Finite Group Databasebeta

← #16857211

Subgroups — #16857211

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×#11758485C211yesmaximal, characteristic, complemented
2C22×#92470C211yesmaximal, characteristic, complemented
3#14521708C211yesmaximal, characteristic, complemented
4D4×D124C211yesmaximal, characteristic, complemented
5#14521438C211yesmaximal, characteristic, complemented
6#14521295C211yesmaximal, characteristic, complemented
7#14521285C211yesmaximal, characteristic, complemented
8#14521265C211yesmaximal, characteristic, complemented
9#14521250C211yesmaximal, characteristic, complemented
10#14521202C211yesmaximal, characteristic, complemented
11#14521196C211yesmaximal, characteristic, complemented
12#14521190C211yesmaximal, characteristic, complemented
13#14521185C211yesmaximal, characteristic, complemented
14#14521151C211yesmaximal, characteristic, complemented
15C31×#435C211yesmaximal, characteristic, complemented
16C2×D4×D31—21no—
17C2×#92495C2211yescharacteristic, complemented
18C22×D124C2211yescharacteristic
19C22×D124C2211yescharacteristic
20C22×Dic62C2211yescharacteristic, complemented
21C2×M16×C31C2211yescharacteristic, complemented
22C22×C8×C31C2211yescharacteristic, complemented
23C4×D4×C31C2211yescharacteristic
24#11758516—21no—
25#11758507C2211yescharacteristic
26#11758507C2211yescharacteristic
27#11758507C2211yescharacteristic, complemented
28#11758501C2211yescharacteristic, complemented
29#11758485—24no—
30C2×D248C2211yescharacteristic
31C2×D248C2211yescharacteristic, complemented
32C2×D248C2211yescharacteristic, complemented
33C2×#92470—24no—
34C2×#92470C2211yescharacteristic
35C2×#92470C2211yescharacteristic
36C2×#92470C2211yescharacteristic, complemented
37C2×#92470C2211yescharacteristic, complemented
38C2×#92470C2211yescharacteristic, complemented
39C2×#92470C2211yescharacteristic, complemented
40#11758473—21no—
41#11758472—21no—
42#11758462—21no—
43#11758459—21no—
44#11758459—21no—
45(C22⋊D31)⋊C4—21no—
46#11758452C2211yescharacteristic, complemented
47#11758451C2211yescharacteristic
48C4×D124C2211yescharacteristic
49C4×Dic62C2211yescharacteristic, complemented
50C31×C4⋊C8C2211yescharacteristic, complemented
51C31×C22⋊C8C2211yescharacteristic, complemented
52C31×C22⋊C8C2211yescharacteristic, complemented
53C31×#97C2211yescharacteristic, complemented
54#11758409C2211yescharacteristic
55#11758409C2211yescharacteristic, complemented
56#11758409C2211yescharacteristic, complemented
57#11758409C2211yescharacteristic, complemented
58#11758406C2211yescharacteristic, complemented
59#11758404C2211yescharacteristic, complemented
60#11758404C2211yescharacteristic, complemented
61C23×D31—21no—
62C23×D31—21no—
63C2×D4×C31D411yescharacteristic
64C22×C4×C31C2311yescharacteristic
65C22×C4×C31C2311yescharacteristic
66C2×C22⋊D31—21no—
67C2×C22⋊D31—21no—
68C2×C22⋊D31—21no—
69C2×C22⋊D31—21no—
70C22×Dic31—21no—
71D4×D31—44no—
72#92495—22no—
73#92495—41no—
74C2×D124—22no—
75C2×D124—22no—
76C2×D124—21no—
77C2×D124—21no—
78C2×D124—21no—
79C2×D124—21no—
80C2×D124—41no—
81C2×D124—41no—
82C2×D124C2311yescharacteristic
83C2×D124C2311yescharacteristic
84C2×D124C2311yescharacteristic
85C2×D124C2311yescharacteristic
86C2×C4×D31—21no—
87C2×C4×D31—21no—
88C2×Dic62—22no—
89C2×Dic62—21no—
90C2×Dic62—41no—
91C2×Dic62C2311yescharacteristic
92C2×Dic62C2311yescharacteristic
93M16×C31—22no—
94C2×C8×C31—22no—
95C2×C8×C31C2311yescharacteristic
96C2×C8×C31C2311yescharacteristic
97C2×C8×C31C2311yescharacteristic
98C2×C8×C31C2311yescharacteristic
99C4⋊C4×C31D411yescharacteristic
100C22⋊C4×C31D411yescharacteristic
101C22⋊C4×C31D411yescharacteristic
102C42×C31C2311yescharacteristic
103C22⋊Dic31—21no—
104(C2×D31)⋊C4—21no—
105(C2×D31)⋊C4—21no—
106(C2×D31)⋊C4—21no—
107(C2×D31)⋊C4—21no—
108(C2×D31)⋊C4—21no—
109(C2×D31)⋊C4—21no—
110C4⋊Dic31C2311yescharacteristic
111C4⋊Dic31C2311yescharacteristic
112Dic31⋊C4—21no—
113Dic31⋊C4—21no—
114C4×Dic31—21no—
115D248—22no—
116D248—22no—
117D248—41no—
118#92470—22no—
119#92470—22no—
120#92470—22no—
121#92470—22no—
122#92470—41no—
123#92470D414yescomplemented
124C23×C31C2×D411yescharacteristic
125C23×C31C2×D411yescharacteristic
126C22×D31—21no—
127C22×D31—21no—
128C22×D31—21no—
129C22×D31—21no—
130C22×D31—42no—
131C22×D31—42no—
132C22×D31—42no—
133C22×D31—42no—
134C22×D31—81no—
135C22×D31—81no—
136D4×C31—22no—
137C2×C4×C31—21no—
138C2×C4×C31—21no—
139C2×C4×C31—21no—
140C2×C4×C31—21no—
141C2×C4×C31C2×D411yescharacteristic
142C2×C4×C31C2×D411yescharacteristic
143C2×C4×C31C2×D411yescharacteristic
144C2×C4×C31C2×D411yescharacteristic
145C2×C4×C31C2411yescharacteristic
146C22⋊D31—42no—
147C22⋊D31—42no—
148C22⋊D31—42no—
149C22⋊D31—42no—
150C2×Dic31—21no—
151C2×Dic31—21no—
152C2×Dic31—21no—
153C2×Dic31—21no—
154C2×Dic31—42no—
155D124—21no—
156D124—21no—
157D124—21no—
158D124—21no—
159D124—21no—
160D124—21no—
161D124—21no—
162D124—42no—
163D124—42no—
164D124—42no—
165D124—81no—
166D124C2×D412yes—
167C4×D31—42no—
168C4×D31—42no—
169Dic62—21no—
170Dic62—21no—
171Dic62—21no—
172Dic62—42no—
173Dic62C2×D412yescomplemented
174C8×C31—21no—
175C8×C31—21no—
176C8×C31—21no—
177C8×C31C2×D412yescomplemented
178#3265—311nomaximal, Sylow
179C22×C31—21no—
180C22×C31—21no—
181C22×C31—21no—
182C22×C31—21no—
183C22×C31—21no—
184C22×C31AGL(1,ℤ/8)12yescomplemented
185C22×C31C22×D411yescharacteristic
186C2×D31—22no—
187C2×D31—41no—
188C2×D31—41no—
189C2×D31—41no—
190C2×D31—41no—
191C2×D31—41no—
192C2×D31—41no—
193C2×D31—41no—
194C2×D31—41no—
195C2×D31—41no—
196C2×D31—41no—
197C2×D31—41no—
198C2×D31—84no—
199C2×D31—84no—
200C4×C31—21no—
201C4×C31—21no—
202C4×C31—21no—
203C4×C31—21no—
204C4×C31C22×D411yescharacteristic
205C4×C31H(5,𝔽2)11yescharacteristic
206Dic31—22no—
207Dic31—41no—
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#460—311no—
219#450—311no—
220#449—311no—
221#448—311no—
222#447—311no—
223#440—311no—
224#435D3111yescharacteristic, complemented
225C2×C31—22no—
226C2×C31—41no—
227C2×C31D4211yescharacteristic
228C2×C31C2×AGL(1,ℤ/8)11yescharacteristic
229C2×C31#57711yescharacteristic
230D31—42no—
231D31—81no—
232D31—81no—
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—311no—
245C2×SD16—311no—
246C2×SD16—624no—
247C2×D8—311no—
248C2×D8—311no—
249C2×D8—311no—
250C2×M16C2×D3111yescharacteristic, complemented
251C22×C8C2×D3111yescharacteristic, complemented
252#127—311no—
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×Q8—311no—
265C4×D4—311no—
266C4×D4—621no—
267C4×D4C2×D3111yescharacteristic
268#107—311no—
269C4⋊C8C2×D3111yescharacteristic, complemented
270#103—311no—
271#103—311no—
272#102—311no—
273#102—311no—
274#102—311no—
275#102—311no—
276C22⋊C8C2×D3111yescharacteristic, complemented
277C22⋊C8C2×D3111yescharacteristic, complemented
278#97C2×D3111yescharacteristic, complemented
279C31#326511yescharacteristic, complemented, Sylow
280C24—621no—
281C24—621no—
282#41—622no—
283#41—1241no—
284C2×Q8—311no—
285C2×Q8—311no—
286C2×Q8—622no—
287C2×Q8—621no—
288C2×Q8—1241no—
289C2×D4—311no—
290C2×D4—311no—
291C2×D4—311no—
292C2×D4—311no—
293C2×D4—622no—
294C2×D4—622no—
295C2×D4—621no—
296C2×D4—621no—
297C2×D4—621no—
298C2×D4—621no—
299C2×D4—621no—
300C2×D4—621no—
301C2×D4—621no—
302C2×D4—621no—
303C2×D4—1241no—
304C2×D4—1241no—
305C2×D4—1244no—
306C2×D4D12411yescharacteristic
307C22×C4—621no—
308C22×C4—621no—
309C22×C4—621no—
310C22×C4C22×D3111yescharacteristic
311C22×C4C22×D3111yescharacteristic
312SD16—314no—
313SD16—622no—
314SD16—622no—
315SD16—622no—
316SD16—622no—
317SD16—1241no—
318D8—622no—
319D8—622no—
320D8—1241no—
321M16—22no—
322C2×C8—22no—
323C2×C8C22×D3111yescharacteristic
324C2×C8C22×D3111yescharacteristic
325C2×C8C22×D3111yescharacteristic
326C2×C8C22×D3111yescharacteristic
327C4⋊C4—311no—
328C4⋊C4—311no—
329C4⋊C4—621no—
330C4⋊C4—621no—
331C4⋊C4D12411yescharacteristic
332C22⋊C4—621no—
333C22⋊C4—621no—
334C22⋊C4—621no—
335C22⋊C4—621no—
336C22⋊C4—621no—
337C22⋊C4—621no—
338C22⋊C4—621no—
339C22⋊C4D12411yescharacteristic
340C22⋊C4D12411yescharacteristic
341C42—621no—
342C42C22×D3111yescharacteristic
343C23—621no—
344C23—621no—
345C23—621no—
346C23—621no—
347C23—1242no—
348C23—1242no—
349C23—1242no—
350C23—1242no—
351C23—2481no—
352C23—2481no—
353C23C2×D12411yescharacteristic
354C23C2×D12411yescharacteristic
355Q8—312no—
356Q8—621no—
357Q8—621no—
358Q8—621no—
359Q8—1242no—
360D4—22no—
361D4—312no—
362D4—621no—
363D4—621no—
364D4—621no—
365D4—621no—
366D4—621no—
367D4—621no—
368D4—621no—
369D4—1242no—
370D4—1242no—
371D4—1242no—
372D4—1242no—
373D4—1242no—
374D4—1242no—
375D4—1242no—
376D4—2481no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—621no—
382C2×C4—621no—
383C2×C4—621no—
384C2×C4—621no—
385C2×C4—1242no—
386C2×C4—1242no—
387C2×C4—1242no—
388C2×C4C2×D12411yescharacteristic
389C2×C4C2×D12411yescharacteristic
390C2×C4C2×D12411yescharacteristic
391C2×C4C2×D12411yescharacteristic
392C2×C4C23×D3111yescharacteristic
393C8—21no—
394C8—21no—
395C8—21no—
396C8D4×D3112yescomplemented
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—622no—
403C22—1241no—
404C22—1241no—
405C22—1241no—
406C22—1241no—
407C22—1241no—
408C22—1241no—
409C22—1241no—
410C22—1241no—
411C22—1241no—
412C22—1241no—
413C22—1241no—
414C22—2484no—
415C22—2484no—
416C22#1175848512yescomplemented
417C22C22×D12411yescharacteristic
418C4—21no—
419C4—21no—
420C4—21no—
421C4—21no—
422C4—622no—
423C4—1241no—
424C4—1241no—
425C4—1241no—
426C4C2×D4×D3111yescharacteristic
427C4#1175856611yescharacteristic
428C2—22no—
429C2—41no—
430C2—1242no—
431C2—2481no—
432C2—2481no—
433C2—2481no—
434C2D4×D12411yescharacteristic
435C2C2×#1175848511yescharacteristic
436C2#1452186111yescharacteristic
subgroupG/H
centerC22C22×D124
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#435C2
socleC22×C31C22×D4
hypercenter#435D31
hypocenterC31#3265
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C4×D4 ◁ #435

Complements

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

NH ≅ G/Ntype
1. C2×#11758485C2semidirect
2. C22×#92470C2semidirect
3. #14521708C2semidirect
4. D4×D124C2semidirect
5. #14521438C2semidirect
6. #14521295C2semidirect
7. #14521285C2semidirect
8. #14521265C2semidirect
9. #14521250C2semidirect
10. #14521202C2semidirect
11. #14521196C2semidirect
12. #14521190C2semidirect
13. #14521185C2semidirect
14. #14521151C2semidirect
15. C31×#435C2semidirect
17. C2×#92495C22semidirect
20. C22×Dic62C22semidirect
21. C2×M16×C31C22semidirect
22. C22×C8×C31C22semidirect
27. #11758507C22semidirect
28. #11758501C22semidirect
31. C2×D248C22semidirect
32. C2×D248C22semidirect
36. C2×#92470C22semidirect
37. C2×#92470C22semidirect
38. C2×#92470C22semidirect
39. C2×#92470C22semidirect
46. #11758452C22semidirect
49. C4×Dic62C22semidirect
50. C31×C4⋊C8C22semidirect
51. C31×C22⋊C8C22semidirect
52. C31×C22⋊C8C22semidirect
53. C31×#97C22semidirect
55. #11758409C22semidirect
56. #11758409C22semidirect
57. #11758409C22semidirect
58. #11758406C22semidirect
59. #11758404C22semidirect
60. #11758404C22semidirect
123. #92470D4semidirect
173. Dic62C2×D4semidirect
177. C8×C31C2×D4semidirect
184. C22×C31AGL(1,ℤ/8)semidirect
224. #435D31semidirect
250. C2×M16C2×D31semidirect
251. C22×C8C2×D31semidirect
269. C4⋊C8C2×D31semidirect
276. C22⋊C8C2×D31semidirect
277. C22⋊C8C2×D31semidirect
278. #97C2×D31semidirect
279. C31#3265semidirect
396. C8D4×D31semidirect
416. C22#11758485semidirect

Maximal subgroups

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

subgroupindex
C2×#117584852
C22×#924702
#145217082
D4×D1242
#145214382
#145212952
#145212852
#145212652
#145212502
#145212022
#145211962
#145211902
#145211852
#145211512
C31×#4352
#326531