Finite Group Databasebeta

← #44560616

Subgroups — #44560616

Order 32128.

394 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#30632295C211yesmaximal, characteristic, complemented
2#30632159C211yesmaximal, characteristic
3#30632030C211yesmaximal, characteristic, complemented
4#30632027C211yesmaximal, characteristic, complemented
5#30631990C211yesmaximal, characteristic, complemented
6#30631985C211yesmaximal, characteristic, complemented
7#30631891C211yesmaximal, characteristic, complemented
8#30631890C211yesmaximal, characteristic, complemented
9#30631839C211yesmaximal, characteristic
10#30631836C211yesmaximal, characteristic
11#30631789C211yesmaximal, characteristic, complemented
12C4×#16860797C211yesmaximal, characteristic, complemented
13C4×#16860798C211yesmaximal, characteristic, complemented
14#30631749C211yesmaximal, characteristic, complemented
15#440×C251C211yesmaximal, characteristic, complemented
16C2×D4×D251C2211yescharacteristic
17C2×#16860799C2211yescharacteristic
18C2×#16860798C2211yescharacteristic
19C2×#16860797C2211yescharacteristic
20#22548713C2211yescharacteristic, complemented
21#22548710—21no—
22#22548709—21no—
23#22548707—21no—
24#22548706C2211yescharacteristic, complemented
25#22548705—21no—
26#22548695C2211yescharacteristic
27#22548691C2211yescharacteristic
28C4×Dic502C2211yescharacteristic, complemented
29C4×D1004C2211yescharacteristic
30D4×Dic251C2211yescharacteristic
31#22548677C2211yescharacteristic
32C4×C22⋊D251—21no—
33(C22⋊D251)⋊C4—21no—
34#22548673C2211yescharacteristic
35#22548672C2211yescharacteristic
36C2×(C2×D251)⋊C4—21no—
37C22⋊C4×D251—21no—
38#22548649C2211yescharacteristic, complemented
39#22548646C2211yescharacteristic, complemented
40#22548646C2211yescharacteristic, complemented
41Dic251⋊C8C2211yescharacteristic, complemented
42Dic251⋊C8C2211yescharacteristic, complemented
43#22548640C2211yescharacteristic, complemented
44#22548639C2211yescharacteristic, complemented
45#22548638C2211yescharacteristic, complemented
46#22548637C2211yescharacteristic
47#22548636C2211yescharacteristic
48#22548635C2211yescharacteristic
49(C2×D251)⋊C8C2211yescharacteristic
50(C2×D251)⋊C8C2211yescharacteristic
51C4×C251⋊C8C2211yescharacteristic, complemented
52C2×SD16×C251C2211yescharacteristic
53C4×Q8×C251C2211yescharacteristic, complemented
54C4×D4×C251C2211yescharacteristic
55#107×C251C2211yescharacteristic, complemented
56#103×C251C2211yescharacteristic, complemented
57#102×C251C2211yescharacteristic
58#97×C251C2211yescharacteristic, complemented
59C23×D251—21no—
60C2×Dic502C2311yescharacteristic
61C2×D1004C2311yescharacteristic
62C2×C22⋊D251—21no—
63D4×D251—22no—
64D4×D251—22no—
65C22×Dic251—21no—
66C2×C4×D251—21no—
67C2×C4×D251—21no—
68C2×C4×D251C2311yescharacteristic
69#16860799—22no—
70#16860798—22no—
71#16860797—22no—
72C2×C251⋊C8C2311yescharacteristic
73C2×C251⋊C8C2311yescharacteristic
74C4⋊Dic251—21no—
75C4⋊Dic251—21no—
76C4⋊Dic251C2311yescharacteristic
77C4⋊Dic251C2311yescharacteristic
78Dic251⋊C4—21no—
79Dic251⋊C4—21no—
80Dic251⋊C4—21no—
81Dic251⋊C4—21no—
82Dic251⋊C4—21no—
83Dic251⋊C4D411yescharacteristic
84Dic251⋊C4D411yescharacteristic
85C22⋊Dic251—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)⋊C4D411yescharacteristic
92(C2×D251)⋊C4D411yescharacteristic
93C4×Dic251—21no—
94C4×Dic251—21no—
95C4×Dic251C2311yescharacteristic
96C2×Q8×C251C2311yescharacteristic
97C2×D4×C251C2311yescharacteristic
98C22×C4×C251—21no—
99SD16×C251—22no—
100C2×C8×C251C2311yescharacteristic
101C2×C8×C251C2311yescharacteristic
102C4⋊C4×C251—21no—
103C4⋊C4×C251C2311yescharacteristic
104C4⋊C4×C251C2311yescharacteristic
105C22⋊C4×C251—21no—
106C42×C251—21no—
107C42×C251C2311yescharacteristic
108C22×D251—21no—
109C22×D251—24no—
110C22×D251—81no—
111C22×D251—81no—
112C22×D251C2×D411yescharacteristic
113Dic502—21no—
114Dic502—21no—
115D1004—21no—
116D1004#4112yes—
117C22⋊D251—24no—
118C2×Dic251—21no—
119C2×Dic251—21no—
120C2×Dic251—21no—
121C2×Dic251—81no—
122C2×Dic251C2×D411yescharacteristic
123C2×Dic251C2×D411yescharacteristic
124C2×Dic251C2×D411yescharacteristic
125C4×D251—21no—
126C4×D251—21no—
127C4×D251—81no—
128C4×D251—81no—
129C251⋊C8—21no—
130C251⋊C8—21no—
131C23×C251—21no—
132Q8×C251—21no—
133Q8×C251—21no—
134D4×C251—21no—
135D4×C251#4112yes—
136C2×C4×C251—21no—
137C2×C4×C251—21no—
138C2×C4×C251—21no—
139C2×C4×C251—81no—
140C2×C4×C251C2×D411yescharacteristic
141C2×C4×C251C2×D411yescharacteristic
142C2×C4×C251C2411yescharacteristic
143C8×C251—21no—
144C8×C251—21no—
145C22×C251—22no—
146C22×C251—41no—
147C22×C251—41no—
148C22×C251C22×D411yescharacteristic
149C2×D251—21no—
150C2×D251—22no—
151C2×D251—21no—
152C2×D251—21no—
153C2×D251—41no—
154C2×D251—41no—
155C2×D251—84no—
156C4×C251—21no—
157C4×C251—21no—
158C4×C251—41no—
159C4×C251—41no—
160C4×C251—41no—
161C4×C251C2×#4111yescharacteristic
162C4×C251C2×#4111yescharacteristic
163Dic251—21no—
164Dic251—21no—
165Dic251—21no—
166Dic251—41no—
167Dic251—41no—
168Dic251—41no—
169C2×C251—42no—
170C2×C251#52511yescharacteristic
171C2×C251#57711yescharacteristic
172C2×C251#57711yescharacteristic
173D251—41no—
174D251—42no—
175C251#312311yescharacteristic, complemented, Sylow
176#3123—2511nomaximal, Sylow
177#541—2511no—
178#518—2511no—
179#485—2511no—
180#485—2511no—
181#478—2511no—
182#478—2511no—
183#461—2511no—
184#461—2511no—
185#450—2511no—
186#450—2511no—
187#440—2511no—
188#440D25111yescharacteristic, complemented
189C4×SD16—2511no—
190C4×SD16—2511no—
191#433—2511no—
192C22×D4—2511no—
193C2×SD16—2511no—
194C2×SD16—2511no—
195C2×SD16—2511no—
196C2×SD16C2×D25111yescharacteristic
197C4⋊Q8—2511no—
198#126—5021no—
199#126—5021no—
200#125—5021no—
201#125—5021no—
202#125—2511no—
203C22⋊Q8—2511no—
204C22⋊Q8—2511no—
205C4×Q8—2511no—
206C4×Q8C2×D25111yescharacteristic, complemented
207C4×D4—5021no—
208C4×D4—5021no—
209C4×D4—2511no—
210C4×D4—2511no—
211C4×D4—2511no—
212C4×D4C2×D25111yescharacteristic
213#117—2511no—
214#117—2511no—
215C2×C22⋊C4—5021no—
216C2×C22⋊C4—5021no—
217#107—2511no—
218#107C2×D25111yescharacteristic, complemented
219#106—2511no—
220#106—2511no—
221C4⋊C8—2511no—
222C4⋊C8—2511no—
223#103—2511no—
224#103—2511no—
225#103—2511no—
226#103C2×D25111yescharacteristic, complemented
227#102—2511no—
228#102—2511no—
229#102—2511no—
230#102C2×D25111yescharacteristic
231C22⋊C8—2511no—
232C22⋊C8—2511no—
233#97C2×D25111yescharacteristic, complemented
234C4×C8—2511no—
235C24—5021no—
236C2×Q8—2511no—
237C2×Q8C22×D25111yescharacteristic
238C2×D4—5022no—
239C2×D4—5022no—
240C2×D4—5021no—
241C2×D4—2511no—
242C2×D4C22×D25111yescharacteristic
243C22×C4—21no—
244C22×C4—5021no—
245C22×C4—5021no—
246C22×C4—5021no—
247C22×C4—2511no—
248SD16—22no—
249SD16—5022no—
250SD16—5022no—
251SD16—5022no—
252C2×C8—2511no—
253C2×C8—2511no—
254C2×C8C22×D25111yescharacteristic
255C2×C8C22×D25111yescharacteristic
256C4⋊C4—21no—
257C4⋊C4—5021no—
258C4⋊C4—5021no—
259C4⋊C4—5021no—
260C4⋊C4—5021no—
261C4⋊C4—5021no—
262C4⋊C4—5021no—
263C4⋊C4—5021no—
264C4⋊C4—2511no—
265C4⋊C4—2511no—
266C4⋊C4—2511no—
267C4⋊C4—2511no—
268C4⋊C4C22×D25111yescharacteristic
269C4⋊C4C22×D25111yescharacteristic
270C22⋊C4—21no—
271C22⋊C4—5021no—
272C22⋊C4—5021no—
273C22⋊C4—5021no—
274C22⋊C4—5021no—
275C22⋊C4—5021no—
276C22⋊C4—5021no—
277C22⋊C4—2511no—
278C22⋊C4—2511no—
279C42—21no—
280C42—5021no—
281C42—5021no—
282C42—2511no—
283C42C22×D25111yescharacteristic
284C23—21no—
285C23—20081no—
286C23—20081no—
287C23—5024no—
288C23—5021no—
289C23—2511no—
290Q8—21no—
291Q8—21no—
292Q8—5021no—
293Q8—5021no—
294D4—21no—
295D4—5024no—
296D4—5021no—
297D4—2512no—
298D4#1686081012yes—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—81no—
303C2×C4—20081no—
304C2×C4—20081no—
305C2×C4—20081no—
306C2×C4—5021no—
307C2×C4—5021no—
308C2×C4—5021no—
309C2×C4—5021no—
310C2×C4—5021no—
311C2×C4—2511no—
312C2×C4—2511no—
313C2×C4—2511no—
314C2×C4D4×D25111yescharacteristic
315C2×C4D4×D25111yescharacteristic
316C2×C4C23×D25111yescharacteristic
317C8—21no—
318C8—21no—
319C8—5021no—
320C8—5021no—
321C22—22no—
322C22—41no—
323C22—41no—
324C22—20084no—
325C22—10041no—
326C22—10041no—
327C22—5021no—
328C22—5022no—
329C22—5021no—
330C22—5021no—
331C22C2×D4×D25111yescharacteristic
332C4—21no—
333C4—21no—
334C4—41no—
335C4—41no—
336C4—41no—
337C4—10041no—
338C4—10041no—
339C4—10041no—
340C4—5021no—
341C4—5021no—
342C4—5021no—
343C4#41×D25111yescharacteristic
344C4C2×#1686081011yescharacteristic
345C2—42no—
346C2—10041no—
347C2—10042no—
348C2#3063219611yescharacteristic
349C2#3063249611yescharacteristic
350C2#3063249711yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#440×C251C2
socleC22×C251C22×D4
prime radicalC2×D4×D251C22
hypercenter#440D251
hypocenterC251#3123
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ #440

Complements

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

NH ≅ G/Ntype
1. #30632295C2semidirect
3. #30632030C2semidirect
4. #30632027C2semidirect
5. #30631990C2semidirect
6. #30631985C2semidirect
7. #30631891C2semidirect
8. #30631890C2semidirect
11. #30631789C2semidirect
12. C4×#16860797C2semidirect
13. C4×#16860798C2semidirect
14. #30631749C2semidirect
15. #440×C251C2semidirect
20. #22548713C22semidirect
24. #22548706C22semidirect
28. C4×Dic502C22semidirect
38. #22548649C22semidirect
39. #22548646C22semidirect
40. #22548646C22semidirect
41. Dic251⋊C8C22semidirect
42. Dic251⋊C8C22semidirect
43. #22548640C22semidirect
44. #22548639C22semidirect
45. #22548638C22semidirect
51. C4×C251⋊C8C22semidirect
53. C4×Q8×C251C22semidirect
55. #107×C251C22semidirect
56. #103×C251C22semidirect
58. #97×C251C22semidirect
175. C251#3123semidirect
188. #440D251semidirect
206. C4×Q8C2×D251semidirect
218. #107C2×D251semidirect
226. #103C2×D251semidirect
233. #97C2×D251semidirect

Maximal subgroups

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

subgroupindex
#306322952
#306321592
#306320302
#306320272
#306319902
#306319852
#306318912
#306318902
#306318392
#306318362
#306317892
C4×#168607972
C4×#168607982
#306317492
#440×C2512
#3123251