Finite Group Databasebeta

← #16849025

Subgroups — #16849025

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×#92497C211yesmaximal, characteristic, complemented
2C2×#11758514C212yesmaximal, complemented
3C2×C22⋊C4×D31C211yesmaximal, characteristic, complemented
4#14520962C211yesmaximal, characteristic, complemented
5#14520961C211yesmaximal, characteristic, complemented
6#14520957C211yesmaximal, characteristic, complemented
7#14520929C212yesmaximal, complemented
8#14520926C212yesmaximal, complemented
9#14520925C212yesmaximal, complemented
10#14520911C211yesmaximal, characteristic, complemented
11C31×#392C211yesmaximal, characteristic, complemented
12C24×D31C2211yescharacteristic
13C22×D4×C31C2212yes—
14C22×D4×C31C2211yescharacteristic, complemented
15C22×C22⋊D31C2212yes—
16C22×C22⋊D31C2212yes—
17C23×Dic31C2212yescomplemented
18C2×#92497—28no—
19C2×#92497—24no—
20C22×C4×D31C2211yescharacteristic
21C22×C4×D31C2211yescharacteristic
22C22×Dic62C2211yescharacteristic, complemented
23C2×C22⋊C4×C31C2211yescharacteristic
24C2×C22⋊C4×C31C2212yescomplemented
25C2×C22⋊Dic31C2211yescharacteristic
26C2×C22⋊Dic31C2212yes—
27C2×C22⋊Dic31C2212yescomplemented
28#11758514—28no—
29C2×(C2×D31)⋊C4C2212yes—
30C2×(C2×D31)⋊C4C2211yescharacteristic
31C2×(C2×D31)⋊C4C2212yes—
32C2×(C2×D31)⋊C4C2211yescharacteristic
33C22⋊C4×D31—24no—
34C31×#95C2211yescharacteristic, complemented
35C31⋊#95C2212yescomplemented
36C31⋊#95C2211yescharacteristic, complemented
37C31⋊#95C2212yescomplemented
38C31⋊#95C2212yescomplemented
39C24×C31C2312yes—
40C24×C31C2311yescharacteristic
41C23×D31—22no—
42C23×D31—22no—
43C23×D31—22no—
44C23×D31—24no—
45C23×D31—42no—
46C23×D31C2311yescharacteristic
47C23×D31C2311yescharacteristic
48C2×D4×C31—24no—
49C2×D4×C31—24no—
50C2×D4×C31—24no—
51C2×D4×C31—22no—
52C2×D4×C31—42no—
53C22×C4×C31C2312yes—
54C22×C4×C31C2311yescharacteristic
55C22×C4×C31C2311yescharacteristic
56C2×C22⋊D31—24no—
57C2×C22⋊D31—24no—
58C2×C22⋊D31—24no—
59C2×C22⋊D31—24no—
60C2×C22⋊D31—42no—
61C2×C22⋊D31D418yescomplemented
62C22×Dic31—24no—
63C22×Dic31—24no—
64C22×Dic31—24no—
65C22×Dic31C2312yes—
66C22×Dic31C2311yescharacteristic
67C22×Dic31C2312yes—
68C22×Dic31C2311yescharacteristic
69#92497—232no—
70C2×C4×D31—24no—
71C2×C4×D31—22no—
72C2×C4×D31—42no—
73C2×C4×D31D414yescomplemented
74C2×Dic62—22no—
75C2×Dic62—24no—
76C22⋊C4×C31—22no—
77C22⋊C4×C31—42no—
78C22⋊Dic31—22no—
79C22⋊Dic31—24no—
80C22⋊Dic31—42no—
81(C2×D31)⋊C4—24no—
82(C2×D31)⋊C4—22no—
83(C2×D31)⋊C4—22no—
84(C2×D31)⋊C4—24no—
85C23×C31—21no—
86C23×C31—22no—
87C23×C31—22no—
88C23×C31—22no—
89C23×C31—22no—
90C23×C31—22no—
91C23×C31—21no—
92C23×C31—22no—
93C23×C31—22no—
94C23×C31—22no—
95C23×C31C2×D414yes—
96C23×C31#4112yescomplemented
97C23×C31C2411yescharacteristic
98C22×D31—21no—
99C22×D31—21no—
100C22×D31—22no—
101C22×D31—28no—
102C22×D31—21no—
103C22×D31—22no—
104C22×D31—21no—
105C22×D31—22no—
106C22×D31—48no—
107C22×D31—48no—
108C22×D31—44no—
109C22×D31—44no—
110C22×D31C2×D414yes—
111C22×D31C2×D412yes—
112C22×D31#4112yescomplemented
113D4×C31—28no—
114D4×C31—48no—
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×C31C2×D412yes—
122C22⋊D31—216no—
123C22⋊D31—48no—
124C2×Dic31—216no—
125C2×Dic31—22no—
126C2×Dic31—22no—
127C2×Dic31—22no—
128C2×Dic31—21no—
129C2×Dic31—22no—
130C2×Dic31—22no—
131C2×Dic31—22no—
132C2×Dic31—22no—
133C2×Dic31C2×D414yes—
134C2×Dic31C2×D412yes—
135C4×D31—28no—
136C4×D31—44no—
137Dic62—28no—
138#2404—311nomaximal, Sylow
139C22×C31—24no—
140C22×C31—22no—
141C22×C31—24no—
142C22×C31—24no—
143C22×C31—24no—
144C22×C31—41no—
145C22×C31—42no—
146C22×C31—41no—
147C22×C31—42no—
148C22×C31—42no—
149C22×C31—42no—
150C22×C31—42no—
151C22×C31C22×D411yescharacteristic
152C22×C31C22×D412yes—
153C22×C31C2×#4111yescharacteristic
154C22×C31C2×#4111yescharacteristic
155C22×C31H(5,𝔽2)12yes—
156C2×D31—22no—
157C2×D31—22no—
158C2×D31—24no—
159C2×D31—24no—
160C2×D31—22no—
161C2×D31—24no—
162C2×D31—416no—
163C2×D31—41no—
164C2×D31—41no—
165C2×D31—42no—
166C2×D31#12014yes—
167C4×C31—22no—
168C4×C31—41no—
169C4×C31—42no—
170Dic31—24no—
171Dic31—22no—
172Dic31—41no—
173Dic31—42no—
174C22×#41—311no—
175C2×#120—312no—
176C22×C22⋊C4—311no—
177#393—311no—
178#393—311no—
179#392—311no—
180#392D3111yescharacteristic, complemented
181#386—312no—
182#386—312no—
183#386—312no—
184#381—311no—
185C2×C31—42no—
186C2×C31—44no—
187C2×C31C2×#12011yescharacteristic
188C2×C31#52511yescharacteristic
189C2×C31#53511yescharacteristic
190C2×C31#54612yes—
191C2×C31#54612yes—
192D31—24no—
193D31—42no—
194C25—311no—
195C2×#41—624no—
196C2×#41—628no—
197C22×Q8—311no—
198C22×D4—312no—
199C22×D4—312no—
200C22×D4C2×D3112yes—
201C22×D4C2×D3111yescharacteristic, complemented
202C23×C4—311no—
203C23×C4—312no—
204C23×C4—311no—
205#120—628no—
206C2×C22⋊C4—311no—
207C2×C22⋊C4—312no—
208C2×C22⋊C4—311no—
209C2×C22⋊C4—311no—
210C2×C22⋊C4—312no—
211C2×C22⋊C4—312no—
212C2×C22⋊C4—312no—
213C2×C22⋊C4—624no—
214C2×C22⋊C4C2×D3111yescharacteristic
215C2×C22⋊C4C2×D3112yescomplemented
216#95—312no—
217#95—311no—
218#95—312no—
219#95—312no—
220#95C2×D3111yescharacteristic, complemented
221C31#240411yescharacteristic, complemented, Sylow
222C24—311no—
223C24—311no—
224C24—622no—
225C24—622no—
226C24—624no—
227C24—622no—
228C24—1242no—
229C24C22×D3112yes—
230C24C22×D3111yescharacteristic
231#41—6232no—
232C2×Q8—624no—
233C2×Q8—622no—
234C2×D4—24no—
235C2×D4—24no—
236C2×D4—24no—
237C2×D4—22no—
238C2×D4—42no—
239C2×D4—318no—
240C2×D4—624no—
241C2×D4—624no—
242C2×D4—624no—
243C2×D4—624no—
244C2×D4—1242no—
245C22×C4—311no—
246C22×C4—312no—
247C22×C4—314no—
248C22×C4—312no—
249C22×C4—311no—
250C22×C4—624no—
251C22×C4—622no—
252C22×C4—624no—
253C22×C4—624no—
254C22×C4—624no—
255C22×C4—1242no—
256C22×C4C22×D3112yes—
257C22×C4C22×D3111yescharacteristic
258C22×C4C22×D3111yescharacteristic
259C22⋊C4—22no—
260C22⋊C4—42no—
261C22⋊C4—622no—
262C22⋊C4—624no—
263C22⋊C4—624no—
264C22⋊C4—622no—
265C22⋊C4—624no—
266C22⋊C4—622no—
267C22⋊C4—1242no—
268C23—21no—
269C23—22no—
270C23—22no—
271C23—22no—
272C23—22no—
273C23—22no—
274C23—22no—
275C23—22no—
276C23—22no—
277C23—21no—
278C23—314no—
279C23—312no—
280C23—312no—
281C23—628no—
282C23—621no—
283C23—622no—
284C23—621no—
285C23—621no—
286C23—622no—
287C23—621no—
288C23—622no—
289C23—1248no—
290C23—1244no—
291C23—1248no—
292C23—1244no—
293C23D4×D3114yes—
294C23#9249712yescomplemented
295C23C23×D3111yescharacteristic
296Q8—628no—
297D4—28no—
298D4—48no—
299D4—6216no—
300D4—1248no—
301C2×C4—22no—
302C2×C4—21no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—312no—
308C2×C4—314no—
309C2×C4—628no—
310C2×C4—6216no—
311C2×C4—622no—
312C2×C4—622no—
313C2×C4—622no—
314C2×C4—621no—
315C2×C4—622no—
316C2×C4—622no—
317C2×C4—622no—
318C2×C4—622no—
319C2×C4—1244no—
320C2×C4D4×D3112yes—
321C22—24no—
322C22—24no—
323C22—24no—
324C22—24no—
325C22—22no—
326C22—41no—
327C22—42no—
328C22—42no—
329C22—42no—
330C22—42no—
331C22—41no—
332C22—42no—
333C22—314no—
334C22—622no—
335C22—624no—
336C22—624no—
337C22—622no—
338C22—624no—
339C22—622no—
340C22—12416no—
341C22—1241no—
342C22—1242no—
343C22—1241no—
344C22C2×D4×D3111yescharacteristic
345C22C2×D4×D3112yes—
346C22C2×#9249711yescharacteristic
347C22#1175856112yes—
348C22#41×D3111yescharacteristic
349C4—22no—
350C4—41no—
351C4—42no—
352C4—622no—
353C4—624no—
354C4—1241no—
355C4—1242no—
356C2—42no—
357C2—44no—
358C2—624no—
359C2—1242no—
360C2#120×D3111yescharacteristic
361C2#1452155911yescharacteristic
362C2#1452161211yescharacteristic
363C2#1452168612yes—
364C2#1452168812yes—
subgroupG/H
center, FrattiniC23C23×D31
commutator, socle, cosocleC23×C31C24
FittingC31×#392C2
hypercenter#392D31
hypocenterC31#2404
LCSG ▷ C23×C31 ▷ C31
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C22×#92497C2semidirect
2. C2×#11758514C2semidirect
3. C2×C22⋊C4×D31C2semidirect
4. #14520962C2semidirect
5. #14520961C2semidirect
6. #14520957C2semidirect
7. #14520929C2semidirect
8. #14520926C2semidirect
9. #14520925C2semidirect
10. #14520911C2semidirect
11. C31×#392C2semidirect
14. C22×D4×C31C22semidirect
17. C23×Dic31C22semidirect
22. C22×Dic62C22semidirect
24. C2×C22⋊C4×C31C22semidirect
27. C2×C22⋊Dic31C22semidirect
34. C31×#95C22semidirect
35. C31⋊#95C22semidirect
36. C31⋊#95C22semidirect
37. C31⋊#95C22semidirect
38. C31⋊#95C22semidirect
61. C2×C22⋊D31D4semidirect
73. C2×C4×D31D4semidirect
96. C23×C31#41semidirect
112. C22×D31#41semidirect
180. #392D31semidirect
201. C22×D4C2×D31semidirect
215. C2×C22⋊C4C2×D31semidirect
220. #95C2×D31semidirect
221. C31#2404semidirect
294. C23#92497semidirect

Maximal subgroups

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

subgroupindex
C22×#924972
C2×#117585142
C2×C22⋊C4×D312
#145209622
#145209612
#145209572
#145209292
#145209262
#145209252
#145209112
C31×#3922
#240431