Finite Group Databasebeta

← #16849020

Subgroups — #16849020

Order 3968.

1564 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 1564 conjugacy classes of subgroups — too many to draw. Only its 181 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×D31C212yesmaximal, complemented
2C2×#11758459C212yesmaximal, complemented
3C2×#11758514C212yesmaximal, complemented
4#14520958C211yesmaximal, characteristic, complemented
5#14520957C212yesmaximal, complemented
6#14520927C212yesmaximal, complemented
7#14520925C212yesmaximal, complemented
8#14520904C211yesmaximal, characteristic, complemented
9C31×#392C211yesmaximal, characteristic, complemented
10C24×D31C2211yescharacteristic
11C24×D31C2212yescomplemented
12C22×D4×C31C2212yescomplemented
13C22×D4×C31C2211yescharacteristic, complemented
14C22×C22⋊D31C2212yes—
15C22×C22⋊D31C2212yescomplemented
16C22×C22⋊D31C2212yescomplemented
17C2×D4×D31—28no—
18C2×D4×D31—28no—
19C2×D4×D31—28no—
20C22×D124C2212yes—
21C22×D124C2211yescharacteristic, complemented
22C22×C4×D31C2212yescomplemented
23C2×C22⋊C4×C31C2212yescomplemented
24C2×C22⋊C4×C31C2211yescharacteristic, complemented
25C2×C22⋊Dic31C2212yescomplemented
26C2×C22⋊Dic31C2211yescharacteristic, complemented
27#11758514—28no—
28C2×(C2×D31)⋊C4C2212yescomplemented
29C2×(C2×D31)⋊C4C2212yescomplemented
30C2×(C2×D31)⋊C4C2212yescomplemented
31C2×(C2×D31)⋊C4C2212yescomplemented
32#11758459—28no—
33C31×#95C2211yescharacteristic, complemented
34C31⋊#95C2211yescharacteristic, complemented
35C31⋊#95C2212yescomplemented
36C24×C31C2311yescharacteristic
37C24×C31C2312yes—
38C23×D31—24no—
39C23×D31—24no—
40C23×D31—24no—
41C23×D31—24no—
42C23×D31—24no—
43C23×D31—24no—
44C23×D31—24no—
45C23×D31—41no—
46C23×D31—41no—
47C23×D31—42no—
48C23×D31—42no—
49C23×D31D414yescomplemented
50C23×D31C2312yes—
51C23×D31C2312yes—
52C2×D4×C31—24no—
53C2×D4×C31—24no—
54C2×D4×C31—24no—
55C2×D4×C31—24no—
56C2×D4×C31—41no—
57C22×C4×C31C2311yescharacteristic
58C22×C4×C31C2311yescharacteristic
59C22×C4×C31C2312yes—
60C2×C22⋊D31—24no—
61C2×C22⋊D31—24no—
62C2×C22⋊D31—24no—
63C2×C22⋊D31—24no—
64C2×C22⋊D31—24no—
65C2×C22⋊D31—24no—
66C2×C22⋊D31—24no—
67C2×C22⋊D31—42no—
68C2×C22⋊D31D418yescomplemented
69C22×Dic31C2312yes—
70C22×Dic31C2312yes—
71D4×D31—264no—
72C2×D124—24no—
73C2×D124—24no—
74C2×D124—24no—
75C2×D124—41no—
76C2×D124D418yescomplemented
77C2×C4×D31—24no—
78C2×C4×D31—24no—
79C2×C4×D31—24no—
80C22⋊C4×C31—24no—
81C22⋊C4×C31—41no—
82C22⋊Dic31—24no—
83C22⋊Dic31—41no—
84(C2×D31)⋊C4—24no—
85(C2×D31)⋊C4—24no—
86(C2×D31)⋊C4—24no—
87(C2×D31)⋊C4—24no—
88C23×C31—22no—
89C23×C31—22no—
90C23×C31—22no—
91C23×C31—22no—
92C23×C31—22no—
93C23×C31—22no—
94C23×C31—21no—
95C23×C31—21no—
96C23×C31—22no—
97C23×C31—22no—
98C23×C31C2×D412yes—
99C23×C31C2×D414yescomplemented
100C23×C31C2411yescharacteristic
101C22×D31—22no—
102C22×D31—22no—
103C22×D31—22no—
104C22×D31—22no—
105C22×D31—216no—
106C22×D31—22no—
107C22×D31—22no—
108C22×D31—22no—
109C22×D31—22no—
110C22×D31—22no—
111C22×D31—22no—
112C22×D31—216no—
113C22×D31—44no—
114C22×D31—44no—
115C22×D31—48no—
116C22×D31—44no—
117C22×D31—48no—
118C22×D31—48no—
119C22×D31—48no—
120C22×D31—48no—
121C22×D31—48no—
122C22×D31—48no—
123C22×D31C2×D414yes—
124C22×D31C2×D414yes—
125C22×D31C2×D414yes—
126C22×D31C2×D414yescomplemented
127D4×C31—216no—
128D4×C31—44no—
129C2×C4×C31—22no—
130C2×C4×C31—22no—
131C2×C4×C31—22no—
132C2×C4×C31—22no—
133C2×C4×C31—21no—
134C2×C4×C31—21no—
135C2×C4×C31C2×D414yescomplemented
136C22⋊D31—216no—
137C22⋊D31—216no—
138C22⋊D31—48no—
139C2×Dic31—22no—
140C2×Dic31—22no—
141C2×Dic31—22no—
142C2×Dic31—22no—
143C2×Dic31—22no—
144C2×Dic31C2×D414yescomplemented
145D124—216no—
146D124—44no—
147C4×D31—216no—
148#2403—311nomaximal, Sylow
149C22×C31—24no—
150C22×C31—24no—
151C22×C31—22no—
152C22×C31—24no—
153C22×C31—24no—
154C22×C31—42no—
155C22×C31—42no—
156C22×C31—41no—
157C22×C31—42no—
158C22×C31—42no—
159C22×C31—42no—
160C22×C31—41no—
161C22×C31C22×D411yescharacteristic
162C22×C31C22×D412yes—
163C22×C31C22×D412yes—
164C22×C31H(5,𝔽2)11yescharacteristic
165C22×C31H(5,𝔽2)11yescharacteristic
166C2×D31—24no—
167C2×D31—24no—
168C2×D31—24no—
169C2×D31—24no—
170C2×D31—24no—
171C2×D31—24no—
172C2×D31—24no—
173C2×D31—24no—
174C2×D31—24no—
175C2×D31—416no—
176C2×D31—432no—
177C2×D31—42no—
178C2×D31—42no—
179C2×D31—42no—
180C2×D31—42no—
181C2×D31#12018yescomplemented
182C4×C31—24no—
183C4×C31—41no—
184C4×C31—41no—
185Dic31—24no—
186Dic31—42no—
187C23×D4—312no—
188C2×#120—312no—
189C2×#120—312no—
190#392—312no—
191#392—311no—
192#392D3111yescharacteristic, complemented
193#386—312no—
194#386—312no—
195#379—311no—
196C2×C31—42no—
197C2×C31—44no—
198C2×C31C2×#12012yes—
199C2×C31#53411yescharacteristic
200C2×C31D4211yescharacteristic
201C2×C31D4212yes—
202C2×C31D4211yescharacteristic
203D31—28no—
204D31—44no—
205C25—311no—
206C25—312no—
207C22×D4—312no—
208C22×D4—312no—
209C22×D4—312no—
210C22×D4—312no—
211C22×D4—311no—
212C22×D4—628no—
213C22×D4—628no—
214C22×D4—628no—
215C22×D4C2×D3112yescomplemented
216C22×D4C2×D3111yescharacteristic, complemented
217C23×C4—312no—
218#120—628no—
219#120—628no—
220C2×C22⋊C4—312no—
221C2×C22⋊C4—312no—
222C2×C22⋊C4—312no—
223C2×C22⋊C4—312no—
224C2×C22⋊C4—312no—
225C2×C22⋊C4—311no—
226C2×C22⋊C4C2×D3112yescomplemented
227C2×C22⋊C4C2×D3111yescharacteristic, complemented
228#95—311no—
229#95—312no—
230#95C2×D3111yescharacteristic, complemented
231C31#240311yescharacteristic, complemented, Sylow
232C24—312no—
233C24—312no—
234C24—314no—
235C24—624no—
236C24—624no—
237C24—624no—
238C24—624no—
239C24—624no—
240C24—624no—
241C24—624no—
242C24—1241no—
243C24—1242no—
244C24—1241no—
245C24—1242no—
246C24C22×D3111yescharacteristic
247C24C22×D3112yes—
248C2×D4—24no—
249C2×D4—24no—
250C2×D4—24no—
251C2×D4—24no—
252C2×D4—41no—
253C2×D4—318no—
254C2×D4—318no—
255C2×D4—624no—
256C2×D4—624no—
257C2×D4—624no—
258C2×D4—624no—
259C2×D4—6264no—
260C2×D4—624no—
261C2×D4—624no—
262C2×D4—624no—
263C2×D4—624no—
264C2×D4—624no—
265C2×D4—624no—
266C2×D4—1241no—
267C2×D4—1242no—
268C22×C4—312no—
269C22×C4—312no—
270C22×C4—624no—
271C22×C4—624no—
272C22×C4—624no—
273C22×C4C22×D3111yescharacteristic
274C22×C4C22×D3111yescharacteristic
275C22×C4C22×D3112yes—
276C22⋊C4—24no—
277C22⋊C4—41no—
278C22⋊C4—624no—
279C22⋊C4—624no—
280C22⋊C4—624no—
281C22⋊C4—624no—
282C22⋊C4—624no—
283C22⋊C4—1241no—
284C23—22no—
285C23—22no—
286C23—22no—
287C23—22no—
288C23—22no—
289C23—22no—
290C23—21no—
291C23—22no—
292C23—21no—
293C23—22no—
294C23—314no—
295C23—314no—
296C23—314no—
297C23—314no—
298C23—6216no—
299C23—6216no—
300C23—622no—
301C23—622no—
302C23—622no—
303C23—622no—
304C23—622no—
305C23—622no—
306C23—622no—
307C23—622no—
308C23—622no—
309C23—622no—
310C23—1244no—
311C23—1248no—
312C23—1248no—
313C23—1248no—
314C23—1244no—
315C23—1248no—
316C23—1248no—
317C23—1244no—
318C23—1248no—
319C23—1248no—
320C23D4×D3112yes—
321C23D4×D3114yescomplemented
322C23C23×D3111yescharacteristic
323D4—216no—
324D4—44no—
325D4—6216no—
326D4—6216no—
327D4—6216no—
328D4—1244no—
329D4—1248no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—314no—
337C2×C4—6216no—
338C2×C4—622no—
339C2×C4—622no—
340C2×C4—622no—
341C2×C4—622no—
342C2×C4—622no—
343C2×C4D4×D3114yescomplemented
344C22—22no—
345C22—24no—
346C22—24no—
347C22—24no—
348C22—24no—
349C22—42no—
350C22—41no—
351C22—42no—
352C22—42no—
353C22—42no—
354C22—41no—
355C22—42no—
356C22—318no—
357C22—624no—
358C22—624no—
359C22—624no—
360C22—624no—
361C22—624no—
362C22—624no—
363C22—624no—
364C22—624no—
365C22—624no—
366C22—12416no—
367C22—12432no—
368C22—1242no—
369C22—1242no—
370C22—1242no—
371C22—1242no—
372C22C2×D4×D3111yescharacteristic
373C22C2×D4×D3112yes—
374C22C2×D4×D3112yes—
375C22#1175856111yescharacteristic
376C22#1175856611yescharacteristic
377C4—24no—
378C4—41no—
379C4—41no—
380C4—624no—
381C4—1242no—
382C2—42no—
383C2—44no—
384C2—628no—
385C2—1244no—
386C2#120×D3112yes—
387C2#1452160711yescharacteristic
388C2#1452168111yescharacteristic
389C2#1452168312yes—
390C2#1452168411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D31
commutator, socle, cosocleC23×C31C24
FittingC31×#392C2
hypercenter#392D31
hypocenterC31#2403
LCSG ▷ C23×C31 ▷ C31
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C22×D4×D31C2semidirect
2. C2×#11758459C2semidirect
3. C2×#11758514C2semidirect
4. #14520958C2semidirect
5. #14520957C2semidirect
6. #14520927C2semidirect
7. #14520925C2semidirect
8. #14520904C2semidirect
9. C31×#392C2semidirect
11. C24×D31C22semidirect
12. C22×D4×C31C22semidirect
13. C22×D4×C31C22semidirect
15. C22×C22⋊D31C22semidirect
16. C22×C22⋊D31C22semidirect
21. C22×D124C22semidirect
22. C22×C4×D31C22semidirect
23. C2×C22⋊C4×C31C22semidirect
24. C2×C22⋊C4×C31C22semidirect
25. C2×C22⋊Dic31C22semidirect
26. C2×C22⋊Dic31C22semidirect
28. C2×(C2×D31)⋊C4C22semidirect
29. C2×(C2×D31)⋊C4C22semidirect
30. C2×(C2×D31)⋊C4C22semidirect
31. C2×(C2×D31)⋊C4C22semidirect
33. C31×#95C22semidirect
34. C31⋊#95C22semidirect
35. C31⋊#95C22semidirect
49. C23×D31D4semidirect
68. C2×C22⋊D31D4semidirect
76. C2×D124D4semidirect
99. C23×C31C2×D4semidirect
126. C22×D31C2×D4semidirect
135. C2×C4×C31C2×D4semidirect
144. C2×Dic31C2×D4semidirect
181. C2×D31#120semidirect
192. #392D31semidirect
215. C22×D4C2×D31semidirect
216. C22×D4C2×D31semidirect
226. C2×C22⋊C4C2×D31semidirect
227. C2×C22⋊C4C2×D31semidirect
230. #95C2×D31semidirect
231. C31#2403semidirect
321. C23D4×D31semidirect
343. C2×C4D4×D31semidirect

Maximal subgroups

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

subgroupindex
C22×D4×D312
C2×#117584592
C2×#117585142
#145209582
#145209572
#145209272
#145209252
#145209042
C31×#3922
#240331