Finite Group Databasebeta

← #16857209

Subgroups — #16857209

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×#11758526C211yesmaximal, characteristic, complemented
2C2×SD16×D31C211yesmaximal, characteristic, complemented
3#14521706C211yesmaximal, characteristic, complemented
4#14521683C211yesmaximal, characteristic, complemented
5#14521437C211yesmaximal, characteristic, complemented
6#14521293C211yesmaximal, characteristic, complemented
7#14521284C211yesmaximal, characteristic, complemented
8#14521263C211yesmaximal, characteristic, complemented
9#14521247C211yesmaximal, characteristic, complemented
10#14521202C211yesmaximal, characteristic, complemented
11#14521189C211yesmaximal, characteristic, complemented
12#14521181C211yesmaximal, characteristic, complemented
13#14521152C211yesmaximal, characteristic, complemented
14#14521123C211yesmaximal, characteristic, complemented
15C31×#449C211yesmaximal, characteristic, complemented
16C2×#41×C31C2211yescharacteristic, complemented
17C2×Q8×D31C2211yescharacteristic, complemented
18C2×D4×D31—21no—
19C2×D4×D31C2211yescharacteristic
20C22×D124C2211yescharacteristic
21C2×SD16×C31C2211yescharacteristic, complemented
22C2×D8×C31C2211yescharacteristic, complemented
23C31×#121C2211yescharacteristic
24#11758526—24no—
25Q8×Dic31C2211yescharacteristic, complemented
26C2×#92481C2211yescharacteristic
27C2×#92481C2211yescharacteristic, complemented
28#11758517C2211yescharacteristic
29#11758515—21no—
30#11758515C2211yescharacteristic, complemented
31#11758514—21no—
32#11758513—21no—
33#11758513C2211yescharacteristic, complemented
34C2×#92480C2211yescharacteristic, complemented
35C2×#92479C2211yescharacteristic
36C2×#92479C2211yescharacteristic, complemented
37C4×C22⋊D31—21no—
38C4×C22⋊D31C2211yescharacteristic
39#11758501—21no—
40#11758501C2211yescharacteristic, complemented
41C2×#92474C2211yescharacteristic, complemented
42SD16×D31—24no—
43C2×#92470C2211yescharacteristic
44C2×#92470C2211yescharacteristic, complemented
45C2×C8×D31C2211yescharacteristic, complemented
46#11758472C2211yescharacteristic
47#11758470C2211yescharacteristic
48#11758461—21no—
49#11758459—21no—
50C31×#103C2211yescharacteristic, complemented
51C31×#102C2211yescharacteristic, complemented
52C31×C22⋊C8C2211yescharacteristic, complemented
53#11758420C2211yescharacteristic, complemented
54#11758409C2211yescharacteristic
55#11758409C2211yescharacteristic, complemented
56(C2×D31)⋊C8C2211yescharacteristic, complemented
57#11758403C2211yescharacteristic, complemented
58Dic31⋊C8C2211yescharacteristic, complemented
59#11758398C2211yescharacteristic, complemented
60#11758395C2211yescharacteristic, complemented
61C23×D31—21no—
62C23×D31—21no—
63#41×C31—22no—
64#41×C31—41no—
65C2×Q8×C31C2311yescharacteristic
66C2×D4×C31—21no—
67C2×D4×C31—21no—
68C2×D4×C31C2311yescharacteristic
69C2×D4×C31C2311yescharacteristic
70C22×C4×C31—21no—
71C22×C4×C31C2311yescharacteristic
72C2×C22⋊D31—21no—
73C2×C22⋊D31—21no—
74C2×C22⋊D31—21no—
75C2×C22⋊D31—21no—
76C2×C22⋊D31D411yescharacteristic
77Q8×D31—22no—
78Q8×D31—41no—
79D4×D31—22no—
80D4×D31—41no—
81D4×D31—44no—
82C2×D124—22no—
83C2×D124—21no—
84C2×D124—21no—
85C2×D124—41no—
86C2×D124C2311yescharacteristic
87C2×D124C2311yescharacteristic
88C2×C4×D31—21no—
89C2×C4×D31—21no—
90C2×C4×D31C2311yescharacteristic
91C2×Dic62—21no—
92C2×Dic62C2311yescharacteristic
93SD16×C31—22no—
94D8×C31—41no—
95C2×C8×C31C2311yescharacteristic
96C2×C8×C31C2311yescharacteristic
97C4⋊C4×C31C2311yescharacteristic
98C22⋊C4×C31—21no—
99C22⋊Dic31—21no—
100C22⋊Dic31—21no—
101C22⋊Dic31—21no—
102C22⋊Dic31—21no—
103C22⋊Dic31D411yescharacteristic
104#92481—22no—
105#92481D414yescomplemented
106#92480—22no—
107#92479—22no—
108#92479—22no—
109(C2×D31)⋊C4—21no—
110(C2×D31)⋊C4—21no—
111(C2×D31)⋊C4D411yescharacteristic
112C4⋊Dic31—21no—
113C4⋊Dic31C2311yescharacteristic
114Dic31⋊C4—21no—
115Dic31⋊C4D411yescharacteristic
116C4×Dic31—21no—
117C4×Dic31C2311yescharacteristic
118#92474—22no—
119C2×C31⋊C8C2311yescharacteristic
120C2×C31⋊C8C2311yescharacteristic
121#92470—22no—
122#92470—41no—
123C8×D31—22no—
124C23×C31—21no—
125C23×C31—21no—
126C23×C31C2×D411yescharacteristic
127C22×D31—21no—
128C22×D31—21no—
129C22×D31—42no—
130C22×D31—42no—
131C22×D31—42no—
132C22×D31—42no—
133C22×D31—81no—
134C22×D31—81no—
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—42no—
144C2×C4×C31—21no—
145C2×C4×C31—21no—
146C2×C4×C31—21no—
147C2×C4×C31—21no—
148C2×C4×C31—42no—
149C2×C4×C31C2×D411yescharacteristic
150C2×C4×C31C2411yescharacteristic
151C22⋊D31—22no—
152C22⋊D31—42no—
153C22⋊D31—42no—
154C22⋊D31—42no—
155C22⋊D31—81no—
156C2×Dic31—21no—
157C2×Dic31—21no—
158C2×Dic31C2×D411yescharacteristic
159C2×Dic31C2×D411yescharacteristic
160C2×Dic31C2×D411yescharacteristic
161D124—21no—
162D124—21no—
163D124—21no—
164D124—42no—
165D124—42no—
166D124C2×D412yes—
167C4×D31—21no—
168C4×D31—21no—
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—21no—
180C22×C31—21no—
181C22×C31—21no—
182C22×C31—41no—
183C22×C31—41no—
184C22×C31—41no—
185C22×C31—41no—
186C22×C31—41no—
187C22×C31—41no—
188C22×C31C22×D411yescharacteristic
189C2×D31—21no—
190C2×D31—21no—
191C2×D31—22no—
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—41no—
203C4×C31C22×D411yescharacteristic
204C4×C31H(5,𝔽2)11yescharacteristic
205Dic31—21no—
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#460—311no—
219#450—311no—
220#449D3111yescharacteristic, complemented
221#448—311no—
222#447—311no—
223#440—311no—
224#435—311no—
225C2×C31—41no—
226C2×C31—81no—
227C2×C31—81no—
228C2×C31D4211yescharacteristic
229C2×C31C2×AGL(1,ℤ/8)11yescharacteristic
230C2×C31#57711yescharacteristic
231D31—22no—
232D31—42no—
233D31—81no—
234C2×#41C2×D3111yescharacteristic, complemented
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—624no—
246C2×SD16C2×D3111yescharacteristic, complemented
247C2×D8—311no—
248C2×D8—311no—
249C2×D8C2×D3111yescharacteristic, complemented
250C2×M16—311no—
251C22×C8—311no—
252#127—311no—
253#124—311no—
254#124—621no—
255C22⋊Q8—311no—
256C22⋊Q8—621no—
257#121—311no—
258#121—311no—
259#121—621no—
260#121—621no—
261#121C2×D3111yescharacteristic
262#120—621no—
263#120—621no—
264C4×Q8—311no—
265C4×D4—311no—
266C4×D4—311no—
267C4×D4—621no—
268#107—311no—
269C4⋊C8—311no—
270#103—311no—
271#103C2×D3111yescharacteristic, complemented
272#102—311no—
273#102—311no—
274#102—311no—
275#102C2×D3111yescharacteristic, complemented
276C22⋊C8—311no—
277C22⋊C8C2×D3111yescharacteristic, complemented
278#97—311no—
279C31#326511yescharacteristic, complemented, Sylow
280C24—621no—
281C24—621no—
282#41—22no—
283#41—41no—
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—622no—
295C2×D4—621no—
296C2×D4—621no—
297C2×D4—621no—
298C2×D4—621no—
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—21no—
308C22×C4—311no—
309C22×C4—621no—
310C22×C4—621no—
311C22×C4C22×D3111yescharacteristic
312SD16—22no—
313SD16—314no—
314SD16—622no—
315SD16—622no—
316SD16—622no—
317SD16—1241no—
318D8—41no—
319D8—622no—
320D8—622no—
321M16—622no—
322C2×C8—311no—
323C2×C8—311no—
324C2×C8—622no—
325C2×C8C22×D3111yescharacteristic
326C2×C8C22×D3111yescharacteristic
327C4⋊C4—311no—
328C4⋊C4—311no—
329C4⋊C4—621no—
330C4⋊C4—621no—
331C4⋊C4C22×D3111yescharacteristic
332C22⋊C4—21no—
333C22⋊C4—311no—
334C22⋊C4—311no—
335C22⋊C4—621no—
336C22⋊C4—621no—
337C22⋊C4—621no—
338C22⋊C4—621no—
339C22⋊C4—621no—
340C22⋊C4—621no—
341C42—311no—
342C42—621no—
343C23—21no—
344C23—21no—
345C23—311no—
346C23—621no—
347C23—621no—
348C23—1242no—
349C23—1242no—
350C23—1242no—
351C23—1242no—
352C23—2481no—
353C23—2481no—
354C23D4×D3111yescharacteristic
355Q8—21no—
356Q8—621no—
357Q8—621no—
358Q8—1242no—
359Q8D4×D3112yescomplemented
360D4—21no—
361D4—21no—
362D4—21no—
363D4—21no—
364D4—42no—
365D4—42no—
366D4—312no—
367D4—622no—
368D4—621no—
369D4—621no—
370D4—621no—
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—42no—
382C2×C4—311no—
383C2×C4—311no—
384C2×C4—311no—
385C2×C4—621no—
386C2×C4—621no—
387C2×C4—621no—
388C2×C4—621no—
389C2×C4—1242no—
390C2×C4—1242no—
391C2×C4D4×D3111yescharacteristic
392C2×C4C23×D3111yescharacteristic
393C8—21no—
394C8—21no—
395C8—312no—
396C8—621no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—312no—
407C22—621no—
408C22—621no—
409C22—622no—
410C22—1241no—
411C22—1241no—
412C22—1241no—
413C22—1241no—
414C22—1241no—
415C22—2484no—
416C22—2484no—
417C22C2×D4×D3111yescharacteristic
418C4—21no—
419C4—22no—
420C4—41no—
421C4—621no—
422C4—621no—
423C4—621no—
424C4—1241no—
425C4—1241no—
426C4C2×D4×D3111yescharacteristic
427C4#1175856111yescharacteristic
428C2—41no—
429C2—81no—
430C2—81no—
431C2—622no—
432C2—1242no—
433C2—2481no—
434C2#1452168111yescharacteristic
435C2AGL(1,ℤ/8)×D3111yescharacteristic
436C2#1452185611yescharacteristic
subgroupG/H
centerC22C2×D4×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#449C2
socleC22×C31C22×D4
hypercenter#449D31
hypocenterC31#3265
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. C2×#11758526C2semidirect
2. C2×SD16×D31C2semidirect
3. #14521706C2semidirect
4. #14521683C2semidirect
5. #14521437C2semidirect
6. #14521293C2semidirect
7. #14521284C2semidirect
8. #14521263C2semidirect
9. #14521247C2semidirect
10. #14521202C2semidirect
11. #14521189C2semidirect
12. #14521181C2semidirect
13. #14521152C2semidirect
14. #14521123C2semidirect
15. C31×#449C2semidirect
16. C2×#41×C31C22semidirect
17. C2×Q8×D31C22semidirect
21. C2×SD16×C31C22semidirect
22. C2×D8×C31C22semidirect
25. Q8×Dic31C22semidirect
27. C2×#92481C22semidirect
30. #11758515C22semidirect
33. #11758513C22semidirect
34. C2×#92480C22semidirect
36. C2×#92479C22semidirect
40. #11758501C22semidirect
41. C2×#92474C22semidirect
44. C2×#92470C22semidirect
45. C2×C8×D31C22semidirect
50. C31×#103C22semidirect
51. C31×#102C22semidirect
52. C31×C22⋊C8C22semidirect
53. #11758420C22semidirect
55. #11758409C22semidirect
56. (C2×D31)⋊C8C22semidirect
57. #11758403C22semidirect
58. Dic31⋊C8C22semidirect
59. #11758398C22semidirect
60. #11758395C22semidirect
105. #92481D4semidirect
137. Q8×C31C2×D4semidirect
177. C31⋊C8C2×D4semidirect
199. C2×D31AGL(1,ℤ/8)semidirect
220. #449D31semidirect
234. C2×#41C2×D31semidirect
246. C2×SD16C2×D31semidirect
249. C2×D8C2×D31semidirect
271. #103C2×D31semidirect
275. #102C2×D31semidirect
277. C22⋊C8C2×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×#117585262
C2×SD16×D312
#145217062
#145216832
#145214372
#145212932
#145212842
#145212632
#145212472
#145212022
#145211892
#145211812
#145211522
#145211232
C31×#4492
#326531