Finite Group Databasebeta

← #24911097

Subgroups — #24911097

Order 10624.

1348 conjugacy classes of subgroups; only the 193 normal subgroups are shown.

normal subgroups only

This group has 1348 conjugacy classes of subgroups — too many to draw. Only its 193 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, complemented
2C2×#14581788C211yesmaximal, characteristic, complemented
3C2×#14581763C211yesmaximal, characteristic, complemented
4C2×#14581760C214yesmaximal, complemented
5#19127609C214yesmaximal, complemented
6#19127604C211yesmaximal, characteristic, complemented
7#19127603C211yesmaximal, characteristic, complemented
8C4×(C2×D83)⋊C4C211yesmaximal, characteristic, complemented
9C2×#127×C83C211yesmaximal, characteristic, complemented
10C24×D83C2212yes—
11C22×D332C2211yescharacteristic
12C22×C22⋊D83C2212yes—
13C22×C22⋊D83C2212yes—
14C22×C22⋊D83C2214yescomplemented
15C2×D4×D83—24no—
16C2×D4×D83—24no—
17C2×D4×D83—24no—
18C22×C4×D83C2211yescharacteristic
19#14581788—24no—
20#14581763—24no—
21#14581760—216no—
22C2×C4⋊Dic83C2211yescharacteristic, complemented
23C2×C22⋊Dic83C2214yescomplemented
24C2×C22⋊Dic83C2212yescomplemented
25C2×(C2×D83)⋊C4C2212yes—
26C2×(C2×D83)⋊C4C2214yescomplemented
27C2×C4×Dic83C2211yescharacteristic, complemented
28C83⋊#95C2212yescomplemented
29C22×D4×C83C2211yescharacteristic
30C22×D4×C83C2211yescharacteristic, complemented
31C22×D4×C83C2214yescomplemented
32#127×C83—24no—
33C2×C42×C83C2211yescharacteristic, complemented
34C23×D83—24no—
35C23×D83—24no—
36C23×D83—24no—
37C23×D83—28no—
38C23×D83—44no—
39C23×D83C2312yes—
40C23×D83C2311yescharacteristic
41C2×D332—22no—
42C2×D332—22no—
43C2×D332D414yescomplemented
44C2×C22⋊D83—24no—
45C2×C22⋊D83—24no—
46C2×C22⋊D83—28no—
47C2×C22⋊D83—24no—
48C2×C22⋊D83—28no—
49C2×C22⋊D83—24no—
50C2×C22⋊D83—28no—
51C2×C22⋊D83—44no—
52D4×D83—232no—
53C22×Dic83C2312yes—
54C22×Dic83C2311yescharacteristic
55C22×Dic83C2312yes—
56C2×C4×D83—22no—
57C2×C4×D83—22no—
58C2×C4×D83D414yescomplemented
59C4⋊Dic83—22no—
60C22⋊Dic83—24no—
61C22⋊Dic83—28no—
62(C2×D83)⋊C4—28no—
63(C2×D83)⋊C4D418yescomplemented
64C4×Dic83—22no—
65C24×C83C2312yes—
66C24×C83C2312yes—
67C2×D4×C83—28no—
68C2×D4×C83—22no—
69C2×D4×C83—22no—
70C2×D4×C83—22no—
71C2×D4×C83—22no—
72C2×D4×C83—22no—
73C2×D4×C83—28no—
74C2×D4×C83—44no—
75C2×D4×C83D414yescomplemented
76C22×C4×C83C2312yes—
77C22×C4×C83C2311yescharacteristic
78C42×C83—22no—
79C23×C83—24no—
80C23×C83—22no—
81C23×C83—22no—
82C23×C83—24no—
83C23×C83—22no—
84C23×C83—22no—
85C23×C83—22no—
86C23×C83—24no—
87C23×C83—24no—
88C23×C83C2×D414yes—
89C23×C83C2411yescharacteristic
90C22×D83—21no—
91C22×D83—21no—
92C22×D83—22no—
93C22×D83—216no—
94C22×D83—24no—
95C22×D83—22no—
96C22×D83—22no—
97C22×D83—24no—
98C22×D83—416no—
99C22×D83—416no—
100C22×D83—48no—
101C22×D83—48no—
102C22×D83C2×D414yes—
103C22×D83C2×D414yes—
104C22×D83C2×D412yes—
105D4×C83—28no—
106D4×C83—416no—
107D4×C83—44no—
108C2×C4×C83—22no—
109C2×C4×C83—22no—
110C2×C4×C83C2×D412yes—
111C2×C4×C83C2×D412yes—
112C2×C4×C83C2×D412yes—
113C2×C4×C83C2×D414yescomplemented
114C22⋊D83—216no—
115C22⋊D83—416no—
116C22⋊D83—48no—
117C2×Dic83—22no—
118C2×Dic83—21no—
119C2×Dic83—21no—
120C2×Dic83—22no—
121C2×Dic83—24no—
122C2×Dic83—22no—
123C2×Dic83C2×D412yes—
124C2×Dic83C2×D414yescomplemented
125D332—28no—
126C4×D83—28no—
127C22×C83—24no—
128C22×C83—28no—
129C22×C83—42no—
130C22×C83—42no—
131C22×C83—44no—
132C22×C83—42no—
133C22×C83—42no—
134C22×C83—42no—
135C22×C83—44no—
136C22×C83—44no—
137C22×C83C22×D411yescharacteristic
138C22×C83C22×D411yescharacteristic
139C22×C83C22×D412yes—
140C22×C83C22×D411yescharacteristic
141C22×C83H(5,𝔽2)12yes—
142C2×D83—22no—
143C2×D83—22no—
144C2×D83—24no—
145C2×D83—24no—
146C2×D83—24no—
147C2×D83—28no—
148C2×D83—432no—
149C2×D83—42no—
150C2×D83—42no—
151C2×D83—44no—
152C2×D83#12714yescomplemented
153C4×C83—24no—
154C4×C83#12014yescomplemented
155Dic83—22no—
156Dic83—24no—
157Dic83—42no—
158C2×C83—44no—
159C2×C83—82no—
160C2×C83C2×#12011yescharacteristic
161C2×C83C2×#12711yescharacteristic
162C2×C83#53511yescharacteristic
163C2×C83D4212yes—
164C2×C83D4212yes—
165D83—24no—
166D83—44no—
167#2433—831nomaximal, Sylow
168C83#243311yescharacteristic, complemented, Sylow
169C23×D4—831no—
170C2×#127—831no—
171C2×#127D8311yescharacteristic, complemented
172C2×#121—831no—
173C2×#120—834no—
174#392—834no—
175#390—831no—
176#390—831no—
177C4×C22⋊C4—831no—
178C25—832no—
179C22×D4—834no—
180C22×D4—832no—
181C22×D4—832no—
182C22×D4—831no—
183C22×D4—1664no—
184C22×D4—1664no—
185C22×D4—1664no—
186C22×D4C2×D8311yescharacteristic
187C22×D4C2×D8311yescharacteristic, complemented
188C22×D4C2×D8314yescomplemented
189C23×C4—831no—
190#127—24no—
191#127—1664no—
192#121—1664no—
193#120—16616no—
194C2×C4⋊C4—831no—
195C2×C22⋊C4—832no—
196C2×C22⋊C4—834no—
197C2×C22⋊C4—832no—
198C2×C22⋊C4—834no—
199C2×C42—831no—
200C2×C42C2×D8311yescharacteristic, complemented
201#95—832no—
202C24—3324no—
203C24—831no—
204C24—832no—
205C24—1668no—
206C24—1664no—
207C24—1664no—
208C24—1664no—
209C24C22×D8312yes—
210C24C22×D8312yes—
211C2×D4—28no—
212C2×D4—22no—
213C2×D4—22no—
214C2×D4—22no—
215C2×D4—22no—
216C2×D4—28no—
217C2×D4—22no—
218C2×D4—44no—
219C2×D4—3324no—
220C2×D4—834no—
221C2×D4—1668no—
222C2×D4—1664no—
223C2×D4—16632no—
224C2×D4—1668no—
225C2×D4—1664no—
226C2×D4—1668no—
227C2×D4—1664no—
228C2×D4—1664no—
229C2×D4—1662no—
230C2×D4—1662no—
231C2×D4C22⋊D8314yescomplemented
232C22×C4—831no—
233C22×C4—832no—
234C22×C4—834no—
235C22×C4—832no—
236C22×C4—1662no—
237C22×C4—1662no—
238C22×C4C22×D8312yes—
239C22×C4C22×D8311yescharacteristic
240C4⋊C4—1662no—
241C22⋊C4—838no—
242C22⋊C4—1668no—
243C22⋊C4—1664no—
244C22⋊C4—1668no—
245C42—22no—
246C42—1662no—
247C23—24no—
248C23—22no—
249C23—22no—
250C23—24no—
251C23—22no—
252C23—24no—
253C23—22no—
254C23—22no—
255C23—24no—
256C23—33216no—
257C23—3328no—
258C23—3328no—
259C23—33216no—
260C23—834no—
261C23—832no—
262C23—834no—
263C23—16616no—
264C23—1664no—
265C23—1662no—
266C23—1662no—
267C23—1664no—
268C23—1661no—
269C23—1661no—
270C23—1662no—
271C23C2×C22⋊D8314yes—
272C23C23×D8311yescharacteristic
273D4—28no—
274D4—416no—
275D4—44no—
276D4—3328no—
277D4—33216no—
278D4—16616no—
279D4—1668no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—834no—
283C2×C4—832no—
284C2×C4—1664no—
285C2×C4—1662no—
286C2×C4—1662no—
287C2×C4—1661no—
288C2×C4—1661no—
289C2×C4—1662no—
290C2×C4—1668no—
291C2×C4D4×D8312yes—
292C2×C4D4×D8312yes—
293C2×C4D4×D8314yescomplemented
294C2×C4C2×C22⋊D8312yes—
295C22—28no—
296C22—24no—
297C22—44no—
298C22—42no—
299C22—42no—
300C22—44no—
301C22—44no—
302C22—42no—
303C22—42no—
304C22—42no—
305C22—33232no—
306C22—3324no—
307C22—3322no—
308C22—3322no—
309C22—834no—
310C22—1668no—
311C22—1664no—
312C22—1664no—
313C22—1664no—
314C22—1662no—
315C22—1662no—
316C22C2×D4×D8312yes—
317C22C2×D4×D8311yescharacteristic
318C22C2×D4×D8311yescharacteristic
319C22C22×C22⋊D8311yescharacteristic
320C22#1458183712yes—
321C4—24no—
322C4—3322no—
323C4—1664no—
324C4—1662no—
325C4#1458176014yescomplemented
326C2—44no—
327C2—82no—
328C2—3324no—
329C2—1664no—
330C2C2×#1458176011yescharacteristic
331C2#127×D8311yescharacteristic
332C2#1912826411yescharacteristic
333C2D4×C22⋊D8312yes—
334C2#1912833612yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
FittingC2×#127×C83C2
hypercenterC2×#127D83
hypocenterC83#2433
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×#127

Complements

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

NH ≅ G/Ntype
1. C22×D4×D83C2semidirect
2. C2×#14581788C2semidirect
3. C2×#14581763C2semidirect
4. C2×#14581760C2semidirect
5. #19127609C2semidirect
6. #19127604C2semidirect
7. #19127603C2semidirect
8. C4×(C2×D83)⋊C4C2semidirect
9. C2×#127×C83C2semidirect
14. C22×C22⋊D83C22semidirect
22. C2×C4⋊Dic83C22semidirect
23. C2×C22⋊Dic83C22semidirect
24. C2×C22⋊Dic83C22semidirect
26. C2×(C2×D83)⋊C4C22semidirect
27. C2×C4×Dic83C22semidirect
28. C83⋊#95C22semidirect
30. C22×D4×C83C22semidirect
31. C22×D4×C83C22semidirect
33. C2×C42×C83C22semidirect
43. C2×D332D4semidirect
58. C2×C4×D83D4semidirect
63. (C2×D83)⋊C4D4semidirect
75. C2×D4×C83D4semidirect
113. C2×C4×C83C2×D4semidirect
124. C2×Dic83C2×D4semidirect
152. C2×D83#127semidirect
154. C4×C83#120semidirect
168. C83#2433semidirect
171. C2×#127D83semidirect
187. C22×D4C2×D83semidirect
188. C22×D4C2×D83semidirect
200. C2×C42C2×D83semidirect
231. C2×D4C22⋊D83semidirect
293. C2×C4D4×D83semidirect
325. C4#14581760semidirect

Maximal subgroups

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

subgroupindex
C22×D4×D832
C2×#145817882
C2×#145817632
C2×#145817602
#191276092
#191276042
#191276032
C4×(C2×D83)⋊C42
C2×#127×C832
#243383