Finite Group Databasebeta

← #24910798

Subgroups — #24910798

Order 10624.

864 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 864 conjugacy classes of subgroups — too many to draw. Only its 173 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
1C22×#13050590C211yesmaximal, characteristic
2C2×#14581775C212yesmaximal, complemented
3#19127621C212yesmaximal, complemented
4#19127618C212yesmaximal, complemented
5#19127580C212yesmaximal, complemented
6#19127572C214yesmaximal, complemented
7#19127564C211yesmaximal, characteristic, complemented
8C22×C4⋊C4×C83C211yesmaximal, characteristic, complemented
9C2×#13050590—28no—
10C2×#13050590—24no—
11C22×Dic166C2211yescharacteristic
12C22×D332C2211yescharacteristic
13C22×C22⋊D83C2212yes—
14C22×C4×D83C2212yes—
15#14581775—28no—
16C2×Dic83⋊C4C2214yes—
17C2×C22⋊Dic83C2212yes—
18C2×(C2×D83)⋊C4C2214yes—
19C2×(C2×D83)⋊C4C2214yes—
20C83⋊#95C2212yes—
21C83⋊#95C2212yes—
22C83⋊#95C2214yes—
23C23×C4×C83C2211yescharacteristic
24C23×C4×C83C2211yescharacteristic
25C23×C4×C83C2211yescharacteristic
26C2×C4⋊C4×C83—24no—
27C2×C4⋊C4×C83C2212yes—
28C2×C4⋊C4×C83C2212yes—
29C23×D83C2312yes—
30#13050590—232no—
31C2×Dic166—22no—
32C2×Dic166—24no—
33C2×D332—24no—
34C2×D332—22no—
35C2×C22⋊D83—24no—
36C2×C22⋊D83—24no—
37C2×C22⋊D83D418yes—
38C22×Dic83C2314yes—
39C22×Dic83C2312yes—
40C2×C4×D83—24no—
41C2×C4×D83—24no—
42C2×C4×D83—24no—
43Dic83⋊C4—28no—
44C22⋊Dic83—24no—
45(C2×D83)⋊C4—28no—
46(C2×D83)⋊C4—28no—
47C24×C83C2311yescharacteristic
48C22×C4×C83—24no—
49C22×C4×C83—22no—
50C22×C4×C83—24no—
51C22×C4×C83—24no—
52C22×C4×C83—22no—
53C22×C4×C83D414yes—
54C22×C4×C83C2312yes—
55C22×C4×C83C2311yescharacteristic
56C22×C4×C83C2311yescharacteristic
57C22×C4×C83C2312yes—
58C4⋊C4×C83—24no—
59C4⋊C4×C83—24no—
60C23×C83—22no—
61C23×C83—22no—
62C23×C83C2×D414yes—
63C23×C83C2×D412yes—
64C23×C83C2411yescharacteristic
65C22×D83—22no—
66C22×D83—22no—
67C22×D83—24no—
68C22×D83—24no—
69C22×D83C2×D414yes—
70C2×C4×C83—28no—
71C2×C4×C83—28no—
72C2×C4×C83—22no—
73C2×C4×C83—22no—
74C2×C4×C83—22no—
75C2×C4×C83—22no—
76C2×C4×C83—22no—
77C2×C4×C83—24no—
78C2×C4×C83—44no—
79C2×C4×C83C2×D412yes—
80C2×C4×C83C2×D412yes—
81C2×C4×C83#4114yescomplemented
82C22⋊D83—216no—
83C2×Dic83—22no—
84C2×Dic83—22no—
85C2×Dic83—24no—
86C2×Dic83—24no—
87C2×Dic83—24no—
88C2×Dic83C2×D414yes—
89D332—28no—
90C4×D83—216no—
91Dic166—28no—
92C22×C83—22no—
93C22×C83—22no—
94C22×C83—28no—
95C22×C83#12014yes—
96C22×C83C22×D412yes—
97C22×C83C22×D411yescharacteristic
98C22×C83C2×#4112yes—
99C22×C832−1+412yes—
100C2×D83—24no—
101C2×D83—28no—
102C2×D83—42no—
103C2×D83—42no—
104C2×D83—44no—
105C4×C83—22no—
106C4×C83—22no—
107C4×C83—24no—
108C4×C83—42no—
109Dic83—24no—
110Dic83—44no—
111C2×C83—24no—
112C2×C83C2×#12011yescharacteristic
113C2×C83#52511yescharacteristic
114C2×C83#53611yescharacteristic
115C2×C83#54714yes—
116D83—44no—
117#2406—831nomaximal, Sylow
118C83#240611yescharacteristic, complemented, Sylow
119C22×#41—831no—
120C2×#123—832no—
121C22×C4⋊C4D8311yescharacteristic, complemented
122#395—832no—
123#394—832no—
124#386—832no—
125#385—834no—
126#381—831no—
127C2×#41—1668no—
128C2×#41—1664no—
129C22×Q8—831no—
130C22×D4—832no—
131C22×D4—831no—
132C23×C4—832no—
133C23×C4C2×D8311yescharacteristic
134C23×C4C2×D8311yescharacteristic
135C23×C4C2×D8311yescharacteristic
136#123—1668no—
137C2×C4⋊C4—24no—
138C2×C4⋊C4—834no—
139C2×C4⋊C4C2×D8312yes—
140C2×C4⋊C4C2×D8312yes—
141C2×C22⋊C4—832no—
142C2×C22⋊C4—834no—
143C2×C22⋊C4—834no—
144#95—832no—
145#95—832no—
146#95—834no—
147C24—832no—
148C24C22×D8311yescharacteristic
149#41—16632no—
150C2×Q8—1664no—
151C2×Q8—1662no—
152C2×D4—838no—
153C2×D4—1664no—
154C2×D4—1664no—
155C2×D4—1664no—
156C2×D4—1662no—
157C22×C4—24no—
158C22×C4—22no—
159C22×C4—24no—
160C22×C4—24no—
161C22×C4—22no—
162C22×C4—832no—
163C22×C4—834no—
164C22×C4—1664no—
165C22×C4—1664no—
166C22×C4—1664no—
167C22×C4C22⋊D8314yes—
168C22×C4C22×D8312yes—
169C22×C4C22×D8311yescharacteristic
170C22×C4C22×D8311yescharacteristic
171C22×C4C22×D8312yes—
172C4⋊C4—24no—
173C4⋊C4—24no—
174C4⋊C4—1668no—
175C22⋊C4—1668no—
176C22⋊C4—1664no—
177C22⋊C4—1668no—
178C23—22no—
179C23—22no—
180C23—834no—
181C23—1662no—
182C23—1662no—
183C23—1664no—
184C23—1664no—
185C23D4×D8314yes—
186C23C2×C22⋊D8312yes—
187C23C23×D8311yescharacteristic
188Q8—1668no—
189D4—1668no—
190D4—16616no—
191C2×C4—28no—
192C2×C4—28no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—24no—
198C2×C4—22no—
199C2×C4—44no—
200C2×C4—834no—
201C2×C4—16616no—
202C2×C4—1664no—
203C2×C4—1664no—
204C2×C4—1664no—
205C2×C4—1662no—
206C2×C4—1662no—
207C2×C4C2×C22⋊D8312yes—
208C2×C4C2×C22⋊D8312yes—
209C2×C4#1305059014yescomplemented
210C22—28no—
211C22—22no—
212C22—22no—
213C22—3324no—
214C22—3322no—
215C22—3322no—
216C22—1668no—
217C22—1664no—
218C22#1458176014yes—
219C22C2×D4×D8312yes—
220C22C22×C22⋊D8311yescharacteristic
221C22C2×#1305059012yes—
222C22#1458184012yes—
223C4—22no—
224C4—22no—
225C4—24no—
226C4—42no—
227C4—3324no—
228C4—1664no—
229C2—24no—
230C2—3324no—
231C2C2×#1458176011yescharacteristic
232C2#1912821211yescharacteristic
233C2#1912827511yescharacteristic
234C2#1912835214yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
FittingC22×C4⋊C4×C83C2
prime radicalC22×#13050590C2
hypercenterC22×C4⋊C4D83
hypocenterC83#2406
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#14581775C2semidirect
3. #19127621C2semidirect
4. #19127618C2semidirect
5. #19127580C2semidirect
6. #19127572C2semidirect
7. #19127564C2semidirect
8. C22×C4⋊C4×C83C2semidirect
81. C2×C4×C83#41semidirect
118. C83#2406semidirect
121. C22×C4⋊C4D83semidirect
209. C2×C4#13050590semidirect

Maximal subgroups

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

subgroupindex
C22×#130505902
C2×#145817752
#191276212
#191276182
#191275802
#191275722
#191275642
C22×C4⋊C4×C832
#240683