Finite Group Databasebeta

← #16855017

Subgroups — #16855017

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×#11758566C211yesmaximal, characteristic, complemented
2C2×#11758485C211yesmaximal, characteristic, complemented
3C2×#11758527C211yesmaximal, characteristic, complemented
4#14521630C211yesmaximal, characteristic, complemented
5#14521214C212yesmaximal, complemented
6#14521201C212yesmaximal, complemented
7#14521197C212yesmaximal, complemented
8#14521184C212yesmaximal, complemented
9C2×#11758397C211yesmaximal, characteristic, complemented
10#14521017C211yesmaximal, characteristic, complemented
11C31×#417C211yesmaximal, characteristic, complemented
12C2×#41×C31C2211yescharacteristic
13#11758566—24no—
14#11758566—24no—
15C2×#92499C2212yes—
16C2×D4×D31—22no—
17C2×D4×D31C2212yes—
18C2×#92495—21no—
19C2×#92495C2211yescharacteristic
20C2×#92495C2211yescharacteristic, complemented
21C22×D124—21no—
22C22×D124C2211yescharacteristic
23C2×M16×C31C2211yescharacteristic, complemented
24C2×C4⋊C4×C31C2211yescharacteristic, complemented
25#11758527—24no—
26C2×#92482C2211yescharacteristic, complemented
27C2×#92481C2211yescharacteristic, complemented
28C2×#92480C2211yescharacteristic, complemented
29C2×#92479C2211yescharacteristic, complemented
30#11758507—22no—
31C22×C31⋊C8C2211yescharacteristic, complemented
32#11758485—24no—
33C2×D248C2212yescomplemented
34C2×#92470C2212yescomplemented
35#11758474—21no—
36#11758474C2212yescomplemented
37#11758472—21no—
38#11758472C2212yescomplemented
39C31×#103C2212yescomplemented
40C31×#102C2212yescomplemented
41(C2×D31)⋊C8C2212yescomplemented
42(C2×D31)⋊C8C2212yescomplemented
43#11758397C2212yescomplemented
44#11758397C2212yescomplemented
45C23×D31—21no—
46C23×D31—22no—
47#41×C31—22no—
48#41×C31D414yescomplemented
49C2×Q8×C31C2311yescharacteristic
50C2×D4×C31—21no—
51C2×D4×C31C2311yescharacteristic
52C22×C4×C31—21no—
53C22×C4×C31—21no—
54C22×C4×C31C2311yescharacteristic
55C2×C22⋊D31—21no—
56C2×C22⋊D31—22no—
57C2×C22⋊D31D412yes—
58#92499—24no—
59#92499—24no—
60D4×D31—216no—
61D4×D31—24no—
62D4×D31—24no—
63#92495—28no—
64#92495—22no—
65#92495—22no—
66#92495—22no—
67#92495—22no—
68C2×D124—28no—
69C2×D124—22no—
70C2×D124—21no—
71C2×D124—22no—
72C2×D124—22no—
73C2×D124D411yescharacteristic
74C2×D124D414yescomplemented
75C2×D124C2311yescharacteristic
76C2×D124C2312yes—
77C2×C4×D31—22no—
78C2×C4×D31—21no—
79C2×C4×D31C2312yes—
80C2×Dic62D411yescharacteristic
81C2×Dic62C2311yescharacteristic
82M16×C31—22no—
83C2×C8×C31C2312yes—
84C4⋊C4×C31—21no—
85C4⋊C4×C31C2312yes—
86#92482—22no—
87#92481—22no—
88#92480—22no—
89#92479—22no—
90(C2×D31)⋊C4—22no—
91(C2×D31)⋊C4—22no—
92C4⋊Dic31—22no—
93C2×C31⋊C8—22no—
94C2×C31⋊C8C2311yescharacteristic
95C2×C31⋊C8C2311yescharacteristic
96D248—24no—
97#92470—24no—
98C23×C31—21no—
99C23×C31C2×D411yescharacteristic
100C22×D31—22no—
101C22×D31—28no—
102C22×D31—24no—
103C22×D31—21no—
104C22×D31—82no—
105C22×D31—82no—
106C22×D31—81no—
107C22×D31—81no—
108C22×D31C2×D412yes—
109Q8×C31—21no—
110Q8×C31C2×D412yes—
111D4×C31—24no—
112D4×C31—21no—
113D4×C31C2×D412yes—
114C2×C4×C31—21no—
115C2×C4×C31—24no—
116C2×C4×C31—22no—
117C2×C4×C31—81no—
118C2×C4×C31C2×D412yes—
119C2×C4×C31C2×D411yescharacteristic
120C2×C4×C31C2×D412yes—
121C2×C4×C31C2411yescharacteristic
122C22⋊D31—28no—
123C22⋊D31—24no—
124C22⋊D31—42no—
125C2×Dic31—21no—
126C2×Dic31C2×D412yes—
127D124—28no—
128D124—22no—
129D124—24no—
130D124—22no—
131D124—21no—
132D124—21no—
133D124—82no—
134D124C2×D414yes—
135C4×D31—28no—
136C4×D31—22no—
137C4×D31—22no—
138C4×D31—42no—
139Dic62—22no—
140Dic62—21no—
141Dic62—21no—
142C8×C31—22no—
143C31⋊C8—21no—
144C31⋊C8—21no—
145#3012—311nomaximal, Sylow
146C22×C31—21no—
147C22×C31—21no—
148C22×C31—22no—
149C22×C31—41no—
150C22×C31—41no—
151C22×C31#12012yes—
152C22×C31C22×D411yescharacteristic
153C2×D31—22no—
154C2×D31—24no—
155C2×D31—22no—
156C2×D31—22no—
157C2×D31—42no—
158C2×D31—42no—
159C2×D31—41no—
160C2×D31—41no—
161C2×D31—41no—
162C2×D31—84no—
163C2×D31—88no—
164C4×C31—22no—
165C4×C31—42no—
166C4×C31#12012yes—
167C4×C31C22×D411yescharacteristic
168C4×C31C22×D411yescharacteristic
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#417D3111yescharacteristic, complemented
180C2×#102—311no—
181#408—311no—
182C2×C31—22no—
183C2×C31—42no—
184C2×C31C2×#12011yescharacteristic
185C2×C31#57611yescharacteristic
186C2×C31#57711yescharacteristic
187D31—42no—
188D31—44no—
189D31—81no—
190H(5,𝔽2)—624no—
191H(5,𝔽2)—624no—
192C2×#41—311no—
193C2×#41—312no—
194C2×#41—311no—
195C2×#41—621no—
196C2×#41C2×D3111yescharacteristic
197C22×D4—312no—
198C22×D4—311no—
199C22×D4—621no—
200C22×D4—622no—
201AGL(1,ℤ/8)—624no—
202#135—624no—
203C2×Q16—311no—
204C2×SD16—311no—
205C2×SD16—311no—
206C2×SD16—312no—
207C2×D8—311no—
208C2×D8—312no—
209C2×M16C2×D3111yescharacteristic, complemented
210C22×C8—311no—
211C22⋊Q8—312no—
212C22⋊Q8—621no—
213#121—312no—
214#121—621no—
215#121—622no—
216C2×C4⋊C4C2×D3111yescharacteristic, complemented
217#103C2×D3112yescomplemented
218#102—312no—
219#102—312no—
220#102C2×D3112yescomplemented
221C22⋊C8—312no—
222C22⋊C8—312no—
223C31#301211yescharacteristic, complemented, Sylow
224C24—622no—
225C24—621no—
226#41—22no—
227#41—628no—
228#41—624no—
229#41—624no—
230#41—622no—
231#41—622no—
232#41—622no—
233#41—622no—
234#41D12414yescomplemented
235C2×Q8—311no—
236C2×Q8—311no—
237C2×Q8C22×D3111yescharacteristic
238C2×D4—21no—
239C2×D4—312no—
240C2×D4—314no—
241C2×D4—311no—
242C2×D4—312no—
243C2×D4—311no—
244C2×D4—622no—
245C2×D4—628no—
246C2×D4—622no—
247C2×D4—622no—
248C2×D4—624no—
249C2×D4—6216no—
250C2×D4—624no—
251C2×D4—621no—
252C2×D4—622no—
253C2×D4—621no—
254C2×D4C22×D3111yescharacteristic
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—312no—
258C22×C4—622no—
259C22×C4—621no—
260C22×C4C22×D3111yescharacteristic
261Q16—622no—
262SD16—624no—
263SD16—622no—
264SD16—622no—
265D8—624no—
266D8—622no—
267M16—22no—
268C2×C8—311no—
269C2×C8—311no—
270C2×C8—622no—
271C2×C8C22×D3112yes—
272C4⋊C4—21no—
273C4⋊C4—622no—
274C4⋊C4C22×D3112yes—
275C22⋊C4—622no—
276C22⋊C4—622no—
277C23—21no—
278C23—312no—
279C23—628no—
280C23—622no—
281C23—624no—
282C23—621no—
283C23—2481no—
284C23—2481no—
285C23—2482no—
286C23—2482no—
287C23D4×D3111yescharacteristic
288Q8—21no—
289Q8—622no—
290Q8—621no—
291Q8—621no—
292Q8C2×D12412yes—
293D4—24no—
294D4—21no—
295D4—314no—
296D4—628no—
297D4—628no—
298D4—624no—
299D4—624no—
300D4—622no—
301D4—622no—
302D4—621no—
303D4—621no—
304D4—1242no—
305D4—2482no—
306D4C2×D12412yes—
307C2×C4—21no—
308C2×C4—24no—
309C2×C4—22no—
310C2×C4—81no—
311C2×C4—312no—
312C2×C4—628no—
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—22no—
322C8—621no—
323C8—621no—
324C22—21no—
325C22—21no—
326C22—22no—
327C22—41no—
328C22—41no—
329C22—622no—
330C22—624no—
331C22—622no—
332C22—622no—
333C22—1242no—
334C22—1242no—
335C22—1241no—
336C22—1241no—
337C22—1241no—
338C22—2484no—
339C22—2488no—
340C22#1175845912yes—
341C22C2×D4×D3111yescharacteristic
342C4—22no—
343C4—42no—
344C4—622no—
345C4—1241no—
346C4#1175845912yes—
347C4C22×D12411yescharacteristic
348C4C2×D4×D3111yescharacteristic
349C2—22no—
350C2—42no—
351C2—1242no—
352C2—1244no—
353C2—2481no—
354C2C2×#1175845911yescharacteristic
355C2#1452185311yescharacteristic
356C2#1452185811yescharacteristic
subgroupG/H
centerC22C2×D4×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#417C2
socleC22×C31C22×D4
hypercenter#417D31
hypocenterC31#3012
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
1. C2×#11758566C2semidirect
2. C2×#11758485C2semidirect
3. C2×#11758527C2semidirect
4. #14521630C2semidirect
5. #14521214C2semidirect
6. #14521201C2semidirect
7. #14521197C2semidirect
8. #14521184C2semidirect
9. C2×#11758397C2semidirect
10. #14521017C2semidirect
11. C31×#417C2semidirect
20. C2×#92495C22semidirect
23. C2×M16×C31C22semidirect
24. C2×C4⋊C4×C31C22semidirect
26. C2×#92482C22semidirect
27. C2×#92481C22semidirect
28. C2×#92480C22semidirect
29. C2×#92479C22semidirect
31. C22×C31⋊C8C22semidirect
33. C2×D248C22semidirect
34. C2×#92470C22semidirect
36. #11758474C22semidirect
38. #11758472C22semidirect
39. C31×#103C22semidirect
40. C31×#102C22semidirect
41. (C2×D31)⋊C8C22semidirect
42. (C2×D31)⋊C8C22semidirect
43. #11758397C22semidirect
44. #11758397C22semidirect
48. #41×C31D4semidirect
74. C2×D124D4semidirect
179. #417D31semidirect
209. C2×M16C2×D31semidirect
216. C2×C4⋊C4C2×D31semidirect
217. #103C2×D31semidirect
220. #102C2×D31semidirect
223. C31#3012semidirect
234. #41D124semidirect

Maximal subgroups

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

subgroupindex
C2×#117585662
C2×#117584852
C2×#117585272
#145216302
#145212142
#145212012
#145211972
#145211842
C2×#117583972
#145210172
C31×#4172
#301231