Finite Group Databasebeta

← #44560054

Subgroups — #44560054

Order 32128.

472 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 133 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×#22548746C211yesmaximal, characteristic, complemented
2C2×#22548721C211yesmaximal, characteristic, complemented
3#30632268C211yesmaximal, characteristic
4#30632261C211yesmaximal, characteristic, complemented
5#30631944C212yesmaximal, complemented
6#30631936C212yesmaximal, complemented
7#30631927C212yesmaximal, complemented
8#30631919C212yesmaximal, complemented
9#30631727C211yesmaximal, characteristic, complemented
10#30631684C211yesmaximal, characteristic, complemented
11C2×#103×C251C211yesmaximal, characteristic, complemented
12C2×#16860810C2211yescharacteristic
13C2×#16860810C2211yescharacteristic
14#22548746—24no—
15C2×#16860800C2212yes—
16C2×#16860797C2212yes—
17#22548721—24no—
18C2×#16860794C2211yescharacteristic
19C2×#16860793C2212yes—
20C2×C8×D251C2212yes—
21#22548713—21no—
22#22548704—21no—
23#22548698—22no—
24#22548695—21no—
25#22548695C2212yes—
26#22548691C2212yes—
27#22548691C2212yes—
28#22548690—21no—
29#22548690C2212yes—
30#22548689—22no—
31#22548674C2211yescharacteristic
32#22548648C2212yes—
33#22548646C2212yes—
34#22548640C2211yescharacteristic
35#22548640C2211yescharacteristic
36#22548637C2211yescharacteristic
37#22548637C2211yescharacteristic
38C22×Q8×C251C2211yescharacteristic
39C22×C8×C251C2211yescharacteristic
40C2×C4⋊C4×C251C2211yescharacteristic
41#103×C251C2212yes—
42#103×C251C2212yes—
43#16860810—22no—
44#16860810—22no—
45#16860810—22no—
46#16860810—22no—
47C2×Dic502D411yescharacteristic
48C2×Dic502C2311yescharacteristic
49C2×D1004—22no—
50C2×D1004D411yescharacteristic
51C2×D1004C2311yescharacteristic
52C2×C22⋊D251—21no—
53C2×C22⋊D251D412yes—
54C2×C4×D251—21no—
55C2×C4×D251C2312yes—
56#16860800—24no—
57#16860797—24no—
58#16860794D414yescomplemented
59#16860793—24no—
60C2×C251⋊C8C2312yes—
61C8×D251—24no—
62C4⋊Dic251—22no—
63C4⋊Dic251C2311yescharacteristic
64C4⋊Dic251C2311yescharacteristic
65Dic251⋊C4—22no—
66Dic251⋊C4—22no—
67C22⋊Dic251—21no—
68(C2×D251)⋊C4—22no—
69(C2×D251)⋊C4—22no—
70(C2×D251)⋊C4—22no—
71(C2×D251)⋊C4—22no—
72C4×Dic251—21no—
73C2×Q8×C251—21no—
74C2×Q8×C251—22no—
75C2×Q8×C251—22no—
76C2×Q8×C251C2312yes—
77C22×C4×C251—21no—
78C22×C4×C251—21no—
79C22×C4×C251C2311yescharacteristic
80C2×C8×C251—22no—
81C2×C8×C251C2311yescharacteristic
82C2×C8×C251C2311yescharacteristic
83C4⋊C4×C251—21no—
84C4⋊C4×C251C2312yes—
85C22×D251—21no—
86C22×D251C2×D412yes—
87Dic502—22no—
88Dic502—21no—
89Dic502—21no—
90D1004—22no—
91D1004—21no—
92D1004—21no—
93D1004—82no—
94C22⋊D251—24no—
95C22⋊D251—42no—
96C2×Dic251—21no—
97C2×Dic251—21no—
98C2×Dic251—21no—
99C2×Dic251C2×D412yes—
100C4×D251—22no—
101C4×D251—22no—
102C4×D251—42no—
103C251⋊C8C2×D414yes—
104C23×C251C2×D411yescharacteristic
105Q8×C251—22no—
106Q8×C251—22no—
107Q8×C251—42no—
108C2×C4×C251—22no—
109C2×C4×C251—22no—
110C2×C4×C251—42no—
111C2×C4×C251—81no—
112C2×C4×C251C2×D411yescharacteristic
113C2×C4×C251C2×D412yes—
114C2×C4×C251#4112yes—
115C2×C4×C251C2411yescharacteristic
116C8×C251—21no—
117C8×C251—21no—
118C22×C251—21no—
119C22×C251—21no—
120C22×C251#12112yes—
121C22×C251C22×D411yescharacteristic
122C2×D251—22no—
123C2×D251—22no—
124C2×D251—22no—
125C2×D251—41no—
126C2×D251—41no—
127C2×D251—41no—
128C4×C251—42no—
129C4×C251—42no—
130C4×C251#12112yes—
131C4×C251C22×D411yescharacteristic
132C4×C251C2×#4111yescharacteristic
133Dic251—22no—
134Dic251—41no—
135Dic251—41no—
136Dic251—41no—
137C2×C251—22no—
138C2×C251C2×#12111yescharacteristic
139C2×C251#57711yescharacteristic
140C2×C251#57811yescharacteristic
141D251—42no—
142D251—81no—
143C251#306311yescharacteristic, complemented, Sylow
144#3063—2511nomaximal, Sylow
145C2×#137—2511no—
146C2×#131—2511no—
147#537—2511no—
148#536—2511no—
149#470—2512no—
150#468—2512no—
151#467—2512no—
152#465—2512no—
153#428—2511no—
154#416—2511no—
155C2×#103D25111yescharacteristic, complemented
156C2×#41—2511no—
157C2×#41—2511no—
158C22×Q8C2×D25111yescharacteristic
159#137—5024no—
160C2×Q16—2512no—
161C2×SD16—2512no—
162#131—5024no—
163C2×M16—2511no—
164C2×M16—2512no—
165C22×C8—2512no—
166C22×C8C2×D25111yescharacteristic
167C4⋊Q8—5021no—
168#124—5021no—
169#123—5022no—
170C22⋊Q8—5021no—
171C22⋊Q8—2512no—
172C22⋊Q8—2512no—
173C22⋊Q8—2512no—
174#121—5021no—
175#121—5022no—
176#121—2512no—
177#117—2511no—
178C2×C4⋊C4C2×D25111yescharacteristic
179#107—2512no—
180#106—2512no—
181#103—2511no—
182#103—2511no—
183#103C2×D25112yes—
184#103C2×D25112yes—
185#102—2511no—
186#102—2511no—
187#41—5022no—
188#41—5022no—
189#41—5022no—
190#41—5022no—
191C2×Q8—21no—
192C2×Q8—22no—
193C2×Q8—22no—
194C2×Q8—2511no—
195C2×Q8—2511no—
196C2×Q8C22×D25112yes—
197C2×D4—5022no—
198C2×D4—5021no—
199C2×D4—2512no—
200C2×D4—2511no—
201C2×D4—2511no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—5021no—
205C22×C4—2512no—
206C22×C4C22×D25111yescharacteristic
207Q16—5024no—
208SD16—5024no—
209M16—5024no—
210M16—2514no—
211C2×C8—22no—
212C2×C8—5024no—
213C2×C8—2512no—
214C2×C8C22×D25111yescharacteristic
215C2×C8C22×D25111yescharacteristic
216C4⋊C4—21no—
217C4⋊C4—5022no—
218C4⋊C4—5022no—
219C4⋊C4—5022no—
220C4⋊C4—2511no—
221C4⋊C4—2511no—
222C4⋊C4C22×D25112yes—
223C22⋊C4—5022no—
224C22⋊C4—5022no—
225C22⋊C4—5022no—
226C22⋊C4—5022no—
227C22⋊C4—5021no—
228C42—5021no—
229C23—5021no—
230C23—2512no—
231C23D4×D25111yescharacteristic
232Q8—22no—
233Q8—22no—
234Q8—42no—
235Q8—5021no—
236Q8—5021no—
237Q8—5022no—
238D4—20082no—
239D4—10042no—
240D4—5024no—
241D4—5021no—
242D4—5021no—
243D4—5022no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—42no—
247C2×C4—81no—
248C2×C4—10042no—
249C2×C4—5021no—
250C2×C4—5021no—
251C2×C4—5021no—
252C2×C4—5022no—
253C2×C4—5022no—
254C2×C4—2512no—
255C2×C4D4×D25112yes—
256C2×C4D4×D25111yescharacteristic
257C2×C4#1686081012yes—
258C2×C4C23×D25111yescharacteristic
259C8—21no—
260C8—21no—
261C8—2514no—
262C22—21no—
263C22—21no—
264C22—10041no—
265C22—10041no—
266C22—10041no—
267C22—5022no—
268C22—5022no—
269C22—5022no—
270C22#2254868812yes—
271C22C2×D4×D25111yescharacteristic
272C4—42no—
273C4—42no—
274C4—10041no—
275C4—10041no—
276C4—10041no—
277C4—5022no—
278C4#2254868812yes—
279C4C2×D4×D25111yescharacteristic
280C4C2×#1686081011yescharacteristic
281C2—22no—
282C2—20081no—
283C2—10042no—
284C2C2×#2254868811yescharacteristic
285C2#3063249311yescharacteristic
286C2#3063250011yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
FittingC2×#103×C251C2
socleC22×C251C22×D4
prime radical#30632268C2
hypercenterC2×#103D251
hypocenterC251#3063
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#103

Complements

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

NH ≅ G/Ntype
1. C2×#22548746C2semidirect
2. C2×#22548721C2semidirect
4. #30632261C2semidirect
5. #30631944C2semidirect
6. #30631936C2semidirect
7. #30631927C2semidirect
8. #30631919C2semidirect
9. #30631727C2semidirect
10. #30631684C2semidirect
11. C2×#103×C251C2semidirect
58. #16860794D4semidirect
143. C251#3063semidirect
155. C2×#103D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225487462
C2×#225487212
#306322682
#306322612
#306319442
#306319362
#306319272
#306319192
#306317272
#306316842
C2×#103×C2512
#3063251