Finite Group Databasebeta

← #44559658

Subgroups — #44559658

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×#22548762C211yesmaximal, characteristic
2C2×#22548746C211yesmaximal, characteristic, complemented
3C22×#16860800C211yesmaximal, characteristic, complemented
4#30631847C214yesmaximal, complemented
5#30631842C214yesmaximal, complemented
6#30631695C211yesmaximal, characteristic, complemented
7#30631681C211yesmaximal, characteristic, complemented
8#30631654C211yesmaximal, characteristic, complemented
9#536×C251C211yesmaximal, characteristic, complemented
10#22548762—24no—
11#22548762—24no—
12C2×#16860810C2212yes—
13C2×#16860810C2211yescharacteristic
14C2×#16860809—22no—
15C2×#16860809—21no—
16C22×Dic502C2212yes—
17C22×Dic502C2211yescharacteristic
18C2×Q8×D251—21no—
19#22548746—24no—
20C2×#16860800—24no—
21C2×#16860800C2214yes—
22C2×#16860800C2212yes—
23C2×#16860797C2212yes—
24C2×#16860794C2211yescharacteristic
25C22×C251⋊C8C2211yescharacteristic
26#22548638C2212yes—
27#22548638C2214yes—
28#22548635C2212yes—
29C22⋊(C251⋊C8)C2214yes—
30C2×#41×C251C2211yescharacteristic
31C22×Q8×C251C2211yescharacteristic
32C4⋊Q8×C251—21no—
33#124×C251—21no—
34#123×C251—22no—
35C22⋊Q8×C251C2214yes—
36#117×C251C2211yescharacteristic
37#16860810—24no—
38#16860810—24no—
39#16860810—22no—
40#16860810D414yes—
41#16860809—216no—
42#16860809—28no—
43C2×Dic502—24no—
44C2×Dic502—24no—
45C2×Dic502—22no—
46C2×Dic502—21no—
47C2×Dic502—22no—
48C2×Dic502D414yes—
49C2×Dic502C2311yescharacteristic
50C2×Dic502C2312yes—
51Q8×D251—28no—
52C2×D1004C2311yescharacteristic
53C2×C22⋊D251—22no—
54C2×C22⋊D251—21no—
55C22×Dic251—22no—
56C22×Dic251—21no—
57C2×C4×D251—22no—
58C2×C4×D251—21no—
59#16860800—24no—
60#16860800—28no—
61#16860797—24no—
62#16860794—22no—
63C2×C251⋊C8—22no—
64C2×C251⋊C8C2312yes—
65C2×C251⋊C8C2312yes—
66#41×C251—22no—
67#41×C251—22no—
68C2×Q8×C251—21no—
69C2×Q8×C251—22no—
70C2×Q8×C251—22no—
71C2×Q8×C251D411yescharacteristic
72C2×Q8×C251C2312yes—
73C2×D4×C251D412yes—
74C2×D4×C251D411yescharacteristic
75C22×C4×C251—21no—
76C22×C4×C251C2312yes—
77C22×C4×C251C2311yescharacteristic
78C4⋊C4×C251—24no—
79C4⋊C4×C251C2312yes—
80C22⋊C4×C251—21no—
81C22⋊C4×C251—24no—
82C42×C251—21no—
83C22×D251—21no—
84Dic502—28no—
85Dic502—24no—
86Dic502—21no—
87Dic502—22no—
88Dic502C2×D412yes—
89Dic502C2×D414yes—
90D1004—21no—
91D1004C2×D412yes—
92C22⋊D251—28no—
93C22⋊D251—24no—
94C2×Dic251—28no—
95C2×Dic251—24no—
96C2×Dic251—22no—
97C2×Dic251—21no—
98C4×D251—28no—
99C4×D251—24no—
100C251⋊C8—22no—
101C251⋊C8—22no—
102C23×C251C2×D412yes—
103C23×C251C2×D411yescharacteristic
104Q8×C251—22no—
105Q8×C251—22no—
106Q8×C251—22no—
107Q8×C251—42no—
108D4×C251—24no—
109D4×C251—22no—
110C2×C4×C251—22no—
111C2×C4×C251—22no—
112C2×C4×C251—22no—
113C2×C4×C251—22no—
114C2×C4×C251—42no—
115C2×C4×C251C2×D412yes—
116C2×C4×C251C2×D412yes—
117C2×C4×C251C2×D412yes—
118C2×C4×C251C2×D411yescharacteristic
119C2×C4×C251C2411yescharacteristic
120C22×C251—22no—
121C22×C251—22no—
122C22×C251—24no—
123C22×C251#12012yes—
124C22×C251C22×D411yescharacteristic
125C2×D251—22no—
126C2×D251—42no—
127C4×C251—22no—
128C4×C251—42no—
129C4×C251—42no—
130C4×C251#12012yes—
131C4×C251C22×D411yescharacteristic
132C4×C251C22×D411yescharacteristic
133Dic251—24no—
134Dic251—22no—
135C2×C251—22no—
136C2×C251—42no—
137C2×C251C2×#12011yescharacteristic
138C2×C251#57812yes—
139D251—42no—
140C251#301111yescharacteristic, complemented, Sylow
141#3011—2511nomaximal, Sylow
142C2×2−1+4—2511no—
143C2×#137—2511no—
144C22×Q16—2511no—
145#536D25111yescharacteristic, complemented
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×#41C2×D25111yescharacteristic
158C22×Q8—5021no—
159C22×Q8—2512no—
160C22×Q8—2511no—
161C22×Q8C2×D25111yescharacteristic
162#137—5024no—
163C2×Q16—5024no—
164C2×Q16—2514no—
165C2×Q16—2512no—
166C2×SD16—2512no—
167C2×M16—2511no—
168C22×C8—2511no—
169C4⋊Q8—21no—
170#124—21no—
171#123—22no—
172C22⋊Q8C2×D25114yes—
173#117C2×D25111yescharacteristic
174#103—2512no—
175#103—2514no—
176#102—2512no—
177C22⋊C8—2514no—
178#41—22no—
179#41—22no—
180#41—5024no—
181#41—5024no—
182#41—50216no—
183#41—5028no—
184#41—5022no—
185#41—2514no—
186C2×Q8—21no—
187C2×Q8—22no—
188C2×Q8—22no—
189C2×Q8—5024no—
190C2×Q8—5024no—
191C2×Q8—5028no—
192C2×Q8—5022no—
193C2×Q8—5022no—
194C2×Q8—5021no—
195C2×Q8—2514no—
196C2×Q8—2511no—
197C2×Q8—2512no—
198C2×Q8C22⋊D25111yescharacteristic
199C2×Q8C22×D25112yes—
200C2×D4—5022no—
201C2×D4—5021no—
202C2×D4—2511no—
203C2×D4C22⋊D25112yes—
204C2×D4C22⋊D25111yescharacteristic
205C22×C4—21no—
206C22×C4—5022no—
207C22×C4—5021no—
208C22×C4—5022no—
209C22×C4—5021no—
210C22×C4C22×D25112yes—
211C22×C4C22×D25111yescharacteristic
212Q16—5028no—
213Q16—5024no—
214SD16—5024no—
215M16—5022no—
216C2×C8—5022no—
217C2×C8—2512no—
218C2×C8—2512no—
219C4⋊C4—24no—
220C4⋊C4C22×D25112yes—
221C22⋊C4—21no—
222C22⋊C4—24no—
223C42—21no—
224C23—5021no—
225C23C2×C22⋊D25112yes—
226C23C2×C22⋊D25111yescharacteristic
227Q8—22no—
228Q8—22no—
229Q8—22no—
230Q8—42no—
231Q8—5028no—
232Q8—5021no—
233Q8—5022no—
234Q8—5024no—
235Q8—2512no—
236Q8—2514no—
237D4—24no—
238D4—22no—
239D4—5028no—
240D4—5024no—
241D4—5021no—
242D4—2512no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—42no—
248C2×C4—5028no—
249C2×C4—5028no—
250C2×C4—5022no—
251C2×C4—5021no—
252C2×C4—5024no—
253C2×C4—5024no—
254C2×C4D4×D25112yes—
255C2×C4D4×D25112yes—
256C2×C4C2×C22⋊D25112yes—
257C2×C4C2×C22⋊D25111yescharacteristic
258C2×C4C23×D25111yescharacteristic
259C8—5022no—
260C8—5022no—
261C22—22no—
262C22—24no—
263C22—22no—
264C22—10042no—
265C22—5022no—
266C22#2254868312yes—
267C22C22×C22⋊D25111yescharacteristic
268C4—22no—
269C4—42no—
270C4—42no—
271C4—5024no—
272C4—5022no—
273C4#2254868312yes—
274C4C2×D4×D25111yescharacteristic
275C4C2×D4×D25111yescharacteristic
276C2—22no—
277C2—42no—
278C2—10042no—
279C2C2×#2254868311yescharacteristic
280C2#3063250212yes—
subgroupG/H
centerC22C22×C22⋊D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#536×C251C2
socleC22×C251C22×D4
prime radicalC2×#22548762C2
hypercenter#536D251
hypocenterC251#3011
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C2×#41 ◁ #536

Complements

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

NH ≅ G/Ntype
2. C2×#22548746C2semidirect
3. C22×#16860800C2semidirect
4. #30631847C2semidirect
5. #30631842C2semidirect
6. #30631695C2semidirect
7. #30631681C2semidirect
8. #30631654C2semidirect
9. #536×C251C2semidirect
140. C251#3011semidirect
145. #536D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225487622
C2×#225487462
C22×#168608002
#306318472
#306318422
#306316952
#306316812
#306316542
#536×C2512
#3011251