Finite Group Databasebeta

← #16857250

Subgroups — #16857250

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×#11758488C211yesmaximal, characteristic, complemented
2C2×#11758493C211yesmaximal, characteristic, complemented
3#14521719C211yesmaximal, characteristic, complemented
4#14521692C211yesmaximal, characteristic, complemented
5#14521292C211yesmaximal, characteristic, complemented
6#14521282C211yesmaximal, characteristic, complemented
7#14521268C211yesmaximal, characteristic, complemented
8#14521241C211yesmaximal, characteristic, complemented
9#14521208C211yesmaximal, characteristic, complemented
10#14521193C211yesmaximal, characteristic, complemented
11#14521191C211yesmaximal, characteristic, complemented
12#14521181C211yesmaximal, characteristic, complemented
13#14521151C211yesmaximal, characteristic, complemented
14#14521122C211yesmaximal, characteristic, complemented
15C31×#496C211yesmaximal, characteristic, complemented
16C2×#92499C2211yescharacteristic
17C2×#92497C2211yescharacteristic
18C2×#92497C2211yescharacteristic, complemented
19C2×D4×D31C2211yescharacteristic
20C2×SD16×C31C2211yescharacteristic, complemented
21C2×SD16×C31C2211yescharacteristic, complemented
22C2×D8×C31C2211yescharacteristic
23C2×D8×C31C2211yescharacteristic
24C31×#127C2211yescharacteristic, complemented
25C31×#124C2211yescharacteristic
26#11758523C2211yescharacteristic
27C2×#92482C2211yescharacteristic, complemented
28#11758517—21no—
29#11758516—21no—
30#11758516—21no—
31#11758515C2211yescharacteristic
32#11758515C2211yescharacteristic
33#11758515C2211yescharacteristic, complemented
34D4×Dic31—21no—
35C2×#92480C2211yescharacteristic
36C2×#92479C2211yescharacteristic
37C2×#92479C2211yescharacteristic
38#11758493—24no—
39#11758488—24no—
40C2×#92470C2211yescharacteristic
41C2×#92469C2211yescharacteristic
42C2×C8×D31C2211yescharacteristic, complemented
43#11758461—21no—
44#11758460—21no—
45#11758459—21no—
46C22⋊C4×D31—21no—
47C4×D124C2211yescharacteristic
48C4×D124C2211yescharacteristic
49C4×Dic62C2211yescharacteristic, complemented
50C31×#97C2211yescharacteristic, complemented
51#11758423C2211yescharacteristic, complemented
52#11758420C2211yescharacteristic
53#11758420C2211yescharacteristic, complemented
54#11758420C2211yescharacteristic, complemented
55#11758409C2211yescharacteristic
56(C2×D31)⋊C8C2211yescharacteristic
57(C2×D31)⋊C8C2211yescharacteristic, complemented
58#11758406C2211yescharacteristic, complemented
59#11758404C2211yescharacteristic, complemented
60C4⋊(C31⋊C8)C2211yescharacteristic, complemented
61C23×D31—21no—
62C2×Q8×C31C2311yescharacteristic
63C2×D4×C31—21no—
64C2×D4×C31—21no—
65C2×D4×C31C2311yescharacteristic
66C2×D4×C31C2311yescharacteristic
67C2×D4×C31C2311yescharacteristic
68C2×C22⋊D31—21no—
69C2×C22⋊D31—21no—
70C2×C22⋊D31—21no—
71C2×C22⋊D31—21no—
72C2×C22⋊D31—21no—
73C2×C22⋊D31—21no—
74C22×Dic31—21no—
75C22×Dic31—21no—
76#92499—22no—
77#92499—41no—
78#92497—22no—
79#92497—22no—
80#92497—41no—
81#92497—41no—
82D4×D31—22no—
83D4×D31—41no—
84C2×D124—21no—
85C2×D124D411yescharacteristic
86C2×D124C2311yescharacteristic
87C2×C4×D31—21no—
88C2×C4×D31—21no—
89C2×C4×D31C2311yescharacteristic
90C2×C4×D31C2311yescharacteristic
91C2×Dic62C2311yescharacteristic
92SD16×C31—22no—
93SD16×C31—41no—
94D8×C31—22no—
95D8×C31—41no—
96C2×C8×C31C2311yescharacteristic
97C2×C8×C31C2311yescharacteristic
98C22⋊C4×C31—21no—
99C22⋊C4×C31—21no—
100C42×C31C2311yescharacteristic
101C22⋊Dic31—21no—
102C22⋊Dic31—21no—
103C22⋊Dic31—21no—
104#92482—22no—
105#92480—22no—
106#92479—22no—
107#92479—22no—
108(C2×D31)⋊C4—21no—
109(C2×D31)⋊C4—21no—
110(C2×D31)⋊C4D411yescharacteristic
111(C2×D31)⋊C4D411yescharacteristic
112C4⋊Dic31D411yescharacteristic
113C4⋊Dic31C2311yescharacteristic
114C4⋊Dic31C2311yescharacteristic
115Dic31⋊C4—21no—
116Dic31⋊C4—21no—
117C4×Dic31—21no—
118C2×C31⋊C8C2311yescharacteristic
119C2×C31⋊C8C2311yescharacteristic
120#92470D414yescomplemented
121#92469—22no—
122C8×D31—22no—
123C23×C31—21no—
124C23×C31—21no—
125C23×C31—21no—
126C22×D31—21no—
127C22×D31—42no—
128C22×D31—42no—
129C22×D31—81no—
130C22×D31C2×D411yescharacteristic
131C22×D31C2×D411yescharacteristic
132Q8×C31—21no—
133Q8×C31—21no—
134D4×C31—21no—
135D4×C31—21no—
136D4×C31—21no—
137D4×C31—21no—
138D4×C31—21no—
139D4×C31—21no—
140D4×C31—42no—
141D4×C31—81no—
142C2×C4×C31—21no—
143C2×C4×C31C2×D411yescharacteristic
144C2×C4×C31C2×D411yescharacteristic
145C2×C4×C31C2411yescharacteristic
146C22⋊D31—42no—
147C22⋊D31—42no—
148C22⋊D31—42no—
149C22⋊D31—81no—
150C22⋊D31—81no—
151C22⋊D31—81no—
152C2×Dic31—21no—
153C2×Dic31—21no—
154C2×Dic31—21no—
155C2×Dic31—42no—
156C2×Dic31—42no—
157C2×Dic31C2×D411yescharacteristic
158C2×Dic31C2×D411yescharacteristic
159D124—21no—
160D124—41no—
161D124—42no—
162D124C2×D412yes—
163C4×D31—21no—
164C4×D31—21no—
165C4×D31—21no—
166C4×D31—21no—
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—41no—
177C22×C31—41no—
178C22×C31—41no—
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—22no—
189C2×D31—21no—
190C2×D31—21no—
191C2×D31—21no—
192C2×D31—21no—
193C2×D31—41no—
194C2×D31—41no—
195C2×D31—84no—
196C4×C31—21no—
197C4×C31—21no—
198C4×C31—41no—
199C4×C31C22×D411yescharacteristic
200C4×C31H(5,𝔽2)11yescharacteristic
201Dic31—22no—
202Dic31—21no—
203Dic31—21no—
204Dic31—41no—
205Dic31—41no—
206C2×AGL(1,ℤ/8)—311no—
207C2×#135—311no—
208#550—311no—
209#546—311no—
210#496D3111yescharacteristic, complemented
211#468—311no—
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—81no—
222C2×C31—81no—
223C2×C31—81no—
224C2×C31D4211yescharacteristic
225C2×C31#57511yescharacteristic
226C2×C31#57611yescharacteristic
227D31—41no—
228D31—41no—
229D31—42no—
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×SD16C2×D3111yescharacteristic, complemented
240C2×SD16C2×D3111yescharacteristic, complemented
241C2×D8—311no—
242C2×D8—311no—
243C2×D8C2×D3111yescharacteristic
244C2×D8C2×D3111yescharacteristic
245C2×M16—311no—
246C22×C8—311no—
247#127—621no—
248#127C2×D3111yescharacteristic, complemented
249#124C2×D3111yescharacteristic
250#123—621no—
251C22⋊Q8—311no—
252#121—311no—
253#121—311no—
254#121—311no—
255#121—621no—
256#121—621no—
257#121—621no—
258#120—621no—
259C4×Q8—311no—
260C4×D4—311no—
261C4×D4—311no—
262C4×D4—621no—
263C2×C22⋊C4—621no—
264#107—311no—
265C4⋊C8—311no—
266#103—311no—
267#103—311no—
268#102—311no—
269#102—311no—
270#102—311no—
271#102—311no—
272C22⋊C8—311no—
273C22⋊C8—311no—
274#97C2×D3111yescharacteristic, complemented
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—21no—
287C2×D4—311no—
288C2×D4—311no—
289C2×D4—621no—
290C2×D4—621no—
291C2×D4—622no—
292C2×D4—621no—
293C2×D4—621no—
294C2×D4—621no—
295C2×D4—621no—
296C2×D4—621no—
297C2×D4—1241no—
298C2×D4C22×D3111yescharacteristic
299C2×D4C22×D3111yescharacteristic
300C2×D4C22×D3111yescharacteristic
301C22×C4—311no—
302C22×C4—311no—
303C22×C4—621no—
304C22×C4—621no—
305C22×C4—621no—
306C22×C4—621no—
307Q16—622no—
308SD16—22no—
309SD16—41no—
310SD16—314no—
311SD16—622no—
312D8—22no—
313D8—41no—
314D8—622no—
315D8—622no—
316M16—622no—
317C2×C8—311no—
318C2×C8—311no—
319C2×C8—622no—
320C2×C8C22×D3111yescharacteristic
321C2×C8C22×D3111yescharacteristic
322C4⋊C4—311no—
323C4⋊C4—311no—
324C4⋊C4—311no—
325C4⋊C4—621no—
326C4⋊C4—621no—
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—621no—
337C42C22×D3111yescharacteristic
338C23—21no—
339C23—21no—
340C23—21no—
341C23—311no—
342C23—311no—
343C23—621no—
344C23—1242no—
345C23—1242no—
346C23—2481no—
347Q8—21no—
348Q8—21no—
349Q8—312no—
350Q8—621no—
351D4—21no—
352D4—21no—
353D4—21no—
354D4—21no—
355D4—21no—
356D4—21no—
357D4—42no—
358D4—81no—
359D4—312no—
360D4—621no—
361D4—1241no—
362D4—1242no—
363D4—1242no—
364D4—1242no—
365D4—1242no—
366D4—2481no—
367D4—2481no—
368D4—2481no—
369C2×C4—21no—
370C2×C4—311no—
371C2×C4—311no—
372C2×C4—621no—
373C2×C4—621no—
374C2×C4—621no—
375C2×C4—621no—
376C2×C4—621no—
377C2×C4—621no—
378C2×C4—621no—
379C2×C4—1242no—
380C2×C4—1242no—
381C2×C4—1242no—
382C2×C4—2481no—
383C2×C4D4×D3111yescharacteristic
384C2×C4D4×D3111yescharacteristic
385C2×C4C23×D3111yescharacteristic
386C8—21no—
387C8—621no—
388C8—621no—
389C8D4×D3112yes—
390C22—41no—
391C22—41no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—621no—
400C22—621no—
401C22—622no—
402C22—621no—
403C22—621no—
404C22—621no—
405C22—621no—
406C22—1241no—
407C22—1241no—
408C22—2484no—
409C22C2×D4×D3111yescharacteristic
410C4—21no—
411C4—21no—
412C4—41no—
413C4—621no—
414C4—621no—
415C4—622no—
416C4—1241no—
417C4—1241no—
418C4C2×D4×D3111yescharacteristic
419C4#1175856111yescharacteristic
420C2—81no—
421C2—81no—
422C2—81no—
423C2—1241no—
424C2—1241no—
425C2—1242no—
426C2#1452168411yescharacteristic
427C2#1452184511yescharacteristic
428C2#1452185311yescharacteristic
subgroupG/H
centerC22C2×D4×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#496C2
socleC22×C31C22×D4
hypercenter#496D31
hypocenterC31#3268
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. C2×#11758488C2semidirect
2. C2×#11758493C2semidirect
3. #14521719C2semidirect
4. #14521692C2semidirect
5. #14521292C2semidirect
6. #14521282C2semidirect
7. #14521268C2semidirect
8. #14521241C2semidirect
9. #14521208C2semidirect
10. #14521193C2semidirect
11. #14521191C2semidirect
12. #14521181C2semidirect
13. #14521151C2semidirect
14. #14521122C2semidirect
15. C31×#496C2semidirect
18. C2×#92497C22semidirect
20. C2×SD16×C31C22semidirect
21. C2×SD16×C31C22semidirect
24. C31×#127C22semidirect
27. C2×#92482C22semidirect
33. #11758515C22semidirect
42. C2×C8×D31C22semidirect
49. C4×Dic62C22semidirect
50. C31×#97C22semidirect
51. #11758423C22semidirect
53. #11758420C22semidirect
54. #11758420C22semidirect
57. (C2×D31)⋊C8C22semidirect
58. #11758406C22semidirect
59. #11758404C22semidirect
60. C4⋊(C31⋊C8)C22semidirect
120. #92470D4semidirect
210. #496D31semidirect
239. C2×SD16C2×D31semidirect
240. C2×SD16C2×D31semidirect
248. #127C2×D31semidirect
274. #97C2×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×#117584882
C2×#117584932
#145217192
#145216922
#145212922
#145212822
#145212682
#145212412
#145212082
#145211932
#145211912
#145211812
#145211512
#145211222
C31×#4962
#326831