22no—
177C2×C4—3324no—
178C2×C4—832no—
179C2×C4—832no—
180C2×C4—1662no—
181C2×C4—1662no—
182C2×C4D4×D8311yescharacteristic
183C2×C4D4×D8311yescharacteristic
184C2×C4C23×D8311yescharacteristic
185C8—21no—
186C8—1662no—
187C8#1305058912yes—
188C22—42no—
189C22—42no—
190C22—42no—
191C22C2×D4×D8311yescharacteristic
192C4—21no—
193C4—21no—
194C4—42no—
195C4—3322no—
196C4—3322no—
197C4—1662no—
198C4—1662no—
199C4C2×#1305058911yescharacteristic
200C4#1458183711yescharacteristic
201C2—82no—
202C2#1912837111yescharacteristic
203C2#1912849811yescharacteristic
204C2#1912851511yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#495×C83C2
socleC22×C83C22×D4
prime radicalC2×D8×C83C22
hypercenter#495D83
hypocenterC83#3402
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #495

Complements

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

NH ≅ G/Ntype
1. #19128420C2semidirect
2. #19128092C2semidirect
3. #19127920C2semidirect
5. #19127909C2semidirect
6. #19127831C2semidirect
7. #19127802C2semidirect
8. C4×Dic332C2semidirect
103. C83#3402semidirect

Maximal subgroups

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

subgroupindex
#191284202
#191280922
#191279202
#191279142
#191279092
#191278312
#191278022
C4×Dic3322
#495×C832
#340283