Finite Group Databasebeta

← #16857199

Subgroups — #16857199

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
2C22×#92481C211yesmaximal, characteristic, complemented
3#14521710C211yesmaximal, characteristic, complemented
4#14521683C211yesmaximal, characteristic, complemented
5#14521435C211yesmaximal, characteristic, complemented
6#14521298C211yesmaximal, characteristic, complemented
7#14521283C211yesmaximal, characteristic, complemented
8#14521267C211yesmaximal, characteristic, complemented
9#14521247C211yesmaximal, characteristic, complemented
10#14521198C211yesmaximal, characteristic, complemented
11#14521197C211yesmaximal, characteristic, complemented
12#14521185C211yesmaximal, characteristic, complemented
13#14521152C211yesmaximal, characteristic, complemented
14#14521117C211yesmaximal, characteristic, complemented
15C31×#448C211yesmaximal, characteristic, complemented
16C22×Q8×C31C2211yescharacteristic, complemented
17C2×#92499C2211yescharacteristic, complemented
18C2×D4×D31—21no—
19C2×D4×D31C2211yescharacteristic
20C22×D124C2211yescharacteristic
21C2×SD16×C31C2211yescharacteristic, complemented
22C2×SD16×C31C2211yescharacteristic, complemented
23C31×#121C2211yescharacteristic
24#11758524—21no—
25#11758524C2211yescharacteristic, complemented
26#11758523—21no—
27#11758523C2211yescharacteristic, complemented
28Q8×Dic31C2211yescharacteristic, complemented
29C2×#92481—24no—
30C2×#92481C2211yescharacteristic
31C2×#92481C2211yescharacteristic
32C2×#92481C2211yescharacteristic, complemented
33C2×#92481C2211yescharacteristic, complemented
34#11758517C2211yescharacteristic
35#11758514—21no—
36C2×#92479C2211yescharacteristic, complemented
37#11758507—21no—
38#11758507C2211yescharacteristic, complemented
39C4×C22⋊D31—21no—
40C4×C22⋊D31C2211yescharacteristic
41C22×C31⋊C8C2211yescharacteristic, complemented
42#11758491—24no—
43C2×D248C2211yescharacteristic
44C2×D248C2211yescharacteristic, complemented
45C2×#92469C2211yescharacteristic, complemented
46#11758472C2211yescharacteristic
47#11758470C2211yescharacteristic
48#11758461—21no—
49#11758459—21no—
50C31×#103C2211yescharacteristic, complemented
51C31×#103C2211yescharacteristic, 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#11758397C2211yescharacteristic, complemented
60#11758395C2211yescharacteristic, complemented
61C23×D31—21no—
62C23×D31—21no—
63C2×Q8×C31—21no—
64C2×Q8×C31—22no—
65C2×Q8×C31—41no—
66C2×Q8×C31C2311yescharacteristic
67C2×Q8×C31C2311yescharacteristic
68C2×D4×C31—21no—
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
77#92499—22no—
78#92499—41no—
79D4×D31—22no—
80D4×D31—41no—
81D4×D31—44no—
82C2×D124—22no—
83C2×D124—21no—
84C2×D124—21no—
85C2×D124—21no—
86C2×D124—41no—
87C2×D124C2311yescharacteristic
88C2×D124C2311yescharacteristic
89C2×D124C2311yescharacteristic
90C2×C4×D31—21no—
91C2×C4×D31—21no—
92C2×C4×D31C2311yescharacteristic
93SD16×C31—22no—
94SD16×C31—41no—
95C2×C8×C31C2311yescharacteristic
96C2×C8×C31C2311yescharacteristic
97C4⋊C4×C31C2311yescharacteristic
98C22⋊C4×C31—21no—
99C22⋊Dic31D411yescharacteristic
100#92481—22no—
101#92481—22no—
102#92481—22no—
103#92481D414yescomplemented
104#92479—22no—
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—
110(C2×D31)⋊C4—21no—
111(C2×D31)⋊C4D411yescharacteristic
112C4⋊Dic31—21no—
113C4⋊Dic31C2311yescharacteristic
114Dic31⋊C4—21no—
115Dic31⋊C4D411yescharacteristic
116C4×Dic31—21no—
117C4×Dic31C2311yescharacteristic
118C2×C31⋊C8—22no—
119C2×C31⋊C8C2311yescharacteristic
120C2×C31⋊C8C2311yescharacteristic
121D248—22no—
122D248—41no—
123#92469—22no—
124C23×C31—21no—
125C23×C31C2×D411yescharacteristic
126C22×D31—21no—
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×C31—21no—
138Q8×C31—21no—
139Q8×C31—42no—
140Q8×C31C2×D412yescomplemented
141D4×C31—21no—
142D4×C31—21no—
143D4×C31—42no—
144C2×C4×C31—21no—
145C2×C4×C31—21no—
146C2×C4×C31—21no—
147C2×C4×C31—21no—
148C2×C4×C31—21no—
149C2×C4×C31—42no—
150C2×C4×C31C2×D411yescharacteristic
151C2×C4×C31C2411yescharacteristic
152C22⋊D31—22no—
153C22⋊D31—42no—
154C22⋊D31—42no—
155C22⋊D31—42no—
156C22⋊D31—81no—
157C2×Dic31—21no—
158C2×Dic31C2×D411yescharacteristic
159C2×Dic31C2×D411yescharacteristic
160C2×Dic31C2×D411yescharacteristic
161D124—21no—
162D124—21no—
163D124—21no—
164D124—21no—
165D124—21no—
166D124—42no—
167D124—42no—
168D124—42no—
169D124C2×D412yes—
170C4×D31—21no—
171C4×D31—21no—
172C4×D31—42no—
173C4×D31—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—41no—
182C22×C31—41no—
183C22×C31—41no—
184C22×C31AGL(1,ℤ/8)12yescomplemented
185C22×C31C22×D411yescharacteristic
186C2×D31—21no—
187C2×D31—22no—
188C2×D31—21no—
189C2×D31—21no—
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—22no—
201C4×C31—21no—
202C4×C31—41no—
203C4×C31—41no—
204C4×C31C22×D411yescharacteristic
205C4×C31H(5,𝔽2)11yescharacteristic
206Dic31—21no—
207Dic31—21no—
208Dic31—21no—
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#448D3111yescharacteristic, complemented
222#447—311no—
223#440—311no—
224#435—311no—
225C2×C31—22no—
226C2×C31—81no—
227C2×C31D4211yescharacteristic
228C2×C31C2×AGL(1,ℤ/8)11yescharacteristic
229C2×C31#57711yescharacteristic
230D31—41no—
231D31—42no—
232D31—81no—
233D31—81no—
234C2×#41—311no—
235C22×Q8C2×D3111yescharacteristic, complemented
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#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#103C2×D3111yescharacteristic, complemented
271#103C2×D3111yescharacteristic, complemented
272#102—311no—
273#102—311no—
274#102—311no—
275#102—311no—
276C22⋊C8—311no—
277C22⋊C8C2×D3111yescharacteristic, complemented
278#97—311no—
279C31#326511yescharacteristic, complemented, Sylow
280C24—621no—
281C24—621no—
282#41—622no—
283#41—1241no—
284C2×Q8—21no—
285C2×Q8—22no—
286C2×Q8—41no—
287C2×Q8C22×D3111yescharacteristic
288C2×Q8C22×D3111yescharacteristic
289C2×D4—21no—
290C2×D4—311no—
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—621no—
300C2×D4—622no—
301C2×D4—621no—
302C2×D4—621no—
303C2×D4—1241no—
304C2×D4—1241no—
305C2×D4—1244no—
306C2×D4C22×D3111yescharacteristic
307C22×C4—21no—
308C22×C4—311no—
309C22×C4—621no—
310C22×C4—621no—
311C22×C4C22×D3111yescharacteristic
312SD16—22no—
313SD16—41no—
314SD16—314no—
315SD16—622no—
316SD16—622no—
317SD16—622no—
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—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—311no—
345C23—621no—
346C23—621no—
347C23—621no—
348C23—1242no—
349C23—1242no—
350C23—1242no—
351C23—1242no—
352C23—2481no—
353C23—2481no—
354C23D4×D3111yescharacteristic
355Q8—21no—
356Q8—21no—
357Q8—21no—
358Q8—42no—
359Q8D4×D3112yescomplemented
360D4—21no—
361D4—21no—
362D4—42no—
363D4—312no—
364D4—622no—
365D4—621no—
366D4—621no—
367D4—621no—
368D4—621no—
369D4—621no—
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—21no—
382C2×C4—42no—
383C2×C4—311no—
384C2×C4—311no—
385C2×C4—311no—
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—41no—
400C22—41no—
401C22—41no—
402C22—621no—
403C22—621no—
404C22—622no—
405C22—621no—
406C22—1241no—
407C22—1241no—
408C22—1241no—
409C22—1241no—
410C22—1241no—
411C22—1241no—
412C22—1241no—
413C22—1241no—
414C22—2484no—
415C22—2484no—
416C22#1175849112yescomplemented
417C22C2×D4×D3111yescharacteristic
418C4—21no—
419C4—22no—
420C4—41no—
421C4—41no—
422C4—621no—
423C4—621no—
424C4—621no—
425C4—1241no—
426C4C2×D4×D3111yescharacteristic
427C4#1175856111yescharacteristic
428C2—22no—
429C2—81no—
430C2—1241no—
431C2—1242no—
432C2—2481no—
433C2—2481no—
434C2#1452168111yescharacteristic
435C2C2×#1175849111yescharacteristic
436C2#1452185811yescharacteristic
subgroupG/H
centerC22C2×D4×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#448C2
socleC22×C31C22×D4
hypercenter#448D31
hypocenterC31#3265
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
1. C2×#11758491C2semidirect
2. C22×#92481C2semidirect
3. #14521710C2semidirect
4. #14521683C2semidirect
5. #14521435C2semidirect
6. #14521298C2semidirect
7. #14521283C2semidirect
8. #14521267C2semidirect
9. #14521247C2semidirect
10. #14521198C2semidirect
11. #14521197C2semidirect
12. #14521185C2semidirect
13. #14521152C2semidirect
14. #14521117C2semidirect
15. C31×#448C2semidirect
16. C22×Q8×C31C22semidirect
17. C2×#92499C22semidirect
21. C2×SD16×C31C22semidirect
22. C2×SD16×C31C22semidirect
25. #11758524C22semidirect
27. #11758523C22semidirect
28. Q8×Dic31C22semidirect
32. C2×#92481C22semidirect
33. C2×#92481C22semidirect
36. C2×#92479C22semidirect
38. #11758507C22semidirect
41. C22×C31⋊C8C22semidirect
44. C2×D248C22semidirect
45. C2×#92469C22semidirect
50. C31×#103C22semidirect
51. C31×#103C22semidirect
52. C31×C22⋊C8C22semidirect
53. #11758420C22semidirect
55. #11758409C22semidirect
56. (C2×D31)⋊C8C22semidirect
57. #11758403C22semidirect
58. Dic31⋊C8C22semidirect
59. #11758397C22semidirect
60. #11758395C22semidirect
103. #92481D4semidirect
140. Q8×C31C2×D4semidirect
177. C31⋊C8C2×D4semidirect
184. C22×C31AGL(1,ℤ/8)semidirect
221. #448D31semidirect
235. C22×Q8C2×D31semidirect
245. C2×SD16C2×D31semidirect
246. C2×SD16C2×D31semidirect
270. #103C2×D31semidirect
271. #103C2×D31semidirect
277. C22⋊C8C2×D31semidirect
279. C31#3265semidirect
359. Q8D4×D31semidirect
416. C22#11758491semidirect

Maximal subgroups

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

subgroupindex
C2×#117584912
C22×#924812
#145217102
#145216832
#145214352
#145212982
#145212832
#145212672
#145212472
#145211982
#145211972
#145211852
#145211522
#145211172
C31×#4482
#326531