Finite Group Databasebeta

← #44559982

Subgroups — #44559982

Order 32128.

548 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 135 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22548741C211yesmaximal, characteristic, complemented
2C2×M16×D251C211yesmaximal, characteristic, complemented
3#30632199C211yesmaximal, characteristic, complemented
4C2×#22548686C211yesmaximal, characteristic, complemented
5#30631939C212yesmaximal, complemented
6#30631932C212yesmaximal, complemented
7#30631922C212yesmaximal, complemented
8#30631915C212yesmaximal, complemented
9#30631730C211yesmaximal, characteristic, complemented
10#30631689C211yesmaximal, characteristic, complemented
11#418×C251C211yesmaximal, characteristic, complemented
12C2×#16860810C2211yescharacteristic
13C22×C22⋊D251—21no—
14C22×C4×D251C2211yescharacteristic
15#22548741—24no—
16C2×#16860798C2212yescomplemented
17C2×#16860795C2212yescomplemented
18C2×#16860794C2211yescharacteristic
19C2×#16860793C2212yescomplemented
20M16×D251—24no—
21C2×C8×D251C2212yescomplemented
22#22548699—21no—
23#22548695C2211yescharacteristic, complemented
24#22548695C2211yescharacteristic, complemented
25#22548690C2211yescharacteristic, complemented
26#22548690C2211yescharacteristic, complemented
27#22548686—21no—
28#22548686—21no—
29#22548686C2212yes—
30#22548686C2212yes—
31#22548685—21no—
32C4×D1004—21no—
33C4×D1004—21no—
34C2×C4⋊Dic251C2211yescharacteristic
35C2×C22⋊Dic251—21no—
36#22548648C2212yescomplemented
37#22548646C2212yescomplemented
38#22548640C2212yescomplemented
39#22548637C2212yescomplemented
40C22×D4×C251C2211yescharacteristic
41C2×M16×C251C2211yescharacteristic
42#117×C251C2211yescharacteristic
43#102×C251C2212yescomplemented
44#102×C251C2212yescomplemented
45C23×D251D411yescharacteristic
46#16860810—22no—
47#16860810—22no—
48C2×Dic502C2311yescharacteristic
49C2×D1004—21no—
50C2×D1004—21no—
51C2×D1004C2311yescharacteristic
52C2×C22⋊D251—22no—
53C2×C22⋊D251—22no—
54C2×C22⋊D251—21no—
55C2×C22⋊D251—44no—
56C22×Dic251—21no—
57C22×Dic251D411yescharacteristic
58C2×C4×D251—22no—
59C2×C4×D251—21no—
60C2×C4×D251—22no—
61C2×C4×D251D411yescharacteristic
62C2×C4×D251D411yescharacteristic
63C2×C4×D251C2311yescharacteristic
64C2×C4×D251C2311yescharacteristic
65#16860798—24no—
66#16860795—24no—
67#16860794D414yescomplemented
68#16860793—24no—
69C2×C251⋊C8C2312yes—
70C8×D251—24no—
71C4⋊Dic251—21no—
72C4⋊Dic251—21no—
73C4⋊Dic251—21no—
74C4⋊Dic251C2312yes—
75C22⋊Dic251—22no—
76C22⋊Dic251—22no—
77(C2×D251)⋊C4—21no—
78(C2×D251)⋊C4—21no—
79(C2×D251)⋊C4—21no—
80(C2×D251)⋊C4—21no—
81C24×C251—21no—
82C2×D4×C251—22no—
83C2×D4×C251—21no—
84C2×D4×C251—22no—
85C2×D4×C251C2312yes—
86C22×C4×C251C2311yescharacteristic
87M16×C251—22no—
88C2×C8×C251C2312yes—
89C4⋊C4×C251C2311yescharacteristic
90C4⋊C4×C251C2311yescharacteristic
91C22⋊C4×C251—21no—
92C42×C251—21no—
93C22×D251—22no—
94C22×D251—22no—
95C22×D251—21no—
96C22×D251—41no—
97C22×D251C2×D411yescharacteristic
98C22×D251C2×D411yescharacteristic
99Dic502—21no—
100Dic502—21no—
101D1004—21no—
102D1004—21no—
103D1004—42no—
104D1004—42no—
105C22⋊D251—44no—
106C22⋊D251—42no—
107C22⋊D251—44no—
108C2×Dic251—22no—
109C2×Dic251—21no—
110C2×Dic251—22no—
111C2×Dic251—42no—
112C2×Dic251C2×D411yescharacteristic
113C2×Dic251C2×D411yescharacteristic
114C4×D251—22no—
115C4×D251—22no—
116C4×D251—21no—
117C4×D251—21no—
118C4×D251—21no—
119C4×D251—21no—
120C4×D251—42no—
121C251⋊C8C2×D414yescomplemented
122C23×C251—22no—
123C23×C251—42no—
124C23×C251—42no—
125C23×C251—42no—
126C23×C251C2×D411yescharacteristic
127D4×C251—22no—
128D4×C251—22no—
129D4×C251—42no—
130C2×C4×C251—21no—
131C2×C4×C251—21no—
132C2×C4×C251C2×D411yescharacteristic
133C2×C4×C251C2×D412yes—
134C2×C4×C251#4112yes—
135C2×C4×C251C2411yescharacteristic
136C8×C251—22no—
137C22×C251—21no—
138C22×C251—21no—
139C22×C251—42no—
140C22×C251—42no—
141C22×C251—42no—
142C22×C251—84no—
143C22×C251#12112yes—
144C22×C251C22×D411yescharacteristic
145C2×D251—21no—
146C2×D251—21no—
147C2×D251—21no—
148C2×D251—21no—
149C2×D251—21no—
150C2×D251—44no—
151C2×D251—41no—
152C2×D251—41no—
153C2×D251—41no—
154C2×D251AGL(1,ℤ/8)12yescomplemented
155C4×C251—41no—
156C4×C251—41no—
157C4×C251#12112yes—
158C4×C251C22×D411yescharacteristic
159C4×C251C2×#4111yescharacteristic
160Dic251—21no—
161Dic251—21no—
162Dic251—41no—
163Dic251—42no—
164C2×C251—22no—
165C2×C251—82no—
166C2×C251C2×#12111yescharacteristic
167C2×C251C2×AGL(1,ℤ/8)11yescharacteristic
168C2×C251#57511yescharacteristic
169D251—22no—
170D251—41no—
171D251—81no—
172C251#305511yescharacteristic, complemented, Sylow
173#3055—2511nomaximal, Sylow
174C2×AGL(1,ℤ/8)—2511no—
175C22×M16—2511no—
176#525—2511no—
177C2×#121—2511no—
178#469—2512no—
179#468—2512no—
180#466—2512no—
181#465—2512no—
182#428—2511no—
183#418D25111yescharacteristic, complemented
184#417—2511no—
185C2×#41—2511no—
186C22×D4—5021no—
187C22×D4C2×D25111yescharacteristic
188C23×C4—2511no—
189AGL(1,ℤ/8)—5024no—
190C2×SD16—2512no—
191C2×D8—2512no—
192C2×M16—5024no—
193C2×M16—2511no—
194C2×M16—2512no—
195C2×M16C2×D25111yescharacteristic
196C22×C8—2512no—
197#123—5021no—
198C22⋊Q8—2511no—
199C22⋊Q8—2511no—
200#121—5021no—
201#121—5021no—
202#121—2512no—
203#121—2512no—
204#121—2511no—
205#121—2511no—
206#120—5021no—
207C4×D4—5021no—
208C4×D4—5021no—
209#117C2×D25111yescharacteristic
210C2×C4⋊C4—2511no—
211C2×C22⋊C4—5021no—
212#107—2512no—
213#106—2512no—
214#103—2512no—
215#102—2512no—
216#102C2×D25112yescomplemented
217#102C2×D25112yescomplemented
218C24—21no—
219C24—2511no—
220#41—5022no—
221#41—5022no—
222C2×Q8—2511no—
223C2×D4—22no—
224C2×D4—21no—
225C2×D4—22no—
226C2×D4—10044no—
227C2×D4—5021no—
228C2×D4—5021no—
229C2×D4—5022no—
230C2×D4—5022no—
231C2×D4—5021no—
232C2×D4—2511no—
233C2×D4C22×D25112yes—
234C22×C4—5021no—
235C22×C4—5021no—
236C22×C4—5022no—
237C22×C4—5022no—
238C22×C4—2511no—
239C22×C4—2511no—
240C22×C4—2511no—
241C22×C4—2511no—
242C22×C4—2511no—
243C22×C4C22×D25111yescharacteristic
244SD16—5024no—
245D8—5024no—
246M16—22no—
247M16—5024no—
248M16—2514no—
249C2×C8—5024no—
250C2×C8—2512no—
251C2×C8C22×D25112yes—
252C4⋊C4—5021no—
253C4⋊C4—5021no—
254C4⋊C4—5021no—
255C4⋊C4—2512no—
256C4⋊C4C22×D25111yescharacteristic
257C4⋊C4C22×D25111yescharacteristic
258C22⋊C4—21no—
259C22⋊C4—5021no—
260C22⋊C4—5021no—
261C22⋊C4—5021no—
262C22⋊C4—5021no—
263C22⋊C4—5022no—
264C22⋊C4—5022no—
265C42—21no—
266C23—22no—
267C23—42no—
268C23—42no—
269C23—42no—
270C23—10041no—
271C23—5022no—
272C23—5021no—
273C23—5022no—
274C23—2511no—
275C23—2511no—
276C23D4×D25111yescharacteristic
277Q8—5021no—
278Q8—5021no—
279D4—22no—
280D4—22no—
281D4—42no—
282D4—10042no—
283D4—10042no—
284D4—10044no—
285D4—10044no—
286D4—10042no—
287D4—5021no—
288D4—5021no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—10042no—
292C2×C4—10042no—
293C2×C4—5021no—
294C2×C4—5022no—
295C2×C4—5022no—
296C2×C4—5021no—
297C2×C4—5021no—
298C2×C4—5021no—
299C2×C4—5021no—
300C2×C4—5022no—
301C2×C4—5022no—
302C2×C4—2511no—
303C2×C4—2511no—
304C2×C4D4×D25112yes—
305C2×C4D4×D25111yescharacteristic
306C2×C4#1686081012yes—
307C2×C4C23×D25111yescharacteristic
308C8—22no—
309C8—2514no—
310C22—21no—
311C22—21no—
312C22—42no—
313C22—42no—
314C22—42no—
315C22—84no—
316C22—10044no—
317C22—10041no—
318C22—10041no—
319C22—10041no—
320C22—5021no—
321C22—5021no—
322C22—5021no—
323C22—5021no—
324C22—5021no—
325C22—2512no—
326C22#2254868812yes—
327C22C2×D4×D25111yescharacteristic
328C4—41no—
329C4—41no—
330C4—10042no—
331C4—10041no—
332C4—5021no—
333C4—5021no—
334C4#2254868812yes—
335C4C2×D4×D25111yescharacteristic
336C4C2×#1686081011yescharacteristic
337C2—22no—
338C2—82no—
339C2—20081no—
340C2—10041no—
341C2—5022no—
342C2C2×#2254868811yescharacteristic
343C2AGL(1,ℤ/8)×D25111yescharacteristic
344C2#3063248311yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#418×C251C2
socleC22×C251C22×D4
hypercenter#418D251
hypocenterC251#3055
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
1. C2×#22548741C2semidirect
2. C2×M16×D251C2semidirect
3. #30632199C2semidirect
4. C2×#22548686C2semidirect
5. #30631939C2semidirect
6. #30631932C2semidirect
7. #30631922C2semidirect
8. #30631915C2semidirect
9. #30631730C2semidirect
10. #30631689C2semidirect
11. #418×C251C2semidirect
16. C2×#16860798C22semidirect
17. C2×#16860795C22semidirect
19. C2×#16860793C22semidirect
21. C2×C8×D251C22semidirect
23. #22548695C22semidirect
24. #22548695C22semidirect
25. #22548690C22semidirect
26. #22548690C22semidirect
36. #22548648C22semidirect
37. #22548646C22semidirect
38. #22548640C22semidirect
39. #22548637C22semidirect
43. #102×C251C22semidirect
44. #102×C251C22semidirect
67. #16860794D4semidirect
121. C251⋊C8C2×D4semidirect
154. C2×D251AGL(1,ℤ/8)semidirect
172. C251#3055semidirect
183. #418D251semidirect
216. #102C2×D251semidirect
217. #102C2×D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225487412
C2×M16×D2512
#306321992
C2×#225486862
#306319392
#306319322
#306319222
#306319152
#306317302
#306316892
#418×C2512
#3055251