Finite Group Databasebeta

← #24920343

Subgroups — #24920343

Order 10624.

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

normal subgroups only

This group has 338 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#19128426C211yesmaximal, characteristic, complemented
2#19128412C211yesmaximal, characteristic
3#19128101C212yesmaximal, complemented
4#19128011C212yesmaximal, complemented
5#19128000C212yesmaximal
6#19127993C211yesmaximal, characteristic, complemented
7#19127978C211yesmaximal, characteristic
8C8×Dic166C211yesmaximal, characteristic, complemented
9C4×#13050579C211yesmaximal, characteristic
10#19127788C212yesmaximal
11#498×C83C211yesmaximal, characteristic, complemented
12C2×#13050579C2211yescharacteristic
13#14581790—21no—
14#14581783C2212yes—
15#14581783C2212yes—
16#14581782—22no—
17#14581772—21no—
18#14581768—22no—
19Q8×Dic83—21no—
20C4×Dic166C2211yescharacteristic
21C4×Dic166C2211yescharacteristic
22#14581757C2212yes—
23C4×D332C2211yescharacteristic
24#14581754C2212yes—
25C4⋊C4×D83—21no—
26#14581725C2212yes—
27#14581725C2212yes—
28#14581724C2211yescharacteristic
29C4⋊(C83⋊C8)C2211yescharacteristic
30Dic83⋊C8C2212yes—
31#14581717C2211yescharacteristic
32#14581715C2212yes—
33#14581714C2211yescharacteristic
34#14581712C2212yes—
35C8×Dic83C2212yes—
36C4⋊Q8×C83C2211yescharacteristic
37C4⋊Q8×C83C2211yescharacteristic
38#106×C83C2212yes—
39#106×C83C2212yes—
40C4×C8×C83C2211yescharacteristic
41C2×Dic166D411yescharacteristic
42C2×Dic166C2311yescharacteristic
43C2×D332C2311yescharacteristic
44C2×C4×D83—21no—
45C2×C4×D83—22no—
46#13050579Q814yescomplemented
47C2×C83⋊C8C2312yes—
48C4⋊Dic83—22no—
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—42no—
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—
91Dic166C2×Q812yes—
92C8×C83—21no—
93C8×C83C2×Q812yes—
94C83⋊C8—22no—
95C22×C83C22×D411yescharacteristic
96C2×D83—22no—
97C2×D83—41no—
98C2×D83—41no—
99C4×C83—21no—
100C4×C83—42no—
101C4×C83—42no—
102C4×C83#13512yes—
103C4×C83C22×Q811yescharacteristic
104C4×C832−1+411yescharacteristic
105Dic83—22no—
106Dic83—22no—
107Dic83—22no—
108Dic83—41no—
109Dic83—41no—
110C2×C83D4×Q811yescharacteristic
111C2×C83C2×#13511yescharacteristic
112C2×C83#57711yescharacteristic
113D83—42no—
114#3381—831nomaximal, Sylow
115C83#338111yescharacteristic, complemented, Sylow
116#557—831no—
117#554—831no—
118#499—832no—
119#498D8311yescharacteristic, complemented
120#479—832no—
121#478—832no—
122#477—831no—
123#474—831no—
124C8×Q8—831no—
125C4×SD16—832no—
126C4×SD16—831no—
127C2×SD16—831no—
128C4⋊Q8—1661no—
129C4⋊Q8C2×D8311yescharacteristic
130C4⋊Q8C2×D8311yescharacteristic
131#125—832no—
132#125—832no—
133#125—1662no—
134C22⋊Q8—1661no—
135C22⋊Q8—1662no—
136C4×Q8—831no—
137C4×Q8—832no—
138C4×Q8—831no—
139C4×Q8—1661no—
140C4×D4—832no—
141C4×D4—831no—
142C2×C4⋊C4—1661no—
143#107—832no—
144#107—832no—
145#106—831no—
146#106C2×D8312yes—
147#106C2×D8312yes—
148C4⋊C8—832no—
149C4⋊C8—831no—
150#103—831no—
151#103—832no—
152#102—831no—
153#102—832no—
154C4×C8—832no—
155C4×C8C2×D8311yescharacteristic
156C2×Q8—21no—
157C2×Q8—21no—
158C2×Q8—831no—
159C2×Q8—831no—
160C2×D4—831no—
161C22×C4—1662no—
162C22×C4—1661no—
163SD16—834no—
164C2×C8—832no—
165C2×C8C22×D8311yescharacteristic
166C2×C8C22×D8311yescharacteristic
167C4⋊C4—21no—
168C4⋊C4—21no—
169C4⋊C4—831no—
170C4⋊C4—832no—
171C4⋊C4—831no—
172C4⋊C4—831no—
173C4⋊C4—1662no—
174C4⋊C4—1662no—
175C4⋊C4—1662no—
176C4⋊C4—1662no—
177C4⋊C4—1662no—
178C4⋊C4—1661no—
179C4⋊C4C22×D8312yes—
180C4⋊C4C22×D8312yes—
181C22⋊C4—1662no—
182C22⋊C4—1661no—
183C42—832no—
184C42—1662no—
185C42—1661no—
186C42C22×D8311yescharacteristic
187C23—1661no—
188Q8—42no—
189Q8—81no—
190Q8—832no—
191Q8—1662no—
192Q8—1661no—
193D4—832no—
194D4—1661no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—3322no—
198C2×C4—832no—
199C2×C4—832no—
200C2×C4—6642no—
201C2×C4—1661no—
202C2×C4—1661no—
203C2×C4—1661no—
204C2×C4D4×D8311yescharacteristic
205C2×C4D4×D8311yescharacteristic
206C2×C4C23×D8311yescharacteristic
207C8—21no—
208C8—1662no—
209C8Q8×D8312yes—
210C22—3321no—
211C22—3321no—
212C22—1662no—
213C22C2×D4×D8311yescharacteristic
214C4—21no—
215C4—42no—
216C4—42no—
217C4—3321no—
218C4—3321no—
219C4—1662no—
220C4—1662no—
221C4—1662no—
222C4#1458181312yes—
223C4C2×Q8×D8311yescharacteristic
224C4#1458184011yescharacteristic
225C2—3322no—
226C2#1912836611yescharacteristic
227C2C2×#1458181311yescharacteristic
228C2#1912850711yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#498×C83C2
socleC22×C83C22×D4
prime radicalC2×D332C23
hypercenter#498D83
hypocenterC83#3381
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #498

Complements

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

NH ≅ G/Ntype
1. #19128426C2semidirect
3. #19128101C2semidirect
4. #19128011C2semidirect
6. #19127993C2semidirect
8. C8×Dic166C2semidirect
11. #498×C83C2semidirect
46. #13050579Q8semidirect
115. C83#3381semidirect
119. #498D83semidirect

Maximal subgroups

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

subgroupindex
#191284262
#191284122
#191281012
#191280112
#191280002
#191279932
#191279782
C8×Dic1662
C4×#130505792
#191277882
#498×C832
#338183