Finite Group Databasebeta

← #16854963

Subgroups — #16854963

Order 3968.

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

normal subgroups only

This group has 806 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
1C2×#11758526C212yesmaximal, complemented
2C2×#11758507C211yesmaximal, characteristic, complemented
3#14521194C214yesmaximal, complemented
4#14521188C212yesmaximal, complemented
5#14521183C212yesmaximal, complemented
6#14521050C212yesmaximal, complemented
7#14521013C211yesmaximal, characteristic, complemented
8C22×#41×C31C211yesmaximal, characteristic, complemented
9C2×#41×C31—28no—
10C2×#41×C31—21no—
11C2×#41×C31—21no—
12C2×#41×C31C2212yes—
13C2×#41×C31C2212yescomplemented
14C22×Q8×C31C2211yescharacteristic, complemented
15C22×D4×C31—21no—
16C22×D4×C31C2211yescharacteristic, complemented
17C23×C4×C31—21no—
18C23×C4×C31C2211yescharacteristic
19C22×C22⋊D31—21no—
20C22×D124C2211yescharacteristic, complemented
21#11758526—28no—
22C2×#92481C2214yescomplemented
23C2×#92479C2214yescomplemented
24#11758507—21no—
25#11758507—21no—
26#11758507C2212yes—
27#11758507C2212yescomplemented
28C2×(C2×D31)⋊C4—21no—
29C2×C4⋊Dic31C2211yescharacteristic, complemented
30C2×#92474C2212yescomplemented
31#11758423C2214yescomplemented
32#11758420C2214yescomplemented
33C22⋊(C31⋊C8)C2212yescomplemented
34C22⋊(C31⋊C8)C2212yescomplemented
35C24×C31—21no—
36C24×C31D411yescharacteristic
37C23×D31—21no—
38#41×C31—28no—
39#41×C31—28no—
40#41×C31—24no—
41#41×C31—24no—
42#41×C31—24no—
43#41×C31D418yescomplemented
44C2×Q8×C31—24no—
45C2×Q8×C31—21no—
46C2×Q8×C31C2312yes—
47C2×D4×C31—28no—
48C2×D4×C31—22no—
49C2×D4×C31—22no—
50C2×D4×C31—24no—
51C2×D4×C31—21no—
52C2×D4×C31C2312yes—
53C22×C4×C31—28no—
54C22×C4×C31—21no—
55C22×C4×C31—22no—
56C22×C4×C31—22no—
57C22×C4×C31—24no—
58C22×C4×C31D411yescharacteristic
59C22×C4×C31D411yescharacteristic
60C22×C4×C31D411yescharacteristic
61C22×C4×C31C2311yescharacteristic
62C22×C4×C31C2311yescharacteristic
63C22×C4×C31C2311yescharacteristic
64C2×C22⋊D31—22no—
65C2×C22⋊D31—22no—
66C2×C22⋊D31—44no—
67C22×Dic31—21no—
68C2×D124—24no—
69C2×D124—21no—
70C2×D124C2312yes—
71#92481—28no—
72#92479—28no—
73(C2×D31)⋊C4—22no—
74(C2×D31)⋊C4—22no—
75C4⋊Dic31—21no—
76C4⋊Dic31C2312yes—
77#92474—24no—
78C2×C31⋊C8C2314yes—
79C23×C31—24no—
80C23×C31—22no—
81C23×C31—22no—
82C23×C31—22no—
83C23×C31—41no—
84C23×C31—42no—
85C23×C31—42no—
86C23×C31C2×D411yescharacteristic
87C23×C31C2×D411yescharacteristic
88C23×C31C2×D411yescharacteristic
89C22×D31—22no—
90C22×D31—42no—
91C22×D31—42no—
92C22×D31—42no—
93Q8×C31—24no—
94Q8×C31—22no—
95Q8×C31C2×D414yescomplemented
96D4×C31—28no—
97D4×C31—24no—
98D4×C31—28no—
99D4×C31—22no—
100D4×C31C2×D414yescomplemented
101C2×C4×C31—24no—
102C2×C4×C31—28no—
103C2×C4×C31—22no—
104C2×C4×C31—22no—
105C2×C4×C31—22no—
106C2×C4×C31—28no—
107C2×C4×C31—22no—
108C2×C4×C31—22no—
109C2×C4×C31—22no—
110C2×C4×C31C2×D411yescharacteristic
111C2×C4×C31C2×D411yescharacteristic
112C2×C4×C31C2×D411yescharacteristic
113C2×C4×C31C2×D414yes—
114C2×C4×C31C2411yescharacteristic
115C22⋊D31—44no—
116C22⋊D31—44no—
117C2×Dic31—22no—
118C2×Dic31—42no—
119D124—24no—
120D124—42no—
121C31⋊C8—24no—
122#3006—311nomaximal, Sylow
123C22×C31—21no—
124C22×C31—21no—
125C22×C31—21no—
126C22×C31—21no—
127C22×C31—21no—
128C22×C31—24no—
129C22×C31—21no—
130C22×C31—21no—
131C22×C31—44no—
132C22×C31—48no—
133C22×C31—42no—
134C22×C31—42no—
135C22×C31#12012yes—
136C22×C31AGL(1,ℤ/8)12yescomplemented
137C22×C31C22×D411yescharacteristic
138C2×D31—42no—
139C2×D31—42no—
140C2×D31—42no—
141C2×D31—84no—
142C4×C31—21no—
143C4×C31—24no—
144C4×C31—21no—
145C4×C31#12012yes—
146C4×C31C22×D412yes—
147Dic31—42no—
148C22×#41D3111yescharacteristic, complemented
149C2×AGL(1,ℤ/8)—312no—
150C2×#121—311no—
151#449—314no—
152#448—312no—
153#447—312no—
154#417—312no—
155#407—311no—
156C2×C31—22no—
157C2×C31—22no—
158C2×C31—41no—
159C2×C31—44no—
160C2×C31C2×#12011yescharacteristic
161C2×C31C2×AGL(1,ℤ/8)11yescharacteristic
162C2×C31#57511yescharacteristic
163D31—82no—
164C2×#41—28no—
165C2×#41—21no—
166C2×#41—21no—
167C2×#41C2×D3112yes—
168C2×#41C2×D3112yescomplemented
169C22×Q8C2×D3111yescharacteristic, complemented
170C22×D4—21no—
171C22×D4—311no—
172C22×D4—621no—
173C22×D4C2×D3111yescharacteristic, complemented
174C23×C4—21no—
175C23×C4C2×D3111yescharacteristic
176AGL(1,ℤ/8)—628no—
177C2×SD16—314no—
178C2×D8—314no—
179C2×M16—312no—
180#121—312no—
181#121—312no—
182#121—621no—
183#121—621no—
184C2×C4⋊C4—311no—
185C2×C22⋊C4—621no—
186#103—314no—
187#102—314no—
188C22⋊C8—312no—
189C22⋊C8—312no—
190C31#300611yescharacteristic, complemented, Sylow
191C24—21no—
192C24—621no—
193C24C22⋊D3111yescharacteristic
194#41—28no—
195#41—28no—
196#41—24no—
197#41—24no—
198#41—24no—
199#41C22⋊D3118yescomplemented
200C2×Q8—24no—
201C2×Q8—21no—
202C2×Q8C22×D3112yes—
203C2×D4—28no—
204C2×D4—22no—
205C2×D4—22no—
206C2×D4—24no—
207C2×D4—21no—
208C2×D4—312no—
209C2×D4—622no—
210C2×D4—622no—
211C2×D4—624no—
212C2×D4—621no—
213C2×D4—1244no—
214C2×D4C22×D3112yes—
215C22×C4—28no—
216C22×C4—21no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—24no—
220C22×C4—621no—
221C22×C4C22⋊D3111yescharacteristic
222C22×C4C22⋊D3111yescharacteristic
223C22×C4C22⋊D3111yescharacteristic
224C22×C4C22×D3111yescharacteristic
225C22×C4C22×D3111yescharacteristic
226C22×C4C22×D3111yescharacteristic
227SD16—628no—
228D8—628no—
229M16—624no—
230C2×C8—314no—
231C4⋊C4—312no—
232C4⋊C4—621no—
233C22⋊C4—622no—
234C22⋊C4—622no—
235C23—24no—
236C23—22no—
237C23—22no—
238C23—22no—
239C23—41no—
240C23—42no—
241C23—42no—
242C23—622no—
243C23—1242no—
244C23—1242no—
245C23—1242no—
246C23C2×C22⋊D3111yescharacteristic
247C23C2×C22⋊D3111yescharacteristic
248C23C2×C22⋊D3111yescharacteristic
249Q8—24no—
250Q8—22no—
251Q8C2×C22⋊D3114yescomplemented
252D4—28no—
253D4—24no—
254D4—28no—
255D4—22no—
256D4—624no—
257D4—1244no—
258D4—1244no—
259D4—1242no—
260D4C2×C22⋊D3114yescomplemented
261C2×C4—24no—
262C2×C4—28no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—28no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—622no—
271C2×C4—1242no—
272C2×C4C2×C22⋊D3111yescharacteristic
273C2×C4C2×C22⋊D3111yescharacteristic
274C2×C4C2×C22⋊D3111yescharacteristic
275C2×C4C2×C22⋊D3114yes—
276C2×C4C23×D3111yescharacteristic
277C8—624no—
278C22—21no—
279C22—21no—
280C22—21no—
281C22—21no—
282C22—21no—
283C22—24no—
284C22—21no—
285C22—21no—
286C22—44no—
287C22—48no—
288C22—42no—
289C22—42no—
290C22—1242no—
291C22—1242no—
292C22—1242no—
293C22—2484no—
294C22#1175852612yescomplemented
295C22#1175853012yes—
296C22C22×C22⋊D3111yescharacteristic
297C4—21no—
298C4—24no—
299C4—21no—
300C4—1242no—
301C4#1175853012yes—
302C4C22×C22⋊D3112yes—
303C2—22no—
304C2—22no—
305C2—41no—
306C2—44no—
307C2—2482no—
308C2C2×#1175853011yescharacteristic
309C2C2×#1175852611yescharacteristic
310C2#1452184711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC22×#41×C31C2
socleC22×C31C22×D4
hypercenterC22×#41D31
hypocenterC31#3006
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×#41

Complements

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

NH ≅ G/Ntype
1. C2×#11758526C2semidirect
2. C2×#11758507C2semidirect
3. #14521194C2semidirect
4. #14521188C2semidirect
5. #14521183C2semidirect
6. #14521050C2semidirect
7. #14521013C2semidirect
8. C22×#41×C31C2semidirect
13. C2×#41×C31C22semidirect
14. C22×Q8×C31C22semidirect
16. C22×D4×C31C22semidirect
20. C22×D124C22semidirect
22. C2×#92481C22semidirect
23. C2×#92479C22semidirect
27. #11758507C22semidirect
29. C2×C4⋊Dic31C22semidirect
30. C2×#92474C22semidirect
31. #11758423C22semidirect
32. #11758420C22semidirect
33. C22⋊(C31⋊C8)C22semidirect
34. C22⋊(C31⋊C8)C22semidirect
43. #41×C31D4semidirect
95. Q8×C31C2×D4semidirect
100. D4×C31C2×D4semidirect
136. C22×C31AGL(1,ℤ/8)semidirect
148. C22×#41D31semidirect
168. C2×#41C2×D31semidirect
169. C22×Q8C2×D31semidirect
173. C22×D4C2×D31semidirect
190. C31#3006semidirect
199. #41C22⋊D31semidirect
251. Q8C2×C22⋊D31semidirect
260. D4C2×C22⋊D31semidirect
294. C22#11758526semidirect

Maximal subgroups

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

subgroupindex
C2×#117585262
C2×#117585072
#145211942
#145211882
#145211832
#145210502
#145210132
C22×#41×C312
#300631