Finite Group Databasebeta

← #44561865

Subgroups — #44561865

Order 32128.

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×#22548743C211yesmaximal, characteristic, complemented
2C22×#16860799C211yesmaximal, characteristic, complemented
3#30632346C211yesmaximal, characteristic, complemented
4D4×D1004C211yesmaximal, characteristic, complemented
5#30632076C211yesmaximal, characteristic, complemented
6#30631933C211yesmaximal, characteristic, complemented
7#30631923C211yesmaximal, characteristic, complemented
8#30631903C211yesmaximal, characteristic, complemented
9#30631888C211yesmaximal, characteristic, complemented
10#30631840C211yesmaximal, characteristic, complemented
11#30631834C211yesmaximal, characteristic, complemented
12#30631828C211yesmaximal, characteristic, complemented
13#30631823C211yesmaximal, characteristic, complemented
14#30631789C211yesmaximal, characteristic, complemented
15#435×C251C211yesmaximal, characteristic, complemented
16C2×#16860810C2211yescharacteristic, complemented
17C22×Dic502C2211yescharacteristic, complemented
18C22×D1004C2211yescharacteristic
19C22×D1004C2211yescharacteristic
20C2×D4×D251—21no—
21#22548743—24no—
22C2×#16860799—24no—
23C2×#16860799C2211yescharacteristic
24C2×#16860799C2211yescharacteristic
25C2×#16860799C2211yescharacteristic, complemented
26C2×#16860799C2211yescharacteristic, complemented
27C2×#16860799C2211yescharacteristic, complemented
28C2×#16860799C2211yescharacteristic, complemented
29C2×D2008C2211yescharacteristic
30C2×D2008C2211yescharacteristic, complemented
31C2×D2008C2211yescharacteristic, complemented
32#22548712C2211yescharacteristic
33#22548703C2211yescharacteristic, complemented
34#22548701—21no—
35#22548694C2211yescharacteristic, complemented
36#22548693—21no—
37#22548690—21no—
38#22548689C2211yescharacteristic
39#22548689C2211yescharacteristic
40#22548689C2211yescharacteristic, complemented
41#22548687—21no—
42#22548685—21no—
43#22548685—21no—
44C4×Dic502C2211yescharacteristic, complemented
45C4×D1004C2211yescharacteristic
46(C22⋊D251)⋊C4—21no—
47#22548649C2211yescharacteristic, complemented
48#22548640C2211yescharacteristic, complemented
49#22548640C2211yescharacteristic, complemented
50#22548637C2211yescharacteristic
51#22548637C2211yescharacteristic, complemented
52#22548637C2211yescharacteristic, complemented
53#22548637C2211yescharacteristic, complemented
54C2×M16×C251C2211yescharacteristic, complemented
55C22×C8×C251C2211yescharacteristic, complemented
56C4×D4×C251C2211yescharacteristic
57C4⋊C8×C251C2211yescharacteristic, complemented
58C22⋊C8×C251C2211yescharacteristic, complemented
59C22⋊C8×C251C2211yescharacteristic, complemented
60#97×C251C2211yescharacteristic, complemented
61C23×D251—21no—
62C23×D251—21no—
63#16860810—22no—
64#16860810—41no—
65C2×Dic502—22no—
66C2×Dic502—21no—
67C2×Dic502—41no—
68C2×Dic502C2311yescharacteristic
69C2×Dic502C2311yescharacteristic
70C2×D1004—22no—
71C2×D1004—22no—
72C2×D1004—21no—
73C2×D1004—21no—
74C2×D1004—21no—
75C2×D1004—21no—
76C2×D1004—41no—
77C2×D1004—41no—
78C2×D1004C2311yescharacteristic
79C2×D1004C2311yescharacteristic
80C2×D1004C2311yescharacteristic
81C2×D1004C2311yescharacteristic
82C2×C22⋊D251—21no—
83C2×C22⋊D251—21no—
84C2×C22⋊D251—21no—
85C2×C22⋊D251—21no—
86D4×D251—44no—
87C22×Dic251—21no—
88C2×C4×D251—21no—
89C2×C4×D251—21no—
90#16860799—22no—
91#16860799—22no—
92#16860799—22no—
93#16860799—22no—
94#16860799—41no—
95#16860799D414yescomplemented
96D2008—22no—
97D2008—22no—
98D2008—41no—
99C4⋊Dic251C2311yescharacteristic
100C4⋊Dic251C2311yescharacteristic
101Dic251⋊C4—21no—
102Dic251⋊C4—21no—
103C22⋊Dic251—21no—
104(C2×D251)⋊C4—21no—
105(C2×D251)⋊C4—21no—
106(C2×D251)⋊C4—21no—
107(C2×D251)⋊C4—21no—
108(C2×D251)⋊C4—21no—
109(C2×D251)⋊C4—21no—
110C4×Dic251—21no—
111C2×D4×C251D411yescharacteristic
112C22×C4×C251C2311yescharacteristic
113C22×C4×C251C2311yescharacteristic
114M16×C251—22no—
115C2×C8×C251—22no—
116C2×C8×C251C2311yescharacteristic
117C2×C8×C251C2311yescharacteristic
118C2×C8×C251C2311yescharacteristic
119C2×C8×C251C2311yescharacteristic
120C4⋊C4×C251D411yescharacteristic
121C22⋊C4×C251D411yescharacteristic
122C22⋊C4×C251D411yescharacteristic
123C42×C251C2311yescharacteristic
124C22×D251—21no—
125C22×D251—21no—
126C22×D251—21no—
127C22×D251—21no—
128C22×D251—42no—
129C22×D251—42no—
130C22×D251—42no—
131C22×D251—42no—
132C22×D251—81no—
133C22×D251—81no—
134Dic502—21no—
135Dic502—21no—
136Dic502—21no—
137Dic502—42no—
138Dic502C2×D412yescomplemented
139D1004—21no—
140D1004—21no—
141D1004—21no—
142D1004—21no—
143D1004—21no—
144D1004—21no—
145D1004—21no—
146D1004—42no—
147D1004—42no—
148D1004—42no—
149D1004—81no—
150D1004C2×D412yes—
151C22⋊D251—42no—
152C22⋊D251—42no—
153C22⋊D251—42no—
154C22⋊D251—42no—
155C2×Dic251—21no—
156C2×Dic251—21no—
157C2×Dic251—21no—
158C2×Dic251—21no—
159C2×Dic251—42no—
160C4×D251—42no—
161C4×D251—42no—
162C23×C251C2×D411yescharacteristic
163C23×C251C2×D411yescharacteristic
164D4×C251—22no—
165C2×C4×C251—21no—
166C2×C4×C251—21no—
167C2×C4×C251—21no—
168C2×C4×C251—21no—
169C2×C4×C251C2×D411yescharacteristic
170C2×C4×C251C2×D411yescharacteristic
171C2×C4×C251C2×D411yescharacteristic
172C2×C4×C251C2×D411yescharacteristic
173C2×C4×C251C2411yescharacteristic
174C8×C251—21no—
175C8×C251—21no—
176C8×C251—21no—
177C8×C251C2×D412yescomplemented
178C22×C251—21no—
179C22×C251—21no—
180C22×C251—21no—
181C22×C251—21no—
182C22×C251—21no—
183C22×C251AGL(1,ℤ/8)12yescomplemented
184C22×C251C22×D411yescharacteristic
185C2×D251—22no—
186C2×D251—41no—
187C2×D251—41no—
188C2×D251—41no—
189C2×D251—41no—
190C2×D251—41no—
191C2×D251—41no—
192C2×D251—41no—
193C2×D251—41no—
194C2×D251—41no—
195C2×D251—41no—
196C2×D251—41no—
197C2×D251—84no—
198C2×D251—84no—
199C4×C251—21no—
200C4×C251—21no—
201C4×C251—21no—
202C4×C251—21no—
203C4×C251C22×D411yescharacteristic
204C4×C251H(5,𝔽2)11yescharacteristic
205Dic251—22no—
206Dic251—41no—
207Dic251—41no—
208Dic251—41no—
209C2×C251—22no—
210C2×C251—41no—
211C2×C251D4211yescharacteristic
212C2×C251C2×AGL(1,ℤ/8)11yescharacteristic
213C2×C251#57711yescharacteristic
214D251—42no—
215D251—81no—
216D251—81no—
217D251—81no—
218C251#326511yescharacteristic, complemented, Sylow
219#3265—2511nomaximal, Sylow
220C2×AGL(1,ℤ/8)—2511no—
221C22×SD16—2511no—
222#548—2511no—
223D42—2511no—
224#496—2511no—
225#468—2511no—
226#466—2511no—
227#463—2511no—
228#460—2511no—
229#450—2511no—
230#449—2511no—
231#448—2511no—
232#447—2511no—
233#440—2511no—
234#435D25111yescharacteristic, complemented
235C2×#41—2511no—
236C22×Q8—2511no—
237C22×D4—5021no—
238C22×D4—2511no—
239C22×D4—2511no—
240AGL(1,ℤ/8)—5024no—
241C2×SD16—5024no—
242C2×SD16—2511no—
243C2×SD16—2511no—
244C2×SD16—2511no—
245C2×SD16—2511no—
246C2×SD16—2511no—
247C2×SD16—2511no—
248C2×D8—2511no—
249C2×D8—2511no—
250C2×D8—2511no—
251C2×M16C2×D25111yescharacteristic, complemented
252C22×C8C2×D25111yescharacteristic, complemented
253#127—2511no—
254#124—5021no—
255#124—2511no—
256C22⋊Q8—5021no—
257C22⋊Q8—2511no—
258#121—5021no—
259#121—5021no—
260#121—2511no—
261#121—2511no—
262#121—2511no—
263#120—5021no—
264#120—5021no—
265C4×Q8—2511no—
266C4×D4—5021no—
267C4×D4—2511no—
268C4×D4C2×D25111yescharacteristic
269#107—2511no—
270C4⋊C8C2×D25111yescharacteristic, complemented
271#103—2511no—
272#103—2511no—
273#102—2511no—
274#102—2511no—
275#102—2511no—
276#102—2511no—
277C22⋊C8C2×D25111yescharacteristic, complemented
278C22⋊C8C2×D25111yescharacteristic, complemented
279#97C2×D25111yescharacteristic, complemented
280C24—5021no—
281C24—5021no—
282#41—10041no—
283#41—5022no—
284C2×Q8—10041no—
285C2×Q8—5022no—
286C2×Q8—5021no—
287C2×Q8—2511no—
288C2×Q8—2511no—
289C2×D4—10041no—
290C2×D4—10041no—
291C2×D4—10044no—
292C2×D4—5022no—
293C2×D4—5022no—
294C2×D4—5021no—
295C2×D4—5021no—
296C2×D4—5021no—
297C2×D4—5021no—
298C2×D4—5021no—
299C2×D4—5021no—
300C2×D4—5021no—
301C2×D4—5021no—
302C2×D4—2511no—
303C2×D4—2511no—
304C2×D4—2511no—
305C2×D4—2511no—
306C2×D4D100411yescharacteristic
307C22×C4—5021no—
308C22×C4—5021no—
309C22×C4—5021no—
310C22×C4C22×D25111yescharacteristic
311C22×C4C22×D25111yescharacteristic
312SD16—10041no—
313SD16—5022no—
314SD16—5022no—
315SD16—5022no—
316SD16—5022no—
317SD16—2514no—
318D8—10041no—
319D8—5022no—
320D8—5022no—
321M16—22no—
322C2×C8—22no—
323C2×C8C22×D25111yescharacteristic
324C2×C8C22×D25111yescharacteristic
325C2×C8C22×D25111yescharacteristic
326C2×C8C22×D25111yescharacteristic
327C4⋊C4—5021no—
328C4⋊C4—5021no—
329C4⋊C4—2511no—
330C4⋊C4—2511no—
331C4⋊C4D100411yescharacteristic
332C22⋊C4—5021no—
333C22⋊C4—5021no—
334C22⋊C4—5021no—
335C22⋊C4—5021no—
336C22⋊C4—5021no—
337C22⋊C4—5021no—
338C22⋊C4—5021no—
339C22⋊C4D100411yescharacteristic
340C22⋊C4D100411yescharacteristic
341C42—5021no—
342C42C22×D25111yescharacteristic
343C23—20081no—
344C23—20081no—
345C23—10042no—
346C23—10042no—
347C23—10042no—
348C23—10042no—
349C23—5021no—
350C23—5021no—
351C23—5021no—
352C23—5021no—
353C23C2×D100411yescharacteristic
354C23C2×D100411yescharacteristic
355Q8—10042no—
356Q8—5021no—
357Q8—5021no—
358Q8—5021no—
359Q8—2512no—
360D4—22no—
361D4—20081no—
362D4—10042no—
363D4—10042no—
364D4—10042no—
365D4—10042no—
366D4—10042no—
367D4—10042no—
368D4—10042no—
369D4—5021no—
370D4—5021no—
371D4—5021no—
372D4—5021no—
373D4—5021no—
374D4—5021no—
375D4—5021no—
376D4—2512no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—10042no—
382C2×C4—10042no—
383C2×C4—10042no—
384C2×C4—5021no—
385C2×C4—5021no—
386C2×C4—5021no—
387C2×C4—5021no—
388C2×C4C2×D100411yescharacteristic
389C2×C4C2×D100411yescharacteristic
390C2×C4C2×D100411yescharacteristic
391C2×C4C2×D100411yescharacteristic
392C2×C4C23×D25111yescharacteristic
393C8—21no—
394C8—21no—
395C8—21no—
396C8D4×D25112yescomplemented
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—20084no—
403C22—20084no—
404C22—10041no—
405C22—10041no—
406C22—10041no—
407C22—10041no—
408C22—10041no—
409C22—10041no—
410C22—10041no—
411C22—10041no—
412C22—10041no—
413C22—10041no—
414C22—10041no—
415C22—5022no—
416C22#2254874312yescomplemented
417C22C22×D100411yescharacteristic
418C4—21no—
419C4—21no—
420C4—21no—
421C4—21no—
422C4—10041no—
423C4—10041no—
424C4—10041no—
425C4—5022no—
426C4C2×D4×D25111yescharacteristic
427C4#2254876111yescharacteristic
428C2—22no—
429C2—41no—
430C2—20081no—
431C2—20081no—
432C2—20081no—
433C2—10042no—
434C2D4×D100411yescharacteristic
435C2C2×#2254874311yescharacteristic
436C2#3063249911yescharacteristic
subgroupG/H
centerC22C22×D1004
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#435×C251C2
socleC22×C251C22×D4
hypercenter#435D251
hypocenterC251#3265
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C4×D4 ◁ #435

Complements

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

NH ≅ G/Ntype
1. C2×#22548743C2semidirect
2. C22×#16860799C2semidirect
3. #30632346C2semidirect
4. D4×D1004C2semidirect
5. #30632076C2semidirect
6. #30631933C2semidirect
7. #30631923C2semidirect
8. #30631903C2semidirect
9. #30631888C2semidirect
10. #30631840C2semidirect
11. #30631834C2semidirect
12. #30631828C2semidirect
13. #30631823C2semidirect
14. #30631789C2semidirect
15. #435×C251C2semidirect
16. C2×#16860810C22semidirect
17. C22×Dic502C22semidirect
25. C2×#16860799C22semidirect
26. C2×#16860799C22semidirect
27. C2×#16860799C22semidirect
28. C2×#16860799C22semidirect
30. C2×D2008C22semidirect
31. C2×D2008C22semidirect
33. #22548703C22semidirect
35. #22548694C22semidirect
40. #22548689C22semidirect
44. C4×Dic502C22semidirect
47. #22548649C22semidirect
48. #22548640C22semidirect
49. #22548640C22semidirect
51. #22548637C22semidirect
52. #22548637C22semidirect
53. #22548637C22semidirect
54. C2×M16×C251C22semidirect
55. C22×C8×C251C22semidirect
57. C4⋊C8×C251C22semidirect
58. C22⋊C8×C251C22semidirect
59. C22⋊C8×C251C22semidirect
60. #97×C251C22semidirect
95. #16860799D4semidirect
138. Dic502C2×D4semidirect
177. C8×C251C2×D4semidirect
183. C22×C251AGL(1,ℤ/8)semidirect
218. C251#3265semidirect
234. #435D251semidirect
251. C2×M16C2×D251semidirect
252. C22×C8C2×D251semidirect
270. C4⋊C8C2×D251semidirect
277. C22⋊C8C2×D251semidirect
278. C22⋊C8C2×D251semidirect
279. #97C2×D251semidirect
396. C8D4×D251semidirect
416. C22#22548743semidirect

Maximal subgroups

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

subgroupindex
C2×#225487432
C22×#168607992
#306323462
D4×D10042
#306320762
#306319332
#306319232
#306319032
#306318882
#306318402
#306318342
#306318282
#306318232
#306317892
#435×C2512
#3265251