Finite Group Databasebeta

← #44561909

Subgroups — #44561909

Order 32128.

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×#22548739C211yesmaximal, characteristic, complemented
2C2×#22548735C211yesmaximal, characteristic, complemented
3#30632355C211yesmaximal, characteristic, complemented
4#30632325C211yesmaximal, characteristic, complemented
5#30632073C211yesmaximal, characteristic, complemented
6#30631936C211yesmaximal, characteristic, complemented
7#30631921C211yesmaximal, characteristic, complemented
8#30631912C211yesmaximal, characteristic, complemented
9#30631880C211yesmaximal, characteristic, complemented
10#30631846C211yesmaximal, characteristic, complemented
11#30631834C211yesmaximal, characteristic, complemented
12#30631822C211yesmaximal, characteristic, complemented
13#30631784C211yesmaximal, characteristic, complemented
14#30631755C211yesmaximal, characteristic, complemented
15#449×C251C211yesmaximal, characteristic, complemented
16C2×#16860810C2211yescharacteristic, complemented
17C2×#16860809C2211yescharacteristic
18C2×D4×D251C2211yescharacteristic
19#22548739—24no—
20#22548735—24no—
21C2×#16860800C2211yescharacteristic, complemented
22C2×#16860799C2211yescharacteristic
23C2×#16860798C2211yescharacteristic
24C2×#16860797C2211yescharacteristic, complemented
25C2×D2008C2211yescharacteristic
26C2×#16860795C2211yescharacteristic
27C2×#16860795C2211yescharacteristic
28C2×#16860793C2211yescharacteristic
29C22×C251⋊C8C2211yescharacteristic, complemented
30#22548711—21no—
31#22548711C2211yescharacteristic, complemented
32#22548704C2211yescharacteristic
33#22548698—21no—
34#22548691C2211yescharacteristic
35#22548690C2211yescharacteristic
36#22548689C2211yescharacteristic
37#22548688—21no—
38#22548687—21no—
39#22548687—21no—
40#22548683—21no—
41#22548680C2211yescharacteristic, complemented
42D4×Dic251C2211yescharacteristic
43C4×C22⋊D251C2211yescharacteristic
44(C22⋊D251)⋊C4—21no—
45C2×(C2×D251)⋊C4—21no—
46#22548648C2211yescharacteristic, complemented
47Dic251⋊C8C2211yescharacteristic, complemented
48#22548640C2211yescharacteristic, complemented
49#22548637C2211yescharacteristic, complemented
50#22548636C2211yescharacteristic, complemented
51#22548635C2211yescharacteristic
52(C2×D251)⋊C8C2211yescharacteristic
53#22548625C2211yescharacteristic, complemented
54C2×#41×C251C2211yescharacteristic
55C2×SD16×C251C2211yescharacteristic, complemented
56C2×D8×C251C2211yescharacteristic
57#121×C251C2211yescharacteristic, complemented
58#103×C251C2211yescharacteristic, complemented
59#102×C251C2211yescharacteristic
60C22⋊C8×C251C2211yescharacteristic, complemented
61C23×D251—21no—
62#16860810—22no—
63#16860810—41no—
64#16860809—22no—
65#16860809—41no—
66C2×Dic502C2311yescharacteristic
67C2×D1004—21no—
68C2×D1004C2311yescharacteristic
69C2×D1004C2311yescharacteristic
70C2×C22⋊D251—21no—
71C2×C22⋊D251—21no—
72C2×C22⋊D251—21no—
73C2×C22⋊D251—21no—
74C2×C22⋊D251—21no—
75C2×C22⋊D251—21no—
76C2×C22⋊D251D411yescharacteristic
77D4×D251—22no—
78D4×D251—41no—
79C22×Dic251—21no—
80C22×Dic251—21no—
81C2×C4×D251—21no—
82C2×C4×D251C2311yescharacteristic
83#16860800—22no—
84#16860799—22no—
85#16860798D414yescomplemented
86#16860797—22no—
87D2008—41no—
88#16860795—22no—
89#16860795—22no—
90#16860793—22no—
91C2×C251⋊C8—22no—
92C2×C251⋊C8C2311yescharacteristic
93C2×C251⋊C8C2311yescharacteristic
94C4⋊Dic251C2311yescharacteristic
95Dic251⋊C4—21no—
96Dic251⋊C4—21no—
97Dic251⋊C4D411yescharacteristic
98C22⋊Dic251—21no—
99C22⋊Dic251D411yescharacteristic
100(C2×D251)⋊C4—21no—
101(C2×D251)⋊C4—21no—
102(C2×D251)⋊C4—21no—
103(C2×D251)⋊C4—21no—
104(C2×D251)⋊C4—21no—
105(C2×D251)⋊C4D411yescharacteristic
106C4×Dic251—21no—
107C4×Dic251C2311yescharacteristic
108#41×C251—22no—
109#41×C251—41no—
110C2×Q8×C251C2311yescharacteristic
111C2×D4×C251—21no—
112C2×D4×C251—21no—
113C2×D4×C251C2311yescharacteristic
114C2×D4×C251C2311yescharacteristic
115C22×C4×C251—21no—
116C22×C4×C251C2311yescharacteristic
117SD16×C251—41no—
118D8×C251—22no—
119C2×C8×C251C2311yescharacteristic
120C2×C8×C251C2311yescharacteristic
121C4⋊C4×C251C2311yescharacteristic
122C22⋊C4×C251—21no—
123C22×D251—21no—
124C22×D251—21no—
125C22×D251—42no—
126C22×D251—42no—
127C22×D251—81no—
128C22×D251C2×D411yescharacteristic
129Dic502—21no—
130Dic502C2×D412yes—
131D1004—21no—
132D1004—21no—
133D1004—21no—
134D1004—21no—
135D1004—81no—
136C22⋊D251—41no—
137C22⋊D251—42no—
138C22⋊D251—42no—
139C22⋊D251—42no—
140C22⋊D251—42no—
141C22⋊D251—81no—
142C22⋊D251—81no—
143C2×Dic251—21no—
144C2×Dic251—21no—
145C2×Dic251—42no—
146C2×Dic251—81no—
147C2×Dic251C2×D411yescharacteristic
148C2×Dic251C2×D411yescharacteristic
149C2×Dic251C2×D411yescharacteristic
150C4×D251—21no—
151C4×D251—21no—
152C4×D251—42no—
153C251⋊C8—21no—
154C251⋊C8C2×D412yes—
155C23×C251—21no—
156C23×C251—21no—
157C23×C251C2×D411yescharacteristic
158Q8×C251—21no—
159Q8×C251—21no—
160D4×C251—21no—
161D4×C251—21no—
162D4×C251—21no—
163D4×C251—42no—
164D4×C251—81no—
165D4×C251C2×D412yes—
166C2×C4×C251—21no—
167C2×C4×C251—21no—
168C2×C4×C251—21no—
169C2×C4×C251—21no—
170C2×C4×C251—42no—
171C2×C4×C251C2×D411yescharacteristic
172C2×C4×C251C2411yescharacteristic
173C8×C251—21no—
174C8×C251—21no—
175C22×C251—21no—
176C22×C251—22no—
177C22×C251—21no—
178C22×C251—21no—
179C22×C251—41no—
180C22×C251—41no—
181C22×C251—41no—
182C22×C251—41no—
183C22×C251—41no—
184C22×C251C22×D411yescharacteristic
185C2×D251—21no—
186C2×D251—21no—
187C2×D251—21no—
188C2×D251—41no—
189C2×D251—41no—
190C2×D251—41no—
191C2×D251—41no—
192C2×D251—41no—
193C2×D251—41no—
194C2×D251—84no—
195C4×C251—21no—
196C4×C251—41no—
197C4×C251—41no—
198C4×C251C22×D411yescharacteristic
199C4×C251H(5,𝔽2)11yescharacteristic
200Dic251—21no—
201Dic251—22no—
202Dic251—21no—
203Dic251—21no—
204Dic251—41no—
205C2×C251—42no—
206C2×C251—41no—
207C2×C251—81no—
208C2×C251D4211yescharacteristic
209C2×C251#57511yescharacteristic
210C2×C251#57611yescharacteristic
211D251—41no—
212D251—81no—
213D251—81no—
214C251#326811yescharacteristic, complemented, Sylow
215#3268—2511nomaximal, Sylow
216C2×AGL(1,ℤ/8)—2511no—
217C2×#135—2511no—
218#550—2511no—
219#546—2511no—
220#496—2511no—
221#468—2511no—
222#466—2511no—
223#464—2511no—
224#459—2511no—
225#452—2511no—
226#449—2511no—
227#449D25111yescharacteristic, complemented
228#447—2511no—
229#440—2511no—
230#435—2511no—
231C2×#41—2511no—
232C2×#41—2511no—
233C2×#41C2×D25111yescharacteristic
234C22×D4—2511no—
235AGL(1,ℤ/8)—5024no—
236#135—5024no—
237C2×Q16—2511no—
238C2×SD16—2511no—
239C2×SD16—2511no—
240C2×SD16—2511no—
241C2×SD16C2×D25111yescharacteristic, complemented
242C2×D8—2511no—
243C2×D8—2511no—
244C2×D8—2511no—
245C2×D8C2×D25111yescharacteristic
246C2×M16—2511no—
247C22×C8—2511no—
248#127—5021no—
249#127—2511no—
250#124—2511no—
251#123—5021no—
252C22⋊Q8—2511no—
253#121—5021no—
254#121—5021no—
255#121—5021no—
256#121—2511no—
257#121—2511no—
258#121C2×D25111yescharacteristic, complemented
259#120—5021no—
260C4×Q8—2511no—
261C4×D4—5021no—
262C4×D4—2511no—
263C4×D4—2511no—
264C2×C22⋊C4—5021no—
265#107—2511no—
266C4⋊C8—2511no—
267#103—2511no—
268#103C2×D25111yescharacteristic, complemented
269#102—2511no—
270#102—2511no—
271#102—2511no—
272#102C2×D25111yescharacteristic
273C22⋊C8—2511no—
274C22⋊C8C2×D25111yescharacteristic, complemented
275#97—2511no—
276C24—5021no—
277#41—22no—
278#41—41no—
279#41—10041no—
280#41—10041no—
281#41—5022no—
282#41—5022no—
283C2×Q8—2511no—
284C2×Q8C22×D25111yescharacteristic
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—10041no—
288C2×D4—5021no—
289C2×D4—5021no—
290C2×D4—5021no—
291C2×D4—5021no—
292C2×D4—5022no—
293C2×D4—5021no—
294C2×D4—5021no—
295C2×D4—5021no—
296C2×D4—2511no—
297C2×D4—2511no—
298C2×D4—2511no—
299C2×D4C22×D25111yescharacteristic
300C2×D4C22×D25111yescharacteristic
301C22×C4—21no—
302C22×C4—5021no—
303C22×C4—5021no—
304C22×C4—5021no—
305C22×C4—2511no—
306C22×C4C22×D25111yescharacteristic
307Q16—5022no—
308SD16—41no—
309SD16—5022no—
310SD16—5022no—
311SD16—2514no—
312D8—22no—
313D8—10041no—
314D8—5022no—
315D8—5022no—
316M16—5022no—
317C2×C8—5022no—
318C2×C8—2511no—
319C2×C8—2511no—
320C2×C8C22×D25111yescharacteristic
321C2×C8C22×D25111yescharacteristic
322C4⋊C4—5021no—
323C4⋊C4—5021no—
324C4⋊C4—2511no—
325C4⋊C4—2511no—
326C4⋊C4C22×D25111yescharacteristic
327C22⋊C4—21no—
328C22⋊C4—5021no—
329C22⋊C4—5021no—
330C22⋊C4—5021no—
331C22⋊C4—5021no—
332C22⋊C4—5021no—
333C22⋊C4—5021no—
334C22⋊C4—2511no—
335C22⋊C4—2511no—
336C42—5021no—
337C42—2511no—
338C23—21no—
339C23—21no—
340C23—20081no—
341C23—10042no—
342C23—10042no—
343C23—5021no—
344C23—5021no—
345C23—2511no—
346C23D4×D25111yescharacteristic
347Q8—21no—
348Q8—21no—
349Q8—5021no—
350Q8—2512no—
351D4—21no—
352D4—21no—
353D4—21no—
354D4—42no—
355D4—81no—
356D4—20081no—
357D4—20081no—
358D4—20081no—
359D4—10042no—
360D4—10042no—
361D4—10041no—
362D4—10042no—
363D4—10042no—
364D4—5021no—
365D4—5021no—
366D4—5021no—
367D4—5021no—
368D4D4×D25112yes—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—42no—
374C2×C4—20081no—
375C2×C4—10042no—
376C2×C4—10042no—
377C2×C4—5021no—
378C2×C4—5021no—
379C2×C4—5021no—
380C2×C4—5021no—
381C2×C4—2511no—
382C2×C4—2511no—
383C2×C4—2511no—
384C2×C4D4×D25111yescharacteristic
385C2×C4C23×D25111yescharacteristic
386C8—21no—
387C8—21no—
388C8—5021no—
389C8—2512no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—22no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—20084no—
400C22—10041no—
401C22—10041no—
402C22—10041no—
403C22—10041no—
404C22—10041no—
405C22—10041no—
406C22—5021no—
407C22—5021no—
408C22—5021no—
409C22C2×D4×D25111yescharacteristic
410C4—21no—
411C4—41no—
412C4—41no—
413C4—10041no—
414C4—5021no—
415C4—5021no—
416C4—5021no—
417C4—5022no—
418C4C2×D4×D25111yescharacteristic
419C4#2254876011yescharacteristic
420C2—41no—
421C2—42no—
422C2—81no—
423C2—20081no—
424C2—20081no—
425C2—10041no—
426C2#3063231911yescharacteristic
427C2#3063248411yescharacteristic
428C2#3063249111yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#449×C251C2
socleC22×C251C22×D4
hypercenter#449D251
hypocenterC251#3268
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. C2×#22548739C2semidirect
2. C2×#22548735C2semidirect
3. #30632355C2semidirect
4. #30632325C2semidirect
5. #30632073C2semidirect
6. #30631936C2semidirect
7. #30631921C2semidirect
8. #30631912C2semidirect
9. #30631880C2semidirect
10. #30631846C2semidirect
11. #30631834C2semidirect
12. #30631822C2semidirect
13. #30631784C2semidirect
14. #30631755C2semidirect
15. #449×C251C2semidirect
16. C2×#16860810C22semidirect
21. C2×#16860800C22semidirect
24. C2×#16860797C22semidirect
29. C22×C251⋊C8C22semidirect
31. #22548711C22semidirect
41. #22548680C22semidirect
46. #22548648C22semidirect
47. Dic251⋊C8C22semidirect
48. #22548640C22semidirect
49. #22548637C22semidirect
50. #22548636C22semidirect
53. #22548625C22semidirect
55. C2×SD16×C251C22semidirect
57. #121×C251C22semidirect
58. #103×C251C22semidirect
60. C22⋊C8×C251C22semidirect
85. #16860798D4semidirect
214. C251#3268semidirect
227. #449D251semidirect
241. C2×SD16C2×D251semidirect
258. #121C2×D251semidirect
268. #103C2×D251semidirect
274. C22⋊C8C2×D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225487392
C2×#225487352
#306323552
#306323252
#306320732
#306319362
#306319212
#306319122
#306318802
#306318462
#306318342
#306318222
#306317842
#306317552
#449×C2512
#3268251