Finite Group Databasebeta

← #24912396

Subgroups — #24912396

Order 10624.

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

normal subgroups only

This group has 760 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×#14581778C211yesmaximal, characteristic, complemented
2C2×#14581769C212yesmaximal, complemented
3C2×#14581764C212yesmaximal, complemented
4C2×D4×Dic83C211yesmaximal, characteristic, complemented
5#19127638C211yesmaximal, characteristic, complemented
6#19127637C211yesmaximal, characteristic, complemented
7#19127609C211yesmaximal, characteristic, complemented
8#19127604C212yesmaximal, complemented
9#19127601C212yesmaximal, complemented
10#19127552C211yesmaximal, characteristic, complemented
11#392×C83C211yesmaximal, characteristic, complemented
12C22×Dic166C2211yescharacteristic
13C22×C22⋊D83C2211yescharacteristic
14C22×C22⋊D83C2212yes—
15C23×Dic83C2212yes—
16#14581778—24no—
17#14581769—28no—
18#14581764—28no—
19D4×Dic83—24no—
20C2×C4⋊Dic83C2212yes—
21C2×C4⋊Dic83C2211yescharacteristic
22C2×Dic83⋊C4C2212yes—
23C2×Dic83⋊C4C2212yes—
24C2×C22⋊Dic83C2212yes—
25C2×C22⋊Dic83C2211yescharacteristic
26C2×C22⋊Dic83C2212yes—
27C2×C22⋊Dic83C2212yes—
28C2×(C2×D83)⋊C4C2211yescharacteristic
29C2×(C2×D83)⋊C4C2212yes—
30C2×(C2×D83)⋊C4C2211yescharacteristic
31C2×C4×Dic83C2212yes—
32C2×C4×Dic83C2211yescharacteristic
33C83⋊#95C2211yescharacteristic
34C22×D4×C83C2211yescharacteristic
35C22×D4×C83C2212yes—
36C2×C22⋊C4×C83C2212yes—
37C2×C22⋊C4×C83C2211yescharacteristic
38#95×C83C2211yescharacteristic
39C23×D83C2311yescharacteristic
40C2×Dic166—24no—
41C2×Dic166—22no—
42C2×C22⋊D83—24no—
43C2×C22⋊D83—24no—
44C2×C22⋊D83—24no—
45C2×C22⋊D83—22no—
46C2×C22⋊D83—42no—
47C22×Dic83—24no—
48C22×Dic83—24no—
49C22×Dic83—24no—
50C22×Dic83C2312yes—
51C22×Dic83C2311yescharacteristic
52C22×Dic83C2312yes—
53C22×Dic83C2311yescharacteristic
54C22×Dic83C2311yescharacteristic
55C4⋊Dic83—24no—
56C4⋊Dic83—22no—
57Dic83⋊C4—24no—
58Dic83⋊C4D418yescomplemented
59C22⋊Dic83—24no—
60C22⋊Dic83—24no—
61C22⋊Dic83—22no—
62C22⋊Dic83—24no—
63(C2×D83)⋊C4—24no—
64(C2×D83)⋊C4—22no—
65(C2×D83)⋊C4—22no—
66C4×Dic83—42no—
67C4×Dic83D414yescomplemented
68C24×C83C2311yescharacteristic
69C24×C83C2312yes—
70C2×D4×C83—24no—
71C2×D4×C83—24no—
72C2×D4×C83—22no—
73C2×D4×C83—24no—
74C2×D4×C83—42no—
75C22×C4×C83C2311yescharacteristic
76C22×C4×C83C2311yescharacteristic
77C22×C4×C83C2312yes—
78C22⋊C4×C83—24no—
79C22⋊C4×C83—22no—
80C23×C83—22no—
81C23×C83—22no—
82C23×C83—22no—
83C23×C83—22no—
84C23×C83—22no—
85C23×C83—22no—
86C23×C83—21no—
87C23×C83—22no—
88C23×C83—22no—
89C23×C83—21no—
90C23×C83—21no—
91C23×C83#4114yes—
92C23×C83C2411yescharacteristic
93C22×D83—21no—
94C22×D83—22no—
95C22×D83—22no—
96C22×D83—21no—
97C22×D83—21no—
98D4×C83—28no—
99D4×C83—48no—
100C2×C4×C83—22no—
101C2×C4×C83—22no—
102C2×C4×C83—22no—
103C2×C4×C83—21no—
104C2×C4×C83—22no—
105C2×C4×C83C2×D412yes—
106C2×C4×C83C2×D414yes—
107C22⋊D83—48no—
108C22⋊D83—44no—
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—48no—
119C2×Dic83C2×D412yes—
120C2×Dic83C2×D414yes—
121C2×Dic83C2×D412yes—
122C2×Dic83C2×D414yes—
123Dic166—44no—
124C22×C83—24no—
125C22×C83—24no—
126C22×C83—24no—
127C22×C83—42no—
128C22×C83—42no—
129C22×C83—42no—
130C22×C83—42no—
131C22×C83—41no—
132C22×C83—41no—
133C22×C83—41no—
134C22×C83—42no—
135C22×C83—42no—
136C22×C83C22×D411yescharacteristic
137C22×C83C22×D412yes—
138C22×C83C2×#4112yes—
139C22×C83H(5,𝔽2)11yescharacteristic
140C22×C832−1+411yescharacteristic
141C2×D83—41no—
142C2×D83—41no—
143C2×D83—41no—
144C2×D83—42no—
145C2×D83—42no—
146C4×C83—22no—
147C4×C83—24no—
148C4×C83—41no—
149Dic83—22no—
150Dic83—24no—
151Dic83—41no—
152Dic83—42no—
153Dic83#12714yes—
154C2×C83—44no—
155C2×C83—81no—
156C2×C83#52511yescharacteristic
157C2×C83C2×#12711yescharacteristic
158C2×C83#53711yescharacteristic
159C2×C83#54612yes—
160C2×C83#54812yes—
161D83—81no—
162#2540—831nomaximal, Sylow
163C83#254011yescharacteristic, complemented, Sylow
164C2×#124—831no—
165C2×C22⋊Q8—832no—
166C2×#121—832no—
167C2×C4×D4—831no—
168#399—831no—
169#399—831no—
170#392—831no—
171#392D8311yescharacteristic, complemented
172#390—832no—
173#390—832no—
174C4×C4⋊C4—831no—
175C22×Q8—831no—
176C22×D4—832no—
177C22×D4—831no—
178C22×D4C2×D8311yescharacteristic
179C22×D4C2×D8312yes—
180C23×C4—832no—
181#124—1664no—
182C22⋊Q8—1668no—
183#121—1668no—
184C4×D4—1664no—
185C2×C4⋊C4—832no—
186C2×C4⋊C4—831no—
187C2×C4⋊C4—832no—
188C2×C4⋊C4—832no—
189C2×C22⋊C4—832no—
190C2×C22⋊C4—832no—
191C2×C22⋊C4—832no—
192C2×C22⋊C4—831no—
193C2×C22⋊C4—832no—
194C2×C22⋊C4—831no—
195C2×C22⋊C4—831no—
196C2×C22⋊C4C2×D8312yes—
197C2×C22⋊C4C2×D8311yescharacteristic
198C2×C42—832no—
199C2×C42—831no—
200#95—831no—
201#95C2×D8311yescharacteristic
202C24—831no—
203C24C22×D8311yescharacteristic
204C24C22×D8312yes—
205C2×Q8—1664no—
206C2×Q8—1662no—
207C2×D4—24no—
208C2×D4—24no—
209C2×D4—22no—
210C2×D4—24no—
211C2×D4—42no—
212C2×D4—3322no—
213C2×D4—1664no—
214C2×D4—1664no—
215C2×D4—1662no—
216C2×D4—1664no—
217C22×C4—831no—
218C22×C4—832no—
219C22×C4—831no—
220C22×C4—831no—
221C22×C4—832no—
222C22×C4—1664no—
223C22×C4—1664no—
224C22×C4—1664no—
225C22×C4C22×D8311yescharacteristic
226C22×C4C22×D8311yescharacteristic
227C22×C4C22×D8312yes—
228C4⋊C4—838no—
229C4⋊C4—1664no—
230C4⋊C4—1662no—
231C4⋊C4—1664no—
232C22⋊C4—24no—
233C22⋊C4—22no—
234C22⋊C4—1662no—
235C22⋊C4—1664no—
236C22⋊C4—1664no—
237C22⋊C4—1664no—
238C22⋊C4—1662no—
239C22⋊C4—1664no—
240C22⋊C4—1662no—
241C42—3322no—
242C42—834no—
243C23—22no—
244C23—22no—
245C23—22no—
246C23—22no—
247C23—22no—
248C23—22no—
249C23—21no—
250C23—22no—
251C23—22no—
252C23—21no—
253C23—21no—
254C23—1661no—
255C23—1662no—
256C23—1662no—
257C23—1661no—
258C23—1661no—
259C23#1305058914yes—
260C23C23×D8311yescharacteristic
261Q8—3324no—
262D4—28no—
263D4—48no—
264D4—3324no—
265D4—3328no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—21no—
270C2×C4—22no—
271C2×C4—3328no—
272C2×C4—832no—
273C2×C4—834no—
274C2×C4—832no—
275C2×C4—834no—
276C2×C4—1662no—
277C2×C4—1661no—
278C2×C4—1662no—
279C2×C4—1662no—
280C2×C4—1662no—
281C2×C4—1662no—
282C2×C4—1661no—
283C2×C4—1662no—
284C2×C4—1661no—
285C2×C4D4×D8312yes—
286C2×C4D4×D8314yes—
287C22—24no—
288C22—24no—
289C22—24no—
290C22—42no—
291C22—42no—
292C22—42no—
293C22—42no—
294C22—42no—
295C22—42no—
296C22—41no—
297C22—41no—
298C22—41no—
299C22—3322no—
300C22—3322no—
301C22—3321no—
302C22—3321no—
303C22—3321no—
304C22C2×D4×D8312yes—
305C22C2×D4×D8311yescharacteristic
306C22C2×#1305058912yes—
307C22#1458183711yescharacteristic
308C22#1458183911yescharacteristic
309C4—24no—
310C4—22no—
311C4—41no—
312C4—3322no—
313C4—3321no—
314C4—834no—
315C4—1664no—
316C4—1662no—
317C2—44no—
318C2—81no—
319C2—6641no—
320C2#1912820711yescharacteristic
321C2#127×D8311yescharacteristic
322C2#1912828011yescharacteristic
323C2#1912833812yes—
324C2#1912835712yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#392×C83C2
hypercenter#392D83
hypocenterC83#2540
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C2×#14581778C2semidirect
2. C2×#14581769C2semidirect
3. C2×#14581764C2semidirect
4. C2×D4×Dic83C2semidirect
5. #19127638C2semidirect
6. #19127637C2semidirect
7. #19127609C2semidirect
8. #19127604C2semidirect
9. #19127601C2semidirect
10. #19127552C2semidirect
11. #392×C83C2semidirect
58. Dic83⋊C4D4semidirect
67. C4×Dic83D4semidirect
163. C83#2540semidirect
171. #392D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145817782
C2×#145817692
C2×#145817642
C2×D4×Dic832
#191276382
#191276372
#191276092
#191276042
#191276012
#191275522
#392×C832
#254083