Finite Group Databasebeta

← #24912623

Subgroups — #24912623

Order 10624.

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

normal subgroups only

This group has 532 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×#14581782C211yesmaximal, characteristic, complemented
2C2×D4×Dic83C211yesmaximal, characteristic
3#19127641C211yesmaximal, characteristic, complemented
4#19127640C211yesmaximal, characteristic, complemented
5#19127615C211yesmaximal, characteristic
6#19127594C212yesmaximal, complemented
7#19127589C212yesmaximal, complemented
8#19127559C212yesmaximal, complemented
9#19127559C212yesmaximal, complemented
10#19127552C211yesmaximal, characteristic, complemented
11#394×C83C211yesmaximal, characteristic
12C23×Dic83C2212yes—
13#14581782—24no—
14D4×Dic83—24no—
15C2×C4⋊Dic83C2211yescharacteristic
16C2×C4⋊Dic83C2211yescharacteristic
17C2×Dic83⋊C4C2212yes—
18C2×Dic83⋊C4C2211yescharacteristic
19C2×Dic83⋊C4C2211yescharacteristic
20C2×Dic83⋊C4C2211yescharacteristic
21C2×Dic83⋊C4C2211yescharacteristic
22C2×C22⋊Dic83C2212yes—
23C2×C22⋊Dic83C2212yes—
24C2×C22⋊Dic83C2212yes—
25C2×C4×Dic83C2212yes—
26C2×C4×Dic83C2211yescharacteristic
27C83⋊#95C2211yescharacteristic
28C83⋊#95C2212yes—
29C83⋊#95C2212yes—
30C83⋊#95C2212yes—
31C83⋊#95C2212yes—
32C22×D4×C83C2211yescharacteristic
33C2×C4⋊C4×C83C2211yescharacteristic
34C2×C22⋊C4×C83C2212yes—
35C2×C22⋊C4×C83C2212yes—
36#95×C83C2211yescharacteristic
37C22×Dic83—24no—
38C22×Dic83—24no—
39C22×Dic83—24no—
40C22×Dic83C2312yes—
41C22×Dic83C2312yes—
42C22×Dic83C2311yescharacteristic
43C22×Dic83C2311yescharacteristic
44C22×Dic83C2311yescharacteristic
45C22×Dic83C2311yescharacteristic
46C4⋊Dic83—22no—
47C4⋊Dic83—22no—
48Dic83⋊C4—22no—
49Dic83⋊C4—22no—
50Dic83⋊C4—22no—
51Dic83⋊C4—22no—
52Dic83⋊C4—24no—
53C22⋊Dic83—24no—
54C22⋊Dic83—24no—
55C22⋊Dic83—42no—
56C4×Dic83—42no—
57C4×Dic83D414yes—
58C24×C83C2312yes—
59C2×D4×C83—24no—
60C2×D4×C83—22no—
61C22×C4×C83C2312yes—
62C22×C4×C83C2311yescharacteristic
63C22×C4×C83C2311yescharacteristic
64C22×C4×C83C2311yescharacteristic
65C4⋊C4×C83—22no—
66C22⋊C4×C83—24no—
67C22⋊C4×C83—42no—
68C23×C83—22no—
69C23×C83—22no—
70C23×C83—22no—
71C23×C83—22no—
72C23×C83—22no—
73C23×C83—22no—
74C23×C83#4114yes—
75C23×C83C2411yescharacteristic
76D4×C83—28no—
77C2×C4×C83—22no—
78C2×C4×C83—21no—
79C2×C4×C83—21no—
80C2×C4×C83—22no—
81C2×C4×C83—22no—
82C2×C4×C83—22no—
83C2×C4×C83—21no—
84C2×C4×C83—22no—
85C2×C4×C83—21no—
86C2×C4×C83C2×D412yes—
87C2×Dic83—21no—
88C2×Dic83—21no—
89C2×Dic83—22no—
90C2×Dic83—21no—
91C2×Dic83—21no—
92C2×Dic83—22no—
93C2×Dic83—22no—
94C2×Dic83—22no—
95C2×Dic83—22no—
96C2×Dic83—22no—
97C2×Dic83—22no—
98C2×Dic83—21no—
99C2×Dic83—21no—
100C2×Dic83—48no—
101C2×Dic83C2×D412yes—
102C2×Dic83C2×D412yes—
103C2×Dic83#4114yes—
104C22×C83—24no—
105C22×C83—24no—
106C22×C83—24no—
107C22×C83—42no—
108C22×C83—42no—
109C22×C83—42no—
110C22×C83—42no—
111C22×C83C22×D411yescharacteristic
112C22×C83C2×#4112yes—
113C22×C83C2×#4112yes—
114C22×C83H(5,𝔽2)11yescharacteristic
115C22×C832−1+411yescharacteristic
116C4×C83—22no—
117C4×C83—41no—
118C4×C83—41no—
119C4×C83—42no—
120Dic83—22no—
121Dic83—41no—
122Dic83—42no—
123Dic83—41no—
124Dic83—42no—
125Dic83#12414yes—
126C2×C83—44no—
127C2×C83#52511yescharacteristic
128C2×C83C2×#12411yescharacteristic
129C2×C83#53711yescharacteristic
130C2×C83#55212yes—
131C2×C83#55312yes—
132#2562—831nomaximal, Sylow
133C83#256211yescharacteristic, complemented, Sylow
134C2×#125—831no—
135C2×C4×D4—831no—
136#400—831no—
137#400—831no—
138#394—831no—
139#394D8311yescharacteristic
140#388—832no—
141#388—832no—
142#380—832no—
143#380—832no—
144C4×C4⋊C4—831no—
145C22×D4C2×D8311yescharacteristic
146C23×C4—832no—
147#125—1664no—
148C4×D4—1664no—
149C2×C4⋊C4—831no—
150C2×C4⋊C4—831no—
151C2×C4⋊C4—831no—
152C2×C4⋊C4—831no—
153C2×C4⋊C4—831no—
154C2×C4⋊C4—831no—
155C2×C4⋊C4—832no—
156C2×C4⋊C4C2×D8311yescharacteristic
157C2×C22⋊C4—832no—
158C2×C22⋊C4—832no—
159C2×C22⋊C4—832no—
160C2×C22⋊C4C2×D8312yes—
161C2×C22⋊C4C2×D8312yes—
162C2×C42—832no—
163C2×C42—831no—
164#95—831no—
165#95—832no—
166#95—832no—
167#95—832no—
168#95—832no—
169#95C2×D8311yescharacteristic
170C24C22×D8312yes—
171C2×D4—24no—
172C2×D4—22no—
173C22×C4—831no—
174C22×C4—831no—
175C22×C4—832no—
176C22×C4—832no—
177C22×C4—831no—
178C22×C4—831no—
179C22×C4—1664no—
180C22×C4—1664no—
181C22×C4—1664no—
182C22×C4C22×D8312yes—
183C22×C4C22×D8311yescharacteristic
184C22×C4C22×D8311yescharacteristic
185C22×C4C22×D8311yescharacteristic
186C4⋊C4—22no—
187C4⋊C4—1662no—
188C4⋊C4—1662no—
189C4⋊C4—1662no—
190C4⋊C4—1662no—
191C4⋊C4—1664no—
192C4⋊C4—1662no—
193C4⋊C4—1662no—
194C22⋊C4—24no—
195C22⋊C4—42no—
196C22⋊C4—3322no—
197C22⋊C4—1664no—
198C22⋊C4—1664no—
199C42—3322no—
200C42—834no—
201C23—22no—
202C23—22no—
203C23—22no—
204C23—22no—
205C23—22no—
206C23—22no—
207C23#1305058914yes—
208C23C23×D8311yescharacteristic
209D4—28no—
210C2×C4—22no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—3328no—
220C2×C4—832no—
221C2×C4—834no—
222C2×C4—832no—
223C2×C4—1662no—
224C2×C4—1661no—
225C2×C4—1661no—
226C2×C4—1662no—
227C2×C4—1662no—
228C2×C4—1662no—
229C2×C4—1662no—
230C2×C4—1662no—
231C2×C4—1662no—
232C2×C4—1661no—
233C2×C4—1661no—
234C2×C4—1661no—
235C2×C4—1661no—
236C2×C4D4×D8312yes—
237C22—24no—
238C22—24no—
239C22—24no—
240C22—42no—
241C22—42no—
242C22—42no—
243C22—42no—
244C22C2×D4×D8311yescharacteristic
245C22#41×D8312yes—
246C22C2×#1305058912yes—
247C22#1458183711yescharacteristic
248C22#1458183911yescharacteristic
249C4—22no—
250C4—42no—
251C4—41no—
252C4—41no—
253C4—3322no—
254C4—3321no—
255C4—3322no—
256C4—3321no—
257C4—834no—
258C4—1662no—
259C2—44no—
260C2#1912820711yescharacteristic
261C2#124×D8311yescharacteristic
262C2#1912828011yescharacteristic
263C2#1912838812yes—
264C2#1912839412yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#394×C83C2
prime radicalC22×D4×C83C22
hypercenter#394D83
hypocenterC83#2562
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C2×#14581782C2semidirect
3. #19127641C2semidirect
4. #19127640C2semidirect
6. #19127594C2semidirect
7. #19127589C2semidirect
8. #19127559C2semidirect
9. #19127559C2semidirect
10. #19127552C2semidirect
133. C83#2562semidirect

Maximal subgroups

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

subgroupindex
C2×#145817822
C2×D4×Dic832
#191276412
#191276402
#191276152
#191275942
#191275892
#191275592
#191275522
#394×C832
#256283