Finite Group Databasebeta

← #24920378

Subgroups — #24920378

Order 10624.

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

normal subgroups only

This group has 340 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
1#19128431C211yesmaximal, characteristic, complemented
2#19128412C211yesmaximal, characteristic
3#19128109C212yesmaximal, complemented
4#19128007C212yesmaximal, complemented
5#19127993C211yesmaximal, characteristic, complemented
6#19127988C211yesmaximal, characteristic
7#19127980C212yesmaximal
8#19127831C211yesmaximal, characteristic, complemented
9#19127800C212yesmaximal
10C4×#13050579C211yesmaximal, characteristic
11#500×C83C211yesmaximal, characteristic, complemented
12C2×#13050579C2211yescharacteristic
13#14581791—21no—
14#14581791C2212yes—
15#14581790—22no—
16#14581783C2212yes—
17#14581772—21no—
18#14581768—22no—
19C4×Dic166C2211yescharacteristic
20C4×Dic166C2211yescharacteristic
21#14581757—21no—
22#14581757C2212yes—
23C4×D332C2211yescharacteristic
24#14581754C2212yes—
25C4⋊C4×D83—21no—
26#14581725C2212yes—
27#14581724C2211yescharacteristic
28#14581723C2212yes—
29C4⋊(C83⋊C8)C2211yescharacteristic
30Dic83⋊C8C2212yes—
31#14581717C2211yescharacteristic
32#14581715C2212yes—
33#14581714C2211yescharacteristic
34#14581712C2212yes—
35#14581702C2212yes—
36C4⋊Q8×C83C2211yescharacteristic
37C4⋊Q8×C83C2211yescharacteristic
38#107×C83C2212yes—
39#107×C83C2212yes—
40C4×C8×C83C2211yescharacteristic
41C2×Dic166—22no—
42C2×Dic166D411yescharacteristic
43C2×Dic166C2311yescharacteristic
44C2×D332C2311yescharacteristic
45C2×C4×D83—21no—
46C2×C4×D83—22no—
47#13050579Q814yescomplemented
48C2×C83⋊C8C2312yes—
49C4⋊Dic83—22no—
50C4⋊Dic83D411yescharacteristic
51C4⋊Dic83C2311yescharacteristic
52C4⋊Dic83C2311yescharacteristic
53Dic83⋊C4—21no—
54Dic83⋊C4—22no—
55Dic83⋊C4—22no—
56Dic83⋊C4—22no—
57Dic83⋊C4D412yes—
58(C2×D83)⋊C4—21no—
59(C2×D83)⋊C4—22no—
60C4×Dic83—21no—
61C4×Dic83—22no—
62C4×Dic83C2312yes—
63C2×Q8×C83—21no—
64C2×Q8×C83—21no—
65C2×C8×C83C2311yescharacteristic
66C2×C8×C83C2311yescharacteristic
67C4⋊C4×C83—21no—
68C4⋊C4×C83—21no—
69C4⋊C4×C83C2312yes—
70C4⋊C4×C83C2312yes—
71C42×C83C2311yescharacteristic
72C22×D83—21no—
73Q8×C83—81no—
74Q8×C83—81no—
75C2×C4×C83—22no—
76C2×C4×C83—22no—
77C2×C4×C83C2×D411yescharacteristic
78C2×C4×C83C2×D411yescharacteristic
79C2×C4×C83C2411yescharacteristic
80C2×Dic83—21no—
81C2×Dic83—21no—
82C2×Dic83—21no—
83C2×Dic83C2×D412yes—
84C2×Dic83C2×D412yes—
85D332—21no—
86D332C2×Q812yes—
87C4×D83—42no—
88C4×D83—82no—
89Dic166—22no—
90Dic166—21no—
91Dic166—82no—
92Dic166C2×Q812yes—
93C8×C83—21no—
94C8×C83C2×Q812yes—
95C83⋊C8—22no—
96C22×C83C22×D411yescharacteristic
97C2×D83—22no—
98C2×D83—41no—
99C2×D83—41no—
100C4×C83—21no—
101C4×C83—42no—
102C4×C83—42no—
103C4×C83#13712yes—
104C4×C83C22×Q811yescharacteristic
105C4×C832−1+411yescharacteristic
106Dic83—22no—
107Dic83—22no—
108Dic83—22no—
109Dic83—41no—
110Dic83—41no—
111C2×C83D4×Q811yescharacteristic
112C2×C83C2×#13711yescharacteristic
113C2×C83#57611yescharacteristic
114D83—42no—
115#3385—831nomaximal, Sylow
116C83#338511yescharacteristic, complemented, Sylow
117Q82—831no—
118#554—831no—
119#501—832no—
120#500D8311yescharacteristic, complemented
121#479—832no—
122#477—831no—
123#476—831no—
124#474—832no—
125#446—831no—
126#440—832no—
127C4×SD16—831no—
128C2×SD16—831no—
129C4⋊Q8—832no—
130C4⋊Q8—1661no—
131C4⋊Q8—1662no—
132C4⋊Q8C2×D8311yescharacteristic
133C4⋊Q8C2×D8311yescharacteristic
134#125—832no—
135C22⋊Q8—1661no—
136C22⋊Q8—1662no—
137C4×Q8—831no—
138C4×Q8—832no—
139C4×Q8—831no—
140C4×Q8—1661no—
141C4×D4—832no—
142C4×D4—831no—
143C2×C4⋊C4—1661no—
144#107—832no—
145#107C2×D8312yes—
146#107C2×D8312yes—
147#106—831no—
148#106—832no—
149C4⋊C8—832no—
150C4⋊C8—831no—
151#103—831no—
152#103—832no—
153#102—831no—
154#102—832no—
155#97—832no—
156C4×C8C2×D8311yescharacteristic
157C2×Q8—21no—
158C2×Q8—21no—
159C2×Q8—831no—
160C2×Q8—831no—
161C2×Q8—1662no—
162C2×D4—831no—
163C22×C4—1662no—
164C22×C4—1661no—
165SD16—834no—
166C2×C8—832no—
167C2×C8C22×D8311yescharacteristic
168C2×C8C22×D8311yescharacteristic
169C4⋊C4—21no—
170C4⋊C4—21no—
171C4⋊C4—831no—
172C4⋊C4—832no—
173C4⋊C4—831no—
174C4⋊C4—831no—
175C4⋊C4—1662no—
176C4⋊C4—1662no—
177C4⋊C4—1662no—
178C4⋊C4—1662no—
179C4⋊C4—1661no—
180C4⋊C4C22×D8312yes—
181C4⋊C4C22×D8312yes—
182C22⋊C4—1662no—
183C22⋊C4—1661no—
184C42—832no—
185C42—1662no—
186C42—1661no—
187C42C22×D8311yescharacteristic
188C23—1661no—
189Q8—81no—
190Q8—81no—
191Q8—832no—
192Q8—6642no—
193Q8—1662no—
194Q8—1661no—
195D4—832no—
196D4—1661no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—3322no—
200C2×C4—832no—
201C2×C4—832no—
202C2×C4—6642no—
203C2×C4—1661no—
204C2×C4—1661no—
205C2×C4—1661no—
206C2×C4D4×D8311yescharacteristic
207C2×C4D4×D8311yescharacteristic
208C2×C4C23×D8311yescharacteristic
209C8—21no—
210C8—1662no—
211C8Q8×D8312yes—
212C22—3321no—
213C22—3321no—
214C22—1662no—
215C22C2×D4×D8311yescharacteristic
216C4—21no—
217C4—42no—
218C4—42no—
219C4—3321no—
220C4—3321no—
221C4—1662no—
222C4—1662no—
223C4—1662no—
224C4#1458182412yes—
225C4C2×Q8×D8311yescharacteristic
226C4#1458184011yescharacteristic
227C2—3322no—
228C2#1912836611yescharacteristic
229C2C2×#1458182411yescharacteristic
230C2#1912850211yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#500×C83C2
socleC22×C83C22×D4
prime radicalC2×D332C23
hypercenter#500D83
hypocenterC83#3385
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #500

Complements

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

NH ≅ G/Ntype
1. #19128431C2semidirect
3. #19128109C2semidirect
4. #19128007C2semidirect
5. #19127993C2semidirect
8. #19127831C2semidirect
11. #500×C83C2semidirect
47. #13050579Q8semidirect
116. C83#3385semidirect
120. #500D83semidirect

Maximal subgroups

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

subgroupindex
#191284312
#191284122
#191281092
#191280072
#191279932
#191279882
#191279802
#191278312
#191278002
C4×#130505792
#500×C832
#338583