Finite Group Databasebeta

← #24911943

Subgroups — #24911943

Order 10624.

512 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 141 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×#14581770C211yesmaximal, characteristic
2C2×C4⋊C4×D83C211yesmaximal, characteristic
3#19127641C211yesmaximal, characteristic, complemented
4#19127640C211yesmaximal, characteristic, complemented
5#19127639C211yesmaximal, characteristic, complemented
6#19127605C212yesmaximal, complemented
7#19127565C212yesmaximal, complemented
8#19127563C212yesmaximal
9#19127563C212yesmaximal
10#19127548C211yesmaximal, characteristic
11#400×C83C211yesmaximal, characteristic, complemented
12C22×Dic166C2211yescharacteristic
13C22×C4×D83C2212yes—
14C22×C4×D83C2211yescharacteristic
15#14581770—24no—
16C2×C4⋊Dic83C2211yescharacteristic
17C2×Dic83⋊C4C2211yescharacteristic
18C2×Dic83⋊C4C2212yes—
19C2×Dic83⋊C4C2211yescharacteristic
20C4⋊C4×D83—24no—
21C2×(C2×D83)⋊C4C2211yescharacteristic
22C2×(C2×D83)⋊C4C2211yescharacteristic
23C2×(C2×D83)⋊C4C2212yes—
24C2×C4×Dic83C2212yes—
25C83⋊#95C2212yes—
26C83⋊#95C2212yes—
27C83⋊#95C2212yes—
28C83⋊#95C2212yes—
29C83⋊#95C2212yes—
30C83⋊#95C2212yes—
31C83⋊#95C2211yescharacteristic
32C2×C4⋊C4×C83C2211yescharacteristic
33C2×C4⋊C4×C83C2211yescharacteristic
34#95×C83C2212yes—
35#95×C83C2212yes—
36#95×C83C2211yescharacteristic
37C23×D83C2311yescharacteristic
38C2×Dic166—22no—
39C2×Dic166—24no—
40C22×Dic83C2312yes—
41C22×Dic83C2311yescharacteristic
42C22×Dic83C2312yes—
43C22×Dic83C2311yescharacteristic
44C22×Dic83C2311yescharacteristic
45C2×C4×D83—24no—
46C2×C4×D83—24no—
47C2×C4×D83—28no—
48C2×C4×D83D414yes—
49C4⋊Dic83—22no—
50Dic83⋊C4—22no—
51Dic83⋊C4—22no—
52Dic83⋊C4—24no—
53(C2×D83)⋊C4—22no—
54(C2×D83)⋊C4—22no—
55(C2×D83)⋊C4—24no—
56C4×Dic83—24no—
57C22×C4×C83C2312yes—
58C22×C4×C83C2311yescharacteristic
59C22×C4×C83C2312yes—
60C22×C4×C83C2311yescharacteristic
61C22×C4×C83C2311yescharacteristic
62C4⋊C4×C83—22no—
63C4⋊C4×C83—22no—
64C23×C83C2411yescharacteristic
65C22×D83—22no—
66C22×D83—21no—
67C22×D83—21no—
68C22×D83C2×D412yes—
69C22×D83#4114yes—
70C2×C4×C83—22no—
71C2×C4×C83—21no—
72C2×C4×C83—21no—
73C2×C4×C83—22no—
74C2×C4×C83—22no—
75C2×C4×C83—22no—
76C2×C4×C83—24no—
77C2×C4×C83—22no—
78C2×C4×C83—21no—
79C2×C4×C83—22no—
80C2×C4×C83—21no—
81C2×C4×C83C2×D412yes—
82C2×Dic83—21no—
83C2×Dic83—21no—
84C2×Dic83—22no—
85C2×Dic83—22no—
86C2×Dic83—22no—
87C2×Dic83—24no—
88C2×Dic83—22no—
89C2×Dic83—21no—
90C2×Dic83—21no—
91C2×Dic83—22no—
92C2×Dic83C2×D412yes—
93C2×Dic83#4114yes—
94C4×D83—28no—
95C4×D83—48no—
96Dic166—44no—
97C22×C83C22×D411yescharacteristic
98C22×C83C2×#4112yes—
99C22×C83C2×#4112yes—
100C22×C832−1+411yescharacteristic
101C22×C832−1+411yescharacteristic
102C2×D83—22no—
103C2×D83—22no—
104C2×D83—24no—
105C2×D83—24no—
106C2×D83#12414yes—
107C4×C83—22no—
108C4×C83—41no—
109C4×C83—42no—
110C4×C83—41no—
111C4×C83—42no—
112Dic83—41no—
113Dic83—42no—
114Dic83—41no—
115Dic83—42no—
116Dic83#12314yes—
117C2×C83C2×#12311yescharacteristic
118C2×C83C2×#12411yescharacteristic
119C2×C83#53611yescharacteristic
120C2×C83#55212yes—
121C2×C83#55212yes—
122D83—24no—
123#2503—831nomaximal, Sylow
124C83#250311yescharacteristic, complemented, Sylow
125C2×C22⋊Q8—831no—
126C22×C4⋊C4—831no—
127#400—831no—
128#400—831no—
129#400—831no—
130#400D8311yescharacteristic, complemented
131#391—832no—
132#382—832no—
133#381—832no—
134#381—832no—
135C4×C22⋊C4—831no—
136C22×Q8—831no—
137C23×C4—831no—
138C23×C4—832no—
139C22⋊Q8—1664no—
140C2×C4⋊C4—832no—
141C2×C4⋊C4—831no—
142C2×C4⋊C4—831no—
143C2×C4⋊C4—831no—
144C2×C4⋊C4—1664no—
145C2×C4⋊C4C2×D8311yescharacteristic
146C2×C4⋊C4C2×D8311yescharacteristic
147C2×C22⋊C4—831no—
148C2×C22⋊C4—831no—
149C2×C22⋊C4—832no—
150C2×C42—832no—
151#95—831no—
152#95—832no—
153#95—832no—
154#95—832no—
155#95—832no—
156#95—832no—
157#95—832no—
158#95C2×D8311yescharacteristic
159#95C2×D8312yes—
160#95C2×D8312yes—
161C24—831no—
162C2×Q8—1662no—
163C2×Q8—1664no—
164C22×C4—831no—
165C22×C4—834no—
166C22×C4—832no—
167C22×C4—831no—
168C22×C4—831no—
169C22×C4—832no—
170C22×C4—1664no—
171C22×C4—1668no—
172C22×C4—1664no—
173C22×C4C22×D8312yes—
174C22×C4C22×D8311yescharacteristic
175C22×C4C22×D8311yescharacteristic
176C22×C4C22×D8312yes—
177C22×C4C22×D8311yescharacteristic
178C4⋊C4—22no—
179C4⋊C4—22no—
180C4⋊C4—1664no—
181C4⋊C4—1662no—
182C4⋊C4—1662no—
183C4⋊C4—1662no—
184C22⋊C4—1662no—
185C22⋊C4—1662no—
186C22⋊C4—1664no—
187C42—1664no—
188C23—832no—
189C23—834no—
190C23—1662no—
191C23—1661no—
192C23—1661no—
193C23C23×D8311yescharacteristic
194Q8—3324no—
195C2×C4—22no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—24no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—3328no—
207C2×C4—832no—
208C2×C4—834no—
209C2×C4—1668no—
210C2×C4—1661no—
211C2×C4—1661no—
212C2×C4—1662no—
213C2×C4—1662no—
214C2×C4—1662no—
215C2×C4—1664no—
216C2×C4—1662no—
217C2×C4—1661no—
218C2×C4—1662no—
219C2×C4—1661no—
220C2×C4D4×D8312yes—
221C22—834no—
222C22—1664no—
223C22—1664no—
224C22—1662no—
225C22—1662no—
226C22C2×D4×D8311yescharacteristic
227C22#41×D8312yes—
228C22#41×D8312yes—
229C22#1458183911yescharacteristic
230C22#1458184011yescharacteristic
231C4—22no—
232C4—42no—
233C4—41no—
234C4—42no—
235C4—41no—
236C4—3322no—
237C4—3321no—
238C4—3322no—
239C4—3321no—
240C4—834no—
241C2—1664no—
242C2#123×D8311yescharacteristic
243C2#124×D8311yescharacteristic
244C2#1912827411yescharacteristic
245C2#1912838312yes—
246C2#1912838712yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#400×C83C2
prime radicalC23×D83C23
hypercenter#400D83
hypocenterC83#2503
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
3. #19127641C2semidirect
4. #19127640C2semidirect
5. #19127639C2semidirect
6. #19127605C2semidirect
7. #19127565C2semidirect
11. #400×C83C2semidirect
124. C83#2503semidirect
130. #400D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145817702
C2×C4⋊C4×D832
#191276412
#191276402
#191276392
#191276052
#191275652
#191275632
#191275482
#400×C832
#250383