Finite Group Databasebeta

← #24910793

Subgroups — #24910793

Order 10624.

864 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 864 conjugacy classes of subgroups — too many to draw. Only its 159 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×#13050589C211yesmaximal, characteristic
2C2×#14581774C212yesmaximal, complemented
3C2×C4⋊C4×D83C211yesmaximal, characteristic, complemented
4#19127622C211yesmaximal, characteristic, complemented
5#19127621C211yesmaximal, characteristic, complemented
6#19127615C211yesmaximal, characteristic, complemented
7#19127577C212yesmaximal, complemented
8#19127576C212yesmaximal, complemented
9#19127573C212yesmaximal, complemented
10#19127563C211yesmaximal, characteristic, complemented
11#394×C83C211yesmaximal, characteristic, complemented
12C2×#13050589—28no—
13C2×#13050589—24no—
14C22×Dic166C2211yescharacteristic
15C22×C22⋊D83C2212yes—
16C23×Dic83C2212yes—
17C22×C4×D83C2211yescharacteristic
18C22×C4×D83C2211yescharacteristic
19C22×C4×D83C2211yescharacteristic
20#14581774—28no—
21C2×C4⋊Dic83C2211yescharacteristic
22C2×C4⋊Dic83C2212yes—
23C2×Dic83⋊C4C2211yescharacteristic
24C2×Dic83⋊C4C2212yes—
25C2×Dic83⋊C4C2211yescharacteristic
26C4⋊C4×D83—24no—
27C2×C22⋊Dic83C2212yes—
28C2×C22⋊Dic83C2212yes—
29C2×(C2×D83)⋊C4C2212yes—
30C83⋊#95C2212yes—
31C83⋊#95C2211yescharacteristic
32C83⋊#95C2212yes—
33C83⋊#95C2212yes—
34C22×D4×C83C2211yescharacteristic
35C2×C4⋊C4×C83C2211yescharacteristic
36C2×C22⋊C4×C83C2212yes—
37C2×C22⋊C4×C83C2212yes—
38#95×C83C2211yescharacteristic
39C23×D83C2311yescharacteristic
40#13050589—232no—
41C2×Dic166—22no—
42C2×Dic166—24no—
43C2×C22⋊D83—24no—
44C2×C22⋊D83—24no—
45C2×C22⋊D83D418yes—
46C22×Dic83—24no—
47C22×Dic83—24no—
48C22×Dic83—24no—
49C22×Dic83C2311yescharacteristic
50C22×Dic83C2312yes—
51C22×Dic83C2311yescharacteristic
52C22×Dic83C2312yes—
53C22×Dic83C2311yescharacteristic
54C2×C4×D83—24no—
55C2×C4×D83—22no—
56C2×C4×D83—22no—
57C2×C4×D83—22no—
58C2×C4×D83—24no—
59C2×C4×D83—22no—
60C2×C4×D83D414yes—
61C4⋊Dic83—24no—
62C4⋊Dic83—22no—
63Dic83⋊C4—22no—
64Dic83⋊C4—24no—
65Dic83⋊C4—22no—
66C22⋊Dic83—24no—
67C22⋊Dic83—24no—
68(C2×D83)⋊C4—24no—
69C24×C83C2312yes—
70C2×D4×C83—24no—
71C2×D4×C83—22no—
72C22×C4×C83C2311yescharacteristic
73C22×C4×C83C2312yes—
74C22×C4×C83C2311yescharacteristic
75C22×C4×C83C2311yescharacteristic
76C4⋊C4×C83—22no—
77C22⋊C4×C83—24no—
78C22⋊C4×C83—24no—
79C23×C83—22no—
80C23×C83—22no—
81C23×C83—22no—
82C23×C83—22no—
83C23×C83—22no—
84C23×C83—22no—
85C23×C83C2×D414yes—
86C23×C83C2411yescharacteristic
87C22×D83—21no—
88C22×D83—21no—
89C22×D83—22no—
90C22×D83C2×D414yes—
91C22×D83C2×D412yes—
92D4×C83—28no—
93C2×C4×C83—21no—
94C2×C4×C83—22no—
95C2×C4×C83—21no—
96C2×C4×C83—22no—
97C2×C4×C83—22no—
98C2×C4×C83—21no—
99C2×C4×C83—22no—
100C2×C4×C83—22no—
101C2×C4×C83C2×D412yes—
102C2×C4×C83#4112yescomplemented
103C22⋊D83—216no—
104C2×Dic83—216no—
105C2×Dic83—21no—
106C2×Dic83—22no—
107C2×Dic83—22no—
108C2×Dic83—22no—
109C2×Dic83—21no—
110C2×Dic83—22no—
111C2×Dic83—22no—
112C2×Dic83—21no—
113C2×Dic83—22no—
114C2×Dic83—22no—
115C2×Dic83C2×D412yes—
116C2×Dic83C2×D414yes—
117C2×Dic83#4112yescomplemented
118C4×D83—28no—
119C4×D83—28no—
120C4×D83—44no—
121Dic166—28no—
122C22×C83—24no—
123C22×C83—24no—
124C22×C83—24no—
125C22×C83—42no—
126C22×C83—42no—
127C22×C83—42no—
128C22×C83—42no—
129C22×C83C22×D411yescharacteristic
130C22×C83C22×D412yes—
131C22×C83C2×#4111yescharacteristic
132C22×C83C2×#4111yescharacteristic
133C22×C832−1+412yes—
134C2×D83—22no—
135C2×D83—22no—
136C2×D83—24no—
137C2×D83—24no—
138C2×D83#12014yes—
139C4×C83—22no—
140C4×C83—22no—
141C4×C83—41no—
142C4×C83—42no—
143Dic83—24no—
144Dic83—22no—
145Dic83—22no—
146Dic83—41no—
147Dic83—42no—
148C2×C83—44no—
149C2×C83C2×#12011yescharacteristic
150C2×C83#52511yescharacteristic
151C2×C83#53611yescharacteristic
152C2×C83#54712yes—
153C2×C83#54712yes—
154D83—24no—
155#2406—831nomaximal, Sylow
156C83#240611yescharacteristic, complemented, Sylow
157C22×#41—831no—
158C2×#123—832no—
159C22×C4⋊C4—831no—
160#395—831no—
161#395—831no—
162#394—831no—
163#394D8311yescharacteristic, complemented
164#386—832no—
165#385—832no—
166#385—832no—
167#381—831no—
168C2×#41—1664no—
169C2×#41—1668no—
170C22×Q8—831no—
171C22×D4—832no—
172C22×D4C2×D8311yescharacteristic
173C23×C4—831no—
174C23×C4—832no—
175C23×C4—831no—
176C23×C4—831no—
177#123—1668no—
178C2×C4⋊C4—831no—
179C2×C4⋊C4—831no—
180C2×C4⋊C4—831no—
181C2×C4⋊C4—832no—
182C2×C4⋊C4—832no—
183C2×C4⋊C4—1664no—
184C2×C4⋊C4C2×D8311yescharacteristic
185C2×C22⋊C4—832no—
186C2×C22⋊C4—832no—
187C2×C22⋊C4—832no—
188C2×C22⋊C4C2×D8312yes—
189C2×C22⋊C4C2×D8312yes—
190#95—832no—
191#95—831no—
192#95—832no—
193#95—832no—
194#95C2×D8311yescharacteristic
195C24—831no—
196C24C22×D8312yes—
197#41—16632no—
198C2×Q8—1664no—
199C2×Q8—1662no—
200C2×D4—24no—
201C2×D4—22no—
202C2×D4—838no—
203C2×D4—1664no—
204C2×D4—1664no—
205C22×C4—831no—
206C22×C4—831no—
207C22×C4—832no—
208C22×C4—834no—
209C22×C4—832no—
210C22×C4—831no—
211C22×C4—1662no—
212C22×C4—1664no—
213C22×C4—1662no—
214C22×C4—1662no—
215C22×C4—1664no—
216C22×C4—1664no—
217C22×C4—1664no—
218C22×C4—1664no—
219C22×C4—1662no—
220C22×C4C22×D8312yes—
221C22×C4C22×D8311yescharacteristic
222C22×C4C22×D8311yescharacteristic
223C22×C4C22×D8311yescharacteristic
224C4⋊C4—22no—
225C4⋊C4—1662no—
226C4⋊C4—1662no—
227C4⋊C4—1664no—
228C4⋊C4—1664no—
229C4⋊C4—1662no—
230C22⋊C4—24no—
231C22⋊C4—24no—
232C22⋊C4—1664no—
233C22⋊C4—1664no—
234C22⋊C4—1664no—
235C23—22no—
236C23—22no—
237C23—22no—
238C23—22no—
239C23—22no—
240C23—22no—
241C23—834no—
242C23—832no—
243C23—1661no—
244C23—1661no—
245C23—1662no—
246C23D4×D8314yes—
247C23C23×D8311yescharacteristic
248Q8—1668no—
249D4—28no—
250D4—16616no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—21no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—3324no—
260C2×C4—832no—
261C2×C4—834no—
262C2×C4—832no—
263C2×C4—1668no—
264C2×C4—1668no—
265C2×C4—1662no—
266C2×C4—1662no—
267C2×C4—1661no—
268C2×C4—1662no—
269C2×C4—1662no—
270C2×C4—16616no—
271C2×C4—1661no—
272C2×C4—1662no—
273C2×C4—1662no—
274C2×C4—1662no—
275C2×C4—1661no—
276C2×C4D4×D8312yes—
277C2×C4#1305058912yescomplemented
278C22—24no—
279C22—24no—
280C22—24no—
281C22—42no—
282C22—42no—
283C22—42no—
284C22—42no—
285C22—834no—
286C22—1664no—
287C22—1664no—
288C22—1662no—
289C22—1662no—
290C22C2×D4×D8312yes—
291C22C2×D4×D8311yescharacteristic
292C22#41×D8311yescharacteristic
293C22C2×#1305058911yescharacteristic
294C22#1458183912yes—
295C4—22no—
296C4—22no—
297C4—42no—
298C4—41no—
299C4—3322no—
300C4—3321no—
301C4—1662no—
302C4—1664no—
303C4—1662no—
304C2—44no—
305C2—1664no—
306C2#120×D8311yescharacteristic
307C2#1912821111yescharacteristic
308C2#1912827111yescharacteristic
309C2#1912834912yes—
310C2#1912835112yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#394×C83C2
prime radicalC22×#13050589C2
hypercenter#394D83
hypocenterC83#2406
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
2. C2×#14581774C2semidirect
3. C2×C4⋊C4×D83C2semidirect
4. #19127622C2semidirect
5. #19127621C2semidirect
6. #19127615C2semidirect
7. #19127577C2semidirect
8. #19127576C2semidirect
9. #19127573C2semidirect
10. #19127563C2semidirect
11. #394×C83C2semidirect
102. C2×C4×C83#41semidirect
117. C2×Dic83#41semidirect
156. C83#2406semidirect
163. #394D83semidirect
277. C2×C4#13050589semidirect

Maximal subgroups

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

subgroupindex
C22×#130505892
C2×#145817742
C2×C4⋊C4×D832
#191276222
#191276212
#191276152
#191275772
#191275762
#191275732
#191275632
#394×C832
#240683