Finite Group Databasebeta

← #44560549

Subgroups — #44560549

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#30632308C211yesmaximal, characteristic, complemented
2#30632028C211yesmaximal, characteristic, complemented
3#30632011C211yesmaximal, characteristic, complemented
4#30631977C211yesmaximal, characteristic, complemented
5#30631967C211yesmaximal, characteristic, complemented
6#30631902C211yesmaximal, characteristic, complemented
7#30631902C211yesmaximal, characteristic, complemented
8#30631838C211yesmaximal, characteristic, complemented
9#30631821C211yesmaximal, characteristic
10#30631797C211yesmaximal, characteristic, complemented
11#30631787C211yesmaximal, characteristic, complemented
12C4×#16860798C211yesmaximal, characteristic, complemented
13C4×#16860795C211yesmaximal, characteristic, complemented
14#30631748C211yesmaximal, characteristic, complemented
15#518×C251C211yesmaximal, characteristic, complemented
16C2×#16860798C2211yescharacteristic
17C2×#16860798C2211yescharacteristic
18C2×#16860795C2211yescharacteristic
19C2×#16860795C2211yescharacteristic
20#22548715C2211yescharacteristic, complemented
21#22548710—21no—
22#22548703C2211yescharacteristic, complemented
23#22548701—21no—
24#22548697—21no—
25#22548694C2211yescharacteristic
26#22548693—21no—
27#22548689C2211yescharacteristic
28C4×Dic502C2211yescharacteristic, complemented
29C4×D1004C2211yescharacteristic, complemented
30#22548648C2211yescharacteristic, complemented
31#22548648C2211yescharacteristic, complemented
32#22548646C2211yescharacteristic, complemented
33#22548646C2211yescharacteristic, complemented
34C4⋊(C251⋊C8)C2211yescharacteristic, complemented
35C4⋊(C251⋊C8)C2211yescharacteristic, complemented
36#22548638C2211yescharacteristic, complemented
37#22548638C2211yescharacteristic, complemented
38#22548636C2211yescharacteristic
39#22548636C2211yescharacteristic
40#22548636C2211yescharacteristic
41#22548636C2211yescharacteristic
42#22548635C2211yescharacteristic, complemented
43#22548635C2211yescharacteristic, complemented
44C22⋊(C251⋊C8)C2211yescharacteristic
45C22⋊(C251⋊C8)C2211yescharacteristic
46#22548626C2211yescharacteristic, complemented
47C4×C251⋊C8C2211yescharacteristic, complemented
48C22×D4×C251C2211yescharacteristic
49C4×D4×C251—21no—
50C4×D4×C251—21no—
51C4×D4×C251C2211yescharacteristic
52C4×D4×C251C2211yescharacteristic
53C4×D4×C251C2211yescharacteristic
54C4×D4×C251C2211yescharacteristic
55#117×C251C2211yescharacteristic
56#117×C251C2211yescharacteristic
57C2×C22⋊C4×C251—21no—
58C2×C22⋊C4×C251—21no—
59C2×Dic502—21no—
60C2×Dic502C2311yescharacteristic
61C2×D1004C2311yescharacteristic
62C2×C22⋊D251—21no—
63C2×C4×D251—21no—
64#16860798—22no—
65#16860798—22no—
66#16860795—22no—
67#16860795—22no—
68C2×C251⋊C8C2311yescharacteristic
69C2×C251⋊C8C2311yescharacteristic
70C2×C251⋊C8C2311yescharacteristic
71C2×C251⋊C8C2311yescharacteristic
72C4⋊Dic251—21no—
73C4⋊Dic251C2311yescharacteristic
74C4⋊Dic251C2311yescharacteristic
75Dic251⋊C4—21no—
76Dic251⋊C4—21no—
77C22⋊Dic251—21no—
78(C2×D251)⋊C4—21no—
79(C2×D251)⋊C4—21no—
80(C2×D251)⋊C4—21no—
81(C2×D251)⋊C4—21no—
82C4×Dic251—21no—
83C24×C251—21no—
84C2×D4×C251—22no—
85C2×D4×C251—22no—
86C2×D4×C251—21no—
87C2×D4×C251C2311yescharacteristic
88C2×D4×C251C2311yescharacteristic
89C22×C4×C251—21no—
90C22×C4×C251—21no—
91C22×C4×C251—21no—
92C22×C4×C251—21no—
93C22×C4×C251C2311yescharacteristic
94C4⋊C4×C251D411yescharacteristic
95C4⋊C4×C251D411yescharacteristic
96C4⋊C4×C251C2311yescharacteristic
97C4⋊C4×C251C2311yescharacteristic
98C22⋊C4×C251—21no—
99C22⋊C4×C251—21no—
100C22⋊C4×C251—21no—
101C22⋊C4×C251—21no—
102C22⋊C4×C251—21no—
103C22⋊C4×C251D411yescharacteristic
104C22⋊C4×C251D411yescharacteristic
105C42×C251—21no—
106C42×C251C2311yescharacteristic
107C42×C251C2311yescharacteristic
108C22×D251—21no—
109Dic502—21no—
110Dic502—21no—
111Dic502—81no—
112D1004—21no—
113D1004—21no—
114C22⋊D251—81no—
115C2×Dic251—21no—
116C2×Dic251—21no—
117C2×Dic251—21no—
118C4×D251—81no—
119C251⋊C8—21no—
120C251⋊C8—21no—
121C251⋊C8—21no—
122C251⋊C8—21no—
123C23×C251—24no—
124C23×C251—21no—
125C23×C251—21no—
126C23×C251—81no—
127C23×C251—81no—
128C23×C251C2×D411yescharacteristic
129D4×C251—24no—
130D4×C251—21no—
131D4×C251—21no—
132D4×C251#4112yes—
133D4×C251#4112yes—
134C2×C4×C251—21no—
135C2×C4×C251—21no—
136C2×C4×C251—21no—
137C2×C4×C251—21no—
138C2×C4×C251—81no—
139C2×C4×C251—81no—
140C2×C4×C251—81no—
141C2×C4×C251—81no—
142C2×C4×C251C2×D411yescharacteristic
143C2×C4×C251C2×D411yescharacteristic
144C2×C4×C251C2×D411yescharacteristic
145C2×C4×C251C2×D411yescharacteristic
146C2×C4×C251C2×D411yescharacteristic
147C2×C4×C251C2411yescharacteristic
148C22×C251—21no—
149C22×C251—22no—
150C22×C251—22no—
151C22×C251—21no—
152C22×C251—21no—
153C22×C251—41no—
154C22×C251—41no—
155C22×C251—41no—
156C22×C251—41no—
157C22×C251—84no—
158C22×C251C22×D411yescharacteristic
159C2×D251—41no—
160C2×D251—41no—
161C2×D251—41no—
162C4×C251—21no—
163C4×C251—21no—
164C4×C251—21no—
165C4×C251—21no—
166C4×C251—21no—
167C4×C251—41no—
168C4×C251—41no—
169C4×C251C2×#4111yescharacteristic
170C4×C251C2×#4111yescharacteristic
171Dic251—41no—
172Dic251—41no—
173Dic251—41no—
174C2×C251—41no—
175C2×C251—42no—
176C2×C251—42no—
177C2×C251#52511yescharacteristic
178C2×C251#57611yescharacteristic
179C2×C251#57711yescharacteristic
180D251—81no—
181C251#311811yescharacteristic, complemented, Sylow
182#3118—2511nomaximal, Sylow
183#542—2511no—
184#518D25111yescharacteristic, complemented
185#485—2511no—
186#482—2511no—
187#476—2511no—
188#474—2511no—
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×D4C2×D25111yescharacteristic
199C2×SD16—2511no—
200C2×SD16—2511no—
201C2×D8—2511no—
202C2×D8—2511no—
203C4⋊Q8—2511no—
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—21no—
213C4×D4—21no—
214C4×D4—2511no—
215C4×D4C2×D25111yescharacteristic
216C4×D4C2×D25111yescharacteristic
217C4×D4C2×D25111yescharacteristic
218C4×D4C2×D25111yescharacteristic
219#117C2×D25111yescharacteristic
220#117C2×D25111yescharacteristic
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223#107—2511no—
224#107—2511no—
225#106—2511no—
226#106—2511no—
227C4⋊C8—2511no—
228C4⋊C8—2511no—
229#103—2511no—
230#103—2511no—
231#102—2511no—
232#102—2511no—
233#102—2511no—
234#102—2511no—
235#102—2511no—
236#102—2511no—
237C22⋊C8—2511no—
238C22⋊C8—2511no—
239#97—2511no—
240C4×C8—2511no—
241C24—21no—
242C2×Q8—5021no—
243C2×Q8—2511no—
244C2×D4—22no—
245C2×D4—22no—
246C2×D4—21no—
247C2×D4—5021no—
248C2×D4—2511no—
249C2×D4C22×D25111yescharacteristic
250C2×D4C22×D25111yescharacteristic
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—5021no—
256C22×C4C22×D25111yescharacteristic
257SD16—5022no—
258SD16—5022no—
259D8—5022no—
260D8—5022no—
261C2×C8—2511no—
262C2×C8—2511no—
263C2×C8—2511no—
264C2×C8—2511no—
265C4⋊C4—5021no—
266C4⋊C4—5021no—
267C4⋊C4—5021no—
268C4⋊C4—2511no—
269C4⋊C4—2511no—
270C4⋊C4C22⋊D25111yescharacteristic
271C4⋊C4C22⋊D25111yescharacteristic
272C4⋊C4C22×D25111yescharacteristic
273C4⋊C4C22×D25111yescharacteristic
274C22⋊C4—21no—
275C22⋊C4—21no—
276C22⋊C4—21no—
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4—5021no—
280C22⋊C4—5021no—
281C22⋊C4—5021no—
282C22⋊C4—5021no—
283C22⋊C4—5021no—
284C22⋊C4C22⋊D25111yescharacteristic
285C22⋊C4C22⋊D25111yescharacteristic
286C42—21no—
287C42—5021no—
288C42C22×D25111yescharacteristic
289C42C22×D25111yescharacteristic
290C23—24no—
291C23—21no—
292C23—21no—
293C23—81no—
294C23—81no—
295C23—5021no—
296C23C2×C22⋊D25111yescharacteristic
297Q8—20081no—
298Q8—5021no—
299Q8—5021no—
300D4—24no—
301D4—21no—
302D4—21no—
303D4—20081no—
304D4—5021no—
305D4—5021no—
306D4#1686081012yes—
307D4#1686081012yes—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4—81no—
316C2×C4—20081no—
317C2×C4—5021no—
318C2×C4—5021no—
319C2×C4—5021no—
320C2×C4C2×C22⋊D25111yescharacteristic
321C2×C4C2×C22⋊D25111yescharacteristic
322C2×C4C2×C22⋊D25111yescharacteristic
323C2×C4C2×C22⋊D25111yescharacteristic
324C2×C4C2×C22⋊D25111yescharacteristic
325C2×C4C23×D25111yescharacteristic
326C8—5021no—
327C8—5021no—
328C8—5021no—
329C8—5021no—
330C22—21no—
331C22—22no—
332C22—22no—
333C22—21no—
334C22—21no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22—84no—
340C22—10041no—
341C22—10041no—
342C22—10041no—
343C22C22×C22⋊D25111yescharacteristic
344C4—21no—
345C4—21no—
346C4—21no—
347C4—21no—
348C4—21no—
349C4—41no—
350C4—41no—
351C4—10041no—
352C4—10041no—
353C4—10041no—
354C4C2×#1686081011yescharacteristic
355C4C2×#1686081011yescharacteristic
356C2—41no—
357C2—42no—
358C2—42no—
359C2—20081no—
360C2#3063219811yescharacteristic
361C2#3063249011yescharacteristic
362C2#3063249811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#518×C251C2
socleC22×C251C22×D4
prime radical#30631821C2
hypercenter#518D251
hypocenterC251#3118
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ #117 ◁ #518

Complements

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

NH ≅ G/Ntype
1. #30632308C2semidirect
2. #30632028C2semidirect
3. #30632011C2semidirect
4. #30631977C2semidirect
5. #30631967C2semidirect
6. #30631902C2semidirect
7. #30631902C2semidirect
8. #30631838C2semidirect
10. #30631797C2semidirect
11. #30631787C2semidirect
12. C4×#16860798C2semidirect
13. C4×#16860795C2semidirect
14. #30631748C2semidirect
15. #518×C251C2semidirect
20. #22548715C22semidirect
22. #22548703C22semidirect
28. C4×Dic502C22semidirect
29. C4×D1004C22semidirect
30. #22548648C22semidirect
31. #22548648C22semidirect
32. #22548646C22semidirect
33. #22548646C22semidirect
34. C4⋊(C251⋊C8)C22semidirect
35. C4⋊(C251⋊C8)C22semidirect
36. #22548638C22semidirect
37. #22548638C22semidirect
42. #22548635C22semidirect
43. #22548635C22semidirect
46. #22548626C22semidirect
47. C4×C251⋊C8C22semidirect
181. C251#3118semidirect
184. #518D251semidirect

Maximal subgroups

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

subgroupindex
#306323082
#306320282
#306320112
#306319772
#306319672
#306319022
#306318382
#306318212
#306317972
#306317872
C4×#168607982
C4×#168607952
#306317482
#518×C2512
#3118251