Finite Group Databasebeta

← #24917846

Subgroups — #24917846

Order 10624.

400 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 117 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#19128257C211yesmaximal, characteristic, complemented
2#124×D83C211yesmaximal, characteristic
3#19128062C212yesmaximal, complemented
4#19128059C212yesmaximal, complemented
5#19128058C211yesmaximal, characteristic, complemented
6#19128056C211yesmaximal, characteristic, complemented
7#19128055C211yesmaximal, characteristic, complemented
8#19127708C212yesmaximal, complemented
9#19127704C212yesmaximal
10C4×#13050573C211yesmaximal, characteristic, complemented
11#488×C83C211yesmaximal, characteristic, complemented
12C2×Q8×D83C2211yescharacteristic
13C2×D4×D83C2211yescharacteristic
14C2×#13050573C2212yes—
15#14581791C2212yescomplemented
16#14581784C2211yescharacteristic, complemented
17#14581781C2211yescharacteristic
18#14581780C2211yescharacteristic
19#14581779C2211yescharacteristic
20#14581778—22no—
21#14581770—22no—
22#14581757—22no—
23#14581750C2212yes—
24C22⋊C4×D83—22no—
25C42×D83C2211yescharacteristic
26#14581717C2212yescomplemented
27#14581716C2212yescomplemented
28#14581715C2212yescomplemented
29#14581714C2212yes—
30#14581713C2212yes—
31#14581712C2212yes—
32#14581702C2212yescomplemented
33C4×C83⋊C8C2211yescharacteristic, complemented
34#125×C83C2211yescharacteristic, complemented
35#124×C83C2211yescharacteristic
36#103×C83C2212yescomplemented
37#102×C83C2212yes—
38C4×C8×C83C2211yescharacteristic, complemented
39C23×D83—21no—
40C2×Dic166—22no—
41C2×Dic166—21no—
42C2×Dic166C2311yescharacteristic
43Q8×D83—24no—
44C2×D332C2311yescharacteristic
45C2×C22⋊D83—21no—
46D4×D83—24no—
47C2×C4×D83—21no—
48C2×C4×D83D412yes—
49C2×C4×D83C2311yescharacteristic
50#13050573—24no—
51C2×C83⋊C8C2312yes—
52C4⋊Dic83C2312yes—
53Dic83⋊C4—22no—
54Dic83⋊C4—22no—
55C22⋊Dic83—22no—
56(C2×D83)⋊C4—22no—
57(C2×D83)⋊C4—22no—
58(C2×D83)⋊C4—22no—
59C4×Dic83—22no—
60C4×Dic83D412yes—
61C4×Dic83C2311yescharacteristic
62C2×Q8×C83C2311yescharacteristic
63C2×D4×C83C2311yescharacteristic
64C2×C8×C83C2312yes—
65C4⋊C4×C83—22no—
66C4⋊C4×C83C2312yes—
67C22⋊C4×C83—22no—
68C42×C83C2311yescharacteristic
69C23×C83—21no—
70C22×D83—21no—
71C22×D83—44no—
72C22×D83—42no—
73C22×D83C2×D411yescharacteristic
74Q8×C83—22no—
75D4×C83—22no—
76C2×C4×C83—21no—
77C2×C4×C83—22no—
78C2×C4×C83C2×D412yes—
79C2×C4×C83C2411yescharacteristic
80C22⋊D83—42no—
81C2×Dic83—22no—
82C2×Dic83—21no—
83C2×Dic83C2×D412yes—
84C2×Dic83C2×D411yescharacteristic
85D332—22no—
86C4×D83—42no—
87C4×D83—42no—
88C4×D83#4114yes—
89Dic166—22no—
90Dic166—44no—
91Dic166—42no—
92C8×C83—22no—
93C83⋊C8—22no—
94C22×C83—41no—
95C22×C83—42no—
96C22×C83C22×D411yescharacteristic
97C2×D83—22no—
98C2×D83—41no—
99C2×D83—42no—
100C2×D83—84no—
101C2×D83#12412yes—
102C4×C83—22no—
103C4×C83—41no—
104C4×C83—42no—
105C4×C83C2×#4112yes—
106Dic83—22no—
107Dic83—41no—
108Dic83—42no—
109Dic83#12412yes—
110C2×C83—81no—
111C2×C83C2×#12411yescharacteristic
112C2×C83#57512yes—
113D83—22no—
114D83—81no—
115#3140—831nomaximal, Sylow
116C83#314011yescharacteristic, complemented, Sylow
117#533—831no—
118C2×#124—831no—
119#489—832no—
120#489—832no—
121#488—831no—
122#488—831no—
123#488—831no—
124#488D8311yescharacteristic, complemented
125#419—832no—
126#418—832no—
127C4×M16—831no—
128C22×Q8—831no—
129C22×D4—831no—
130C2×M16—832no—
131C4⋊Q8—832no—
132#125—831no—
133#125C2×D8311yescharacteristic, complemented
134#124—831no—
135#124—831no—
136#124—831no—
137#124—1662no—
138#124C2×D8311yescharacteristic
139C22⋊Q8—1662no—
140C4×Q8—1662no—
141#117—832no—
142C2×C22⋊C4—1662no—
143C2×C42—831no—
144#103—832no—
145#103—832no—
146#103—832no—
147#103C2×D8312yescomplemented
148#102—832no—
149#102—832no—
150#102—832no—
151#102C2×D8312yes—
152#97—832no—
153C4×C8—831no—
154C4×C8C2×D8311yescharacteristic, complemented
155C24—1661no—
156C2×Q8—831no—
157C2×Q8—1662no—
158C2×Q8—1661no—
159C2×Q8—1664no—
160C2×Q8C22×D8311yescharacteristic
161C2×D4—831no—
162C2×D4—1664no—
163C2×D4—1661no—
164C2×D4C22×D8311yescharacteristic
165C22×C4—832no—
166C22×C4—831no—
167C22×C4—1661no—
168M16—1664no—
169C2×C8—832no—
170C2×C8C22×D8312yes—
171C4⋊C4—22no—
172C4⋊C4—832no—
173C4⋊C4—1662no—
174C4⋊C4—1662no—
175C4⋊C4C22×D8312yes—
176C22⋊C4—22no—
177C22⋊C4—1662no—
178C22⋊C4—1662no—
179C22⋊C4—1662no—
180C22⋊C4—1662no—
181C42—832no—
182C42—831no—
183C42—1662no—
184C42C22×D8311yescharacteristic
185C23—21no—
186C23—3322no—
187C23—3324no—
188C23—831no—
189C23—1661no—
190Q8—22no—
191Q8—3322no—
192Q8—3324no—
193Q8—1662no—
194D4—22no—
195D4—3322no—
196D4—1662no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—3322no—
200C2×C4—3322no—
201C2×C4—831no—
202C2×C4—834no—
203C2×C4—832no—
204C2×C4—1662no—
205C2×C4—1661no—
206C2×C4D4×D8312yes—
207C2×C4C23×D8311yescharacteristic
208C8—22no—
209C8—1662no—
210C22—41no—
211C22—42no—
212C22—3321no—
213C22—3322no—
214C22—832no—
215C22—6644no—
216C22—1662no—
217C22C2×D4×D8311yescharacteristic
218C4—22no—
219C4—41no—
220C4—42no—
221C4—3321no—
222C4—3322no—
223C4—832no—
224C4—1662no—
225C4#41×D8312yes—
226C2—81no—
227C2—6641no—
228C2—1662no—
229C2#124×D8311yescharacteristic
230C2#1912849812yes—
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#488×C83C2
socleC22×C83C22×D4
prime radicalC2×D4×D83C22
hypercenter#488D83
hypocenterC83#3140
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #488

Complements

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

NH ≅ G/Ntype
1. #19128257C2semidirect
3. #19128062C2semidirect
4. #19128059C2semidirect
5. #19128058C2semidirect
6. #19128056C2semidirect
7. #19128055C2semidirect
8. #19127708C2semidirect
10. C4×#13050573C2semidirect
11. #488×C83C2semidirect
15. #14581791C22semidirect
16. #14581784C22semidirect
26. #14581717C22semidirect
27. #14581716C22semidirect
28. #14581715C22semidirect
32. #14581702C22semidirect
33. C4×C83⋊C8C22semidirect
34. #125×C83C22semidirect
36. #103×C83C22semidirect
38. C4×C8×C83C22semidirect
116. C83#3140semidirect
124. #488D83semidirect
133. #125C2×D83semidirect
147. #103C2×D83semidirect
154. C4×C8C2×D83semidirect

Maximal subgroups

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

subgroupindex
#191282572
#124×D832
#191280622
#191280592
#191280582
#191280562
#191280552
#191277082
#191277042
C4×#130505732
#488×C832
#314083