Finite Group Databasebeta

← #44559656

Subgroups — #44559656

Order 32128.

700 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 700 conjugacy classes of subgroups — too many to draw. Only its 149 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×#22548763C211yesmaximal, characteristic
2C2×#22548746C211yesmaximal, characteristic, complemented
3#30632261C211yesmaximal, characteristic, complemented
4#30631848C214yesmaximal, complemented
5#30631841C214yesmaximal, complemented
6#30631696C211yesmaximal, characteristic, complemented
7#30631684C211yesmaximal, characteristic, complemented
8#30631653C211yesmaximal, characteristic, complemented
9C22×Q16×C251C211yesmaximal, characteristic, complemented
10#22548763—24no—
11#22548763—24no—
12C2×#16860811—21no—
13C2×#16860811C2212yes—
14C2×#16860810—22no—
15C2×#16860810C2211yescharacteristic
16C2×#16860810C2211yescharacteristic
17C2×Q8×D251—21no—
18C2×Q8×D251C2212yes—
19#22548746—24no—
20C2×#16860800C2212yes—
21C2×#16860797C2212yes—
22C2×#16860794C2211yescharacteristic
23#22548713—21no—
24#22548704—21no—
25#22548698—22no—
26#22548691C2214yes—
27#22548674C2211yescharacteristic
28#22548640C2212yes—
29#22548639C2214yes—
30#22548637C2212yes—
31(C2×D251)⋊C8C2214yes—
32C22×Q8×C251C2211yescharacteristic
33C22×Q8×C251C2211yescharacteristic
34C2×Q16×C251—24no—
35C2×Q16×C251C2214yes—
36C22×C8×C251C2211yescharacteristic
37#16860811—28no—
38#16860811—24no—
39#16860811—24no—
40#16860810—216no—
41#16860810—22no—
42#16860810—22no—
43#16860810—22no—
44#16860810D414yes—
45C2×Dic502—22no—
46C2×Dic502D411yescharacteristic
47C2×Dic502C2311yescharacteristic
48Q8×D251—28no—
49Q8×D251—24no—
50Q8×D251—24no—
51C2×D1004—22no—
52C2×D1004D411yescharacteristic
53C2×D1004C2311yescharacteristic
54C2×C22⋊D251—21no—
55C2×C22⋊D251D412yes—
56C2×C4×D251—22no—
57C2×C4×D251—22no—
58C2×C4×D251—21no—
59C2×C4×D251C2312yes—
60#16860800—24no—
61#16860797—24no—
62#16860794—22no—
63C2×C251⋊C8C2312yes—
64C4⋊Dic251C2312yes—
65Dic251⋊C4—24no—
66C22⋊Dic251—21no—
67(C2×D251)⋊C4—24no—
68C4×Dic251—21no—
69C2×Q8×C251—21no—
70C2×Q8×C251—22no—
71C2×Q8×C251—21no—
72C2×Q8×C251—22no—
73C2×Q8×C251—22no—
74C2×Q8×C251D414yes—
75C2×Q8×C251C2312yes—
76C2×Q8×C251C2312yes—
77C22×C4×C251—21no—
78C22×C4×C251—21no—
79C22×C4×C251C2311yescharacteristic
80Q16×C251—28no—
81C2×C8×C251—22no—
82C2×C8×C251C2312yes—
83C22×D251—21no—
84C22×D251C2×D412yes—
85Dic502—28no—
86Dic502—22no—
87Dic502—21no—
88Dic502C2×D412yes—
89D1004—28no—
90D1004—22no—
91D1004—21no—
92D1004C2×D412yes—
93C22⋊D251—24no—
94C22⋊D251—24no—
95C2×Dic251—21no—
96C2×Dic251—22no—
97C2×Dic251C2×D412yes—
98C4×D251—28no—
99C4×D251—28no—
100C4×D251—24no—
101C4×D251—22no—
102C4×D251—22no—
103C251⋊C8—22no—
104C23×C251C2×D411yescharacteristic
105Q8×C251—24no—
106Q8×C251—22no—
107Q8×C251—22no—
108Q8×C251—22no—
109Q8×C251—42no—
110Q8×C251C2×D414yes—
111C2×C4×C251—24no—
112C2×C4×C251—22no—
113C2×C4×C251—22no—
114C2×C4×C251—42no—
115C2×C4×C251C2×D411yescharacteristic
116C2×C4×C251C2×D412yes—
117C2×C4×C251C2×D412yes—
118C2×C4×C251C2411yescharacteristic
119C8×C251—22no—
120C22×C251—22no—
121C22×C251#12012yes—
122C22×C251C22×D411yescharacteristic
123C2×D251—22no—
124C2×D251—22no—
125C2×D251—24no—
126C2×D251—42no—
127C4×C251—24no—
128C4×C251—42no—
129C4×C251#12012yes—
130C4×C251C22×D411yescharacteristic
131C4×C251C22×D411yescharacteristic
132Dic251—22no—
133Dic251—22no—
134Dic251—42no—
135C2×C251—22no—
136C2×C251C2×#12011yescharacteristic
137C2×C251#57812yes—
138D251—42no—
139D251—42no—
140C251#301111yescharacteristic, complemented, Sylow
141#3011—2511nomaximal, Sylow
142C2×2−1+4—2511no—
143C2×#137—2511no—
144C22×Q16D25111yescharacteristic, complemented
145#536—2511no—
146#452—2514no—
147#451—2514no—
148#419—2511no—
149#416—2511no—
150#408—2511no—
1512−1+4—5024no—
1522−1+4—5024no—
153C2×#41—5022no—
154C2×#41—5021no—
155C2×#41—2512no—
156C2×#41—2511no—
157C2×#41—2511no—
158C22×Q8—5021no—
159C22×Q8—2512no—
160C22×Q8C2×D25111yescharacteristic
161C22×Q8C2×D25111yescharacteristic
162#137—5024no—
163C2×Q16—24no—
164C2×Q16—2512no—
165C2×Q16C2×D25114yes—
166C2×SD16—2512no—
167C2×M16—2511no—
168C22×C8C2×D25111yescharacteristic
169C4⋊Q8—5021no—
170#124—5021no—
171#123—5022no—
172C22⋊Q8—2514no—
173#117—2511no—
174#103—2514no—
175#103—2512no—
176#102—2512no—
177C22⋊C8—2514no—
178#41—50216no—
179#41—5028no—
180#41—5024no—
181#41—5024no—
182#41—5022no—
183#41—5022no—
184#41—5022no—
185#41—2514no—
186C2×Q8—21no—
187C2×Q8—22no—
188C2×Q8—21no—
189C2×Q8—22no—
190C2×Q8—22no—
191C2×Q8—5022no—
192C2×Q8—5028no—
193C2×Q8—5024no—
194C2×Q8—5024no—
195C2×Q8—2511no—
196C2×Q8—2511no—
197C2×Q8C22⋊D25114yes—
198C2×Q8C22×D25112yes—
199C2×Q8C22×D25112yes—
200C2×D4—5022no—
201C2×D4—5021no—
202C2×D4—2512no—
203C2×D4—2511no—
204C2×D4—2511no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—5022no—
208C22×C4—5022no—
209C22×C4—5021no—
210C22×C4—2512no—
211C22×C4C22×D25111yescharacteristic
212Q16—28no—
213Q16—5024no—
214SD16—5024no—
215M16—5022no—
216C2×C8—22no—
217C2×C8—2512no—
218C2×C8C22×D25112yes—
219C4⋊C4—5024no—
220C4⋊C4—2512no—
221C22⋊C4—5024no—
222C22⋊C4—5021no—
223C42—5021no—
224C23—5021no—
225C23—2512no—
226C23D4×D25111yescharacteristic
227Q8—22no—
228Q8—24no—
229Q8—22no—
230Q8—22no—
231Q8—42no—
232Q8—5028no—
233Q8—5021no—
234Q8—5022no—
235Q8—2512no—
236Q8C2×C22⋊D25114yes—
237D4—5028no—
238D4—5024no—
239D4—5024no—
240D4—5021no—
241D4—5022no—
242D4—2512no—
243C2×C4—22no—
244C2×C4—24no—
245C2×C4—22no—
246C2×C4—42no—
247C2×C4—5028no—
248C2×C4—5028no—
249C2×C4—5022no—
250C2×C4—5021no—
251C2×C4—5022no—
252C2×C4—5022no—
253C2×C4—5024no—
254C2×C4—2512no—
255C2×C4D4×D25112yes—
256C2×C4D4×D25111yescharacteristic
257C2×C4C2×C22⋊D25112yes—
258C2×C4C23×D25111yescharacteristic
259C8—22no—
260C8—5022no—
261C22—22no—
262C22—10042no—
263C22—5024no—
264C22—5022no—
265C22—5022no—
266C22#2254868312yes—
267C22C2×D4×D25111yescharacteristic
268C4—24no—
269C4—42no—
270C4—10042no—
271C4—5022no—
272C4—5022no—
273C4#2254868312yes—
274C4C2×D4×D25111yescharacteristic
275C4C22×C22⋊D25111yescharacteristic
276C2—22no—
277C2—10042no—
278C2—10042no—
279C2C2×#2254868311yescharacteristic
280C2#3063250012yes—
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
FittingC22×Q16×C251C2
socleC22×C251C22×D4
prime radicalC2×#22548763C2
hypercenterC22×Q16D251
hypocenterC251#3011
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×Q16

Complements

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

NH ≅ G/Ntype
2. C2×#22548746C2semidirect
3. #30632261C2semidirect
4. #30631848C2semidirect
5. #30631841C2semidirect
6. #30631696C2semidirect
7. #30631684C2semidirect
8. #30631653C2semidirect
9. C22×Q16×C251C2semidirect
140. C251#3011semidirect
144. C22×Q16D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225487632
C2×#225487462
#306322612
#306318482
#306318412
#306316962
#306316842
#306316532
C22×Q16×C2512
#3011251