Finite Group Databasebeta

← #24911113

Subgroups — #24911113

Order 10624.

1088 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 1088 conjugacy classes of subgroups — too many to draw. Only its 173 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
1C22×D4×D83C211yesmaximal, characteristic
2C2×#14581780C211yesmaximal, characteristic, complemented
3C2×#14581779C211yesmaximal, characteristic, complemented
4C2×#14581772C211yesmaximal, characteristic, complemented
5#19127617C214yesmaximal, complemented
6#19127602C211yesmaximal, characteristic, complemented
7#19127556C214yesmaximal, complemented
8C4×(C2×D83)⋊C4C211yesmaximal, characteristic, complemented
9#390×C83C211yesmaximal, characteristic, complemented
10C24×D83C2212yes—
11C22×Dic166C2211yescharacteristic, complemented
12C22×D332C2211yescharacteristic
13C22×C22⋊D83C2212yes—
14C2×D4×D83—24no—
15C2×D4×D83—24no—
16C2×D4×D83—24no—
17C22×C4×D83C2211yescharacteristic
18#14581780—24no—
19#14581779—24no—
20#14581772—24no—
21C2×C4⋊Dic83C2212yescomplemented
22C2×C22⋊Dic83C2214yescomplemented
23C2×(C2×D83)⋊C4C2212yes—
24C2×(C2×D83)⋊C4C2212yes—
25C2×(C2×D83)⋊C4C2214yescomplemented
26C2×(C2×D83)⋊C4C2214yescomplemented
27C2×C4×Dic83C2211yescharacteristic, complemented
28C83⋊#95C2212yescomplemented
29C22×D4×C83C2211yescharacteristic
30C2×C4⋊C4×C83C2211yescharacteristic, complemented
31C2×C22⋊C4×C83C2214yescomplemented
32C2×C42×C83C2211yescharacteristic, complemented
33C23×D83—24no—
34C23×D83—24no—
35C23×D83—24no—
36C23×D83—28no—
37C23×D83—44no—
38C23×D83C2312yes—
39C23×D83C2311yescharacteristic
40C2×Dic166—22no—
41C2×Dic166—22no—
42C2×Dic166—22no—
43C2×D332—22no—
44C2×D332—22no—
45C2×D332D414yes—
46C2×C22⋊D83—24no—
47C2×C22⋊D83—24no—
48C2×C22⋊D83—24no—
49D4×D83—232no—
50C22×Dic83C2312yes—
51C22×Dic83C2312yes—
52C22×Dic83C2311yescharacteristic
53C2×C4×D83—22no—
54C2×C4×D83—22no—
55C2×C4×D83D414yes—
56C4⋊Dic83—24no—
57C22⋊Dic83—28no—
58(C2×D83)⋊C4—24no—
59(C2×D83)⋊C4—28no—
60(C2×D83)⋊C4—24no—
61(C2×D83)⋊C4—44no—
62C4×Dic83—22no—
63C24×C83C2312yes—
64C2×D4×C83—22no—
65C2×D4×C83—22no—
66C2×D4×C83D414yes—
67C22×C4×C83C2312yes—
68C22×C4×C83C2312yes—
69C22×C4×C83C2311yescharacteristic
70C4⋊C4×C83—22no—
71C22⋊C4×C83—44no—
72C42×C83—22no—
73C23×C83—22no—
74C23×C83—22no—
75C23×C83—24no—
76C23×C83—24no—
77C23×C83C2×D414yes—
78C23×C83C2411yescharacteristic
79C22×D83—21no—
80C22×D83—21no—
81C22×D83—22no—
82C22×D83—216no—
83C22×D83—22no—
84C22×D83—22no—
85C22×D83—24no—
86C22×D83—24no—
87C22×D83—416no—
88C22×D83—416no—
89C22×D83—48no—
90C22×D83—48no—
91C22×D83C2×D414yes—
92C22×D83C2×D412yes—
93C22×D83#4114yes—
94D4×C83—28no—
95C2×C4×C83—22no—
96C2×C4×C83—22no—
97C2×C4×C83—22no—
98C2×C4×C83—22no—
99C2×C4×C83—24no—
100C2×C4×C83C2×D412yes—
101C2×C4×C83C2×D412yes—
102C2×C4×C83C2×D412yes—
103C22⋊D83—216no—
104C2×Dic83—21no—
105C2×Dic83—21no—
106C2×Dic83—22no—
107C2×Dic83—22no—
108C2×Dic83—22no—
109C2×Dic83—22no—
110C2×Dic83—22no—
111C2×Dic83—22no—
112C2×Dic83C2×D412yes—
113D332—28no—
114C4×D83—28no—
115Dic166—44no—
116C22×C83—24no—
117C22×C83—28no—
118C22×C83—42no—
119C22×C83—42no—
120C22×C83—44no—
121C22×C83C22×D411yescharacteristic
122C22×C83C22×D411yescharacteristic
123C22×C83C22×D411yescharacteristic
124C22×C83C2×#4112yes—
125C22×C83H(5,𝔽2)12yes—
126C2×D83—22no—
127C2×D83—22no—
128C2×D83—24no—
129C2×D83—24no—
130C2×D83—24no—
131C2×D83—28no—
132C2×D83—432no—
133C2×D83—42no—
134C2×D83—42no—
135C2×D83—44no—
136C2×D83#12414yescomplemented
137C4×C83—42no—
138C4×C83—42no—
139C4×C83#12014yes—
140Dic83—22no—
141Dic83—42no—
142Dic83—42no—
143C2×C83—44no—
144C2×C83C2×#12011yescharacteristic
145C2×C83C2×#12411yescharacteristic
146C2×C83#53511yescharacteristic
147C2×C83#54612yes—
148C2×C83#54612yes—
149D83—24no—
150D83—44no—
151#2435—831nomaximal, Sylow
152C83#243511yescharacteristic, complemented, Sylow
153C23×D4—831no—
154C2×#124—831no—
155C2×#124—831no—
156C2×C22⋊Q8—831no—
157#394—834no—
158#390—831no—
159#390D8311yescharacteristic, complemented
160#379—834no—
161C4×C22⋊C4—831no—
162C25—832no—
163C22×Q8—831no—
164C22×D4—832no—
165C22×D4—831no—
166C22×D4—1664no—
167C22×D4—1664no—
168C22×D4—1664no—
169C22×D4C2×D8311yescharacteristic
170C23×C4—831no—
171#124—1664no—
172#124—1664no—
173C22⋊Q8—1664no—
174C2×C4⋊C4—832no—
175C2×C4⋊C4C2×D8311yescharacteristic, complemented
176C2×C22⋊C4—834no—
177C2×C22⋊C4—834no—
178C2×C22⋊C4—832no—
179C2×C22⋊C4—834no—
180C2×C22⋊C4—832no—
181C2×C22⋊C4C2×D8314yescomplemented
182C2×C42—831no—
183C2×C42C2×D8311yescharacteristic, complemented
184#95—832no—
185C24—3324no—
186C24—831no—
187C24—832no—
188C24—1668no—
189C24—1664no—
190C24—1664no—
191C24—1664no—
192C24C22×D8312yes—
193C2×Q8—1662no—
194C2×Q8—1662no—
195C2×Q8—1662no—
196C2×D4—22no—
197C2×D4—22no—
198C2×D4—834no—
199C2×D4—1664no—
200C2×D4—16632no—
201C2×D4—1664no—
202C2×D4—1664no—
203C2×D4—1662no—
204C2×D4—1662no—
205C2×D4D33214yes—
206C22×C4—831no—
207C22×C4—834no—
208C22×C4—832no—
209C22×C4—832no—
210C22×C4—1662no—
211C22×C4—1662no—
212C22×C4C22×D8312yes—
213C22×C4C22×D8312yes—
214C22×C4C22×D8311yescharacteristic
215C4⋊C4—22no—
216C4⋊C4—1664no—
217C22⋊C4—44no—
218C22⋊C4—3324no—
219C22⋊C4—1664no—
220C22⋊C4—1668no—
221C22⋊C4—1664no—
222C22⋊C4—1668no—
223C42—22no—
224C42—1662no—
225C23—22no—
226C23—22no—
227C23—24no—
228C23—24no—
229C23—33216no—
230C23—3328no—
231C23—3328no—
232C23—33216no—
233C23—832no—
234C23—834no—
235C23—834no—
236C23—16616no—
237C23—1662no—
238C23—1662no—
239C23—1664no—
240C23—1664no—
241C23—1661no—
242C23—1661no—
243C23—1662no—
244C23C2×D33214yes—
245C23C23×D8311yescharacteristic
246Q8—3324no—
247D4—28no—
248D4—16616no—
249D4—1668no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—24no—
255C2×C4—832no—
256C2×C4—1662no—
257C2×C4—1662no—
258C2×C4—1662no—
259C2×C4—1662no—
260C2×C4—1662no—
261C2×C4—1662no—
262C2×C4—1661no—
263C2×C4—1661no—
264C2×C4—1668no—
265C2×C4D4×D8312yes—
266C2×C4D4×D8312yes—
267C2×C4C2×D33212yes—
268C22—28no—
269C22—24no—
270C22—44no—
271C22—42no—
272C22—42no—
273C22—33232no—
274C22—3324no—
275C22—3322no—
276C22—3322no—
277C22—834no—
278C22—1668no—
279C22—1664no—
280C22—1664no—
281C22—1664no—
282C22—1662no—
283C22—1662no—
284C22C2×D4×D8311yescharacteristic
285C22C2×D4×D8311yescharacteristic
286C22C22×D33211yescharacteristic
287C22#41×D8312yes—
288C22#1458183712yes—
289C4—42no—
290C4—42no—
291C4—3322no—
292C4—3322no—
293C4—1662no—
294C4#1458176214yes—
295C2—44no—
296C2—3324no—
297C2—1664no—
298C2C2×#1458176211yescharacteristic
299C2#124×D8311yescharacteristic
300C2#1912826411yescharacteristic
301C2#1912834612yes—
302C2#1912834812yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#390×C83C2
prime radicalC22×D4×D83C2
hypercenter#390D83
hypocenterC83#2435
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
2. C2×#14581780C2semidirect
3. C2×#14581779C2semidirect
4. C2×#14581772C2semidirect
5. #19127617C2semidirect
6. #19127602C2semidirect
7. #19127556C2semidirect
8. C4×(C2×D83)⋊C4C2semidirect
9. #390×C83C2semidirect
11. C22×Dic166C22semidirect
21. C2×C4⋊Dic83C22semidirect
22. C2×C22⋊Dic83C22semidirect
25. C2×(C2×D83)⋊C4C22semidirect
26. C2×(C2×D83)⋊C4C22semidirect
27. C2×C4×Dic83C22semidirect
28. C83⋊#95C22semidirect
30. C2×C4⋊C4×C83C22semidirect
31. C2×C22⋊C4×C83C22semidirect
32. C2×C42×C83C22semidirect
136. C2×D83#124semidirect
152. C83#2435semidirect
159. #390D83semidirect
175. C2×C4⋊C4C2×D83semidirect
181. C2×C22⋊C4C2×D83semidirect
183. C2×C42C2×D83semidirect

Maximal subgroups

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

subgroupindex
C22×D4×D832
C2×#145817802
C2×#145817792
C2×#145817722
#191276172
#191276022
#191275562
C4×(C2×D83)⋊C42
#390×C832
#243583