Finite Group Databasebeta

← #16855019

Subgroups — #16855019

Order 3968.

764 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11758566C211yesmaximal, characteristic, complemented
2C2×#11758491C211yesmaximal, characteristic, complemented
3C2×#11758496C211yesmaximal, characteristic, complemented
4#14521629C211yesmaximal, characteristic, complemented
5#14521210C211yesmaximal, characteristic, complemented
6#14521199C211yesmaximal, characteristic, complemented
7#14521198C211yesmaximal, characteristic, complemented
8#14521197C211yesmaximal, characteristic, complemented
9#14521194C211yesmaximal, characteristic, complemented
10#14521185C211yesmaximal, characteristic, complemented
11#14521184C211yesmaximal, characteristic, complemented
12#14521049C211yesmaximal, characteristic, complemented
13#102×D31C211yesmaximal, characteristic, complemented
14#14521014C211yesmaximal, characteristic, complemented
15C31×#452C211yesmaximal, characteristic, complemented
16C2×#41×C31C2211yescharacteristic
17#11758566—24no—
18#11758566—24no—
19C2×#92499C2211yescharacteristic
20C2×#92499C2211yescharacteristic
21C2×#92499C2211yescharacteristic, complemented
22C2×D4×D31—21no—
23C2×D4×D31—21no—
24C2×D4×D31C2211yescharacteristic
25C2×D4×D31C2211yescharacteristic
26C2×#92495—21no—
27C2×#92495C2211yescharacteristic
28C22×D124—21no—
29C22×D124C2211yescharacteristic
30C2×Q16×C31C2211yescharacteristic, complemented
31C2×SD16×C31C2211yescharacteristic, complemented
32C31×C22⋊Q8C2211yescharacteristic, complemented
33#11758523C2211yescharacteristic, complemented
34C2×#92481C2211yescharacteristic, complemented
35C2×#92481C2211yescharacteristic, complemented
36C2×#92481C2211yescharacteristic, complemented
37C2×#92479C2211yescharacteristic, complemented
38#11758507C2211yescharacteristic, complemented
39#11758496—24no—
40#11758491—24no—
41C2×D248C2211yescharacteristic, complemented
42C2×D248C2211yescharacteristic, complemented
43C2×#92469C2211yescharacteristic, complemented
44C2×C8×D31C2211yescharacteristic, complemented
45#11758473—21no—
46#11758472—21no—
47#11758472C2211yescharacteristic, complemented
48C4⋊C4×D31C2211yescharacteristic, complemented
49#11758461—21no—
50#11758461—21no—
51C31×#103C2211yescharacteristic, complemented
52C31×#102C2211yescharacteristic, complemented
53C31×C22⋊C8C2211yescharacteristic, complemented
54#11758423C2211yescharacteristic, complemented
55#11758420C2211yescharacteristic, complemented
56C22⋊(C31⋊C8)C2211yescharacteristic, complemented
57#11758409C2211yescharacteristic, complemented
58#11758409C2211yescharacteristic, complemented
59(C2×D31)⋊C8C2211yescharacteristic, complemented
60(C2×D31)⋊C8C2211yescharacteristic, complemented
61#11758397C2211yescharacteristic, complemented
62#11758397C2211yescharacteristic, complemented
63C23×D31—21no—
64C23×D31—21no—
65C23×D31—21no—
66#41×C31—22no—
67#41×C31—22no—
68C2×Q8×C31C2311yescharacteristic
69C2×Q8×C31C2311yescharacteristic
70C2×D4×C31—21no—
71C2×D4×C31C2311yescharacteristic
72C22×C4×C31—21no—
73C22×C4×C31C2311yescharacteristic
74C2×C22⋊D31—21no—
75C2×C22⋊D31—21no—
76C2×C22⋊D31—21no—
77C2×C22⋊D31D411yescharacteristic
78C2×C22⋊D31D411yescharacteristic
79#92499—22no—
80#92499—22no—
81#92499—22no—
82#92499—22no—
83#92499—22no—
84#92499D414yescomplemented
85D4×D31—28no—
86D4×D31—28no—
87D4×D31—22no—
88D4×D31—22no—
89D4×D31—22no—
90D4×D31D414yescomplemented
91#92495—28no—
92#92495—22no—
93#92495—22no—
94C2×D124—28no—
95C2×D124—21no—
96C2×D124—21no—
97C2×D124—21no—
98C2×D124—22no—
99C2×D124—22no—
100C2×D124—21no—
101C2×D124—21no—
102C2×D124D411yescharacteristic
103C2×D124C2311yescharacteristic
104C2×D124C2311yescharacteristic
105C2×D124C2311yescharacteristic
106C2×C4×D31—21no—
107C2×C4×D31—21no—
108C2×C4×D31—21no—
109C2×C4×D31—21no—
110C2×C4×D31C2311yescharacteristic
111C2×C4×D31C2311yescharacteristic
112C2×Dic62D411yescharacteristic
113Q16×C31—22no—
114SD16×C31—22no—
115C2×C8×C31C2311yescharacteristic
116C2×C8×C31C2311yescharacteristic
117C4⋊C4×C31—21no—
118C4⋊C4×C31C2311yescharacteristic
119C22⋊C4×C31—21no—
120#92481—22no—
121#92481—22no—
122#92481—22no—
123#92479—22no—
124(C2×D31)⋊C4—21no—
125(C2×D31)⋊C4—21no—
126(C2×D31)⋊C4—21no—
127C4⋊Dic31C2311yescharacteristic
128Dic31⋊C4—21no—
129Dic31⋊C4—21no—
130C2×C31⋊C8C2311yescharacteristic
131C2×C31⋊C8C2311yescharacteristic
132D248—22no—
133D248—22no—
134#92469—22no—
135C8×D31—22no—
136C23×C31—21no—
137C23×C31C2×D411yescharacteristic
138C22×D31—21no—
139C22×D31—21no—
140C22×D31—24no—
141C22×D31—24no—
142C22×D31—24no—
143C22×D31—21no—
144C22×D31—81no—
145C22×D31—81no—
146C22×D31—81no—
147C22×D31—81no—
148C22×D31—81no—
149C22×D31—81no—
150C22×D31C2×D411yescharacteristic
151C22×D31C2×D411yescharacteristic
152Q8×C31—21no—
153Q8×C31—21no—
154Q8×C31—21no—
155Q8×C31C2×D412yes—
156D4×C31—24no—
157D4×C31—21no—
158D4×C31C2×D412yes—
159C2×C4×C31—24no—
160C2×C4×C31—21no—
161C2×C4×C31—21no—
162C2×C4×C31—21no—
163C2×C4×C31—21no—
164C2×C4×C31—21no—
165C2×C4×C31C2×D411yescharacteristic
166C2×C4×C31C2411yescharacteristic
167C22⋊D31—24no—
168C22⋊D31—22no—
169C22⋊D31—22no—
170C22⋊D31—24no—
171C22⋊D31—81no—
172C2×Dic31—21no—
173C2×Dic31C2×D411yescharacteristic
174C2×Dic31C2×D411yescharacteristic
175D124—24no—
176D124—24no—
177D124—22no—
178D124—24no—
179D124—21no—
180D124—21no—
181D124—21no—
182D124—21no—
183D124—42no—
184D124—81no—
185D124C2×D412yes—
186D124C2×D412yes—
187C4×D31—24no—
188C4×D31—21no—
189C4×D31—21no—
190C4×D31—24no—
191C4×D31—42no—
192C4×D31—81no—
193C4×D31C2×D412yes—
194C4×D31C2×D412yes—
195Dic62—22no—
196C8×C31—21no—
197C8×C31—21no—
198C31⋊C8—21no—
199C31⋊C8—21no—
200#3012—311nomaximal, Sylow
201C22×C31—21no—
202C22×C31—22no—
203C22×C31—21no—
204C22×C31—21no—
205C22×C31—41no—
206C22×C31—41no—
207C22×C31C22×D411yescharacteristic
208C2×D31—21no—
209C2×D31—21no—
210C2×D31—21no—
211C2×D31—22no—
212C2×D31—22no—
213C2×D31—21no—
214C2×D31—21no—
215C2×D31—41no—
216C2×D31—41no—
217C2×D31—41no—
218C2×D31—41no—
219C2×D31—41no—
220C2×D31—41no—
221C2×D31—41no—
222C2×D31—84no—
223C2×D31—84no—
224C2×D31—84no—
225C2×D31#12012yes—
226C4×C31—21no—
227C4×C31—22no—
228C4×C31—41no—
229C4×C31—41no—
230C4×C31C22×D411yescharacteristic
231C4×C31C22×D411yescharacteristic
232Dic31—21no—
233Dic31—41no—
234Dic31#12012yes—
235C2×H(5,𝔽2)—311no—
236C2×AGL(1,ℤ/8)—311no—
237C2×#135—311no—
238#537—311no—
239#452—311no—
240#452D3111yescharacteristic, complemented
241#450—311no—
242#450—311no—
243#449—311no—
244#449—311no—
245#447—311no—
246#447—311no—
247#417—311no—
248C2×#102—311no—
249#408—311no—
250C2×C31—41no—
251C2×C31—42no—
252C2×C31C2×#12011yescharacteristic
253C2×C31#57611yescharacteristic
254C2×C31#57711yescharacteristic
255D31—22no—
256D31—41no—
257D31—42no—
258D31—42no—
259D31—81no—
260H(5,𝔽2)—624no—
261H(5,𝔽2)—624no—
262C2×#41—311no—
263C2×#41—311no—
264C2×#41—311no—
265C2×#41—311no—
266C2×#41—621no—
267C2×#41C2×D3111yescharacteristic
268C22×D4—311no—
269C22×D4—311no—
270C22×D4—311no—
271C22×D4—621no—
272C22×D4—621no—
273C22×D4—621no—
274AGL(1,ℤ/8)—624no—
275#135—624no—
276C2×Q16C2×D3111yescharacteristic, complemented
277C2×SD16—311no—
278C2×SD16—311no—
279C2×SD16—311no—
280C2×SD16C2×D3111yescharacteristic, complemented
281C2×D8—311no—
282C2×D8—311no—
283C2×D8—311no—
284C2×M16—311no—
285C22×C8—311no—
286C22⋊Q8—311no—
287C22⋊Q8—621no—
288C22⋊Q8C2×D3111yescharacteristic, complemented
289#121—311no—
290#121—311no—
291#121—621no—
292#121—621no—
293#121—621no—
294C2×C4⋊C4—311no—
295#103—311no—
296#103C2×D3111yescharacteristic, complemented
297#102—311no—
298#102—311no—
299#102—311no—
300#102—311no—
301#102—311no—
302#102C2×D3111yescharacteristic, complemented
303C22⋊C8—311no—
304C22⋊C8—311no—
305C22⋊C8—311no—
306C22⋊C8C2×D3111yescharacteristic, complemented
307C31#301211yescharacteristic, complemented, Sylow
308C24—621no—
309C24—621no—
310C24—621no—
311#41—22no—
312#41—22no—
313#41—314no—
314#41—622no—
315#41—622no—
316#41—628no—
317#41—622no—
318#41—622no—
319#41—622no—
320#41—622no—
321#41—622no—
322C2×Q8—311no—
323C2×Q8C22×D3111yescharacteristic
324C2×Q8C22×D3111yescharacteristic
325C2×D4—21no—
326C2×D4—314no—
327C2×D4—311no—
328C2×D4—311no—
329C2×D4—311no—
330C2×D4—311no—
331C2×D4—311no—
332C2×D4—311no—
333C2×D4—622no—
334C2×D4—622no—
335C2×D4—621no—
336C2×D4—621no—
337C2×D4—628no—
338C2×D4—621no—
339C2×D4—621no—
340C2×D4—622no—
341C2×D4—621no—
342C2×D4—621no—
343C2×D4—622no—
344C2×D4—628no—
345C2×D4—628no—
346C2×D4—622no—
347C2×D4—621no—
348C2×D4—621no—
349C2×D4C22×D3111yescharacteristic
350C22×C4—21no—
351C22×C4—311no—
352C22×C4—311no—
353C22×C4—621no—
354C22×C4—621no—
355C22×C4—621no—
356C22×C4—621no—
357C22×C4C22×D3111yescharacteristic
358Q16—22no—
359SD16—22no—
360SD16—622no—
361SD16—622no—
362SD16—622no—
363D8—622no—
364D8—622no—
365D8—622no—
366M16—622no—
367C2×C8—311no—
368C2×C8—311no—
369C2×C8—622no—
370C2×C8C22×D3111yescharacteristic
371C2×C8C22×D3111yescharacteristic
372C4⋊C4—21no—
373C4⋊C4—311no—
374C4⋊C4—621no—
375C4⋊C4—621no—
376C4⋊C4C22×D3111yescharacteristic
377C22⋊C4—21no—
378C22⋊C4—621no—
379C22⋊C4—621no—
380C22⋊C4—621no—
381C23—21no—
382C23—311no—
383C23—311no—
384C23—624no—
385C23—624no—
386C23—624no—
387C23—621no—
388C23—621no—
389C23—621no—
390C23—2481no—
391C23—2481no—
392C23—2481no—
393C23—2481no—
394C23—2481no—
395C23—2481no—
396C23D4×D3111yescharacteristic
397Q8—21no—
398Q8—21no—
399Q8—21no—
400Q8—622no—
401Q8D4×D3112yes—
402D4—24no—
403D4—21no—
404D4—312no—
405D4—312no—
406D4—622no—
407D4—624no—
408D4—624no—
409D4—624no—
410D4—622no—
411D4—624no—
412D4—622no—
413D4—624no—
414D4—621no—
415D4—621no—
416D4—621no—
417D4—621no—
418D4—1242no—
419D4—2481no—
420D4—2481no—
421D4D4×D3112yes—
422C2×C4—24no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—311no—
429C2×C4—312no—
430C2×C4—312no—
431C2×C4—311no—
432C2×C4—624no—
433C2×C4—624no—
434C2×C4—621no—
435C2×C4—621no—
436C2×C4—621no—
437C2×C4—1242no—
438C2×C4—2481no—
439C2×C4D4×D3111yescharacteristic
440C2×C4C23×D3111yescharacteristic
441C8—21no—
442C8—21no—
443C8—621no—
444C8—621no—
445C22—21no—
446C22—21no—
447C22—22no—
448C22—21no—
449C22—41no—
450C22—41no—
451C22—312no—
452C22—621no—
453C22—621no—
454C22—621no—
455C22—622no—
456C22—622no—
457C22—621no—
458C22—621no—
459C22—1241no—
460C22—1241no—
461C22—1241no—
462C22—1241no—
463C22—1241no—
464C22—1241no—
465C22—1241no—
466C22—2484no—
467C22—2484no—
468C22—2484no—
469C22C2×D4×D3111yescharacteristic
470C4—21no—
471C4—22no—
472C4—41no—
473C4—41no—
474C4—312no—
475C4—621no—
476C4—1241no—
477C4C2×D4×D3111yescharacteristic
478C4C2×D4×D3111yescharacteristic
479C2—41no—
480C2—42no—
481C2—622no—
482C2—1241no—
483C2—1242no—
484C2—1242no—
485C2—2481no—
486C2#120×D3111yescharacteristic
487C2#1452185111yescharacteristic
488C2#1452185811yescharacteristic
subgroupG/H
centerC22C2×D4×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#452C2
socleC22×C31C22×D4
hypercenter#452D31
hypocenterC31#3012
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
1. C2×#11758566C2semidirect
2. C2×#11758491C2semidirect
3. C2×#11758496C2semidirect
4. #14521629C2semidirect
5. #14521210C2semidirect
6. #14521199C2semidirect
7. #14521198C2semidirect
8. #14521197C2semidirect
9. #14521194C2semidirect
10. #14521185C2semidirect
11. #14521184C2semidirect
12. #14521049C2semidirect
13. #102×D31C2semidirect
14. #14521014C2semidirect
15. C31×#452C2semidirect
21. C2×#92499C22semidirect
30. C2×Q16×C31C22semidirect
31. C2×SD16×C31C22semidirect
32. C31×C22⋊Q8C22semidirect
33. #11758523C22semidirect
34. C2×#92481C22semidirect
35. C2×#92481C22semidirect
36. C2×#92481C22semidirect
37. C2×#92479C22semidirect
38. #11758507C22semidirect
41. C2×D248C22semidirect
42. C2×D248C22semidirect
43. C2×#92469C22semidirect
44. C2×C8×D31C22semidirect
47. #11758472C22semidirect
48. C4⋊C4×D31C22semidirect
51. C31×#103C22semidirect
52. C31×#102C22semidirect
53. C31×C22⋊C8C22semidirect
54. #11758423C22semidirect
55. #11758420C22semidirect
56. C22⋊(C31⋊C8)C22semidirect
57. #11758409C22semidirect
58. #11758409C22semidirect
59. (C2×D31)⋊C8C22semidirect
60. (C2×D31)⋊C8C22semidirect
61. #11758397C22semidirect
62. #11758397C22semidirect
84. #92499D4semidirect
90. D4×D31D4semidirect
240. #452D31semidirect
276. C2×Q16C2×D31semidirect
280. C2×SD16C2×D31semidirect
288. C22⋊Q8C2×D31semidirect
296. #103C2×D31semidirect
302. #102C2×D31semidirect
306. C22⋊C8C2×D31semidirect
307. C31#3012semidirect

Maximal subgroups

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

subgroupindex
C2×#117585662
C2×#117584912
C2×#117584962
#145216292
#145212102
#145211992
#145211982
#145211972
#145211942
#145211852
#145211842
#145210492
#102×D312
#145210142
C31×#4522
#301231