Finite Group Databasebeta

← #24911917

Subgroups — #24911917

Order 10624.

484 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 153 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
1C2×#14581751C211yesmaximal, characteristic
2#19127629C214yesmaximal
3#19127608C211yesmaximal, characteristic, complemented
4#19127608C211yesmaximal, characteristic, complemented
5#19127582C214yesmaximal, complemented
6#19127570C211yesmaximal, characteristic, complemented
7#19127566C211yesmaximal, characteristic, complemented
8C4×C22⋊Dic83C211yesmaximal, characteristic
9C2×C22⋊Q8×C83C211yesmaximal, characteristic
10#14581751—24no—
11C2×C4⋊Dic83C2211yescharacteristic
12C2×C4⋊Dic83C2211yescharacteristic
13C2×Dic83⋊C4C2214yes—
14C2×C22⋊Dic83C2212yes—
15C2×C22⋊Dic83C2212yes—
16C2×C4×Dic83C2211yescharacteristic
17C2×C4×Dic83C2211yescharacteristic
18C2×C4×Dic83C2212yes—
19C83⋊#95C2214yes—
20C83⋊#95C2212yes—
21C83⋊#95C2214yes—
22C83⋊#95C2214yes—
23C22×Q8×C83C2211yescharacteristic
24C23×C4×C83C2211yescharacteristic
25C22⋊Q8×C83—24no—
26C2×C4⋊C4×C83C2211yescharacteristic
27C2×C4⋊C4×C83C2212yes—
28C2×C22⋊C4×C83C2212yes—
29C22×Dic83C2312yes—
30C22×Dic83C2312yes—
31C22×Dic83C2312yes—
32C22×Dic83C2312yes—
33C4⋊Dic83—22no—
34C4⋊Dic83—22no—
35Dic83⋊C4—28no—
36C22⋊Dic83—24no—
37C22⋊Dic83—24no—
38C4×Dic83—24no—
39C4×Dic83—22no—
40C4×Dic83—22no—
41C24×C83C2311yescharacteristic
42C2×Q8×C83—22no—
43C2×Q8×C83—22no—
44C2×Q8×C83—22no—
45C22×C4×C83—22no—
46C22×C4×C83—22no—
47C22×C4×C83D414yes—
48C22×C4×C83C2312yes—
49C22×C4×C83C2312yes—
50C22×C4×C83C2311yescharacteristic
51C22×C4×C83C2311yescharacteristic
52C4⋊C4×C83—22no—
53C4⋊C4×C83—24no—
54C22⋊C4×C83—24no—
55C23×C83—21no—
56C23×C83—21no—
57C23×C83—22no—
58C23×C83—22no—
59C23×C83C2×D412yes—
60C23×C83C2411yescharacteristic
61Q8×C83—44no—
62C2×C4×C83—21no—
63C2×C4×C83—21no—
64C2×C4×C83—28no—
65C2×C4×C83—22no—
66C2×C4×C83—22no—
67C2×C4×C83—22no—
68C2×C4×C83—22no—
69C2×C4×C83—24no—
70C2×C4×C83C2×D412yes—
71C2×C4×C83C2×D412yes—
72C2×C4×C83#4112yes—
73C2×C4×C83#4112yes—
74C2×Dic83—22no—
75C2×Dic83—22no—
76C2×Dic83—22no—
77C2×Dic83—22no—
78C2×Dic83—22no—
79C2×Dic83—22no—
80C2×Dic83—24no—
81C2×Dic83—24no—
82C2×Dic83—22no—
83C2×Dic83#4114yes—
84C22×C83—22no—
85C22×C83—22no—
86C22×C83—22no—
87C22×C83—42no—
88C22×C83—42no—
89C22×C83C22×D411yescharacteristic
90C22×C83C2×#4112yes—
91C22×C83C2×#4111yescharacteristic
92C22×C83C2×#4111yescharacteristic
93C22×C832−1+412yes—
94C4×C83—22no—
95C4×C83—42no—
96C4×C83—42no—
97C4×C83#12314yes—
98Dic83—24no—
99Dic83—42no—
100Dic83—42no—
101Dic83—42no—
102C2×C83—42no—
103C2×C83C2×#12311yescharacteristic
104C2×C83#53211yescharacteristic
105C2×C83#53611yescharacteristic
106C2×C83#55212yes—
107C2×C83#55612yes—
108#2499—831nomaximal, Sylow
109C83#249911yescharacteristic, complemented, Sylow
110C2×C22⋊Q8D8311yescharacteristic
111C2×#117—831no—
112#397—834no—
113#391—831no—
114#391—831no—
115#387—834no—
116#384—831no—
117#382—831no—
118C4×C22⋊C4—831no—
119C22×Q8C2×D8311yescharacteristic
120C23×C4C2×D8311yescharacteristic
121C22⋊Q8—24no—
122#117—1664no—
123C2×C4⋊C4—831no—
124C2×C4⋊C4—831no—
125C2×C4⋊C4—834no—
126C2×C4⋊C4C2×D8311yescharacteristic
127C2×C4⋊C4C2×D8312yes—
128C2×C22⋊C4—832no—
129C2×C22⋊C4—832no—
130C2×C22⋊C4C2×D8312yes—
131C2×C42—832no—
132C2×C42—831no—
133C2×C42—831no—
134#95—832no—
135#95—834no—
136#95—834no—
137#95—834no—
138C24C22×D8311yescharacteristic
139C2×Q8—22no—
140C2×Q8—22no—
141C2×Q8—22no—
142C22×C4—22no—
143C22×C4—22no—
144C22×C4—832no—
145C22×C4—832no—
146C22×C4—832no—
147C22×C4—832no—
148C22×C4C22⋊D8314yes—
149C22×C4C22×D8312yes—
150C22×C4C22×D8312yes—
151C22×C4C22×D8311yescharacteristic
152C22×C4C22×D8311yescharacteristic
153C4⋊C4—22no—
154C4⋊C4—24no—
155C4⋊C4—1668no—
156C4⋊C4—1662no—
157C4⋊C4—1662no—
158C22⋊C4—24no—
159C22⋊C4—1664no—
160C22⋊C4—1664no—
161C42—1664no—
162C42—1662no—
163C42—1662no—
164C23—22no—
165C23—21no—
166C23—21no—
167C23—22no—
168C23C2×C22⋊D8312yes—
169C23C23×D8311yescharacteristic
170Q8—44no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—28no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—24no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—834no—
180C2×C4—1662no—
181C2×C4—1662no—
182C2×C4—1662no—
183C2×C4—1664no—
184C2×C4—1662no—
185C2×C4—1662no—
186C2×C4—1664no—
187C2×C4—1662no—
188C2×C4—1662no—
189C2×C4C2×C22⋊D8312yes—
190C2×C4C2×C22⋊D8312yes—
191C2×C4#1305058912yes—
192C2×C4#1305058912yes—
193C22—22no—
194C22—22no—
195C22—22no—
196C22—42no—
197C22—42no—
198C22C22×C22⋊D8311yescharacteristic
199C22#41×D8312yes—
200C22C2×#1305058911yescharacteristic
201C22C2×#1305058911yescharacteristic
202C22#1458184012yes—
203C4—22no—
204C4—42no—
205C4—42no—
206C4—3322no—
207C4—3322no—
208C4—3322no—
209C4—1664no—
210C4#1458177314yes—
211C2—42no—
212C2C2×#1458177311yescharacteristic
213C2#1912824711yescharacteristic
214C2#1912827511yescharacteristic
215C2#1912838512yes—
216C2#1912842212yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
FittingC2×C22⋊Q8×C83C2
prime radicalC24×C83C23
hypercenterC2×C22⋊Q8D83
hypocenterC83#2499
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. #19127608C2semidirect
4. #19127608C2semidirect
5. #19127582C2semidirect
6. #19127570C2semidirect
7. #19127566C2semidirect
109. C83#2499semidirect

Maximal subgroups

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

subgroupindex
C2×#145817512
#191276292
#191276082
#191275822
#191275702
#191275662
C4×C22⋊Dic832
C2×C22⋊Q8×C832
#249983