Finite Group Databasebeta

← #24912940

Subgroups — #24912940

Order 10624.

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

normal subgroups only

This group has 596 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×#14581764C211yesmaximal, characteristic
2C22×C4⋊Dic83C211yesmaximal, characteristic, complemented
3#19127643C212yesmaximal, complemented
4#19127617C212yesmaximal, complemented
5#19127594C212yesmaximal, complemented
6#19127593C212yesmaximal, complemented
7#19127581C211yesmaximal, characteristic, complemented
8#19127580C211yesmaximal, characteristic, complemented
9#19127578C211yesmaximal, characteristic, complemented
10C22⋊C4×Dic83C211yesmaximal, characteristic, complemented
11#386×C83C211yesmaximal, characteristic, complemented
12C22×C22⋊D83C2211yescharacteristic
13C22×C22⋊D83C2211yescharacteristic
14C23×Dic83C2211yescharacteristic
15C23×Dic83C2211yescharacteristic
16#14581764—24no—
17C2×C4⋊Dic83—24no—
18C2×C4⋊Dic83C2212yes—
19C2×C4⋊Dic83C2212yes—
20C2×Dic83⋊C4C2211yescharacteristic
21C2×C22⋊Dic83C2211yescharacteristic
22C2×C22⋊Dic83C2211yescharacteristic
23C2×C22⋊Dic83C2212yes—
24C2×(C2×D83)⋊C4C2212yes—
25C2×(C2×D83)⋊C4C2211yescharacteristic
26C2×(C2×D83)⋊C4C2212yes—
27C2×C4×Dic83C2212yes—
28C83⋊#95C2212yes—
29C83⋊#95C2212yes—
30C83⋊#95C2212yes—
31C83⋊#95C2212yes—
32C22×D4×C83C2211yescharacteristic
33C23×C4×C83C2211yescharacteristic
34C2×C22⋊C4×C83C2211yescharacteristic
35C2×C22⋊C4×C83C2212yes—
36#95×C83C2212yes—
37C23×D83C2311yescharacteristic
38C2×C22⋊D83—22no—
39C2×C22⋊D83—22no—
40C2×C22⋊D83—22no—
41C2×C22⋊D83—22no—
42C2×C22⋊D83—42no—
43C22×Dic83—22no—
44C22×Dic83—42no—
45C22×Dic83—42no—
46C22×Dic83D414yes—
47C22×Dic83C2312yes—
48C22×Dic83C2311yescharacteristic
49C22×Dic83C2312yes—
50C22×Dic83C2311yescharacteristic
51C22×Dic83C2311yescharacteristic
52C4⋊Dic83—24no—
53C4⋊Dic83—24no—
54Dic83⋊C4—22no—
55C22⋊Dic83—22no—
56C22⋊Dic83—22no—
57C22⋊Dic83—42no—
58(C2×D83)⋊C4—24no—
59(C2×D83)⋊C4—22no—
60(C2×D83)⋊C4—42no—
61C4×Dic83—42no—
62C24×C83C2311yescharacteristic
63C24×C83C2311yescharacteristic
64C2×D4×C83—22no—
65C2×D4×C83—22no—
66C2×D4×C83—22no—
67C22×C4×C83—22no—
68C22×C4×C83—24no—
69C22×C4×C83C2311yescharacteristic
70C22×C4×C83C2312yes—
71C22×C4×C83C2312yes—
72C22⋊C4×C83—22no—
73C22⋊C4×C83—24no—
74C23×C83—22no—
75C23×C83—22no—
76C23×C83—21no—
77C23×C83—22no—
78C23×C83—21no—
79C23×C83—21no—
80C23×C83—22no—
81C23×C83C2×D412yes—
82C23×C83#4112yes—
83C23×C83#4112yes—
84C23×C83C2411yescharacteristic
85C22×D83—21no—
86C22×D83—22no—
87C22×D83—21no—
88C22×D83—21no—
89C22×D83—22no—
90D4×C83—44no—
91C2×C4×C83—28no—
92C2×C4×C83—22no—
93C2×C4×C83—22no—
94C2×C4×C83—21no—
95C2×C4×C83—22no—
96C2×C4×C83—22no—
97C2×C4×C83—21no—
98C2×C4×C83—21no—
99C2×C4×C83—22no—
100C2×C4×C83#4114yes—
101C22⋊D83—44no—
102C22⋊D83—44no—
103C2×Dic83—22no—
104C2×Dic83—22no—
105C2×Dic83—21no—
106C2×Dic83—22no—
107C2×Dic83—22no—
108C2×Dic83—22no—
109C2×Dic83—21no—
110C2×Dic83—21no—
111C2×Dic83—22no—
112C2×Dic83—22no—
113C2×Dic83—22no—
114C2×Dic83—44no—
115C2×Dic83—44no—
116C2×Dic83C2×D412yes—
117C2×Dic83C2×D412yes—
118C22×C83—22no—
119C22×C83—22no—
120C22×C83—24no—
121C22×C83—24no—
122C22×C83—41no—
123C22×C83—41no—
124C22×C83—41no—
125C22×C83—42no—
126C22×C83—42no—
127C22×C83#12414yes—
128C22×C83C22×D411yescharacteristic
129C22×C83C2×#4112yes—
130C22×C83C2×#4111yescharacteristic
131C22×C83C2×#4111yescharacteristic
132C22×C83H(5,𝔽2)12yes—
133C2×D83—41no—
134C2×D83—41no—
135C2×D83—41no—
136C2×D83—42no—
137C2×D83—42no—
138C4×C83—24no—
139C4×C83—41no—
140C4×C83—42no—
141Dic83—22no—
142Dic83—22no—
143Dic83—41no—
144Dic83—42no—
145Dic83—42no—
146C2×C83—24no—
147C2×C83—81no—
148C2×C83#52511yescharacteristic
149C2×C83C2×#12411yescharacteristic
150C2×C83#53411yescharacteristic
151C2×C83#55312yes—
152C2×C83#55312yes—
153D83—81no—
154#2591—831nomaximal, Sylow
155C83#259111yescharacteristic, complemented, Sylow
156C2×#121—831no—
157C22×C4⋊C4—831no—
158#400—832no—
159#394—832no—
160#388—832no—
161#388—832no—
162#386—831no—
163#386—831no—
164#386—831no—
165#386D8311yescharacteristic, complemented
166C4×C22⋊C4—831no—
167C22×D4—831no—
168C22×D4—831no—
169C22×D4C2×D8311yescharacteristic
170C23×C4—831no—
171C23×C4—831no—
172C23×C4C2×D8311yescharacteristic
173#121—1664no—
174C2×C4⋊C4—832no—
175C2×C4⋊C4—832no—
176C2×C4⋊C4—831no—
177C2×C4⋊C4—1664no—
178C2×C22⋊C4—832no—
179C2×C22⋊C4—832no—
180C2×C22⋊C4—831no—
181C2×C22⋊C4—831no—
182C2×C22⋊C4—831no—
183C2×C22⋊C4—832no—
184C2×C22⋊C4C2×D8311yescharacteristic
185C2×C22⋊C4C2×D8312yes—
186C2×C42—832no—
187#95—832no—
188#95—832no—
189#95—832no—
190#95—832no—
191#95C2×D8312yes—
192C24—831no—
193C24C22×D8311yescharacteristic
194C24C22×D8311yescharacteristic
195C2×D4—22no—
196C2×D4—22no—
197C2×D4—22no—
198C2×D4—3322no—
199C2×D4—1662no—
200C2×D4—1662no—
201C2×D4—1662no—
202C2×D4—1662no—
203C22×C4—22no—
204C22×C4—24no—
205C22×C4—3322no—
206C22×C4—3322no—
207C22×C4—834no—
208C22×C4—831no—
209C22×C4—831no—
210C22×C4—831no—
211C22×C4—832no—
212C22×C4—832no—
213C22×C4—1662no—
214C22×C4C22×D8311yescharacteristic
215C22×C4C22×D8312yes—
216C22×C4C22×D8312yes—
217C4⋊C4—1664no—
218C4⋊C4—1664no—
219C4⋊C4—1662no—
220C22⋊C4—22no—
221C22⋊C4—24no—
222C22⋊C4—3322no—
223C22⋊C4—3322no—
224C22⋊C4—1662no—
225C22⋊C4—1662no—
226C22⋊C4—1662no—
227C22⋊C4—1664no—
228C42—3322no—
229C23—22no—
230C23—22no—
231C23—21no—
232C23—22no—
233C23—21no—
234C23—21no—
235C23—22no—
236C23—1661no—
237C23—1662no—
238C23—1661no—
239C23—1661no—
240C23—1662no—
241C23D4×D8312yes—
242C23#1305058912yes—
243C23#1305059012yes—
244C23C23×D8311yescharacteristic
245D4—44no—
246D4—3324no—
247D4—3324no—
248C2×C4—28no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—3324no—
258C2×C4—3324no—
259C2×C4—832no—
260C2×C4—832no—
261C2×C4—1661no—
262C2×C4—1662no—
263C2×C4—1662no—
264C2×C4—1661no—
265C2×C4—1662no—
266C2×C4—1661no—
267C2×C4—1662no—
268C2×C4—1662no—
269C2×C4—1662no—
270C2×C4—1662no—
271C2×C4—1662no—
272C2×C4#1305058914yes—
273C22—24no—
274C22—24no—
275C22—22no—
276C22—22no—
277C22—42no—
278C22—42no—
279C22—41no—
280C22—41no—
281C22—41no—
282C22—3322no—
283C22—3322no—
284C22—3321no—
285C22—3321no—
286C22—3321no—
287C22#1458177814yes—
288C22C2×D4×D8311yescharacteristic
289C22C2×#1305058912yes—
290C22C2×#1305058911yescharacteristic
291C22C2×#1305059011yescharacteristic
292C22#1458183812yes—
293C4—24no—
294C4—42no—
295C4—41no—
296C4—3322no—
297C4—3321no—
298C4—3322no—
299C4—1662no—
300C4—1662no—
301C2—24no—
302C2—81no—
303C2—6641no—
304C2#1912820711yescharacteristic
305C2C2×#1458177811yescharacteristic
306C2#1912826211yescharacteristic
307C2#1912839712yes—
308C2#1912840312yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#386×C83C2
prime radicalC2×#14581764C2
hypercenter#386D83
hypocenterC83#2591
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
2. C22×C4⋊Dic83C2semidirect
3. #19127643C2semidirect
4. #19127617C2semidirect
5. #19127594C2semidirect
6. #19127593C2semidirect
7. #19127581C2semidirect
8. #19127580C2semidirect
9. #19127578C2semidirect
10. C22⋊C4×Dic83C2semidirect
11. #386×C83C2semidirect
155. C83#2591semidirect
165. #386D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145817642
C22×C4⋊Dic832
#191276432
#191276172
#191275942
#191275932
#191275812
#191275802
#191275782
C22⋊C4×Dic832
#386×C832
#259183