Finite Group Databasebeta

← #24911780

Subgroups — #24911780

Order 10624.

584 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 181 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×C4⋊Dic83C211yesmaximal, characteristic, complemented
2#19127640C212yesmaximal, complemented
3#19127637C212yesmaximal, complemented
4#19127599C212yesmaximal, complemented
5#19127581C212yesmaximal
6#19127573C212yesmaximal, complemented
7#19127561C212yesmaximal, complemented
8#19127561C211yesmaximal, characteristic, complemented
9C2×C4×D4×C83C211yesmaximal, characteristic
10C23×Dic83C2212yes—
11C2×C4⋊Dic83—24no—
12C2×C4⋊Dic83C2212yes—
13C2×C4⋊Dic83C2212yes—
14C2×C4⋊Dic83C2212yes—
15C2×Dic83⋊C4C2214yes—
16C2×C22⋊Dic83C2212yes—
17C2×C22⋊Dic83C2214yes—
18C83⋊#95C2214yes—
19C83⋊#95C2212yes—
20C83⋊#95C2214yes—
21C22×D4×C83C2211yescharacteristic
22C23×C4×C83C2211yescharacteristic
23C23×C4×C83C2211yescharacteristic
24C4×D4×C83—24no—
25C2×C4⋊C4×C83C2211yescharacteristic
26C2×C22⋊C4×C83C2211yescharacteristic
27C2×C22⋊C4×C83C2211yescharacteristic
28C2×C42×C83C2211yescharacteristic
29C22×Dic83—24no—
30C22×Dic83—44no—
31C22×Dic83C2314yes—
32C22×Dic83C2314yes—
33C4⋊Dic83—24no—
34C4⋊Dic83—24no—
35C4⋊Dic83—24no—
36Dic83⋊C4—44no—
37C22⋊Dic83—24no—
38C22⋊Dic83—44no—
39C24×C83C2311yescharacteristic
40C24×C83C2311yescharacteristic
41C2×D4×C83—22no—
42C2×D4×C83—22no—
43C2×D4×C83Q814yes—
44C22×C4×C83—24no—
45C22×C4×C83—22no—
46C22×C4×C83—24no—
47C22×C4×C83D414yes—
48C22×C4×C83C2311yescharacteristic
49C22×C4×C83C2311yescharacteristic
50C22×C4×C83C2311yescharacteristic
51C22×C4×C83C2311yescharacteristic
52C22×C4×C83C2311yescharacteristic
53C4⋊C4×C83—22no—
54C22⋊C4×C83—22no—
55C22⋊C4×C83—22no—
56C42×C83—22no—
57C23×C83—22no—
58C23×C83—22no—
59C23×C83—21no—
60C23×C83—22no—
61C23×C83—21no—
62C23×C83—22no—
63C23×C83C2×D412yes—
64C23×C83C2×Q812yes—
65C23×C83C2×Q812yes—
66C23×C83#4112yes—
67C23×C83C2411yescharacteristic
68D4×C83—28no—
69C2×C4×C83—21no—
70C2×C4×C83—22no—
71C2×C4×C83—21no—
72C2×C4×C83—28no—
73C2×C4×C83—21no—
74C2×C4×C83—21no—
75C2×C4×C83—22no—
76C2×C4×C83—22no—
77C2×C4×C83—44no—
78C2×C4×C83C2×D412yes—
79C2×C4×C83C2×D412yes—
80C2×C4×C83C2×Q812yes—
81C2×C4×C83#4114yes—
82C2×Dic83—24no—
83C2×Dic83—24no—
84C2×Dic83—24no—
85C2×Dic83—24no—
86C2×Dic83—24no—
87C2×Dic83—24no—
88C2×Dic83—48no—
89C22×C83—22no—
90C22×C83—22no—
91C22×C83—24no—
92C22×C83—24no—
93C22×C83—22no—
94C22×C83—24no—
95C22×C83—41no—
96C22×C83—41no—
97C22×C83—42no—
98C22×C83C22⋊Q814yes—
99C22×C83C22×D411yescharacteristic
100C22×C83C22×Q811yescharacteristic
101C22×C83C2×#4112yes—
102C22×C83C2×#4111yescharacteristic
103C22×C83H(5,𝔽2)12yes—
104C4×C83—22no—
105C4×C83—22no—
106C4×C83—41no—
107C4×C83—41no—
108C4×C83#12314yes—
109Dic83—44no—
110Dic83—44no—
111C2×C83—24no—
112C2×C83—42no—
113C2×C83C2×C22⋊Q811yescharacteristic
114C2×C83C2×#12311yescharacteristic
115C2×C83#53511yescharacteristic
116C2×C83#55312yes—
117C2×C83#55412yes—
118#2487—831nomaximal, Sylow
119C83#248711yescharacteristic, complemented, Sylow
120C2×C4×D4D8311yescharacteristic
121C22×C4⋊C4—831no—
122#400—832no—
123#399—832no—
124#389—832no—
125#386—832no—
126#385—832no—
127#380—832no—
128#380—831no—
129C22×D4C2×D8311yescharacteristic
130C23×C4—832no—
131C23×C4C2×D8311yescharacteristic
132C23×C4C2×D8311yescharacteristic
133C4×D4—24no—
134C2×C4⋊C4—834no—
135C2×C4⋊C4—832no—
136C2×C4⋊C4—832no—
137C2×C4⋊C4—832no—
138C2×C4⋊C4—1664no—
139C2×C4⋊C4C2×D8311yescharacteristic
140C2×C22⋊C4—834no—
141C2×C22⋊C4—832no—
142C2×C22⋊C4C2×D8311yescharacteristic
143C2×C22⋊C4C2×D8311yescharacteristic
144C2×C42C2×D8311yescharacteristic
145#95—834no—
146#95—832no—
147#95—834no—
148C24C22×D8311yescharacteristic
149C24C22×D8311yescharacteristic
150C2×D4—22no—
151C2×D4—22no—
152C2×D4Dic16614yes—
153C22×C4—24no—
154C22×C4—22no—
155C22×C4—24no—
156C22×C4—3324no—
157C22×C4—834no—
158C22×C4—834no—
159C22×C4—1664no—
160C22×C4C22⋊D8314yes—
161C22×C4C22×D8311yescharacteristic
162C22×C4C22×D8311yescharacteristic
163C22×C4C22×D8311yescharacteristic
164C22×C4C22×D8311yescharacteristic
165C22×C4C22×D8311yescharacteristic
166C4⋊C4—22no—
167C4⋊C4—3324no—
168C4⋊C4—1664no—
169C4⋊C4—1664no—
170C4⋊C4—1664no—
171C22⋊C4—22no—
172C22⋊C4—22no—
173C22⋊C4—3324no—
174C22⋊C4—1664no—
175C42—22no—
176C23—22no—
177C23—22no—
178C23—21no—
179C23—22no—
180C23—21no—
181C23—22no—
182C23C2×C22⋊D8312yes—
183C23C2×Dic16612yes—
184C23C2×Dic16612yes—
185C23#1305059012yes—
186C23C23×D8311yescharacteristic
187D4—28no—
188C2×C4—21no—
189C2×C4—22no—
190C2×C4—21no—
191C2×C4—28no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—44no—
197C2×C4—3328no—
198C2×C4—1664no—
199C2×C4—1664no—
200C2×C4—1664no—
201C2×C4—1664no—
202C2×C4—1664no—
203C2×C4—1664no—
204C2×C4C2×C22⋊D8312yes—
205C2×C4C2×C22⋊D8312yes—
206C2×C4C2×Dic16612yes—
207C2×C4#1305058914yes—
208C22—24no—
209C22—24no—
210C22—22no—
211C22—22no—
212C22—24no—
213C22—22no—
214C22—42no—
215C22—41no—
216C22—41no—
217C22#1458177114yes—
218C22C22×C22⋊D8311yescharacteristic
219C22C22×Dic16611yescharacteristic
220C22C2×#1305058912yes—
221C22C2×#1305059011yescharacteristic
222C22#1458183812yes—
223C4—22no—
224C4—22no—
225C4—41no—
226C4—41no—
227C4—3324no—
228C4—3324no—
229C4#1458177314yes—
230C2—24no—
231C2—42no—
232C2C2×#1458177111yescharacteristic
233C2C2×#1458177311yescharacteristic
234C2#1912826511yescharacteristic
235C2#1912839712yes—
236C2#1912840812yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
FittingC2×C4×D4×C83C2
prime radicalC22×D4×C83C22
hypercenterC2×C4×D4D83
hypocenterC83#2487
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. C22×C4⋊Dic83C2semidirect
2. #19127640C2semidirect
3. #19127637C2semidirect
4. #19127599C2semidirect
6. #19127573C2semidirect
7. #19127561C2semidirect
8. #19127561C2semidirect
119. C83#2487semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊Dic832
#191276402
#191276372
#191275992
#191275812
#191275732
#191275612
C2×C4×D4×C832
#248783