Finite Group Databasebeta

← #16849031

Subgroups — #16849031

Order 3968.

1056 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 1056 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×#92499C211yesmaximal, characteristic, complemented
2C2×#11758459C212yesmaximal, complemented
3C2×C22⋊C4×D31C211yesmaximal, characteristic, complemented
4#14520960C211yesmaximal, characteristic, complemented
5#14520959C211yesmaximal, characteristic, complemented
6#14520958C211yesmaximal, characteristic, complemented
7#14520927C212yesmaximal, complemented
8#14520927C212yesmaximal, complemented
9#14520925C212yesmaximal, complemented
10#14520911C211yesmaximal, characteristic, complemented
11C31×#393C211yesmaximal, characteristic, complemented
12C24×D31C2211yescharacteristic
13C22×Q8×C31C2211yescharacteristic, complemented
14C22×C22⋊D31C2212yes—
15C2×#92499—28no—
16C2×#92499—24no—
17C22×D124C2212yes—
18C22×D124C2212yes—
19C22×D124C2211yescharacteristic, complemented
20C22×C4×D31C2211yescharacteristic
21C22×C4×D31C2211yescharacteristic
22C22×C4×D31C2212yescomplemented
23C2×C22⋊C4×C31C2212yes—
24C2×C22⋊C4×C31C2211yescharacteristic
25C2×C22⋊Dic31C2211yescharacteristic
26C2×(C2×D31)⋊C4C2212yes—
27C2×(C2×D31)⋊C4C2211yescharacteristic
28C2×(C2×D31)⋊C4C2212yes—
29C2×(C2×D31)⋊C4C2211yescharacteristic
30C2×(C2×D31)⋊C4C2212yescomplemented
31C2×(C2×D31)⋊C4C2212yescomplemented
32#11758459—28no—
33C22⋊C4×D31—24no—
34C31×#95C2211yescharacteristic, complemented
35C31×#95C2212yescomplemented
36C31⋊#95C2212yescomplemented
37C31⋊#95C2211yescharacteristic, complemented
38C31⋊#95C2212yescomplemented
39C24×C31C2311yescharacteristic
40C23×D31—22no—
41C23×D31—22no—
42C23×D31—22no—
43C23×D31—24no—
44C23×D31—42no—
45C23×D31C2312yes—
46C23×D31C2311yescharacteristic
47C23×D31C2311yescharacteristic
48C2×Q8×C31—24no—
49C2×Q8×C31—22no—
50C22×C4×C31C2312yes—
51C22×C4×C31C2311yescharacteristic
52C22×C4×C31C2311yescharacteristic
53C22×C4×C31C2312yes—
54C2×C22⋊D31—24no—
55C2×C22⋊D31—24no—
56C2×C22⋊D31—42no—
57C22×Dic31C2312yes—
58C22×Dic31C2311yescharacteristic
59C22×Dic31C2311yescharacteristic
60#92499—232no—
61C2×D124—24no—
62C2×D124—24no—
63C2×D124—24no—
64C2×D124—22no—
65C2×D124—24no—
66C2×D124—24no—
67C2×D124—42no—
68C2×D124D418yescomplemented
69C2×C4×D31—24no—
70C2×C4×D31—24no—
71C2×C4×D31—24no—
72C2×C4×D31—24no—
73C2×C4×D31—22no—
74C2×C4×D31—42no—
75C2×C4×D31D414yescomplemented
76C22⋊C4×C31—22no—
77C22⋊C4×C31—24no—
78C22⋊Dic31—22no—
79(C2×D31)⋊C4—24no—
80(C2×D31)⋊C4—22no—
81(C2×D31)⋊C4—24no—
82(C2×D31)⋊C4—22no—
83(C2×D31)⋊C4—42no—
84(C2×D31)⋊C4—42no—
85C23×C31—21no—
86C23×C31—22no—
87C23×C31—21no—
88C23×C31—22no—
89C23×C31#4112yescomplemented
90C23×C31C2411yescharacteristic
91C22×D31—21no—
92C22×D31—21no—
93C22×D31—22no—
94C22×D31—28no—
95C22×D31—21no—
96C22×D31—22no—
97C22×D31—22no—
98C22×D31—22no—
99C22×D31—22no—
100C22×D31—22no—
101C22×D31—21no—
102C22×D31—22no—
103C22×D31—22no—
104C22×D31—22no—
105C22×D31—48no—
106C22×D31—48no—
107C22×D31—44no—
108C22×D31—44no—
109C22×D31C2×D414yes—
110C22×D31C2×D414yes—
111C22×D31C2×D412yes—
112C22×D31#4112yescomplemented
113Q8×C31—28no—
114C2×C4×C31—22no—
115C2×C4×C31—22no—
116C2×C4×C31—22no—
117C2×C4×C31—21no—
118C2×C4×C31—22no—
119C2×C4×C31—22no—
120C2×C4×C31—22no—
121C2×C4×C31—22no—
122C2×C4×C31C2×D414yes—
123C2×C4×C31C2×D412yes—
124C22⋊D31—48no—
125C2×Dic31—22no—
126C2×Dic31—22no—
127C2×Dic31—21no—
128C2×Dic31—22no—
129C2×Dic31—22no—
130C2×Dic31—22no—
131C2×Dic31C2×D412yes—
132D124—28no—
133D124—216no—
134D124—48no—
135C4×D31—216no—
136C4×D31—28no—
137C4×D31—44no—
138#2404—311nomaximal, Sylow
139C22×C31—22no—
140C22×C31—24no—
141C22×C31—41no—
142C22×C31—41no—
143C22×C31—42no—
144C22×C31C22×D411yescharacteristic
145C22×C31C22×D412yes—
146C22×C31C2×#4111yescharacteristic
147C22×C31C2×#4111yescharacteristic
148C22×C31H(5,𝔽2)12yes—
149C2×D31—22no—
150C2×D31—22no—
151C2×D31—24no—
152C2×D31—24no—
153C2×D31—24no—
154C2×D31—22no—
155C2×D31—24no—
156C2×D31—24no—
157C2×D31—24no—
158C2×D31—416no—
159C2×D31—41no—
160C2×D31—42no—
161C2×D31—42no—
162C2×D31—41no—
163C2×D31—42no—
164C2×D31—42no—
165C2×D31—42no—
166C2×D31#12014yes—
167C4×C31—24no—
168C4×C31—22no—
169C4×C31—41no—
170C4×C31—42no—
171Dic31—22no—
172Dic31—41no—
173Dic31—42no—
174C22×#41—311no—
175C2×#120—312no—
176C22×C22⋊C4—311no—
177#393—311no—
178#393D3111yescharacteristic, complemented
179#392—311no—
180#392—311no—
181#386—312no—
182#386—312no—
183#386—312no—
184#381—311no—
185C2×C31—42no—
186C2×C31C2×#12011yescharacteristic
187C2×C31#52511yescharacteristic
188C2×C31#53511yescharacteristic
189C2×C31#54612yes—
190C2×C31#54612yes—
191D31—24no—
192D31—44no—
193D31—42no—
194C25—311no—
195C2×#41—624no—
196C2×#41—628no—
197C22×Q8C2×D3111yescharacteristic, complemented
198C22×D4—312no—
199C22×D4—312no—
200C22×D4—311no—
201C22×D4—312no—
202C23×C4—311no—
203C23×C4—312no—
204C23×C4—311no—
205#120—628no—
206C2×C22⋊C4—311no—
207C2×C22⋊C4—311no—
208C2×C22⋊C4—311no—
209C2×C22⋊C4—312no—
210C2×C22⋊C4—312no—
211C2×C22⋊C4—312no—
212C2×C22⋊C4—312no—
213C2×C22⋊C4—624no—
214C2×C22⋊C4C2×D3111yescharacteristic
215C2×C22⋊C4C2×D3112yes—
216#95—312no—
217#95—311no—
218#95—312no—
219#95C2×D3111yescharacteristic, complemented
220#95C2×D3112yescomplemented
221C31#240411yescharacteristic, complemented, Sylow
222C24—311no—
223C24—311no—
224C24—312no—
225C24—622no—
226C24—622no—
227C24—624no—
228C24—622no—
229C24—1242no—
230C24C22×D3111yescharacteristic
231#41—6232no—
232C2×Q8—22no—
233C2×Q8—24no—
234C2×D4—318no—
235C2×D4—624no—
236C2×D4—624no—
237C2×D4—624no—
238C2×D4—624no—
239C2×D4—624no—
240C2×D4—624no—
241C2×D4—624no—
242C2×D4—622no—
243C2×D4—1242no—
244C2×D4—1242no—
245C22×C4—311no—
246C22×C4—314no—
247C22×C4—312no—
248C22×C4—311no—
249C22×C4—624no—
250C22×C4—624no—
251C22×C4—624no—
252C22×C4—622no—
253C22×C4—624no—
254C22×C4—1242no—
255C22×C4C22×D3112yes—
256C22×C4C22×D3111yescharacteristic
257C22×C4C22×D3111yescharacteristic
258C22×C4C22×D3112yes—
259C22⋊C4—22no—
260C22⋊C4—24no—
261C22⋊C4—622no—
262C22⋊C4—624no—
263C22⋊C4—622no—
264C22⋊C4—622no—
265C22⋊C4—624no—
266C22⋊C4—1242no—
267C22⋊C4—1242no—
268C23—21no—
269C23—22no—
270C23—21no—
271C23—22no—
272C23—314no—
273C23—312no—
274C23—314no—
275C23—312no—
276C23—628no—
277C23—621no—
278C23—621no—
279C23—622no—
280C23—621no—
281C23—622no—
282C23—622no—
283C23—622no—
284C23—622no—
285C23—621no—
286C23—622no—
287C23—622no—
288C23—622no—
289C23—622no—
290C23—1248no—
291C23—1244no—
292C23—1248no—
293C23—1244no—
294C23#9249912yescomplemented
295C23C23×D3111yescharacteristic
296Q8—28no—
297D4—6216no—
298D4—628no—
299D4—1248no—
300D4—1248no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—21no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—312no—
310C2×C4—628no—
311C2×C4—6216no—
312C2×C4—622no—
313C2×C4—621no—
314C2×C4—622no—
315C2×C4—622no—
316C2×C4—622no—
317C2×C4—622no—
318C2×C4—1244no—
319C2×C4D4×D3112yes—
320C2×C4D4×D3114yes—
321C22—24no—
322C22—22no—
323C22—41no—
324C22—42no—
325C22—41no—
326C22—314no—
327C22—622no—
328C22—624no—
329C22—624no—
330C22—622no—
331C22—624no—
332C22—624no—
333C22—624no—
334C22—624no—
335C22—622no—
336C22—12416no—
337C22—1241no—
338C22—1242no—
339C22—1242no—
340C22—1242no—
341C22—1242no—
342C22—1241no—
343C22—1242no—
344C22C2×D4×D3111yescharacteristic
345C22C2×D4×D3112yes—
346C22C2×#9249911yescharacteristic
347C22#41×D3111yescharacteristic
348C22#1175856612yes—
349C4—22no—
350C4—24no—
351C4—41no—
352C4—42no—
353C4—622no—
354C4—1241no—
355C4—1242no—
356C2—42no—
357C2—624no—
358C2—1242no—
359C2—1244no—
360C2#120×D3111yescharacteristic
361C2#1452156211yescharacteristic
362C2#1452161711yescharacteristic
363C2#1452169012yes—
364C2#1452169212yes—
subgroupG/H
center, FrattiniC23C23×D31
commutator, socle, cosocleC23×C31C24
FittingC31×#393C2
hypercenter#393D31
hypocenterC31#2404
LCSG ▷ C23×C31 ▷ C31
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. C22×#92499C2semidirect
2. C2×#11758459C2semidirect
3. C2×C22⋊C4×D31C2semidirect
4. #14520960C2semidirect
5. #14520959C2semidirect
6. #14520958C2semidirect
7. #14520927C2semidirect
8. #14520927C2semidirect
9. #14520925C2semidirect
10. #14520911C2semidirect
11. C31×#393C2semidirect
13. C22×Q8×C31C22semidirect
19. C22×D124C22semidirect
22. C22×C4×D31C22semidirect
30. C2×(C2×D31)⋊C4C22semidirect
31. C2×(C2×D31)⋊C4C22semidirect
34. C31×#95C22semidirect
35. C31×#95C22semidirect
36. C31⋊#95C22semidirect
37. C31⋊#95C22semidirect
38. C31⋊#95C22semidirect
68. C2×D124D4semidirect
75. C2×C4×D31D4semidirect
89. C23×C31#41semidirect
112. C22×D31#41semidirect
178. #393D31semidirect
197. C22×Q8C2×D31semidirect
219. #95C2×D31semidirect
220. #95C2×D31semidirect
221. C31#2404semidirect
294. C23#92499semidirect

Maximal subgroups

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

subgroupindex
C22×#924992
C2×#117584592
C2×C22⋊C4×D312
#145209602
#145209592
#145209582
#145209272
#145209252
#145209112
C31×#3932
#240431