Finite Group Databasebeta

← #16857201

Subgroups — #16857201

Order 3968.

546 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11758491C211yesmaximal, characteristic, complemented
2C2×SD16×D31C211yesmaximal, characteristic, complemented
3#14521711C211yesmaximal, characteristic, complemented
4#14521684C211yesmaximal, characteristic, complemented
5#14521436C211yesmaximal, characteristic, complemented
6#14521294C211yesmaximal, characteristic, complemented
7#14521284C211yesmaximal, characteristic, complemented
8#14521265C211yesmaximal, characteristic, complemented
9#14521201C211yesmaximal, characteristic, complemented
10#14521193C211yesmaximal, characteristic, complemented
11#14521186C211yesmaximal, characteristic, complemented
12#14521184C211yesmaximal, characteristic, complemented
13#14521147C211yesmaximal, characteristic, complemented
14#14521119C211yesmaximal, characteristic, complemented
15C31×#460C211yesmaximal, characteristic, complemented
16C2×#92499C2211yescharacteristic, complemented
17C2×Q8×D31C2211yescharacteristic, complemented
18C2×D4×D31—21no—
19C2×D4×D31C2211yescharacteristic
20C2×D4×D31C2211yescharacteristic
21C2×SD16×C31C2211yescharacteristic, complemented
22C2×SD16×C31C2211yescharacteristic, complemented
23C31×#127C2211yescharacteristic
24C4×Q8×C31C2211yescharacteristic, complemented
25C2×#92481C2211yescharacteristic
26#11758515—21no—
27#11758515C2211yescharacteristic
28#11758515C2211yescharacteristic
29#11758514—21no—
30#11758514—21no—
31C2×#92480C2211yescharacteristic, complemented
32C2×#92480C2211yescharacteristic, complemented
33C2×#92479C2211yescharacteristic, complemented
34C2×#92479C2211yescharacteristic, complemented
35#11758491—24no—
36SD16×D31—24no—
37C2×D248C2211yescharacteristic
38C2×#92470C2211yescharacteristic
39C2×#92469C2211yescharacteristic, complemented
40C2×C8×D31C2211yescharacteristic, complemented
41#11758474—21no—
42#11758474C2211yescharacteristic, complemented
43#11758472—21no—
44#11758472C2211yescharacteristic, complemented
45#11758452—21no—
46#11758452C2211yescharacteristic, complemented
47C4×D124—21no—
48C4×D124C2211yescharacteristic
49C4×D124C2211yescharacteristic
50C31×C4⋊C8C2211yescharacteristic, complemented
51C31×#102C2211yescharacteristic, complemented
52C31×#102C2211yescharacteristic, complemented
53#11758423C2211yescharacteristic, complemented
54#11758409C2211yescharacteristic, complemented
55(C2×D31)⋊C8C2211yescharacteristic, complemented
56(C2×D31)⋊C8C2211yescharacteristic, complemented
57#11758404C2211yescharacteristic, complemented
58#11758397C2211yescharacteristic
59#11758395C2211yescharacteristic, complemented
60#11758391C2211yescharacteristic, complemented
61C23×D31—21no—
62C23×D31—21no—
63C2×Q8×C31C2311yescharacteristic
64C2×D4×C31—21no—
65C2×D4×C31—21no—
66C2×D4×C31C2311yescharacteristic
67C2×D4×C31C2311yescharacteristic
68C2×C22⋊D31—21no—
69C2×C22⋊D31—21no—
70C2×C22⋊D31—21no—
71C2×C22⋊D31—21no—
72#92499—22no—
73#92499—41no—
74Q8×D31—22no—
75Q8×D31—41no—
76D4×D31—22no—
77D4×D31—22no—
78D4×D31—41no—
79D4×D31—41no—
80D4×D31—44no—
81C2×D124—21no—
82C2×D124—21no—
83C2×D124D411yescharacteristic
84C2×D124C2311yescharacteristic
85C2×D124C2311yescharacteristic
86C2×C4×D31—21no—
87C2×C4×D31—21no—
88C2×C4×D31—21no—
89C2×C4×D31C2311yescharacteristic
90C2×C4×D31C2311yescharacteristic
91C2×Dic62—21no—
92C2×Dic62C2311yescharacteristic
93SD16×C31—22no—
94SD16×C31—22no—
95C2×C8×C31C2311yescharacteristic
96C2×C8×C31C2311yescharacteristic
97C4⋊C4×C31—21no—
98C4⋊C4×C31C2311yescharacteristic
99C42×C31—21no—
100C42×C31C2311yescharacteristic
101C22⋊Dic31—21no—
102C22⋊Dic31—21no—
103#92481D414yescomplemented
104#92480—22no—
105#92480—41no—
106#92479—22no—
107#92479—41no—
108(C2×D31)⋊C4—21no—
109(C2×D31)⋊C4—21no—
110(C2×D31)⋊C4—21no—
111(C2×D31)⋊C4—21no—
112(C2×D31)⋊C4—21no—
113(C2×D31)⋊C4D411yescharacteristic
114(C2×D31)⋊C4D411yescharacteristic
115C4⋊Dic31—21no—
116C4⋊Dic31D411yescharacteristic
117C4⋊Dic31C2311yescharacteristic
118C2×C31⋊C8C2311yescharacteristic
119C2×C31⋊C8C2311yescharacteristic
120D248—22no—
121#92470—22no—
122#92469—22no—
123C8×D31—22no—
124C23×C31—21no—
125C23×C31—21no—
126C22×D31—21no—
127C22×D31—21no—
128C22×D31—42no—
129C22×D31—42no—
130C22×D31—42no—
131C22×D31—42no—
132C22×D31—81no—
133C22×D31—81no—
134C22×D31C2×D411yescharacteristic
135C22×D31C2×D411yescharacteristic
136Q8×C31—21no—
137Q8×C31C2×D412yescomplemented
138D4×C31—21no—
139D4×C31—21no—
140D4×C31—21no—
141D4×C31—21no—
142D4×C31—42no—
143D4×C31—81no—
144C2×C4×C31—21no—
145C2×C4×C31—21no—
146C2×C4×C31C2×D411yescharacteristic
147C2×C4×C31C2×D411yescharacteristic
148C2×C4×C31C2411yescharacteristic
149C22⋊D31—42no—
150C22⋊D31—42no—
151C22⋊D31—42no—
152C22⋊D31—42no—
153C2×Dic31—21no—
154C2×Dic31—21no—
155C2×Dic31C2×D411yescharacteristic
156C2×Dic31C2×D411yescharacteristic
157D124—22no—
158D124—21no—
159D124—21no—
160D124—21no—
161D124—42no—
162D124—42no—
163D124C2×D412yes—
164C4×D31—21no—
165C4×D31—21no—
166C4×D31—21no—
167C4×D31—21no—
168C4×D31—42no—
169C4×D31—42no—
170C4×D31—42no—
171Dic62—21no—
172Dic62—21no—
173Dic62—42no—
174C8×C31—21no—
175C8×C31—21no—
176C31⋊C8—21no—
177C31⋊C8C2×D412yescomplemented
178#3265—311nomaximal, Sylow
179C22×C31—41no—
180C22×C31—41no—
181C22×C31—41no—
182C22×C31—41no—
183C22×C31—41no—
184C22×C31—41no—
185C22×C31C22×D411yescharacteristic
186C2×D31—21no—
187C2×D31—21no—
188C2×D31—21no—
189C2×D31—22no—
190C2×D31—21no—
191C2×D31—21no—
192C2×D31—41no—
193C2×D31—41no—
194C2×D31—41no—
195C2×D31—41no—
196C2×D31—41no—
197C2×D31—84no—
198C2×D31—84no—
199C2×D31AGL(1,ℤ/8)12yescomplemented
200C4×C31—22no—
201C4×C31—21no—
202C4×C31—21no—
203C4×C31—41no—
204C4×C31C22×D411yescharacteristic
205C4×C31H(5,𝔽2)11yescharacteristic
206Dic31—21no—
207Dic31—21no—
208Dic31—41no—
209Dic31—41no—
210C2×AGL(1,ℤ/8)—311no—
211C22×SD16—311no—
212#548—311no—
213D42—311no—
214#496—311no—
215#468—311no—
216#466—311no—
217#463—311no—
218#460D3111yescharacteristic, complemented
219#450—311no—
220#449—311no—
221#448—311no—
222#447—311no—
223#440—311no—
224#435—311no—
225C2×C31—81no—
226C2×C31—81no—
227C2×C31D4211yescharacteristic
228C2×C31C2×AGL(1,ℤ/8)11yescharacteristic
229C2×C31#57711yescharacteristic
230D31—22no—
231D31—41no—
232D31—42no—
233D31—81no—
234C2×#41—311no—
235C22×Q8—311no—
236C22×D4—311no—
237C22×D4—311no—
238C22×D4—621no—
239AGL(1,ℤ/8)—624no—
240C2×SD16—311no—
241C2×SD16—311no—
242C2×SD16—311no—
243C2×SD16—311no—
244C2×SD16—624no—
245C2×SD16C2×D3111yescharacteristic, complemented
246C2×SD16C2×D3111yescharacteristic, complemented
247C2×D8—311no—
248C2×D8—311no—
249C2×D8—311no—
250C2×M16—311no—
251C22×C8—311no—
252#127C2×D3111yescharacteristic
253#124—311no—
254#124—621no—
255C22⋊Q8—311no—
256C22⋊Q8—621no—
257#121—311no—
258#121—311no—
259#121—311no—
260#121—621no—
261#121—621no—
262#120—621no—
263#120—621no—
264C4×Q8C2×D3111yescharacteristic, complemented
265C4×D4—311no—
266C4×D4—311no—
267C4×D4—621no—
268#107—311no—
269C4⋊C8C2×D3111yescharacteristic, complemented
270#103—311no—
271#103—311no—
272#102—311no—
273#102—311no—
274#102C2×D3111yescharacteristic, complemented
275#102C2×D3111yescharacteristic, complemented
276C22⋊C8—311no—
277C22⋊C8—311no—
278#97—311no—
279C31#326511yescharacteristic, complemented, Sylow
280C24—621no—
281C24—621no—
282#41—622no—
283#41—1241no—
284C2×Q8—311no—
285C2×Q8—621no—
286C2×Q8—622no—
287C2×Q8—1241no—
288C2×Q8C22×D3111yescharacteristic
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—311no—
292C2×D4—311no—
293C2×D4—311no—
294C2×D4—621no—
295C2×D4—621no—
296C2×D4—621no—
297C2×D4—621no—
298C2×D4—622no—
299C2×D4—622no—
300C2×D4—621no—
301C2×D4—621no—
302C2×D4—1241no—
303C2×D4—1241no—
304C2×D4—1244no—
305C2×D4C22×D3111yescharacteristic
306C2×D4C22×D3111yescharacteristic
307C22×C4—311no—
308C22×C4—311no—
309C22×C4—621no—
310C22×C4—621no—
311C22×C4—621no—
312SD16—22no—
313SD16—22no—
314SD16—314no—
315SD16—622no—
316SD16—622no—
317SD16—1241no—
318D8—622no—
319D8—622no—
320D8—1241no—
321M16—622no—
322C2×C8—311no—
323C2×C8—311no—
324C2×C8—622no—
325C2×C8C22×D3111yescharacteristic
326C2×C8C22×D3111yescharacteristic
327C4⋊C4—21no—
328C4⋊C4—311no—
329C4⋊C4—311no—
330C4⋊C4—621no—
331C4⋊C4C22×D3111yescharacteristic
332C22⋊C4—311no—
333C22⋊C4—311no—
334C22⋊C4—621no—
335C22⋊C4—621no—
336C22⋊C4—621no—
337C22⋊C4—621no—
338C22⋊C4—621no—
339C22⋊C4—621no—
340C22⋊C4—621no—
341C42—21no—
342C42C22×D3111yescharacteristic
343C23—21no—
344C23—21no—
345C23—311no—
346C23—311no—
347C23—621no—
348C23—621no—
349C23—1242no—
350C23—1242no—
351C23—1242no—
352C23—1242no—
353C23—2481no—
354C23—2481no—
355Q8—21no—
356Q8—621no—
357Q8—621no—
358Q8—1242no—
359Q8D4×D3112yescomplemented
360D4—21no—
361D4—21no—
362D4—21no—
363D4—21no—
364D4—42no—
365D4—81no—
366D4—312no—
367D4—622no—
368D4—621no—
369D4—621no—
370D4—621no—
371D4—1242no—
372D4—1242no—
373D4—1242no—
374D4—1242no—
375D4—1242no—
376D4—1242no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—311no—
380C2×C4—311no—
381C2×C4—621no—
382C2×C4—621no—
383C2×C4—621no—
384C2×C4—621no—
385C2×C4—621no—
386C2×C4—621no—
387C2×C4—1242no—
388C2×C4—1242no—
389C2×C4—1242no—
390C2×C4D4×D3111yescharacteristic
391C2×C4D4×D3111yescharacteristic
392C2×C4C23×D3111yescharacteristic
393C8—21no—
394C8—21no—
395C8—312no—
396C8—621no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—312no—
404C22—621no—
405C22—621no—
406C22—621no—
407C22—622no—
408C22—621no—
409C22—621no—
410C22—1241no—
411C22—1241no—
412C22—1241no—
413C22—1241no—
414C22—1241no—
415C22—2484no—
416C22—2484no—
417C22C2×D4×D3111yescharacteristic
418C4—22no—
419C4—21no—
420C4—21no—
421C4—41no—
422C4—621no—
423C4—621no—
424C4—1241no—
425C4—1241no—
426C4C2×D4×D3111yescharacteristic
427C4#1175856611yescharacteristic
428C2—81no—
429C2—81no—
430C2—622no—
431C2—1241no—
432C2—1242no—
433C2—2481no—
434C2#1452168311yescharacteristic
435C2AGL(1,ℤ/8)×D3111yescharacteristic
436C2#1452185511yescharacteristic
subgroupG/H
centerC22C2×D4×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#460C2
socleC22×C31C22×D4
hypercenter#460D31
hypocenterC31#3265
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
1. C2×#11758491C2semidirect
2. C2×SD16×D31C2semidirect
3. #14521711C2semidirect
4. #14521684C2semidirect
5. #14521436C2semidirect
6. #14521294C2semidirect
7. #14521284C2semidirect
8. #14521265C2semidirect
9. #14521201C2semidirect
10. #14521193C2semidirect
11. #14521186C2semidirect
12. #14521184C2semidirect
13. #14521147C2semidirect
14. #14521119C2semidirect
15. C31×#460C2semidirect
16. C2×#92499C22semidirect
17. C2×Q8×D31C22semidirect
21. C2×SD16×C31C22semidirect
22. C2×SD16×C31C22semidirect
24. C4×Q8×C31C22semidirect
31. C2×#92480C22semidirect
32. C2×#92480C22semidirect
33. C2×#92479C22semidirect
34. C2×#92479C22semidirect
39. C2×#92469C22semidirect
40. C2×C8×D31C22semidirect
42. #11758474C22semidirect
44. #11758472C22semidirect
46. #11758452C22semidirect
50. C31×C4⋊C8C22semidirect
51. C31×#102C22semidirect
52. C31×#102C22semidirect
53. #11758423C22semidirect
54. #11758409C22semidirect
55. (C2×D31)⋊C8C22semidirect
56. (C2×D31)⋊C8C22semidirect
57. #11758404C22semidirect
59. #11758395C22semidirect
60. #11758391C22semidirect
103. #92481D4semidirect
137. Q8×C31C2×D4semidirect
177. C31⋊C8C2×D4semidirect
199. C2×D31AGL(1,ℤ/8)semidirect
218. #460D31semidirect
245. C2×SD16C2×D31semidirect
246. C2×SD16C2×D31semidirect
264. C4×Q8C2×D31semidirect
269. C4⋊C8C2×D31semidirect
274. #102C2×D31semidirect
275. #102C2×D31semidirect
279. C31#3265semidirect
359. Q8D4×D31semidirect

Maximal subgroups

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

subgroupindex
C2×#117584912
C2×SD16×D312
#145217112
#145216842
#145214362
#145212942
#145212842
#145212652
#145212012
#145211932
#145211862
#145211842
#145211472
#145211192
C31×#4602
#326531