Finite Group Databasebeta

← #16854981

Subgroups — #16854981

Order 3968.

784 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 784 conjugacy classes of subgroups — too many to draw. Only its 149 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
1C2×#11758566C211yesmaximal, characteristic, complemented
2C2×#11758485C211yesmaximal, characteristic, complemented
3C22×D248C211yesmaximal, characteristic, complemented
4#14521613C211yesmaximal, characteristic, complemented
5#14521196C214yesmaximal, complemented
6#14521185C214yesmaximal, complemented
7#14521055C211yesmaximal, characteristic, complemented
8#14521046C211yesmaximal, characteristic, complemented
9C31×#408C211yesmaximal, characteristic, complemented
10C2×#41×C31C2211yescharacteristic
11#11758566—24no—
12#11758566—24no—
13C2×#92499—21no—
14C2×D4×D31—21no—
15C2×D4×D31—22no—
16C2×#92495C2211yescharacteristic
17C2×#92495C2212yescomplemented
18C22×D124C2211yescharacteristic
19C22×D124C2211yescharacteristic, complemented
20C22×D124C2212yescomplemented
21C2×M16×C31C2211yescharacteristic, complemented
22C22×C8×C31C2211yescharacteristic, complemented
23#11758524—21no—
24#11758517—21no—
25#11758514—22no—
26#11758507C2214yescomplemented
27#11758503C2211yescharacteristic, complemented
28#11758485—24no—
29C2×D248—24no—
30C2×D248C2212yescomplemented
31C2×D248C2214yescomplemented
32C2×#92470C2212yescomplemented
33C31×C22⋊C8C2214yescomplemented
34#11758409C2212yescomplemented
35#11758409C2214yescomplemented
36#11758404C2212yescomplemented
37C23×D31—21no—
38C23×D31—22no—
39C23×D31—21no—
40#41×C31—22no—
41#41×C31—22no—
42C2×Q8×C31D411yescharacteristic
43C2×D4×C31D412yes—
44C2×D4×C31D411yescharacteristic
45C22×C4×C31C2312yes—
46C22×C4×C31C2311yescharacteristic
47C2×C22⋊D31—21no—
48C2×C22⋊D31—22no—
49C2×C22⋊D31—24no—
50#92499—28no—
51D4×D31—216no—
52D4×D31—28no—
53#92495—22no—
54#92495—24no—
55#92495—24no—
56#92495D414yescomplemented
57C2×D124—21no—
58C2×D124—22no—
59C2×D124—22no—
60C2×D124—24no—
61C2×D124—24no—
62C2×D124—22no—
63C2×D124—22no—
64C2×D124—21no—
65C2×D124D414yescomplemented
66C2×D124C2312yes—
67C2×D124C2312yes—
68C2×D124C2311yescharacteristic
69C2×C4×D31—21no—
70C2×C4×D31—22no—
71C2×Dic62C2311yescharacteristic
72M16×C31—22no—
73C2×C8×C31—22no—
74C2×C8×C31C2312yes—
75C2×C8×C31C2312yes—
76C22⋊Dic31—21no—
77(C2×D31)⋊C4—24no—
78C4⋊Dic31C2312yes—
79C4×Dic31—21no—
80D248—28no—
81D248—24no—
82#92470—24no—
83C23×C31C2×D412yes—
84C23×C31C2×D411yescharacteristic
85C22×D31—21no—
86C22×D31—22no—
87C22×D31—28no—
88C22×D31—24no—
89C22×D31—22no—
90C22×D31—42no—
91C22×D31—82no—
92C22×D31—84no—
93C22×D31—82no—
94Q8×C31—22no—
95D4×C31—24no—
96D4×C31—22no—
97C2×C4×C31—22no—
98C2×C4×C31—22no—
99C2×C4×C31C2×D412yes—
100C2×C4×C31C2×D411yescharacteristic
101C2×C4×C31C2×D412yes—
102C2×C4×C31C2×D412yes—
103C2×C4×C31C2411yescharacteristic
104C22⋊D31—28no—
105C22⋊D31—24no—
106C22⋊D31—84no—
107C2×Dic31—21no—
108C2×Dic31—22no—
109D124—28no—
110D124—24no—
111D124—21no—
112D124—22no—
113D124—22no—
114D124—22no—
115D124—42no—
116D124C2×D412yes—
117D124C2×D414yes—
118C4×D31—28no—
119C4×D31—24no—
120Dic62—21no—
121Dic62C2×D412yes—
122C8×C31—22no—
123C8×C31—22no—
124#3008—311nomaximal, Sylow
125C22×C31—22no—
126C22×C31—22no—
127C22×C31—24no—
128C22×C31#12012yes—
129C22×C31C22×D411yescharacteristic
130C2×D31—24no—
131C2×D31—22no—
132C2×D31—44no—
133C2×D31—42no—
134C2×D31—42no—
135C2×D31—44no—
136C2×D31—84no—
137C2×D31—88no—
138C2×D31—84no—
139C4×C31—22no—
140C4×C31#12012yes—
141C4×C31C22×D411yescharacteristic
142C4×C31C22×D411yescharacteristic
143Dic31—22no—
144Dic31—42no—
145C2×H(5,𝔽2)—311no—
146C2×AGL(1,ℤ/8)—311no—
147C22×D8—311no—
148#535—311no—
149#449—314no—
150#447—314no—
151#418—311no—
152#416—311no—
153#408D3111yescharacteristic, complemented
154C2×C31—22no—
155C2×C31—42no—
156C2×C31C2×#12011yescharacteristic
157C2×C31#57612yes—
158D31—44no—
159D31—42no—
160D31—82no—
161H(5,𝔽2)—624no—
162H(5,𝔽2)—624no—
163C2×#41—311no—
164C2×#41—312no—
165C2×#41—621no—
166C2×#41C2×D3111yescharacteristic
167C22×D4—311no—
168C22×D4—311no—
169C22×D4—312no—
170C22×D4—622no—
171C22×D4—621no—
172AGL(1,ℤ/8)—624no—
173C2×SD16—312no—
174C2×D8—314no—
175C2×D8—312no—
176C2×D8—624no—
177C2×M16C2×D3111yescharacteristic, complemented
178C22×C8C2×D3111yescharacteristic, complemented
179#127—621no—
180#124—621no—
181#121—314no—
182#120—622no—
183#117—311no—
184#103—312no—
185#102—312no—
186#102—314no—
187C22⋊C8C2×D3114yescomplemented
188C31#300811yescharacteristic, complemented, Sylow
189C24—622no—
190C24—621no—
191C24—621no—
192#41—22no—
193#41—22no—
194#41—314no—
195#41—624no—
196#41—624no—
197#41—628no—
198#41—622no—
199C2×Q8—311no—
200C2×Q8D12411yescharacteristic
201C2×D4—314no—
202C2×D4—312no—
203C2×D4—311no—
204C2×D4—312no—
205C2×D4—624no—
206C2×D4—624no—
207C2×D4—6216no—
208C2×D4—628no—
209C2×D4—622no—
210C2×D4—621no—
211C2×D4—624no—
212C2×D4—622no—
213C2×D4—621no—
214C2×D4—621no—
215C2×D4—622no—
216C2×D4—622no—
217C2×D4—622no—
218C2×D4D12412yes—
219C2×D4D12411yescharacteristic
220C22×C4—622no—
221C22×C4—621no—
222C22×C4C22×D3112yes—
223C22×C4C22×D3111yescharacteristic
224SD16—624no—
225D8—628no—
226D8—624no—
227M16—22no—
228C2×C8—22no—
229C2×C8C22×D3112yes—
230C2×C8C22×D3112yes—
231C4⋊C4—312no—
232C22⋊C4—621no—
233C22⋊C4—624no—
234C42—621no—
235C23—628no—
236C23—622no—
237C23—621no—
238C23—624no—
239C23—622no—
240C23—1242no—
241C23—2482no—
242C23—2482no—
243C23—2484no—
244C23C2×D12412yes—
245C23C2×D12411yescharacteristic
246Q8—22no—
247Q8—312no—
248Q8—621no—
249D4—24no—
250D4—22no—
251D4—314no—
252D4—312no—
253D4—628no—
254D4—628no—
255D4—624no—
256D4—624no—
257D4—621no—
258D4—622no—
259D4—622no—
260D4—622no—
261D4—1242no—
262D4—2484no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—628no—
266C2×C4—621no—
267C2×C4—624no—
268C2×C4—622no—
269C2×C4C2×D12412yes—
270C2×C4C2×D12411yescharacteristic
271C2×C4D4×D3112yes—
272C2×C4D4×D3112yes—
273C2×C4C23×D3111yescharacteristic
274C8—22no—
275C8—22no—
276C22—22no—
277C22—22no—
278C22—24no—
279C22—624no—
280C22—622no—
281C22—1244no—
282C22—1242no—
283C22—1242no—
284C22—1244no—
285C22—2484no—
286C22—2484no—
287C22—2488no—
288C22#1175845912yes—
289C22C22×D12411yescharacteristic
290C4—22no—
291C4—622no—
292C4—1242no—
293C4#1175845912yes—
294C4C2×D4×D3111yescharacteristic
295C4C2×D4×D3111yescharacteristic
296C2—22no—
297C2—42no—
298C2—1244no—
299C2—1242no—
300C2—2482no—
301C2C2×#1175845911yescharacteristic
302C2#1452185412yes—
subgroupG/H
centerC22C22×D124
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#408C2
socleC22×C31C22×D4
hypercenter#408D31
hypocenterC31#3008
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C2×#41 ◁ #408

Complements

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

NH ≅ G/Ntype
1. C2×#11758566C2semidirect
2. C2×#11758485C2semidirect
3. C22×D248C2semidirect
4. #14521613C2semidirect
5. #14521196C2semidirect
6. #14521185C2semidirect
7. #14521055C2semidirect
8. #14521046C2semidirect
9. C31×#408C2semidirect
17. C2×#92495C22semidirect
19. C22×D124C22semidirect
20. C22×D124C22semidirect
21. C2×M16×C31C22semidirect
22. C22×C8×C31C22semidirect
26. #11758507C22semidirect
27. #11758503C22semidirect
30. C2×D248C22semidirect
31. C2×D248C22semidirect
32. C2×#92470C22semidirect
33. C31×C22⋊C8C22semidirect
34. #11758409C22semidirect
35. #11758409C22semidirect
36. #11758404C22semidirect
56. #92495D4semidirect
65. C2×D124D4semidirect
153. #408D31semidirect
177. C2×M16C2×D31semidirect
178. C22×C8C2×D31semidirect
187. C22⋊C8C2×D31semidirect
188. C31#3008semidirect

Maximal subgroups

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

subgroupindex
C2×#117585662
C2×#117584852
C22×D2482
#145216132
#145211962
#145211852
#145210552
#145210462
C31×#4082
#300831