Finite Group Databasebeta

← #44559219

Subgroups — #44559219

Order 32128.

492 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×SD16×D251C211yesmaximal, characteristic
2#30632163C211yesmaximal, characteristic, complemented
3#30632159C211yesmaximal, characteristic
4#30631790C211yesmaximal, characteristic, complemented
5#30631787C211yesmaximal, characteristic, complemented
6#30631785C211yesmaximal, characteristic, complemented
7#30631784C211yesmaximal, characteristic, complemented
8C4×#16860799C211yesmaximal, characteristic, complemented
9SD16×Dic251C211yesmaximal, characteristic, complemented
10#30631774C211yesmaximal, characteristic, complemented
11#30631722C211yesmaximal, characteristic, complemented
12#30631694C211yesmaximal, characteristic, complemented
13#30631690C211yesmaximal, characteristic
14#30631641C211yesmaximal, characteristic, complemented
15C4×SD16×C251C211yesmaximal, characteristic, complemented
16C2×Q8×D251C2211yescharacteristic
17C2×D4×D251C2211yescharacteristic
18C2×#16860799C2211yescharacteristic
19C2×#16860798C2211yescharacteristic
20C2×#16860797C2211yescharacteristic
21SD16×D251C418yescomplemented
22C2×#16860793C2211yescharacteristic
23C2×C8×D251C2211yescharacteristic
24Q8×Dic251C2211yescharacteristic, complemented
25C4×Dic502—21no—
26C4×Dic502C2211yescharacteristic, complemented
27#22548680—21no—
28#22548680C2211yescharacteristic, complemented
29C4×D1004C2211yescharacteristic
30D4×Dic251C2211yescharacteristic
31#22548677C2211yescharacteristic
32C4×C22⋊D251—21no—
33(C22⋊D251)⋊C4—21no—
34#22548673—21no—
35#22548673C2211yescharacteristic
36#22548673C2211yescharacteristic
37#22548672—21no—
38#22548672C2211yescharacteristic
39C2×(C2×D251)⋊C4—21no—
40C22⋊C4×D251—21no—
41#22548648C2211yescharacteristic, complemented
42#22548648C2211yescharacteristic, complemented
43#22548647C2211yescharacteristic, complemented
44#22548640C2211yescharacteristic, complemented
45#22548639C2211yescharacteristic, complemented
46#22548638C2211yescharacteristic, complemented
47#22548637C2211yescharacteristic
48#22548636C2211yescharacteristic
49#22548635C2211yescharacteristic
50#22548626C2211yescharacteristic, complemented
51#22548625C2211yescharacteristic, complemented
52C8×Dic251C2211yescharacteristic, complemented
53C2×SD16×C251C2211yescharacteristic
54C4×Q8×C251C2211yescharacteristic, complemented
55C4×D4×C251C2211yescharacteristic
56#106×C251C2211yescharacteristic, complemented
57#103×C251C2211yescharacteristic, complemented
58#102×C251C2211yescharacteristic
59C4×C8×C251C2211yescharacteristic, complemented
60C23×D251—21no—
61C2×Dic502—21no—
62C2×Dic502C2311yescharacteristic
63Q8×D251—22no—
64Q8×D251C2×C414yescomplemented
65C2×D1004C2311yescharacteristic
66C2×C22⋊D251—21no—
67D4×D251—22no—
68D4×D251C2×C414yes—
69C22×Dic251—21no—
70C2×C4×D251—21no—
71C2×C4×D251—21no—
72C2×C4×D251—21no—
73C2×C4×D251C2311yescharacteristic
74#16860799C2×C414yescomplemented
75#16860798C2×C414yescomplemented
76#16860797C2×C414yescomplemented
77#16860793—22no—
78C2×C251⋊C8C2311yescharacteristic
79C2×C251⋊C8C2311yescharacteristic
80C8×D251C2×C414yescomplemented
81C4⋊Dic251—21no—
82C4⋊Dic251C2311yescharacteristic
83C4⋊Dic251C2311yescharacteristic
84Dic251⋊C4—21no—
85Dic251⋊C4—21no—
86Dic251⋊C4D411yescharacteristic
87Dic251⋊C4D411yescharacteristic
88C22⋊Dic251—21no—
89(C2×D251)⋊C4—21no—
90(C2×D251)⋊C4—21no—
91(C2×D251)⋊C4—21no—
92(C2×D251)⋊C4—21no—
93(C2×D251)⋊C4D411yescharacteristic
94(C2×D251)⋊C4D411yescharacteristic
95C4×Dic251—21no—
96C4×Dic251—21no—
97C4×Dic251—21no—
98C4×Dic251—21no—
99C4×Dic251—21no—
100C4×Dic251C2311yescharacteristic
101C2×Q8×C251C2311yescharacteristic
102C2×D4×C251C2311yescharacteristic
103C22×C4×C251—21no—
104SD16×C251C2×C414yescomplemented
105C2×C8×C251C2311yescharacteristic
106C2×C8×C251C2311yescharacteristic
107C4⋊C4×C251—21no—
108C4⋊C4×C251C2311yescharacteristic
109C4⋊C4×C251C2311yescharacteristic
110C22⋊C4×C251—21no—
111C42×C251—21no—
112C42×C251C2311yescharacteristic
113C22×D251—21no—
114C22×D251—24no—
115C22×D251—81no—
116C22×D251—81no—
117C22×D251C2×D411yescharacteristic
118Dic502—21no—
119Dic502—24no—
120Dic502C22×C412yes—
121D1004—21no—
122D1004C22×C412yes—
123C22⋊D251—24no—
124C2×Dic251—21no—
125C2×Dic251—21no—
126C2×Dic251—21no—
127C2×Dic251—81no—
128C2×Dic251C2×D411yescharacteristic
129C2×Dic251C2×D411yescharacteristic
130C2×Dic251C2×D411yescharacteristic
131C4×D251—24no—
132C4×D251—81no—
133C4×D251—81no—
134C4×D251C22×C412yes—
135C4×D251#4112yes—
136C251⋊C8—21no—
137C251⋊C8C22×C412yes—
138C23×C251—21no—
139Q8×C251—21no—
140Q8×C251C22×C412yes—
141D4×C251—21no—
142D4×C251C22×C412yes—
143C2×C4×C251—21no—
144C2×C4×C251—21no—
145C2×C4×C251—21no—
146C2×C4×C251—81no—
147C2×C4×C251C2×D411yescharacteristic
148C2×C4×C251C2×D411yescharacteristic
149C2×C4×C251C2411yescharacteristic
150C8×C251—21no—
151C8×C251C22×C412yes—
152C22×C251—22no—
153C22×C251—41no—
154C22×C251—41no—
155C22×C251C22×D411yescharacteristic
156C2×D251—21no—
157C2×D251—21no—
158C2×D251—22no—
159C2×D251—41no—
160C2×D251—41no—
161C2×D251—84no—
162C2×D251C4×D412yes—
163C4×C251—22no—
164C4×C251—21no—
165C4×C251—21no—
166C4×C251—41no—
167C4×C251—41no—
168C4×C251C23×C411yescharacteristic
169C4×C251C2×#4111yescharacteristic
170Dic251—22no—
171Dic251—21no—
172Dic251—21no—
173Dic251—41no—
174Dic251—41no—
175Dic251C4×D412yes—
176C2×C251—42no—
177C2×C251C2×C4×D411yescharacteristic
178C2×C251#57711yescharacteristic
179C2×C251#57711yescharacteristic
180D251—22no—
181D251—42no—
182C251#294911yescharacteristic, complemented, Sylow
183#2949—2511nomaximal, Sylow
184C22×SD16—2511no—
185#519—2511no—
186#518—2511no—
187#440—2511no—
188#440—2511no—
189#440—2511no—
190#440—2511no—
191C4×SD16—2511no—
192C4×SD16—2511no—
193C4×SD16—2511no—
194C4×SD16D25111yescharacteristic, complemented
195#427—2511no—
196#419—2511no—
197#418—2511no—
198#405—2511no—
199C22×Q8—2511no—
200C22×D4—2511no—
201C2×SD16—2511no—
202C2×SD16—2511no—
203C2×SD16—2518no—
204C2×SD16—2511no—
205C2×SD16C2×D25111yescharacteristic
206C2×M16—2511no—
207C22×C8—2511no—
208C4×Q8—5021no—
209C4×Q8—5021no—
210C4×Q8—2511no—
211C4×Q8—2511no—
212C4×Q8—2511no—
213C4×Q8C2×D25111yescharacteristic, complemented
214C4×D4—5021no—
215C4×D4—5021no—
216C4×D4—2511no—
217C4×D4—2511no—
218C4×D4—2511no—
219C4×D4C2×D25111yescharacteristic
220#117—5021no—
221#117—5021no—
222#117—2511no—
223#117—2511no—
224#117—2511no—
225C2×C22⋊C4—5021no—
226C2×C22⋊C4—5021no—
227#107—2511no—
228#107—2511no—
229#106—2511no—
230#106C2×D25111yescharacteristic, complemented
231#103—2511no—
232#103—2511no—
233#103—2511no—
234#103C2×D25111yescharacteristic, complemented
235#102—2511no—
236#102—2511no—
237#102—2511no—
238#102C2×D25111yescharacteristic
239#97—2511no—
240#97—2511no—
241C4×C8—2511no—
242C4×C8C2×D25111yescharacteristic, complemented
243C24—5021no—
244C2×Q8—5021no—
245C2×Q8—5022no—
246C2×Q8—2514no—
247C2×Q8—2511no—
248C2×Q8C22×D25111yescharacteristic
249C2×D4—5022no—
250C2×D4—5021no—
251C2×D4—2514no—
252C2×D4—2511no—
253C2×D4C22×D25111yescharacteristic
254C22×C4—21no—
255C22×C4—5021no—
256C22×C4—5021no—
257C22×C4—5021no—
258C22×C4—5021no—
259C22×C4—2511no—
260SD16—2514no—
261SD16—2514no—
262SD16—2514no—
263SD16C4×D25114yescomplemented
264M16—5022no—
265C2×C8—2511no—
266C2×C8—2511no—
267C2×C8—2514no—
268C2×C8C22×D25111yescharacteristic
269C2×C8C22×D25111yescharacteristic
270C4⋊C4—21no—
271C4⋊C4—5021no—
272C4⋊C4—5021no—
273C4⋊C4—5021no—
274C4⋊C4—2511no—
275C4⋊C4—2511no—
276C4⋊C4—2511no—
277C4⋊C4—2511no—
278C4⋊C4C22×D25111yescharacteristic
279C4⋊C4C22×D25111yescharacteristic
280C22⋊C4—21no—
281C22⋊C4—5021no—
282C22⋊C4—5021no—
283C22⋊C4—5021no—
284C22⋊C4—5021no—
285C22⋊C4—5021no—
286C22⋊C4—2511no—
287C22⋊C4—2511no—
288C42—21no—
289C42—5021no—
290C42—5021no—
291C42—5021no—
292C42—5021no—
293C42—5021no—
294C42—2511no—
295C42C22×D25111yescharacteristic
296C23—21no—
297C23—20081no—
298C23—20081no—
299C23—5024no—
300C23—5021no—
301C23—2511no—
302Q8—21no—
303Q8—5024no—
304Q8—5021no—
305Q8—2512no—
306Q8C2×C4×D25112yes—
307D4—21no—
308D4—5024no—
309D4—5021no—
310D4—2512no—
311D4C2×C4×D25112yes—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—81no—
316C2×C4—20081no—
317C2×C4—20081no—
318C2×C4—20081no—
319C2×C4—5024no—
320C2×C4—5021no—
321C2×C4—5021no—
322C2×C4—5021no—
323C2×C4—2511no—
324C2×C4—2511no—
325C2×C4—2511no—
326C2×C4—2512no—
327C2×C4—2512no—
328C2×C4D4×D25111yescharacteristic
329C2×C4D4×D25111yescharacteristic
330C2×C4C23×D25111yescharacteristic
331C8—21no—
332C8—5021no—
333C8—2512no—
334C8C2×C4×D25112yes—
335C22—22no—
336C22—41no—
337C22—41no—
338C22—20084no—
339C22—10041no—
340C22—10041no—
341C22—5022no—
342C22—5021no—
343C22—5021no—
344C22—2512no—
345C22C2×D4×D25111yescharacteristic
346C4—21no—
347C4—21no—
348C4—22no—
349C4—41no—
350C4—41no—
351C4—10041no—
352C4—10041no—
353C4—5021no—
354C4—5021no—
355C4—5022no—
356C4—2512no—
357C4C22×C4×D25111yescharacteristic
358C4#41×D25111yescharacteristic
359C2—42no—
360C2—10042no—
361C2—5022no—
362C2C4×D4×D25111yescharacteristic
363C2#3063249311yescharacteristic
364C2#3063249411yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutatorC4×C251C23×C4
FrattiniC2×C4C23×D251
FittingC4×SD16×C251C2
socleC22×C251C22×D4
prime radicalC2×D4×D251C22
cosocleC2×C4×C251C24
hypercenterC4×SD16D251
hypocenterC251#2949
LCSG ▷ C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ C4×SD16

Complements

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

NH ≅ G/Ntype
2. #30632163C2semidirect
4. #30631790C2semidirect
5. #30631787C2semidirect
6. #30631785C2semidirect
7. #30631784C2semidirect
8. C4×#16860799C2semidirect
9. SD16×Dic251C2semidirect
10. #30631774C2semidirect
11. #30631722C2semidirect
12. #30631694C2semidirect
14. #30631641C2semidirect
15. C4×SD16×C251C2semidirect
21. SD16×D251C4semidirect
24. Q8×Dic251C22semidirect
26. C4×Dic502C22semidirect
28. #22548680C22semidirect
41. #22548648C22semidirect
42. #22548648C22semidirect
43. #22548647C22semidirect
44. #22548640C22semidirect
45. #22548639C22semidirect
46. #22548638C22semidirect
50. #22548626C22semidirect
51. #22548625C22semidirect
52. C8×Dic251C22semidirect
54. C4×Q8×C251C22semidirect
56. #106×C251C22semidirect
57. #103×C251C22semidirect
59. C4×C8×C251C22semidirect
64. Q8×D251C2×C4semidirect
74. #16860799C2×C4semidirect
75. #16860798C2×C4semidirect
76. #16860797C2×C4semidirect
80. C8×D251C2×C4semidirect
104. SD16×C251C2×C4semidirect
182. C251#2949semidirect
194. C4×SD16D251semidirect
213. C4×Q8C2×D251semidirect
230. #106C2×D251semidirect
234. #103C2×D251semidirect
242. C4×C8C2×D251semidirect
263. SD16C4×D251semidirect

Maximal subgroups

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

subgroupindex
C2×SD16×D2512
#306321632
#306321592
#306317902
#306317872
#306317852
#306317842
C4×#168607992
SD16×Dic2512
#306317742
#306317222
#306316942
#306316902
#306316412
C4×SD16×C2512
#2949251