Finite Group Databasebeta

← #44560556

Subgroups — #44560556

Order 32128.

406 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#30632311C211yesmaximal, characteristic, complemented
2#30632159C211yesmaximal, characteristic, complemented
3#30632030C211yesmaximal, characteristic, complemented
4#30632014C211yesmaximal, characteristic, complemented
5#30631974C211yesmaximal, characteristic, complemented
6#30631903C211yesmaximal, characteristic, complemented
7#30631902C211yesmaximal, characteristic, complemented
8#30631839C211yesmaximal, characteristic, complemented
9#30631822C211yesmaximal, characteristic
10#30631796C211yesmaximal, characteristic, complemented
11#30631786C211yesmaximal, characteristic, complemented
12#30631778C211yesmaximal, characteristic, complemented
13#30631769C211yesmaximal, characteristic, complemented
14#30631747C211yesmaximal, characteristic, complemented
15#474×C251C211yesmaximal, characteristic, complemented
16C2×D4×D251C2211yescharacteristic
17C2×#16860799C2211yescharacteristic
18C2×#16860798C2211yescharacteristic
19C2×D2008C2211yescharacteristic
20C2×#16860795C2211yescharacteristic
21#22548710—21no—
22#22548704—21no—
23#22548703C2211yescharacteristic, complemented
24#22548700—21no—
25#22548695C2211yescharacteristic
26#22548691—21no—
27#22548690C2211yescharacteristic
28#22548680C2211yescharacteristic, complemented
29C4×D1004C2211yescharacteristic
30D4×Dic251C2211yescharacteristic
31#22548677C2211yescharacteristic
32#22548677C2211yescharacteristic, complemented
33C4×C22⋊D251—21no—
34(C22⋊D251)⋊C4—21no—
35#22548673C2211yescharacteristic
36#22548672C2211yescharacteristic
37C2×(C2×D251)⋊C4—21no—
38C22⋊C4×D251—21no—
39#22548646C2211yescharacteristic, complemented
40#22548646C2211yescharacteristic, complemented
41C4⋊(C251⋊C8)C2211yescharacteristic, complemented
42#22548640C2211yescharacteristic, complemented
43#22548638C2211yescharacteristic, complemented
44#22548637C2211yescharacteristic, complemented
45#22548635C2211yescharacteristic
46#22548635C2211yescharacteristic
47#22548635C2211yescharacteristic, complemented
48(C2×D251)⋊C8C2211yescharacteristic
49(C2×D251)⋊C8C2211yescharacteristic
50#22548625C2211yescharacteristic, complemented
51C8×Dic251C2211yescharacteristic, complemented
52C4⋊Q8×C251C2211yescharacteristic, complemented
53C4×D4×C251C2211yescharacteristic
54#107×C251C2211yescharacteristic, complemented
55#107×C251C2211yescharacteristic, complemented
56C4⋊C8×C251C2211yescharacteristic, complemented
57#102×C251C2211yescharacteristic
58#102×C251C2211yescharacteristic
59C23×D251—21no—
60C2×Dic502C2311yescharacteristic
61C2×D1004—21no—
62C2×D1004C2311yescharacteristic
63C2×D1004C2311yescharacteristic
64C2×C22⋊D251—21no—
65D4×D251—22no—
66D4×D251—22no—
67C22×Dic251—21no—
68C2×C4×D251—21no—
69C2×C4×D251—21no—
70C2×C4×D251—21no—
71C2×C4×D251C2311yescharacteristic
72#16860799—22no—
73#16860798—22no—
74D2008—22no—
75#16860795—22no—
76C2×C251⋊C8C2311yescharacteristic
77C2×C251⋊C8C2311yescharacteristic
78C4⋊Dic251—21no—
79C4⋊Dic251C2311yescharacteristic
80Dic251⋊C4—21no—
81Dic251⋊C4D411yescharacteristic
82Dic251⋊C4D411yescharacteristic
83C22⋊Dic251—21no—
84(C2×D251)⋊C4—21no—
85(C2×D251)⋊C4—21no—
86(C2×D251)⋊C4—21no—
87(C2×D251)⋊C4—21no—
88(C2×D251)⋊C4—21no—
89(C2×D251)⋊C4—21no—
90(C2×D251)⋊C4—21no—
91(C2×D251)⋊C4—21no—
92(C2×D251)⋊C4D411yescharacteristic
93(C2×D251)⋊C4D411yescharacteristic
94C4×Dic251—21no—
95C4×Dic251—21no—
96C4×Dic251C2311yescharacteristic
97C2×Q8×C251—21no—
98C2×D4×C251C2311yescharacteristic
99C22×C4×C251—21no—
100C2×C8×C251C2311yescharacteristic
101C2×C8×C251C2311yescharacteristic
102C4⋊C4×C251—21no—
103C4⋊C4×C251C2311yescharacteristic
104C4⋊C4×C251C2311yescharacteristic
105C4⋊C4×C251C2311yescharacteristic
106C22⋊C4×C251—21no—
107C42×C251C2311yescharacteristic
108C22×D251—21no—
109C22×D251—24no—
110C22×D251—21no—
111C22×D251—81no—
112C22×D251—81no—
113C22×D251C2×D411yescharacteristic
114Dic502—21no—
115Dic502—21no—
116D1004—21no—
117D1004—21no—
118D1004—21no—
119D1004—81no—
120D1004#4112yes—
121C22⋊D251—24no—
122C2×Dic251—21no—
123C2×Dic251—21no—
124C2×Dic251—81no—
125C2×Dic251C2×D411yescharacteristic
126C2×Dic251C2×D411yescharacteristic
127C2×Dic251C2×D411yescharacteristic
128C4×D251—21no—
129C4×D251—21no—
130C4×D251—81no—
131C4×D251—81no—
132C4×D251—81no—
133C251⋊C8—21no—
134C251⋊C8—21no—
135C23×C251—21no—
136Q8×C251—81no—
137D4×C251—21no—
138D4×C251#4112yes—
139C2×C4×C251—21no—
140C2×C4×C251—21no—
141C2×C4×C251—21no—
142C2×C4×C251—81no—
143C2×C4×C251C2×D411yescharacteristic
144C2×C4×C251C2×D411yescharacteristic
145C2×C4×C251C2411yescharacteristic
146C8×C251—21no—
147C8×C251—21no—
148C22×C251—22no—
149C22×C251—41no—
150C22×C251—41no—
151C22×C251C22×D411yescharacteristic
152C2×D251—21no—
153C2×D251—22no—
154C2×D251—21no—
155C2×D251—21no—
156C2×D251—41no—
157C2×D251—41no—
158C2×D251—41no—
159C2×D251—41no—
160C2×D251—41no—
161C2×D251—84no—
162C4×C251—21no—
163C4×C251—21no—
164C4×C251—41no—
165C4×C251—41no—
166C4×C251—41no—
167C4×C251C2×#4111yescharacteristic
168C4×C251C2×#4111yescharacteristic
169Dic251—21no—
170Dic251—21no—
171Dic251—21no—
172Dic251—41no—
173Dic251—41no—
174C2×C251—42no—
175C2×C251#52511yescharacteristic
176C2×C251#57611yescharacteristic
177C2×C251#57711yescharacteristic
178D251—41no—
179D251—42no—
180D251—81no—
181C251#311811yescharacteristic, complemented, Sylow
182#3118—2511nomaximal, Sylow
183#542—2511no—
184#518—2511no—
185#485—2511no—
186#482—2511no—
187#476—2511no—
188#474D25111yescharacteristic, complemented
189#463—2511no—
190#463—2511no—
191#450—2511no—
192#447—2511no—
193#442—2511no—
194#440—2511no—
195C4×SD16—2511no—
196C4×D8—2511no—
197#433—2511no—
198C22×D4—2511no—
199C2×SD16—2511no—
200C2×SD16—2511no—
201C2×D8—2511no—
202C2×D8—2511no—
203C4⋊Q8C2×D25111yescharacteristic, complemented
204#126—5021no—
205#124—5021no—
206#124—2511no—
207#123—5021no—
208C22⋊Q8—5021no—
209C22⋊Q8—2511no—
210#121—2511no—
211C4×Q8—2511no—
212C4×D4—5021no—
213C4×D4—5021no—
214C4×D4—2511no—
215C4×D4—2511no—
216C4×D4—2511no—
217C4×D4—2511no—
218C4×D4C2×D25111yescharacteristic
219#117—2511no—
220#117—2511no—
221C2×C22⋊C4—5021no—
222C2×C22⋊C4—5021no—
223#107C2×D25111yescharacteristic, complemented
224#107C2×D25111yescharacteristic, complemented
225#106—2511no—
226#106—2511no—
227C4⋊C8—2511no—
228C4⋊C8C2×D25111yescharacteristic, complemented
229#103—2511no—
230#103—2511no—
231#102—2511no—
232#102—2511no—
233#102—2511no—
234#102—2511no—
235#102C2×D25111yescharacteristic
236#102C2×D25111yescharacteristic
237C22⋊C8—2511no—
238C22⋊C8—2511no—
239#97—2511no—
240C4×C8—2511no—
241C24—5021no—
242C2×Q8—21no—
243C2×Q8—2511no—
244C2×D4—5021no—
245C2×D4—5022no—
246C2×D4—5022no—
247C2×D4—5021no—
248C2×D4—2511no—
249C2×D4—2511no—
250C2×D4C22×D25111yescharacteristic
251C22×C4—21no—
252C22×C4—5021no—
253C22×C4—5021no—
254C22×C4—5021no—
255C22×C4—5021no—
256C22×C4—2511no—
257SD16—5022no—
258SD16—5022no—
259D8—5022no—
260D8—5022no—
261C2×C8—2511no—
262C2×C8—2511no—
263C2×C8C22×D25111yescharacteristic
264C2×C8C22×D25111yescharacteristic
265C4⋊C4—21no—
266C4⋊C4—5021no—
267C4⋊C4—5021no—
268C4⋊C4—2511no—
269C4⋊C4—2511no—
270C4⋊C4—2511no—
271C4⋊C4C22×D25111yescharacteristic
272C4⋊C4C22×D25111yescharacteristic
273C4⋊C4C22×D25111yescharacteristic
274C22⋊C4—21no—
275C22⋊C4—5021no—
276C22⋊C4—5021no—
277C22⋊C4—5021no—
278C22⋊C4—5021no—
279C22⋊C4—5021no—
280C22⋊C4—5021no—
281C22⋊C4—5021no—
282C22⋊C4—5021no—
283C22⋊C4—5021no—
284C22⋊C4—2511no—
285C22⋊C4—2511no—
286C42—5021no—
287C42—5021no—
288C42—2511no—
289C42C22×D25111yescharacteristic
290C23—21no—
291C23—20081no—
292C23—20081no—
293C23—5024no—
294C23—5021no—
295C23—5021no—
296C23—2511no—
297Q8—81no—
298Q8—5021no—
299Q8—5021no—
300D4—21no—
301D4—20081no—
302D4—5024no—
303D4—5021no—
304D4—5021no—
305D4—5021no—
306D4—2512no—
307D4#1686081112yes—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—81no—
312C2×C4—20081no—
313C2×C4—20081no—
314C2×C4—20081no—
315C2×C4—20081no—
316C2×C4—5021no—
317C2×C4—5021no—
318C2×C4—5021no—
319C2×C4—5021no—
320C2×C4—2511no—
321C2×C4—2511no—
322C2×C4—2511no—
323C2×C4D4×D25111yescharacteristic
324C2×C4D4×D25111yescharacteristic
325C2×C4C23×D25111yescharacteristic
326C8—21no—
327C8—21no—
328C8—5021no—
329C8—5021no—
330C22—22no—
331C22—41no—
332C22—41no—
333C22—20084no—
334C22—10041no—
335C22—10041no—
336C22—10041no—
337C22—10041no—
338C22—10041no—
339C22—5021no—
340C22—5022no—
341C22—5021no—
342C22—5021no—
343C22C2×D4×D25111yescharacteristic
344C4—21no—
345C4—21no—
346C4—41no—
347C4—41no—
348C4—41no—
349C4—10041no—
350C4—10041no—
351C4—5021no—
352C4—5021no—
353C4—5021no—
354C4#41×D25111yescharacteristic
355C4C2×#1686081111yescharacteristic
356C2—42no—
357C2—20081no—
358C2—10041no—
359C2—10042no—
360C2#3063220011yescharacteristic
361C2#3063248811yescharacteristic
362C2#3063249611yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#474×C251C2
socleC22×C251C22×D4
prime radical#30631822C2
hypercenter#474D251
hypocenterC251#3118
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ #474

Complements

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

NH ≅ G/Ntype
1. #30632311C2semidirect
2. #30632159C2semidirect
3. #30632030C2semidirect
4. #30632014C2semidirect
5. #30631974C2semidirect
6. #30631903C2semidirect
7. #30631902C2semidirect
8. #30631839C2semidirect
10. #30631796C2semidirect
11. #30631786C2semidirect
12. #30631778C2semidirect
13. #30631769C2semidirect
14. #30631747C2semidirect
15. #474×C251C2semidirect
23. #22548703C22semidirect
28. #22548680C22semidirect
32. #22548677C22semidirect
39. #22548646C22semidirect
40. #22548646C22semidirect
41. C4⋊(C251⋊C8)C22semidirect
42. #22548640C22semidirect
43. #22548638C22semidirect
44. #22548637C22semidirect
47. #22548635C22semidirect
50. #22548625C22semidirect
51. C8×Dic251C22semidirect
52. C4⋊Q8×C251C22semidirect
54. #107×C251C22semidirect
55. #107×C251C22semidirect
56. C4⋊C8×C251C22semidirect
181. C251#3118semidirect
188. #474D251semidirect
203. C4⋊Q8C2×D251semidirect
223. #107C2×D251semidirect
224. #107C2×D251semidirect
228. C4⋊C8C2×D251semidirect

Maximal subgroups

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

subgroupindex
#306323112
#306321592
#306320302
#306320142
#306319742
#306319032
#306319022
#306318392
#306318222
#306317962
#306317862
#306317782
#306317692
#306317472
#474×C2512
#3118251