Finite Group Databasebeta

← #16855020

Subgroups — #16855020

Order 3968.

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

normal subgroups only

This group has 764 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×#11758561C211yesmaximal, characteristic, complemented
2C2×#11758509C211yesmaximal, characteristic, complemented
3C2×#11758481C211yesmaximal, characteristic, complemented
4#14521630C211yesmaximal, characteristic, complemented
5#14521212C212yesmaximal, complemented
6#14521201C212yesmaximal, complemented
7#14521195C212yesmaximal, complemented
8#14521184C212yesmaximal, complemented
9#14521048C211yesmaximal, characteristic, complemented
10#14521015C211yesmaximal, characteristic, complemented
11C2×C31×#102C211yesmaximal, characteristic, complemented
12C22×D4×C31C2211yescharacteristic
13C22×C22⋊D31—22no—
14#11758561—24no—
15#11758561—24no—
16C2×#92497—21no—
17C2×#92497C2212yes—
18C2×D4×D31—21no—
19C2×D4×D31C2212yes—
20C2×#92495C2211yescharacteristic
21C2×#92495C2211yescharacteristic
22C2×#92495C2211yescharacteristic, complemented
23C22×C8×C31C2211yescharacteristic, complemented
24C2×C4⋊C4×C31C2211yescharacteristic, complemented
25C2×#92480C2212yescomplemented
26#11758509—24no—
27C2×#92479C2212yescomplemented
28#11758507—22no—
29C2×#92474C2211yescharacteristic, complemented
30C2×Dic124C2211yescharacteristic, complemented
31#11758481—24no—
32C2×D248C2211yescharacteristic, complemented
33C2×#92470C2211yescharacteristic, complemented
34C2×#92470C2211yescharacteristic, complemented
35#11758474—21no—
36#11758474C2212yescomplemented
37#11758472—21no—
38#11758472C2212yescomplemented
39C31×#102C2212yescomplemented
40C31×#102C2212yescomplemented
41(C2×D31)⋊C8C2212yescomplemented
42(C2×D31)⋊C8C2212yescomplemented
43#11758398C2212yescomplemented
44#11758397C2212yescomplemented
45C24×C31—21no—
46C23×D31—22no—
47C2×D4×C31—21no—
48C2×D4×C31—22no—
49C2×D4×C31D414yescomplemented
50C2×D4×C31C2312yes—
51C22×C4×C31—21no—
52C22×C4×C31C2311yescharacteristic
53C2×C22⋊D31—216no—
54C2×C22⋊D31—21no—
55C2×C22⋊D31—22no—
56C2×C22⋊D31—21no—
57C2×C22⋊D31—22no—
58C2×C22⋊D31D412yes—
59C22×Dic31—22no—
60#92497—28no—
61#92497—24no—
62#92497—24no—
63D4×D31—28no—
64D4×D31—24no—
65D4×D31—24no—
66#92495—22no—
67#92495—22no—
68#92495—22no—
69#92495—22no—
70#92495—22no—
71#92495D414yescomplemented
72C2×D124—22no—
73C2×D124D411yescharacteristic
74C2×D124C2311yescharacteristic
75C2×D124C2311yescharacteristic
76C2×C4×D31—21no—
77C2×C4×D31—21no—
78C2×C4×D31C2312yes—
79C2×Dic62D411yescharacteristic
80C2×Dic62C2311yescharacteristic
81C2×Dic62C2311yescharacteristic
82C2×C8×C31—22no—
83C2×C8×C31C2311yescharacteristic
84C2×C8×C31C2311yescharacteristic
85C4⋊C4×C31—21no—
86C4⋊C4×C31C2312yes—
87#92480—24no—
88#92479—24no—
89(C2×D31)⋊C4—22no—
90(C2×D31)⋊C4—22no—
91C4⋊Dic31—22no—
92#92474—22no—
93C2×C31⋊C8C2312yes—
94Dic124—22no—
95D248—22no—
96#92470—22no—
97#92470—22no—
98C23×C31—22no—
99C23×C31—24no—
100C23×C31—81no—
101C23×C31—81no—
102C23×C31C2×D411yescharacteristic
103C22×D31—21no—
104C22×D31—28no—
105C22×D31—21no—
106C22×D31—82no—
107C22×D31—82no—
108C22×D31C2×D412yes—
109D4×C31—24no—
110D4×C31—22no—
111D4×C31C2×D414yes—
112C2×C4×C31—22no—
113C2×C4×C31—81no—
114C2×C4×C31C2×D412yes—
115C2×C4×C31C2×D411yescharacteristic
116C2×C4×C31C2×D412yes—
117C2×C4×C31C2411yescharacteristic
118C22⋊D31—28no—
119C22⋊D31—28no—
120C22⋊D31—24no—
121C22⋊D31—24no—
122C22⋊D31—42no—
123C2×Dic31—28no—
124C2×Dic31—21no—
125C2×Dic31—21no—
126C2×Dic31C2×D412yes—
127D124—22no—
128D124—21no—
129D124—21no—
130D124—21no—
131D124—82no—
132D124C2×D412yes—
133C4×D31—24no—
134C4×D31—22no—
135C4×D31—22no—
136C4×D31—42no—
137Dic62—22no—
138Dic62—21no—
139Dic62—21no—
140Dic62—21no—
141Dic62C2×D412yes—
142C8×C31—21no—
143C8×C31—21no—
144C31⋊C8—22no—
145#3012—311nomaximal, Sylow
146C22×C31—21no—
147C22×C31—21no—
148C22×C31—24no—
149C22×C31—42no—
150C22×C31—42no—
151C22×C31—84no—
152C22×C31#12012yes—
153C22×C31C22×D411yescharacteristic
154C2×D31—22no—
155C2×D31—22no—
156C2×D31—22no—
157C2×D31—22no—
158C2×D31—41no—
159C2×D31—41no—
160C2×D31—41no—
161C2×D31—41no—
162C2×D31—41no—
163C2×D31—88no—
164C4×C31—42no—
165C4×C31#12012yes—
166C4×C31C22×D411yescharacteristic
167C4×C31C22×D411yescharacteristic
168Dic31—22no—
169Dic31—22no—
170Dic31—41no—
171C2×H(5,𝔽2)—311no—
172C2×AGL(1,ℤ/8)—311no—
173C2×#135—311no—
174#537—311no—
175#452—312no—
176#450—312no—
177#449—312no—
178#447—312no—
179#417—311no—
180C2×#102D3111yescharacteristic, complemented
181#408—311no—
182C2×C31—22no—
183C2×C31—44no—
184C2×C31C2×#12011yescharacteristic
185C2×C31#57611yescharacteristic
186C2×C31#57711yescharacteristic
187D31—42no—
188D31—42no—
189D31—81no—
190H(5,𝔽2)—624no—
191H(5,𝔽2)—624no—
192C2×#41—311no—
193C2×#41—312no—
194C2×#41—311no—
195C2×#41—311no—
196C2×#41—621no—
197C22×D4—312no—
198C22×D4—621no—
199C22×D4—622no—
200C22×D4C2×D3111yescharacteristic
201AGL(1,ℤ/8)—624no—
202#135—624no—
203C2×Q16—311no—
204C2×SD16—312no—
205C2×SD16—311no—
206C2×SD16—311no—
207C2×D8—312no—
208C2×D8—311no—
209C2×M16—311no—
210C22×C8C2×D3111yescharacteristic, complemented
211C22⋊Q8—312no—
212C22⋊Q8—621no—
213#121—312no—
214#121—621no—
215#121—622no—
216C2×C4⋊C4C2×D3111yescharacteristic, complemented
217#103—312no—
218#102—312no—
219#102C2×D3112yescomplemented
220#102C2×D3112yescomplemented
221C22⋊C8—312no—
222C22⋊C8—312no—
223C31#301211yescharacteristic, complemented, Sylow
224C24—21no—
225C24—622no—
226#41—314no—
227#41—624no—
228#41—628no—
229#41—624no—
230#41—622no—
231#41—622no—
232#41—622no—
233#41—622no—
234#41—622no—
235C2×Q8—311no—
236C2×Q8—311no—
237C2×Q8—311no—
238C2×D4—21no—
239C2×D4—22no—
240C2×D4—312no—
241C2×D4—311no—
242C2×D4—311no—
243C2×D4—311no—
244C2×D4—622no—
245C2×D4—622no—
246C2×D4—624no—
247C2×D4—628no—
248C2×D4—622no—
249C2×D4—6216no—
250C2×D4—621no—
251C2×D4—624no—
252C2×D4—621no—
253C2×D4D12414yescomplemented
254C2×D4C22×D3112yes—
255C22×C4—21no—
256C22×C4—312no—
257C22×C4—622no—
258C22×C4—621no—
259C22×C4—621no—
260C22×C4C22×D3111yescharacteristic
261Q16—622no—
262SD16—622no—
263SD16—622no—
264SD16—624no—
265D8—622no—
266D8—624no—
267M16—622no—
268C2×C8—22no—
269C2×C8—312no—
270C2×C8C22×D3111yescharacteristic
271C2×C8C22×D3111yescharacteristic
272C4⋊C4—21no—
273C4⋊C4—622no—
274C4⋊C4C22×D3112yes—
275C22⋊C4—622no—
276C22⋊C4—622no—
277C23—22no—
278C23—24no—
279C23—81no—
280C23—81no—
281C23—312no—
282C23—628no—
283C23—621no—
284C23—621no—
285C23—2482no—
286C23—2482no—
287C23D4×D3111yescharacteristic
288Q8—312no—
289Q8—622no—
290Q8—621no—
291Q8—621no—
292Q8—621no—
293D4—24no—
294D4—22no—
295D4—312no—
296D4—628no—
297D4—628no—
298D4—624no—
299D4—624no—
300D4—622no—
301D4—621no—
302D4—621no—
303D4—621no—
304D4—1242no—
305D4—2482no—
306D4C2×D12414yes—
307C2×C4—22no—
308C2×C4—81no—
309C2×C4—312no—
310C2×C4—628no—
311C2×C4—621no—
312C2×C4—624no—
313C2×C4—622no—
314C2×C4—622no—
315C2×C4—621no—
316C2×C4—1242no—
317C2×C4C2×D12412yes—
318C2×C4D4×D3111yescharacteristic
319C2×C4D4×D3112yes—
320C2×C4C23×D3111yescharacteristic
321C8—21no—
322C8—21no—
323C8—622no—
324C22—21no—
325C22—21no—
326C22—24no—
327C22—42no—
328C22—42no—
329C22—84no—
330C22—622no—
331C22—622no—
332C22—622no—
333C22—622no—
334C22—1241no—
335C22—1241no—
336C22—1241no—
337C22—1241no—
338C22—1241no—
339C22—2488no—
340C22#1175845912yes—
341C22C2×D4×D3111yescharacteristic
342C4—42no—
343C4—622no—
344C4—622no—
345C4—1241no—
346C4#1175845912yes—
347C4C22×D12411yescharacteristic
348C4C2×D4×D3111yescharacteristic
349C2—22no—
350C2—44no—
351C2—1242no—
352C2—1242no—
353C2—2481no—
354C2C2×#1175845911yescharacteristic
355C2#1452185011yescharacteristic
356C2#1452185511yescharacteristic
subgroupG/H
centerC22C2×D4×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC2×C31×#102C2
socleC22×C31C22×D4
hypercenterC2×#102D31
hypocenterC31#3012
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
1. C2×#11758561C2semidirect
2. C2×#11758509C2semidirect
3. C2×#11758481C2semidirect
4. #14521630C2semidirect
5. #14521212C2semidirect
6. #14521201C2semidirect
7. #14521195C2semidirect
8. #14521184C2semidirect
9. #14521048C2semidirect
10. #14521015C2semidirect
11. C2×C31×#102C2semidirect
22. C2×#92495C22semidirect
23. C22×C8×C31C22semidirect
24. C2×C4⋊C4×C31C22semidirect
25. C2×#92480C22semidirect
27. C2×#92479C22semidirect
29. C2×#92474C22semidirect
30. C2×Dic124C22semidirect
32. C2×D248C22semidirect
33. C2×#92470C22semidirect
34. C2×#92470C22semidirect
36. #11758474C22semidirect
38. #11758472C22semidirect
39. C31×#102C22semidirect
40. C31×#102C22semidirect
41. (C2×D31)⋊C8C22semidirect
42. (C2×D31)⋊C8C22semidirect
43. #11758398C22semidirect
44. #11758397C22semidirect
49. C2×D4×C31D4semidirect
71. #92495D4semidirect
180. C2×#102D31semidirect
210. C22×C8C2×D31semidirect
216. C2×C4⋊C4C2×D31semidirect
219. #102C2×D31semidirect
220. #102C2×D31semidirect
223. C31#3012semidirect
253. C2×D4D124semidirect

Maximal subgroups

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

subgroupindex
C2×#117585612
C2×#117585092
C2×#117584812
#145216302
#145212122
#145212012
#145211952
#145211842
#145210482
#145210152
C2×C31×#1022
#301231