Finite Group Databasebeta

← #16857258

Subgroups — #16857258

Order 3968.

510 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11758485C211yesmaximal, characteristic, complemented
2C2×#11758493C211yesmaximal, characteristic, complemented
3#14521717C211yesmaximal, characteristic, complemented
4#14521690C211yesmaximal, characteristic, complemented
5#14521435C211yesmaximal, characteristic, complemented
6#14521286C211yesmaximal, characteristic, complemented
7#14521274C211yesmaximal, characteristic, complemented
8#14521243C211yesmaximal, characteristic, complemented
9#14521209C211yesmaximal, characteristic, complemented
10#14521197C211yesmaximal, characteristic, complemented
11#14521195C211yesmaximal, characteristic, complemented
12#14521182C211yesmaximal, characteristic, complemented
13#14521148C211yesmaximal, characteristic, complemented
14#14521118C211yesmaximal, characteristic, complemented
15C31×#468C211yesmaximal, characteristic, complemented
16C2×#92499C2211yescharacteristic
17C2×#92497C2211yescharacteristic, complemented
18C2×#92495C2211yescharacteristic
19C22×D124C2211yescharacteristic
20C2×SD16×C31C2211yescharacteristic, complemented
21C2×M16×C31C2211yescharacteristic
22C31×C22⋊Q8C2211yescharacteristic
23C31×#121C2211yescharacteristic
24#11758524C2211yescharacteristic
25C2×#92482C2211yescharacteristic, complemented
26C2×#92481C2211yescharacteristic, complemented
27#11758517—21no—
28#11758517C2211yescharacteristic, complemented
29#11758516—21no—
30C2×#92479C2211yescharacteristic
31C2×#92479C2211yescharacteristic
32C4×C22⋊D31C2211yescharacteristic
33#11758493—24no—
34#11758485—24no—
35C2×D248C2211yescharacteristic
36C2×D248C2211yescharacteristic
37C2×#92470C2211yescharacteristic
38C2×#92470C2211yescharacteristic
39C2×C8×D31C2211yescharacteristic, complemented
40#11758472—21no—
41#11758472C2211yescharacteristic
42#11758472C2211yescharacteristic, complemented
43#11758471—21no—
44#11758470C2211yescharacteristic
45#11758464C2211yescharacteristic, complemented
46#11758461—21no—
47#11758459—21no—
48(C22⋊D31)⋊C4—21no—
49C22⋊C4×D31—21no—
50C31×#107C2211yescharacteristic, complemented
51C31×#103C2211yescharacteristic, complemented
52C31×#102C2211yescharacteristic, complemented
53C22⋊(C31⋊C8)C2211yescharacteristic
54(C2×D31)⋊C8C2211yescharacteristic, complemented
55#11758403C2211yescharacteristic, complemented
56Dic31⋊C8C2211yescharacteristic, complemented
57#11758398C2211yescharacteristic, complemented
58#11758397C2211yescharacteristic
59#11758397C2211yescharacteristic
60#11758397C2211yescharacteristic, complemented
61C23×D31—21no—
62C2×Q8×C31C2311yescharacteristic
63C2×D4×C31—21no—
64C2×D4×C31C2311yescharacteristic
65C22×C4×C31C2311yescharacteristic
66C2×C22⋊D31—21no—
67C2×C22⋊D31—21no—
68C2×C22⋊D31—21no—
69C2×C22⋊D31—21no—
70C2×C22⋊D31D411yescharacteristic
71C22×Dic31—21no—
72#92499—22no—
73#92499—41no—
74#92497—22no—
75#92497—41no—
76#92495—22no—
77#92495—41no—
78C2×D124—22no—
79C2×D124—21no—
80C2×D124—21no—
81C2×D124—21no—
82C2×D124—21no—
83C2×D124—41no—
84C2×D124C2311yescharacteristic
85C2×D124C2311yescharacteristic
86C2×D124C2311yescharacteristic
87C2×C4×D31—21no—
88C2×C4×D31—21no—
89C2×C4×D31—21no—
90C2×C4×D31C2311yescharacteristic
91C2×Dic62C2311yescharacteristic
92SD16×C31—22no—
93M16×C31—22no—
94C2×C8×C31C2311yescharacteristic
95C2×C8×C31C2311yescharacteristic
96C4⋊C4×C31—21no—
97C4⋊C4×C31C2311yescharacteristic
98C4⋊C4×C31C2311yescharacteristic
99C22⋊C4×C31—21no—
100C22⋊C4×C31—21no—
101C22⋊Dic31D411yescharacteristic
102#92482—22no—
103#92481—41no—
104#92479—22no—
105#92479—41no—
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
112Dic31⋊C4—21no—
113Dic31⋊C4D411yescharacteristic
114C4×Dic31—21no—
115C4×Dic31C2311yescharacteristic
116C2×C31⋊C8C2311yescharacteristic
117C2×C31⋊C8C2311yescharacteristic
118D248—22no—
119D248—22no—
120#92470—22no—
121#92470D414yescomplemented
122C8×D31—22no—
123C23×C31—21no—
124C23×C31C2×D411yescharacteristic
125C22×D31—21no—
126C22×D31—21no—
127C22×D31—21no—
128C22×D31—42no—
129C22×D31—42no—
130C22×D31—81no—
131C22×D31C2×D411yescharacteristic
132Q8×C31—21no—
133Q8×C31—21no—
134D4×C31—21no—
135D4×C31—21no—
136D4×C31—81no—
137C2×C4×C31—21no—
138C2×C4×C31—21no—
139C2×C4×C31—21no—
140C2×C4×C31—21no—
141C2×C4×C31—21no—
142C2×C4×C31C2×D411yescharacteristic
143C2×C4×C31C2411yescharacteristic
144C22⋊D31—41no—
145C22⋊D31—42no—
146C22⋊D31—42no—
147C22⋊D31—42no—
148C22⋊D31—81no—
149C2×Dic31—21no—
150C2×Dic31—42no—
151C2×Dic31C2×D411yescharacteristic
152C2×Dic31C2×D411yescharacteristic
153C2×Dic31C2×D411yescharacteristic
154D124—21no—
155D124—21no—
156D124—21no—
157D124—21no—
158D124—21no—
159D124—42no—
160D124—42no—
161D124—81no—
162D124—81no—
163D124C2×D412yes—
164C4×D31—21no—
165C4×D31—21no—
166C4×D31—42no—
167C4×D31—42no—
168C4×D31—81no—
169Dic62—21no—
170Dic62C2×D412yes—
171C8×C31—21no—
172C8×C31C2×D412yes—
173C31⋊C8—21no—
174C31⋊C8—21no—
175#3268—311nomaximal, Sylow
176C22×C31—21no—
177C22×C31—21no—
178C22×C31—21no—
179C22×C31—41no—
180C22×C31—41no—
181C22×C31—41no—
182C22×C31C22×D411yescharacteristic
183C2×D31—21no—
184C2×D31—22no—
185C2×D31—21no—
186C2×D31—21no—
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—84no—
196C4×C31—21no—
197C4×C31—41no—
198C4×C31—41no—
199C4×C31—41no—
200C4×C31C22×D411yescharacteristic
201C4×C31H(5,𝔽2)11yescharacteristic
202Dic31—22no—
203Dic31—21no—
204Dic31—21no—
205Dic31—21no—
206C2×AGL(1,ℤ/8)—311no—
207C2×#135—311no—
208#550—311no—
209#546—311no—
210#496—311no—
211#468D3111yescharacteristic, complemented
212#466—311no—
213#464—311no—
214#459—311no—
215#452—311no—
216#449—311no—
217#449—311no—
218#447—311no—
219#440—311no—
220#435—311no—
221C2×C31—41no—
222C2×C31—81no—
223C2×C31D4211yescharacteristic
224C2×C31#57511yescharacteristic
225C2×C31#57611yescharacteristic
226D31—41no—
227D31—42no—
228D31—81no—
229D31—81no—
230C2×#41—311no—
231C2×#41—311no—
232C2×#41—311no—
233C22×D4—311no—
234AGL(1,ℤ/8)—624no—
235#135—624no—
236C2×Q16—311no—
237C2×SD16—311no—
238C2×SD16—311no—
239C2×SD16—311no—
240C2×SD16C2×D3111yescharacteristic, complemented
241C2×D8—311no—
242C2×D8—311no—
243C2×D8—311no—
244C2×D8—311no—
245C2×M16C2×D3111yescharacteristic
246C22×C8—311no—
247#127—311no—
248#127—621no—
249#124—311no—
250#123—621no—
251C22⋊Q8C2×D3111yescharacteristic
252#121—311no—
253#121—311no—
254#121—621no—
255#121—621no—
256#121—621no—
257#121C2×D3111yescharacteristic
258#120—621no—
259C4×Q8—311no—
260C4×D4—311no—
261C4×D4—311no—
262C4×D4—621no—
263C2×C22⋊C4—621no—
264#107C2×D3111yescharacteristic, complemented
265C4⋊C8—311no—
266#103—311no—
267#103C2×D3111yescharacteristic, complemented
268#102—311no—
269#102—311no—
270#102—311no—
271#102C2×D3111yescharacteristic, complemented
272C22⋊C8—311no—
273C22⋊C8—311no—
274#97—311no—
275C31#326811yescharacteristic, complemented, Sylow
276C24—621no—
277#41—622no—
278#41—622no—
279#41—622no—
280#41—1241no—
281#41—1241no—
282#41—1241no—
283C2×Q8—311no—
284C2×Q8C22×D3111yescharacteristic
285C2×D4—21no—
286C2×D4—311no—
287C2×D4—311no—
288C2×D4—311no—
289C2×D4—311no—
290C2×D4—622no—
291C2×D4—621no—
292C2×D4—621no—
293C2×D4—621no—
294C2×D4—621no—
295C2×D4—621no—
296C2×D4—621no—
297C2×D4—621no—
298C2×D4—621no—
299C2×D4—1241no—
300C2×D4C22×D3111yescharacteristic
301C22×C4—311no—
302C22×C4—621no—
303C22×C4—621no—
304C22×C4—621no—
305C22×C4—621no—
306C22×C4C22×D3111yescharacteristic
307Q16—622no—
308SD16—22no—
309SD16—314no—
310SD16—622no—
311SD16—1241no—
312D8—622no—
313D8—622no—
314D8—622no—
315D8—1241no—
316M16—22no—
317C2×C8—311no—
318C2×C8—311no—
319C2×C8—622no—
320C2×C8C22×D3111yescharacteristic
321C2×C8C22×D3111yescharacteristic
322C4⋊C4—21no—
323C4⋊C4—311no—
324C4⋊C4—621no—
325C4⋊C4C22×D3111yescharacteristic
326C4⋊C4C22×D3111yescharacteristic
327C22⋊C4—21no—
328C22⋊C4—21no—
329C22⋊C4—311no—
330C22⋊C4—311no—
331C22⋊C4—621no—
332C22⋊C4—621no—
333C22⋊C4—621no—
334C22⋊C4—621no—
335C22⋊C4—621no—
336C42—311no—
337C42—621no—
338C23—21no—
339C23—311no—
340C23—621no—
341C23—621no—
342C23—621no—
343C23—1242no—
344C23—1242no—
345C23—2481no—
346C23D4×D3111yescharacteristic
347Q8—21no—
348Q8—21no—
349Q8—312no—
350Q8—621no—
351D4—21no—
352D4—21no—
353D4—81no—
354D4—312no—
355D4—621no—
356D4—621no—
357D4—621no—
358D4—621no—
359D4—621no—
360D4—1242no—
361D4—1242no—
362D4—1241no—
363D4—1242no—
364D4—1242no—
365D4—1242no—
366D4—2481no—
367D4—2481no—
368D4—2481no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—311no—
375C2×C4—311no—
376C2×C4—311no—
377C2×C4—621no—
378C2×C4—621no—
379C2×C4—621no—
380C2×C4—1242no—
381C2×C4—1242no—
382C2×C4—1242no—
383C2×C4—2481no—
384C2×C4D4×D3111yescharacteristic
385C2×C4C23×D3111yescharacteristic
386C8—21no—
387C8—621no—
388C8—621no—
389C8D4×D3112yes—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—621no—
397C22—622no—
398C22—621no—
399C22—621no—
400C22—1241no—
401C22—1241no—
402C22—1241no—
403C22—1241no—
404C22—1241no—
405C22—1241no—
406C22—1241no—
407C22—1241no—
408C22—2484no—
409C22C2×D4×D3111yescharacteristic
410C4—21no—
411C4—41no—
412C4—41no—
413C4—41no—
414C4—621no—
415C4—622no—
416C4—621no—
417C4—621no—
418C4C2×D4×D3111yescharacteristic
419C4#1175856611yescharacteristic
420C2—41no—
421C2—81no—
422C2—1241no—
423C2—1242no—
424C2—2481no—
425C2—2481no—
426C2#1452168311yescharacteristic
427C2#1452184811yescharacteristic
428C2#1452185311yescharacteristic
subgroupG/H
centerC22C2×D4×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#468C2
socleC22×C31C22×D4
hypercenter#468D31
hypocenterC31#3268
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
1. C2×#11758485C2semidirect
2. C2×#11758493C2semidirect
3. #14521717C2semidirect
4. #14521690C2semidirect
5. #14521435C2semidirect
6. #14521286C2semidirect
7. #14521274C2semidirect
8. #14521243C2semidirect
9. #14521209C2semidirect
10. #14521197C2semidirect
11. #14521195C2semidirect
12. #14521182C2semidirect
13. #14521148C2semidirect
14. #14521118C2semidirect
15. C31×#468C2semidirect
17. C2×#92497C22semidirect
20. C2×SD16×C31C22semidirect
25. C2×#92482C22semidirect
26. C2×#92481C22semidirect
28. #11758517C22semidirect
39. C2×C8×D31C22semidirect
42. #11758472C22semidirect
45. #11758464C22semidirect
50. C31×#107C22semidirect
51. C31×#103C22semidirect
52. C31×#102C22semidirect
54. (C2×D31)⋊C8C22semidirect
55. #11758403C22semidirect
56. Dic31⋊C8C22semidirect
57. #11758398C22semidirect
60. #11758397C22semidirect
121. #92470D4semidirect
211. #468D31semidirect
240. C2×SD16C2×D31semidirect
264. #107C2×D31semidirect
267. #103C2×D31semidirect
271. #102C2×D31semidirect
275. C31#3268semidirect

Maximal subgroups

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

subgroupindex
C2×#117584852
C2×#117584932
#145217172
#145216902
#145214352
#145212862
#145212742
#145212432
#145212092
#145211972
#145211952
#145211822
#145211482
#145211182
C31×#4682
#326831