Finite Group Databasebeta

← #24911339

Subgroups — #24911339

Order 10624.

660 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 660 conjugacy classes of subgroups — too many to draw. Only its 149 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×#14581773C212yesmaximal, complemented
2C2×D4×Dic83C212yesmaximal, complemented
3#19127637C211yesmaximal, characteristic, complemented
4#19127615C212yesmaximal, complemented
5#19127598C212yesmaximal, complemented
6#19127595C212yesmaximal, complemented
7#19127576C212yesmaximal, complemented
8#19127573C211yesmaximal, characteristic, complemented
9#392×C83C211yesmaximal, characteristic
10C23×Dic83C2211yescharacteristic
11C23×Dic83C2212yes—
12#14581773—28no—
13D4×Dic83—28no—
14C2×C4⋊Dic83C2212yes—
15C2×C4⋊Dic83C2211yescharacteristic
16C2×Dic83⋊C4C2212yes—
17C2×Dic83⋊C4C2212yes—
18C2×Dic83⋊C4C2212yes—
19C2×C22⋊Dic83C2211yescharacteristic
20C2×C22⋊Dic83C2212yes—
21C2×C22⋊Dic83C2212yes—
22C2×C22⋊Dic83C2212yes—
23C2×C22⋊Dic83C2212yes—
24C2×C4×Dic83C2212yes—
25C83⋊#95C2212yes—
26C83⋊#95C2212yes—
27C83⋊#95C2211yescharacteristic
28C22×D4×C83C2212yes—
29C22×D4×C83C2211yescharacteristic
30C2×C22⋊C4×C83C2212yes—
31C2×C22⋊C4×C83C2211yescharacteristic
32#95×C83C2211yescharacteristic
33C22×Dic83—24no—
34C22×Dic83—24no—
35C22×Dic83—24no—
36C22×Dic83—41no—
37C22×Dic83—42no—
38C22×Dic83C2312yes—
39C22×Dic83C2312yes—
40C22×Dic83C2312yes—
41C22×Dic83C2312yes—
42C4⋊Dic83—24no—
43C4⋊Dic83—41no—
44Dic83⋊C4—24no—
45Dic83⋊C4—24no—
46Dic83⋊C4—24no—
47C22⋊Dic83—24no—
48C22⋊Dic83—22no—
49C22⋊Dic83—24no—
50C22⋊Dic83—24no—
51C22⋊Dic83D418yescomplemented
52C4×Dic83—24no—
53C24×C83C2312yes—
54C24×C83C2311yescharacteristic
55C2×D4×C83—24no—
56C2×D4×C83—24no—
57C2×D4×C83—24no—
58C2×D4×C83—24no—
59C2×D4×C83—41no—
60C22×C4×C83C2311yescharacteristic
61C22×C4×C83C2311yescharacteristic
62C22×C4×C83C2312yes—
63C22⋊C4×C83—24no—
64C22⋊C4×C83—41no—
65C23×C83—22no—
66C23×C83—22no—
67C23×C83—22no—
68C23×C83—22no—
69C23×C83—22no—
70C23×C83—22no—
71C23×C83—21no—
72C23×C83—22no—
73C23×C83—21no—
74C23×C83—22no—
75C23×C83C2×D414yes—
76C23×C83#4112yes—
77C23×C83C2411yescharacteristic
78D4×C83—216no—
79D4×C83—44no—
80C2×C4×C83—22no—
81C2×C4×C83—22no—
82C2×C4×C83—22no—
83C2×C4×C83—22no—
84C2×C4×C83—21no—
85C2×C4×C83—21no—
86C2×C4×C83#4114yes—
87C2×Dic83—22no—
88C2×Dic83—22no—
89C2×Dic83—22no—
90C2×Dic83—22no—
91C2×Dic83—22no—
92C2×Dic83—22no—
93C2×Dic83—22no—
94C2×Dic83—22no—
95C2×Dic83—22no—
96C2×Dic83—22no—
97C2×Dic83—44no—
98C2×Dic83—48no—
99C2×Dic83C2×D414yes—
100C2×Dic83C2×D414yes—
101C22×C83—24no—
102C22×C83—24no—
103C22×C83—22no—
104C22×C83—24no—
105C22×C83—24no—
106C22×C83—42no—
107C22×C83—42no—
108C22×C83—41no—
109C22×C83—42no—
110C22×C83—42no—
111C22×C83—42no—
112C22×C83—41no—
113C22×C83C22×D412yes—
114C22×C83C2×#4111yescharacteristic
115C22×C83C2×#4112yes—
116C22×C83H(5,𝔽2)11yescharacteristic
117C22×C832−1+411yescharacteristic
118C4×C83—24no—
119C4×C83—41no—
120C4×C83—41no—
121Dic83—24no—
122Dic83—24no—
123Dic83—42no—
124Dic83—42no—
125C2×C83—44no—
126C2×C83—42no—
127C2×C83#52512yes—
128C2×C83#53911yescharacteristic
129C2×C83D4211yescharacteristic
130C2×C83#54712yes—
131C2×C83#55911yescharacteristic
132#2455—831nomaximal, Sylow
133C83#245511yescharacteristic, complemented, Sylow
134C2×#123—832no—
135C2×C4×D4—832no—
136#399—831no—
137#394—832no—
138#392D8311yescharacteristic
139#389—832no—
140#388—832no—
141#385—831no—
142#385—832no—
143C22×D4C2×D8312yes—
144C22×D4C2×D8311yescharacteristic
145C23×C4—831no—
146C23×C4—832no—
147#123—1668no—
148C4×D4—1668no—
149C2×C4⋊C4—831no—
150C2×C4⋊C4—832no—
151C2×C4⋊C4—832no—
152C2×C4⋊C4—832no—
153C2×C4⋊C4—832no—
154C2×C22⋊C4—832no—
155C2×C22⋊C4—832no—
156C2×C22⋊C4—832no—
157C2×C22⋊C4—832no—
158C2×C22⋊C4—831no—
159C2×C22⋊C4C2×D8312yes—
160C2×C22⋊C4C2×D8311yescharacteristic
161C2×C42—832no—
162#95—832no—
163#95—832no—
164#95—831no—
165#95C2×D8311yescharacteristic
166C24C22×D8312yes—
167C24C22×D8311yescharacteristic
168C2×D4—24no—
169C2×D4—24no—
170C2×D4—24no—
171C2×D4—24no—
172C2×D4—41no—
173C22×C4—3322no—
174C22×C4—3321no—
175C22×C4—832no—
176C22×C4—832no—
177C22×C4—832no—
178C22×C4—832no—
179C22×C4—1664no—
180C22×C4—1664no—
181C22×C4—1664no—
182C22×C4C22×D8311yescharacteristic
183C22×C4C22×D8311yescharacteristic
184C22×C4C22×D8312yes—
185C4⋊C4—3321no—
186C4⋊C4—1664no—
187C4⋊C4—1664no—
188C4⋊C4—1664no—
189C4⋊C4—1664no—
190C22⋊C4—24no—
191C22⋊C4—41no—
192C22⋊C4—838no—
193C22⋊C4—1664no—
194C22⋊C4—1664no—
195C22⋊C4—1664no—
196C22⋊C4—1662no—
197C42—1664no—
198C23—22no—
199C23—22no—
200C23—22no—
201C23—22no—
202C23—22no—
203C23—22no—
204C23—21no—
205C23—22no—
206C23—22no—
207C23—21no—
208C23D4×D8314yes—
209C23#1305058912yes—
210C23C23×D8311yescharacteristic
211D4—216no—
212D4—44no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—3328no—
220C2×C4—3324no—
221C2×C4—834no—
222C2×C4—834no—
223C2×C4—1662no—
224C2×C4—1662no—
225C2×C4—1662no—
226C2×C4—1662no—
227C2×C4—1662no—
228C2×C4—1662no—
229C2×C4—1662no—
230C2×C4—1662no—
231C2×C4—1662no—
232C2×C4—1662no—
233C2×C4#1305058914yes—
234C22—24no—
235C22—24no—
236C22—24no—
237C22—24no—
238C22—22no—
239C22—41no—
240C22—42no—
241C22—42no—
242C22—42no—
243C22—41no—
244C22—42no—
245C22—42no—
246C22C2×D4×D8312yes—
247C22C2×#1305058912yes—
248C22C2×#1305058911yescharacteristic
249C22#1458183711yescharacteristic
250C22#1458183911yescharacteristic
251C4—24no—
252C4—41no—
253C4—41no—
254C4—3322no—
255C4—3322no—
256C4—1664no—
257C4—1664no—
258C2—44no—
259C2—42no—
260C2#1912820712yes—
261C2#1912829611yescharacteristic
262C2#1912833311yescharacteristic
263C2#1912835112yes—
264C2#1912843711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting, prime radical#392×C83C2
hypercenter#392D83
hypocenterC83#2455
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C2×#14581773C2semidirect
2. C2×D4×Dic83C2semidirect
3. #19127637C2semidirect
4. #19127615C2semidirect
5. #19127598C2semidirect
6. #19127595C2semidirect
7. #19127576C2semidirect
8. #19127573C2semidirect
51. C22⋊Dic83D4semidirect
133. C83#2455semidirect

Maximal subgroups

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

subgroupindex
C2×#145817732
C2×D4×Dic832
#191276372
#191276152
#191275982
#191275952
#191275762
#191275732
#392×C832
#245583