Finite Group Databasebeta

← #16849036

Subgroups — #16849036

Order 3968.

1116 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 1116 conjugacy classes of subgroups — too many to draw. Only its 169 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
1C23×C22⋊D31C211yesmaximal, characteristic
2C2×#11758460C212yesmaximal, complemented
3C2×C22⋊C4×D31C211yesmaximal, characteristic, complemented
4#14520965C212yesmaximal, complemented
5#14520964C212yesmaximal, complemented
6#14520925C212yesmaximal, complemented
7#14520924C211yesmaximal, characteristic, complemented
8#14520921C211yesmaximal, characteristic, complemented
9#14520905C211yesmaximal, characteristic, complemented
10#14520904C211yesmaximal, characteristic, complemented
11C31×#379C211yesmaximal, characteristic, complemented
12C25×C31C2211yescharacteristic
13C24×D31C2211yescharacteristic
14C22×C22⋊D31—24no—
15C22×C22⋊D31—28no—
16C22×C22⋊D31C2212yes—
17C22×C22⋊D31C2212yes—
18C23×Dic31C2211yescharacteristic
19C22×C4×D31C2211yescharacteristic, complemented
20C22×C4×D31C2211yescharacteristic, complemented
21C2×C22⋊C4×C31C2211yescharacteristic
22C2×C22⋊C4×C31C2211yescharacteristic
23C2×C22⋊C4×C31C2212yescomplemented
24C2×C22⋊C4×C31C2212yescomplemented
25C2×C22⋊Dic31C2211yescharacteristic
26C2×C22⋊Dic31C2211yescharacteristic
27C2×C22⋊Dic31C2212yescomplemented
28C2×C22⋊Dic31C2212yescomplemented
29C2×(C2×D31)⋊C4C2212yescomplemented
30C2×(C2×D31)⋊C4C2211yescharacteristic, complemented
31C2×(C2×D31)⋊C4C2211yescharacteristic, complemented
32C2×(C2×D31)⋊C4C2212yescomplemented
33C2×C4⋊Dic31C2212yescomplemented
34C2×Dic31⋊C4C2212yescomplemented
35#11758460—28no—
36C22⋊C4×D31—24no—
37C31⋊#95C2212yescomplemented
38C31⋊#95C2212yescomplemented
39C24×C31—24no—
40C24×C31—22no—
41C24×C31—22no—
42C24×C31—22no—
43C24×C31—42no—
44C24×C31C2312yes—
45C24×C31C2311yescharacteristic
46C23×D31—22no—
47C23×D31—22no—
48C23×D31—24no—
49C23×D31—42no—
50C23×D31D414yes—
51C23×D31C2311yescharacteristic
52C23×D31C2311yescharacteristic
53C22×C4×C31C2311yescharacteristic
54C22×C4×C31C2312yes—
55C22×C4×C31C2311yescharacteristic
56C2×C22⋊D31—232no—
57C2×C22⋊D31—24no—
58C2×C22⋊D31—24no—
59C2×C22⋊D31—24no—
60C2×C22⋊D31—24no—
61C2×C22⋊D31—24no—
62C2×C22⋊D31D418yes—
63C22×Dic31—22no—
64C22×Dic31—24no—
65C22×Dic31C2311yescharacteristic
66C22×Dic31C2312yes—
67C22×Dic31C2311yescharacteristic
68C22×Dic31C2312yes—
69C2×C4×D31—22no—
70C2×C4×D31—22no—
71C2×C4×D31—24no—
72C2×C4×D31—42no—
73C22⋊C4×C31—22no—
74C22⋊C4×C31—24no—
75C22⋊C4×C31—22no—
76C22⋊C4×C31—42no—
77C22⋊Dic31—22no—
78C22⋊Dic31—24no—
79C22⋊Dic31—22no—
80C22⋊Dic31—42no—
81(C2×D31)⋊C4—22no—
82(C2×D31)⋊C4—22no—
83(C2×D31)⋊C4—24no—
84(C2×D31)⋊C4—42no—
85C4⋊Dic31—24no—
86Dic31⋊C4—24no—
87C23×C31—22no—
88C23×C31—22no—
89C23×C31—28no—
90C23×C31—22no—
91C23×C31—22no—
92C23×C31—22no—
93C23×C31—22no—
94C23×C31—22no—
95C23×C31—22no—
96C23×C31—48no—
97C23×C31—48no—
98C23×C31—44no—
99C23×C31—44no—
100C23×C31C2×D414yes—
101C23×C31C2×D412yes—
102C23×C31#4112yes—
103C23×C31#4112yes—
104C23×C31C2411yescharacteristic
105C22×D31—21no—
106C22×D31—21no—
107C22×D31—22no—
108C22×D31—28no—
109C22×D31—22no—
110C22×D31—21no—
111C22×D31—21no—
112C22×D31—22no—
113C22×D31—28no—
114C22×D31—44no—
115C22×D31—48no—
116C22×D31—48no—
117C22×D31C2×D412yes—
118C22×D31C2×D414yes—
119C22×D31C2×D412yes—
120C2×C4×C31—22no—
121C2×C4×C31—22no—
122C2×C4×C31—21no—
123C2×C4×C31—21no—
124C2×C4×C31—22no—
125C2×C4×C31—22no—
126C2×C4×C31—22no—
127C22⋊D31—216no—
128C22⋊D31—216no—
129C2×Dic31—28no—
130C2×Dic31—22no—
131C2×Dic31—22no—
132C2×Dic31—22no—
133C2×Dic31—22no—
134C2×Dic31—21no—
135C2×Dic31—21no—
136C2×Dic31—22no—
137C2×Dic31—22no—
138C2×Dic31—22no—
139C2×Dic31C2×D414yes—
140C4×D31—44no—
141C4×D31—44no—
142#2405—311nomaximal, Sylow
143C22×C31—22no—
144C22×C31—22no—
145C22×C31—24no—
146C22×C31—24no—
147C22×C31—24no—
148C22×C31—24no—
149C22×C31—24no—
150C22×C31—416no—
151C22×C31—42no—
152C22×C31—42no—
153C22×C31—42no—
154C22×C31—42no—
155C22×C31#12314yescomplemented
156C22×C31C22×D411yescharacteristic
157C22×C31C22×D412yes—
158C22×C31C2×#4111yescharacteristic
159C22×C31C2×#4111yescharacteristic
160C22×C31H(5,𝔽2)12yes—
161C2×D31—22no—
162C2×D31—22no—
163C2×D31—24no—
164C2×D31—24no—
165C2×D31—22no—
166C2×D31—22no—
167C2×D31—22no—
168C2×D31—416no—
169C2×D31—42no—
170C2×D31—42no—
171C2×D31#12014yes—
172C4×C31—41no—
173C4×C31—41no—
174C4×C31—42no—
175Dic31—24no—
176Dic31—41no—
177Dic31—41no—
178Dic31—42no—
179C23×D4—311no—
180C2×#123—312no—
181C22×C22⋊C4—311no—
182#394—312no—
183#394—312no—
184#386—312no—
185#385—311no—
186#385—311no—
187#379—311no—
188#379—311no—
189#379D3111yescharacteristic, complemented
190C2×C31—24no—
191C2×C31—44no—
192C2×C31C2×#12011yescharacteristic
193C2×C31C2×#12311yescharacteristic
194C2×C31#53411yescharacteristic
195C2×C31#54612yes—
196C2×C31#54612yes—
197D31—24no—
198D31—42no—
199C25—311no—
200C25C2×D3111yescharacteristic
201C22×D4—312no—
202C22×D4—312no—
203C22×D4—628no—
204C22×D4—624no—
205C23×C4—311no—
206C23×C4—311no—
207C23×C4—311no—
208#123—628no—
209C2×C4⋊C4—312no—
210C2×C4⋊C4—312no—
211C2×C22⋊C4—311no—
212C2×C22⋊C4—312no—
213C2×C22⋊C4—311no—
214C2×C22⋊C4—311no—
215C2×C22⋊C4—311no—
216C2×C22⋊C4—312no—
217C2×C22⋊C4—312no—
218C2×C22⋊C4—312no—
219C2×C22⋊C4—624no—
220C2×C22⋊C4C2×D3111yescharacteristic
221C2×C22⋊C4C2×D3111yescharacteristic
222C2×C22⋊C4C2×D3112yescomplemented
223C2×C22⋊C4C2×D3112yescomplemented
224#95—312no—
225#95—312no—
226C31#240511yescharacteristic, complemented, Sylow
227C24—24no—
228C24—22no—
229C24—22no—
230C24—22no—
231C24—42no—
232C24—311no—
233C24—311no—
234C24—314no—
235C24—624no—
236C24—622no—
237C24—622no—
238C24—1242no—
239C24C22×D3112yes—
240C24C22×D3111yescharacteristic
241C2×D4—318no—
242C2×D4—624no—
243C2×D4—624no—
244C2×D4—624no—
245C2×D4—624no—
246C2×D4—6232no—
247C2×D4—624no—
248C22×C4—312no—
249C22×C4—311no—
250C22×C4—312no—
251C22×C4—311no—
252C22×C4—622no—
253C22×C4—622no—
254C22×C4—624no—
255C22×C4—624no—
256C22×C4—622no—
257C22×C4—1242no—
258C22×C4C22×D3111yescharacteristic
259C22×C4C22×D3112yes—
260C22×C4C22×D3111yescharacteristic
261C4⋊C4—624no—
262C4⋊C4—624no—
263C22⋊C4—22no—
264C22⋊C4—24no—
265C22⋊C4—22no—
266C22⋊C4—42no—
267C22⋊C4—622no—
268C22⋊C4—622no—
269C22⋊C4—624no—
270C22⋊C4—622no—
271C22⋊C4—622no—
272C22⋊C4—624no—
273C22⋊C4—1242no—
274C22⋊C4—1242no—
275C23—22no—
276C23—22no—
277C23—28no—
278C23—22no—
279C23—22no—
280C23—22no—
281C23—22no—
282C23—22no—
283C23—22no—
284C23—48no—
285C23—48no—
286C23—44no—
287C23—44no—
288C23—314no—
289C23—312no—
290C23—312no—
291C23—628no—
292C23—621no—
293C23—621no—
294C23—622no—
295C23—628no—
296C23—621no—
297C23—622no—
298C23—621no—
299C23—622no—
300C23—1244no—
301C23—1248no—
302C23—1248no—
303C23#9249512yes—
304C23D4×D3114yes—
305C23D4×D3112yes—
306C23#9249712yes—
307C23C23×D3111yescharacteristic
308D4—6216no—
309D4—6216no—
310C2×C4—22no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—314no—
318C2×C4—628no—
319C2×C4—622no—
320C2×C4—622no—
321C2×C4—622no—
322C2×C4—622no—
323C2×C4—621no—
324C2×C4—621no—
325C2×C4—622no—
326C2×C4—622no—
327C2×C4—622no—
328C2×C4—1244no—
329C2×C4—1244no—
330C22—22no—
331C22—24no—
332C22—24no—
333C22—22no—
334C22—24no—
335C22—24no—
336C22—24no—
337C22—416no—
338C22—42no—
339C22—42no—
340C22—42no—
341C22—42no—
342C22—314no—
343C22—622no—
344C22—624no—
345C22—624no—
346C22—622no—
347C22—622no—
348C22—622no—
349C22—622no—
350C22—12416no—
351C22—1242no—
352C22—1242no—
353C22#1175846014yescomplemented
354C22C2×#9249511yescharacteristic
355C22C2×D4×D3111yescharacteristic
356C22C2×D4×D3112yes—
357C22C2×#9249711yescharacteristic
358C22#1175856112yes—
359C4—41no—
360C4—42no—
361C4—41no—
362C4—624no—
363C4—1241no—
364C4—1242no—
365C4—1241no—
366C2—24no—
367C2—44no—
368C2—624no—
369C2—1242no—
370C2#120×D3111yescharacteristic
371C2C2×#1175846011yescharacteristic
372C2#1452160911yescharacteristic
373C2#1452168612yes—
374C2#1452168912yes—
subgroupG/H
center, FrattiniC23C23×D31
commutator, socle, cosocleC23×C31C24
FittingC31×#379C2
prime radicalC23×C22⋊D31C2
hypercenter#379D31
hypocenterC31#2405
LCSG ▷ C23×C31 ▷ C31
UCSC1 ◁ C23 ◁ #379

Complements

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

NH ≅ G/Ntype
2. C2×#11758460C2semidirect
3. C2×C22⋊C4×D31C2semidirect
4. #14520965C2semidirect
5. #14520964C2semidirect
6. #14520925C2semidirect
7. #14520924C2semidirect
8. #14520921C2semidirect
9. #14520905C2semidirect
10. #14520904C2semidirect
11. C31×#379C2semidirect
19. C22×C4×D31C22semidirect
20. C22×C4×D31C22semidirect
23. C2×C22⋊C4×C31C22semidirect
24. C2×C22⋊C4×C31C22semidirect
27. C2×C22⋊Dic31C22semidirect
28. C2×C22⋊Dic31C22semidirect
29. C2×(C2×D31)⋊C4C22semidirect
30. C2×(C2×D31)⋊C4C22semidirect
31. C2×(C2×D31)⋊C4C22semidirect
32. C2×(C2×D31)⋊C4C22semidirect
33. C2×C4⋊Dic31C22semidirect
34. C2×Dic31⋊C4C22semidirect
37. C31⋊#95C22semidirect
38. C31⋊#95C22semidirect
155. C22×C31#123semidirect
189. #379D31semidirect
222. C2×C22⋊C4C2×D31semidirect
223. C2×C22⋊C4C2×D31semidirect
226. C31#2405semidirect
353. C22#11758460semidirect

Maximal subgroups

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

subgroupindex
C23×C22⋊D312
C2×#117584602
C2×C22⋊C4×D312
#145209652
#145209642
#145209252
#145209242
#145209212
#145209052
#145209042
C31×#3792
#240531