Finite Group Databasebeta

← #24912425

Subgroups — #24912425

Order 10624.

724 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 724 conjugacy classes of subgroups — too many to draw. Only its 161 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×#14581764C212yesmaximal, complemented
2C2×C4×C22⋊D83C211yesmaximal, characteristic, complemented
3#19127632C211yesmaximal, characteristic, complemented
4#19127627C211yesmaximal, characteristic, complemented
5#19127615C211yesmaximal, characteristic, complemented
6#19127609C211yesmaximal, characteristic, complemented
7#19127589C212yesmaximal, complemented
8#19127581C212yesmaximal, complemented
9#19127577C212yesmaximal, complemented
10C2×C83⋊#95C211yesmaximal, characteristic, complemented
11C2×#121×C83C211yesmaximal, characteristic, complemented
12C22×C22⋊D83C2212yes—
13C22×C22⋊D83C2211yescharacteristic
14C23×Dic83C2212yes—
15C22×C4×D83C2211yescharacteristic
16#14581764—28no—
17C4×C22⋊D83—24no—
18C2×Dic83⋊C4C2212yes—
19C2×Dic83⋊C4C2211yescharacteristic
20C2×C22⋊Dic83C2211yescharacteristic
21C2×C22⋊Dic83C2212yes—
22C2×C22⋊Dic83C2212yes—
23C2×C22⋊Dic83C2212yes—
24C2×(C2×D83)⋊C4C2212yes—
25C2×(C2×D83)⋊C4C2211yescharacteristic
26C2×C4×Dic83C2211yescharacteristic
27C83⋊#95C2212yes—
28C83⋊#95C2212yes—
29C83⋊#95C2212yes—
30C83⋊#95C2211yescharacteristic
31C83⋊#95C2211yescharacteristic
32C22×D4×C83C2212yes—
33C22×D4×C83C2211yescharacteristic
34C23×C4×C83C2211yescharacteristic
35#121×C83—24no—
36C2×C4⋊C4×C83C2211yescharacteristic
37C2×C22⋊C4×C83C2212yes—
38C23×D83C2311yescharacteristic
39C2×C22⋊D83—22no—
40C2×C22⋊D83—22no—
41C2×C22⋊D83—24no—
42C2×C22⋊D83—24no—
43C2×C22⋊D83—42no—
44C2×C22⋊D83D414yescomplemented
45C22×Dic83—24no—
46C22×Dic83—24no—
47C22×Dic83—42no—
48C22×Dic83C2311yescharacteristic
49C22×Dic83C2312yes—
50C22×Dic83C2311yescharacteristic
51C22×Dic83C2311yescharacteristic
52C22×Dic83C2312yes—
53C2×C4×D83—22no—
54C2×C4×D83—42no—
55Dic83⋊C4—22no—
56Dic83⋊C4—24no—
57C22⋊Dic83—22no—
58C22⋊Dic83—24no—
59C22⋊Dic83—24no—
60C22⋊Dic83—42no—
61(C2×D83)⋊C4—24no—
62(C2×D83)⋊C4—22no—
63C4×Dic83—22no—
64C24×C83C2312yes—
65C24×C83C2311yescharacteristic
66C2×D4×C83—24no—
67C2×D4×C83—24no—
68C2×D4×C83—24no—
69C2×D4×C83—22no—
70C2×D4×C83—42no—
71C22×C4×C83—42no—
72C22×C4×C83D414yescomplemented
73C22×C4×C83C2312yes—
74C22×C4×C83C2311yescharacteristic
75C22×C4×C83C2311yescharacteristic
76C4⋊C4×C83—22no—
77C22⋊C4×C83—24no—
78C23×C83—22no—
79C23×C83—22no—
80C23×C83—22no—
81C23×C83—22no—
82C23×C83—22no—
83C23×C83—22no—
84C23×C83—22no—
85C23×C83—22no—
86C23×C83—22no—
87C23×C83C2×D412yes—
88C23×C83C2×D412yes—
89C23×C83#4112yes—
90C23×C83C2411yescharacteristic
91C22×D83—21no—
92C22×D83—22no—
93C22×D83—21no—
94C22×D83—22no—
95C22×D83C2×D412yes—
96D4×C83—44no—
97D4×C83—48no—
98C2×C4×C83—22no—
99C2×C4×C83—22no—
100C2×C4×C83—22no—
101C2×C4×C83—22no—
102C2×C4×C83—22no—
103C2×C4×C83—44no—
104C2×C4×C83C2×D412yes—
105C2×C4×C83C2×D412yes—
106C22⋊D83—28no—
107C22⋊D83—48no—
108C2×Dic83—21no—
109C2×Dic83—22no—
110C2×Dic83—21no—
111C2×Dic83—22no—
112C2×Dic83—21no—
113C2×Dic83—22no—
114C2×Dic83—21no—
115C2×Dic83—22no—
116C2×Dic83—22no—
117C2×Dic83—22no—
118C2×Dic83—22no—
119C2×Dic83—48no—
120C2×Dic83C2×D412yes—
121C2×Dic83#4114yes—
122C4×D83—44no—
123C22×C83—22no—
124C22×C83—22no—
125C22×C83—24no—
126C22×C83—24no—
127C22×C83—42no—
128C22×C83—42no—
129C22×C83—42no—
130C22×C83—42no—
131C22×C83—42no—
132C22×C83—42no—
133C22×C83—42no—
134C22×C83#12114yes—
135C22×C83C22×D411yescharacteristic
136C22×C83C22×D411yescharacteristic
137C22×C83C2×#4112yes—
138C22×C83C2×#4111yescharacteristic
139C22×C83H(5,𝔽2)12yes—
140C2×D83—22no—
141C2×D83—24no—
142C2×D83—41no—
143C2×D83—41no—
144C2×D83—42no—
145C4×C83—22no—
146C4×C83—22no—
147C4×C83—42no—
148Dic83—22no—
149Dic83—24no—
150Dic83—41no—
151Dic83—41no—
152Dic83—42no—
153C2×C83—24no—
154C2×C83—82no—
155C2×C83C2×#12111yescharacteristic
156C2×C83#53211yescharacteristic
157C2×C83#53411yescharacteristic
158C2×C83#54612yes—
159C2×C83#55312yes—
160D83—42no—
161#2543—831nomaximal, Sylow
162C83#254311yescharacteristic, complemented, Sylow
163C2×#121—832no—
164C2×#121D8311yescharacteristic, complemented
165C2×C4×D4—831no—
166#398—831no—
167#397—831no—
168#394—831no—
169#392—831no—
170#388—832no—
171#386—832no—
172#386—832no—
173C2×#95—831no—
174C22×D4—832no—
175C22×D4—831no—
176C22×D4C2×D8312yes—
177C22×D4C2×D8311yescharacteristic
178C23×C4—831no—
179C23×C4—832no—
180C23×C4C2×D8311yescharacteristic
181#121—24no—
182#121—1668no—
183C4×D4—1664no—
184C2×C4⋊C4—831no—
185C2×C4⋊C4—832no—
186C2×C4⋊C4C2×D8311yescharacteristic
187C2×C22⋊C4—832no—
188C2×C22⋊C4—831no—
189C2×C22⋊C4—832no—
190C2×C22⋊C4—832no—
191C2×C22⋊C4—831no—
192C2×C22⋊C4—832no—
193C2×C22⋊C4C2×D8312yes—
194C2×C42—831no—
195#95—832no—
196#95—831no—
197#95—831no—
198#95—832no—
199#95—832no—
200C24—831no—
201C24C22×D8312yes—
202C24C22×D8311yescharacteristic
203C2×D4—24no—
204C2×D4—24no—
205C2×D4—24no—
206C2×D4—22no—
207C2×D4—42no—
208C2×D4—3322no—
209C2×D4—834no—
210C2×D4—1664no—
211C2×D4—1664no—
212C2×D4—1662no—
213C2×D4—1662no—
214C22×C4—42no—
215C22×C4—3322no—
216C22×C4—3322no—
217C22×C4—832no—
218C22×C4—831no—
219C22×C4—832no—
220C22×C4—831no—
221C22×C4—831no—
222C22×C4—1664no—
223C22×C4—1664no—
224C22×C4—1662no—
225C22×C4C22⋊D8314yescomplemented
226C22×C4C22×D8312yes—
227C22×C4C22×D8311yescharacteristic
228C22×C4C22×D8311yescharacteristic
229C4⋊C4—22no—
230C4⋊C4—1664no—
231C4⋊C4—1662no—
232C22⋊C4—24no—
233C22⋊C4—3322no—
234C22⋊C4—1664no—
235C22⋊C4—1664no—
236C22⋊C4—1664no—
237C22⋊C4—1662no—
238C22⋊C4—1662no—
239C42—1662no—
240C23—22no—
241C23—22no—
242C23—22no—
243C23—22no—
244C23—22no—
245C23—22no—
246C23—22no—
247C23—22no—
248C23—22no—
249C23—832no—
250C23—1661no—
251C23—1662no—
252C23—1661no—
253C23—1662no—
254C23D4×D8312yes—
255C23C2×C22⋊D8312yes—
256C23#1305058912yes—
257C23C23×D8311yescharacteristic
258D4—44no—
259D4—48no—
260D4—3328no—
261D4—1668no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—44no—
268C2×C4—3324no—
269C2×C4—3328no—
270C2×C4—832no—
271C2×C4—834no—
272C2×C4—1662no—
273C2×C4—1661no—
274C2×C4—1661no—
275C2×C4—1662no—
276C2×C4—1662no—
277C2×C4—1662no—
278C2×C4—1662no—
279C2×C4—1662no—
280C2×C4—1661no—
281C2×C4—1661no—
282C2×C4—1662no—
283C2×C4C2×C22⋊D8312yes—
284C2×C4C2×C22⋊D8312yes—
285C22—24no—
286C22—24no—
287C22—22no—
288C22—22no—
289C22—42no—
290C22—42no—
291C22—42no—
292C22—42no—
293C22—42no—
294C22—42no—
295C22—42no—
296C22—3322no—
297C22—3321no—
298C22—3321no—
299C22—1664no—
300C22—1662no—
301C22#1458176314yes—
302C22C2×D4×D8311yescharacteristic
303C22C22×C22⋊D8311yescharacteristic
304C22#41×D8312yes—
305C22C2×#1305058911yescharacteristic
306C22#1458183712yes—
307C4—22no—
308C4—22no—
309C4—42no—
310C4—3322no—
311C4—3321no—
312C4—3321no—
313C4—1664no—
314C4—1662no—
315C2—24no—
316C2—82no—
317C2—3322no—
318C2C2×#1458176311yescharacteristic
319C2#1912824711yescharacteristic
320C2#1912826011yescharacteristic
321C2#1912834012yes—
322C2#1912839412yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
FittingC2×#121×C83C2
hypercenterC2×#121D83
hypocenterC83#2543
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C2×#14581764C2semidirect
2. C2×C4×C22⋊D83C2semidirect
3. #19127632C2semidirect
4. #19127627C2semidirect
5. #19127615C2semidirect
6. #19127609C2semidirect
7. #19127589C2semidirect
8. #19127581C2semidirect
9. #19127577C2semidirect
10. C2×C83⋊#95C2semidirect
11. C2×#121×C83C2semidirect
44. C2×C22⋊D83D4semidirect
72. C22×C4×C83D4semidirect
162. C83#2543semidirect
164. C2×#121D83semidirect
225. C22×C4C22⋊D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145817642
C2×C4×C22⋊D832
#191276322
#191276272
#191276152
#191276092
#191275892
#191275812
#191275772
C2×C83⋊#952
C2×#121×C832
#254383