Finite Group Databasebeta

← #44559863

Subgroups — #44559863

Order 32128.

526 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22548739C211yesmaximal, characteristic, complemented
2C2×D8×D251C211yesmaximal, characteristic, complemented
3#30632250C211yesmaximal, characteristic, complemented
4#30632159C211yesmaximal, characteristic, complemented
5#30631903C211yesmaximal, characteristic, complemented
6#30631902C211yesmaximal, characteristic, complemented
7#30631901C211yesmaximal, characteristic, complemented
8#30631900C211yesmaximal, characteristic, complemented
9#30631881C211yesmaximal, characteristic, complemented
10#30631880C211yesmaximal, characteristic, complemented
11#30631879C211yesmaximal, characteristic, complemented
12#30631712C211yesmaximal, characteristic, complemented
13#30631690C211yesmaximal, characteristic, complemented
14#30631680C211yesmaximal, characteristic, complemented
15#459×C251C211yesmaximal, characteristic, complemented
16C2×#16860809C2211yescharacteristic
17C2×D4×D251C2211yescharacteristic
18C2×D4×D251C2211yescharacteristic
19#22548739—24no—
20C2×#16860799C2211yescharacteristic, complemented
21C2×#16860798C2211yescharacteristic, complemented
22C2×D2008C2211yescharacteristic
23C2×#16860795C2211yescharacteristic
24C2×#16860795C2211yescharacteristic
25C2×#16860795C2211yescharacteristic, complemented
26D8×D251—24no—
27C2×#16860793C2211yescharacteristic
28C2×C8×D251C2211yescharacteristic
29#22548711C2211yescharacteristic, complemented
30#22548703C2211yescharacteristic, complemented
31#22548702C2211yescharacteristic, complemented
32#22548687—21no—
33#22548687—21no—
34#22548683—21no—
35#22548683—21no—
36C4×D1004C2211yescharacteristic, complemented
37D4×Dic251C2211yescharacteristic, complemented
38#22548677C2211yescharacteristic, complemented
39C4×C22⋊D251—21no—
40(C22⋊D251)⋊C4—21no—
41#22548673C2211yescharacteristic
42#22548672C2211yescharacteristic
43C2×(C2×D251)⋊C4—21no—
44C22⋊C4×D251—21no—
45C4⋊(C251⋊C8)C2211yescharacteristic, complemented
46Dic251⋊C8C2211yescharacteristic, complemented
47Dic251⋊C8C2211yescharacteristic, complemented
48#22548640C2211yescharacteristic, complemented
49#22548638C2211yescharacteristic, complemented
50#22548637C2211yescharacteristic, complemented
51#22548636C2211yescharacteristic, complemented
52#22548636C2211yescharacteristic, complemented
53#22548635C2211yescharacteristic, complemented
54C2×D8×C251C2211yescharacteristic
55C2×D8×C251C2211yescharacteristic, complemented
56#127×C251C2211yescharacteristic, complemented
57C4×D4×C251C2211yescharacteristic, complemented
58C4⋊C8×C251C2211yescharacteristic, complemented
59#102×C251C2211yescharacteristic, complemented
60#102×C251C2211yescharacteristic, complemented
61C23×D251—21no—
62C23×D251—21no—
63#16860809—22no—
64#16860809—22no—
65C2×Dic502C2311yescharacteristic
66C2×D1004C2311yescharacteristic
67C2×D1004C2311yescharacteristic
68C2×C22⋊D251—21no—
69C2×C22⋊D251—21no—
70C2×C22⋊D251—21no—
71C2×C22⋊D251—21no—
72C2×C22⋊D251—21no—
73D4×D251—22no—
74D4×D251—22no—
75D4×D251—22no—
76D4×D251D414yescomplemented
77C22×Dic251—21no—
78C22×Dic251—21no—
79C2×C4×D251—21no—
80C2×C4×D251—21no—
81C2×C4×D251C2311yescharacteristic
82#16860799—22no—
83#16860798—22no—
84D2008—22no—
85#16860795—22no—
86#16860795—22no—
87#16860795—22no—
88#16860793—22no—
89C2×C251⋊C8C2311yescharacteristic
90C2×C251⋊C8C2311yescharacteristic
91C8×D251—22no—
92C4⋊Dic251C2311yescharacteristic
93Dic251⋊C4D411yescharacteristic
94Dic251⋊C4D411yescharacteristic
95C22⋊Dic251—21no—
96C22⋊Dic251—21no—
97C22⋊Dic251—21no—
98(C2×D251)⋊C4—21no—
99(C2×D251)⋊C4—21no—
100(C2×D251)⋊C4—21no—
101(C2×D251)⋊C4—21no—
102(C2×D251)⋊C4—21no—
103(C2×D251)⋊C4D411yescharacteristic
104(C2×D251)⋊C4D411yescharacteristic
105C4×Dic251—21no—
106C4×Dic251C2311yescharacteristic
107C2×D4×C251—21no—
108C2×D4×C251—21no—
109C2×D4×C251C2311yescharacteristic
110C2×D4×C251C2311yescharacteristic
111C2×D4×C251C2311yescharacteristic
112C22×C4×C251—21no—
113D8×C251—22no—
114D8×C251—22no—
115C2×C8×C251C2311yescharacteristic
116C2×C8×C251C2311yescharacteristic
117C4⋊C4×C251C2311yescharacteristic
118C22⋊C4×C251—21no—
119C42×C251C2311yescharacteristic
120C22×D251—21no—
121C22×D251—24no—
122C22×D251—21no—
123C22×D251—42no—
124C22×D251—81no—
125C22×D251—81no—
126C22×D251—81no—
127C22×D251—81no—
128C22×D251C2×D411yescharacteristic
129Dic502—21no—
130Dic502—21no—
131D1004—21no—
132D1004—21no—
133D1004—21no—
134D1004C2×D412yes—
135C22⋊D251—24no—
136C22⋊D251—42no—
137C22⋊D251—42no—
138C22⋊D251—81no—
139C22⋊D251—81no—
140C2×Dic251—21no—
141C2×Dic251—21no—
142C2×Dic251—42no—
143C2×Dic251—81no—
144C2×Dic251C2×D411yescharacteristic
145C2×Dic251C2×D411yescharacteristic
146C2×Dic251C2×D411yescharacteristic
147C4×D251—81no—
148C4×D251—81no—
149C4×D251C2×D412yes—
150C4×D251#4112yes—
151C251⋊C8—21no—
152C251⋊C8—21no—
153C23×C251—21no—
154C23×C251—21no—
155C23×C251—21no—
156D4×C251—21no—
157D4×C251—21no—
158D4×C251—21no—
159D4×C251—21no—
160D4×C251—21no—
161D4×C251—81no—
162D4×C251—81no—
163D4×C251C2×D412yes—
164C2×C4×C251—21no—
165C2×C4×C251—81no—
166C2×C4×C251C2×D411yescharacteristic
167C2×C4×C251C2×D411yescharacteristic
168C2×C4×C251C2411yescharacteristic
169C8×C251—21no—
170C8×C251—21no—
171C22×C251—22no—
172C22×C251—41no—
173C22×C251—41no—
174C22×C251—41no—
175C22×C251—41no—
176C22×C251—41no—
177C22×C251—41no—
178C22×C251—41no—
179C22×C251—41no—
180C22×C251C22×D411yescharacteristic
181C2×D251—21no—
182C2×D251—21no—
183C2×D251—22no—
184C2×D251—41no—
185C2×D251—41no—
186C2×D251—41no—
187C2×D251—41no—
188C2×D251—41no—
189C2×D251—84no—
190C2×D251—84no—
191C2×D251#12112yes—
192C4×C251—21no—
193C4×C251—21no—
194C4×C251—41no—
195C4×C251C22×D411yescharacteristic
196C4×C251C2×#4111yescharacteristic
197Dic251—21no—
198Dic251—21no—
199Dic251—41no—
200Dic251—41no—
201Dic251#12112yes—
202C2×C251—42no—
203C2×C251—81no—
204C2×C251—81no—
205C2×C251C2×#12111yescharacteristic
206C2×C251#57611yescharacteristic
207C2×C251#57611yescharacteristic
208D251—22no—
209D251—42no—
210D251—81no—
211C251#304111yescharacteristic, complemented, Sylow
212#3041—2511nomaximal, Sylow
213C2×AGL(1,ℤ/8)—2511no—
214C22×D8—2511no—
215#535—2511no—
216#518—2511no—
217#463—2511no—
218#463—2511no—
219#463—2511no—
220#463—2511no—
221#459—2511no—
222#459—2511no—
223#459—2511no—
224#459D25111yescharacteristic, complemented
225#424—2511no—
226#418—2511no—
227#416—2511no—
228C2×#41—2511no—
229C22×D4—2511no—
230C22×D4—2511no—
231AGL(1,ℤ/8)—5024no—
232C2×SD16—2511no—
233C2×SD16—2511no—
234C2×D8—5024no—
235C2×D8—2511no—
236C2×D8—2511no—
237C2×D8—2511no—
238C2×D8—2511no—
239C2×D8C2×D25111yescharacteristic
240C2×D8C2×D25111yescharacteristic, complemented
241C2×M16—2511no—
242C22×C8—2511no—
243#127—2511no—
244#127C2×D25111yescharacteristic, complemented
245#124—2511no—
246#124—2511no—
247#121—5021no—
248#121—5021no—
249#120—5021no—
250#120—5021no—
251C4×D4—5021no—
252C4×D4—5021no—
253C4×D4—2511no—
254C4×D4—2511no—
255C4×D4—2511no—
256C4×D4C2×D25111yescharacteristic, complemented
257#117—2511no—
258#117—2511no—
259C2×C22⋊C4—5021no—
260C2×C22⋊C4—5021no—
261C4⋊C8—2511no—
262C4⋊C8—2511no—
263C4⋊C8—2511no—
264C4⋊C8C2×D25111yescharacteristic, complemented
265#103—2511no—
266#103—2511no—
267#102—2511no—
268#102—2511no—
269#102—2511no—
270#102—2511no—
271#102C2×D25111yescharacteristic, complemented
272#102C2×D25111yescharacteristic, complemented
273C24—5021no—
274C24—5021no—
275#41—5022no—
276#41—5022no—
277C2×Q8—2511no—
278C2×D4—21no—
279C2×D4—21no—
280C2×D4—5021no—
281C2×D4—5022no—
282C2×D4—5022no—
283C2×D4—5022no—
284C2×D4—5021no—
285C2×D4—5021no—
286C2×D4—5021no—
287C2×D4—5021no—
288C2×D4—2514no—
289C2×D4—2511no—
290C2×D4—2511no—
291C2×D4C22×D25111yescharacteristic
292C2×D4C22×D25111yescharacteristic
293C2×D4C22×D25111yescharacteristic
294C22×C4—21no—
295C22×C4—5021no—
296C22×C4—5021no—
297C22×C4—5021no—
298C22×C4—5021no—
299C22×C4—2511no—
300SD16—5022no—
301SD16—5022no—
302D8—22no—
303D8—22no—
304D8—5022no—
305D8—5022no—
306D8—5022no—
307D8—5022no—
308M16—5022no—
309C2×C8—5022no—
310C2×C8—2511no—
311C2×C8—2511no—
312C2×C8C22×D25111yescharacteristic
313C2×C8C22×D25111yescharacteristic
314C4⋊C4—2511no—
315C4⋊C4—2511no—
316C4⋊C4—2511no—
317C4⋊C4C22×D25111yescharacteristic
318C22⋊C4—21no—
319C22⋊C4—5021no—
320C22⋊C4—5021no—
321C22⋊C4—5021no—
322C22⋊C4—5021no—
323C22⋊C4—5021no—
324C22⋊C4—5021no—
325C22⋊C4—5021no—
326C22⋊C4—5021no—
327C22⋊C4—2511no—
328C22⋊C4—2511no—
329C42—5021no—
330C42—2511no—
331C42C22×D25111yescharacteristic
332C23—21no—
333C23—21no—
334C23—21no—
335C23—20081no—
336C23—20081no—
337C23—20081no—
338C23—20081no—
339C23—10042no—
340C23—5024no—
341C23—5021no—
342C23—5021no—
343C23—2511no—
344Q8—5021no—
345Q8—5021no—
346D4—21no—
347D4—21no—
348D4—21no—
349D4—21no—
350D4—21no—
351D4—81no—
352D4—81no—
353D4—20081no—
354D4—20081no—
355D4—10042no—
356D4—10042no—
357D4—5024no—
358D4—5021no—
359D4—5021no—
360D4—5021no—
361D4—2512no—
362D4D4×D25112yes—
363C2×C4—21no—
364C2×C4—81no—
365C2×C4—20081no—
366C2×C4—20081no—
367C2×C4—20081no—
368C2×C4—10042no—
369C2×C4—5021no—
370C2×C4—5021no—
371C2×C4—2511no—
372C2×C4—2511no—
373C2×C4—2511no—
374C2×C4—2512no—
375C2×C4—2512no—
376C2×C4D4×D25111yescharacteristic
377C2×C4D4×D25111yescharacteristic
378C2×C4C23×D25111yescharacteristic
379C8—21no—
380C8—21no—
381C8—5021no—
382C8—5021no—
383C22—22no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22—41no—
391C22—41no—
392C22—20084no—
393C22—20084no—
394C22—10041no—
395C22—10041no—
396C22—10041no—
397C22—10041no—
398C22—10041no—
399C22—5022no—
400C22—5021no—
401C22—5021no—
402C22—2512no—
403C22C2×D4×D25111yescharacteristic
404C4—21no—
405C4—21no—
406C4—41no—
407C4—10041no—
408C4—10041no—
409C4—5021no—
410C4—5021no—
411C4—2512no—
412C4C2×D4×D25111yescharacteristic
413C4#41×D25111yescharacteristic
414C2—42no—
415C2—81no—
416C2—81no—
417C2—20081no—
418C2—10042no—
419C2—5022no—
420C2#121×D25111yescharacteristic
421C2#3063248811yescharacteristic
422C2#3063249111yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#459×C251C2
socleC22×C251C22×D4
hypercenter#459D251
hypocenterC251#3041
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. C2×#22548739C2semidirect
2. C2×D8×D251C2semidirect
3. #30632250C2semidirect
4. #30632159C2semidirect
5. #30631903C2semidirect
6. #30631902C2semidirect
7. #30631901C2semidirect
8. #30631900C2semidirect
9. #30631881C2semidirect
10. #30631880C2semidirect
11. #30631879C2semidirect
12. #30631712C2semidirect
13. #30631690C2semidirect
14. #30631680C2semidirect
15. #459×C251C2semidirect
20. C2×#16860799C22semidirect
21. C2×#16860798C22semidirect
25. C2×#16860795C22semidirect
29. #22548711C22semidirect
30. #22548703C22semidirect
31. #22548702C22semidirect
36. C4×D1004C22semidirect
37. D4×Dic251C22semidirect
38. #22548677C22semidirect
45. C4⋊(C251⋊C8)C22semidirect
46. Dic251⋊C8C22semidirect
47. Dic251⋊C8C22semidirect
48. #22548640C22semidirect
49. #22548638C22semidirect
50. #22548637C22semidirect
51. #22548636C22semidirect
52. #22548636C22semidirect
53. #22548635C22semidirect
55. C2×D8×C251C22semidirect
56. #127×C251C22semidirect
57. C4×D4×C251C22semidirect
58. C4⋊C8×C251C22semidirect
59. #102×C251C22semidirect
60. #102×C251C22semidirect
76. D4×D251D4semidirect
211. C251#3041semidirect
224. #459D251semidirect
240. C2×D8C2×D251semidirect
244. #127C2×D251semidirect
256. C4×D4C2×D251semidirect
264. C4⋊C8C2×D251semidirect
271. #102C2×D251semidirect
272. #102C2×D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225487392
C2×D8×D2512
#306322502
#306321592
#306319032
#306319022
#306319012
#306319002
#306318812
#306318802
#306318792
#306317122
#306316902
#306316802
#459×C2512
#3041251