Finite Group Databasebeta

← #44560310

Subgroups — #44560310

Order 32128.

410 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 121 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
1C22⋊Q8×D251C211yesmaximal, characteristic
2C2×#22548673C211yesmaximal, characteristic
3#30632030C212yesmaximal, complemented
4#30632026C212yesmaximal, complemented
5#30632022C211yesmaximal, characteristic, complemented
6#30632018C211yesmaximal, characteristic, complemented
7#30632015C211yesmaximal, characteristic, complemented
8#30631728C212yesmaximal, complemented
9#30631694C212yesmaximal, complemented
10#30631649C211yesmaximal, characteristic
11#484×C251C211yesmaximal, characteristic, complemented
12C2×Q8×D251C2211yescharacteristic
13C22×C4×D251C2211yescharacteristic
14C2×#16860793C2212yes—
15#22548695C2211yescharacteristic
16#22548694C2211yescharacteristic
17#22548693—21no—
18#22548692—21no—
19#22548691C2211yescharacteristic
20#22548673—21no—
21#22548673—21no—
22#22548673C2212yes—
23#22548673C2212yes—
24C2×C4⋊Dic251C2211yescharacteristic
25C4⋊C4×D251—21no—
26C4⋊C4×D251C2211yescharacteristic
27C2×(C2×D251)⋊C4—21no—
28C22⋊C4×D251—21no—
29C2×C4×Dic251—21no—
30#22548648C2212yescomplemented
31#22548647C2212yescomplemented
32#22548646C2212yescomplemented
33#22548640C2212yescomplemented
34#22548639C2212yescomplemented
35#22548638C2212yescomplemented
36C22⋊(C251⋊C8)C2211yescharacteristic
37(C2×D251)⋊C8C2212yes—
38C22⋊Q8×C251C2211yescharacteristic
39C2×C4⋊C4×C251C2211yescharacteristic
40#107×C251C2212yescomplemented
41#103×C251C2212yescomplemented
42C22⋊C8×C251C2211yescharacteristic
43C23×D251D411yescharacteristic
44C2×Dic502—21no—
45C2×Dic502C2311yescharacteristic
46Q8×D251—24no—
47C22×Dic251—21no—
48C22×Dic251D411yescharacteristic
49C2×C4×D251—24no—
50C2×C4×D251—21no—
51C2×C4×D251—21no—
52C2×C4×D251D411yescharacteristic
53C2×C4×D251D411yescharacteristic
54C2×C4×D251C2311yescharacteristic
55C2×C4×D251C2311yescharacteristic
56#16860793—24no—
57C2×C251⋊C8C2312yes—
58C4⋊Dic251—21no—
59C4⋊Dic251—21no—
60C4⋊Dic251C2312yes—
61C4⋊Dic251C2311yescharacteristic
62Dic251⋊C4—21no—
63Dic251⋊C4—21no—
64Dic251⋊C4—21no—
65C22⋊Dic251—21no—
66(C2×D251)⋊C4—21no—
67(C2×D251)⋊C4—21no—
68(C2×D251)⋊C4—22no—
69(C2×D251)⋊C4—22no—
70C4×Dic251—22no—
71C4×Dic251—22no—
72C2×Q8×C251C2311yescharacteristic
73C22×C4×C251—21no—
74C22×C4×C251C2311yescharacteristic
75C2×C8×C251C2312yes—
76C4⋊C4×C251—21no—
77C4⋊C4×C251—21no—
78C4⋊C4×C251C2312yes—
79C4⋊C4×C251C2311yescharacteristic
80C22⋊C4×C251—21no—
81C22×D251—22no—
82C22×D251—22no—
83C22×D251—41no—
84C22×D251C2×D411yescharacteristic
85C22×D251C2×D411yescharacteristic
86Dic502—22no—
87Dic502—42no—
88C2×Dic251—22no—
89C2×Dic251—22no—
90C2×Dic251—21no—
91C2×Dic251—21no—
92C2×Dic251—42no—
93C2×Dic251C2×D411yescharacteristic
94C2×Dic251C2×D411yescharacteristic
95C4×D251—22no—
96C4×D251—22no—
97C4×D251—22no—
98C4×D251—42no—
99C4×D251—42no—
100C4×D251#4114yes—
101C251⋊C8—22no—
102C23×C251C2×D411yescharacteristic
103Q8×C251—22no—
104C2×C4×C251—22no—
105C2×C4×C251—21no—
106C2×C4×C251—21no—
107C2×C4×C251—22no—
108C2×C4×C251—42no—
109C2×C4×C251C2×D411yescharacteristic
110C2×C4×C251C2411yescharacteristic
111C8×C251—22no—
112C22×C251—21no—
113C22×C251—21no—
114C22×C251#13712yes—
115C22×C251C22×D411yescharacteristic
116C2×D251—21no—
117C2×D251—21no—
118C2×D251—21no—
119C2×D251—21no—
120C2×D251—21no—
121C2×D251—44no—
122C2×D251#12312yes—
123C4×C251—21no—
124C4×C251—41no—
125C4×C251—42no—
126C4×C251—41no—
127C4×C251C2×#4112yes—
128Dic251—21no—
129Dic251—41no—
130Dic251—42no—
131Dic251—41no—
132Dic251#12312yes—
133C2×C251—22no—
134C2×C251C2×#12311yescharacteristic
135C2×C251C2×#13711yescharacteristic
136C2×C251#57511yescharacteristic
137D251—22no—
138D251—41no—
139C251#309511yescharacteristic, complemented, Sylow
140#3095—2511nomaximal, Sylow
141C2×C22⋊Q8—2511no—
142C2×#117—2511no—
143#485—2512no—
144#485—2512no—
145#484—2511no—
146#484D25111yescharacteristic, complemented
147#483—2511no—
148#483—2511no—
149#428—2512no—
150#419—2512no—
151#407—2511no—
152C22×Q8—2511no—
153C23×C4—2511no—
154C2×M16—2512no—
155C22⋊Q8—5021no—
156C22⋊Q8—5021no—
157C22⋊Q8—2511no—
158C22⋊Q8—2511no—
159C22⋊Q8—2511no—
160C22⋊Q8C2×D25111yescharacteristic
161#117—5021no—
162#117—5021no—
163#117—2512no—
164#117—2512no—
165C2×C4⋊C4—5021no—
166C2×C4⋊C4—2511no—
167C2×C4⋊C4—2511no—
168C2×C4⋊C4C2×D25111yescharacteristic
169C2×C22⋊C4—5021no—
170C2×C22⋊C4—5021no—
171C2×C42—5021no—
172#107—2512no—
173#107C2×D25112yescomplemented
174#106—2512no—
175#106—2512no—
176#103—2512no—
177#103—2512no—
178#103—2512no—
179#103C2×D25112yescomplemented
180C22⋊C8—2511no—
181C22⋊C8—2512no—
182C22⋊C8C2×D25111yescharacteristic
183C24—2511no—
184C2×Q8—5021no—
185C2×Q8—5024no—
186C2×Q8—2511no—
187C2×Q8C22×D25111yescharacteristic
188C22×C4—21no—
189C22×C4—5021no—
190C22×C4—5021no—
191C22×C4—5024no—
192C22×C4—5021no—
193C22×C4—2511no—
194C22×C4—2511no—
195C22×C4—2511no—
196C22×C4—2511no—
197C22×C4—2511no—
198C22×C4C22×D25111yescharacteristic
199M16—5024no—
200C2×C8—2512no—
201C2×C8C22×D25112yes—
202C4⋊C4—21no—
203C4⋊C4—21no—
204C4⋊C4—5021no—
205C4⋊C4—5021no—
206C4⋊C4—5021no—
207C4⋊C4—5021no—
208C4⋊C4—5021no—
209C4⋊C4—2512no—
210C4⋊C4—2511no—
211C4⋊C4C22×D25112yes—
212C4⋊C4C22×D25111yescharacteristic
213C22⋊C4—21no—
214C22⋊C4—5022no—
215C22⋊C4—5022no—
216C22⋊C4—5021no—
217C22⋊C4—5021no—
218C22⋊C4—5021no—
219C42—5022no—
220C42—5022no—
221C23—10041no—
222C23—5022no—
223C23—5022no—
224C23—2511no—
225C23—2511no—
226C23D4×D25111yescharacteristic
227Q8—22no—
228Q8—10042no—
229Q8—5022no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—42no—
235C2×C4—10042no—
236C2×C4—10042no—
237C2×C4—10042no—
238C2×C4—5022no—
239C2×C4—5022no—
240C2×C4—5022no—
241C2×C4—5021no—
242C2×C4—5021no—
243C2×C4—5022no—
244C2×C4—5022no—
245C2×C4—2511no—
246C2×C4—2511no—
247C2×C4—2514no—
248C2×C4D4×D25111yescharacteristic
249C2×C4C23×D25111yescharacteristic
250C8—22no—
251C8—5022no—
252C22—21no—
253C22—21no—
254C22—10044no—
255C22—5021no—
256C22—5021no—
257C22—5021no—
258C22—5021no—
259C22—5021no—
260C22—2512no—
261C22#2254874712yes—
262C22C2×D4×D25111yescharacteristic
263C4—21no—
264C4—42no—
265C4—41no—
266C4—41no—
267C4—10042no—
268C4—10041no—
269C4—10041no—
270C4—5021no—
271C4—2512no—
272C4#41×D25112yes—
273C2—22no—
274C2—10041no—
275C2—5022no—
276C2#123×D25111yescharacteristic
277C2C2×#2254874711yescharacteristic
278C2#3063248311yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#484×C251C2
socleC22×C251C22×D4
prime radicalC23×D251D4
hypercenter#484D251
hypocenterC251#3095
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #484

Complements

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

NH ≅ G/Ntype
3. #30632030C2semidirect
4. #30632026C2semidirect
5. #30632022C2semidirect
6. #30632018C2semidirect
7. #30632015C2semidirect
8. #30631728C2semidirect
9. #30631694C2semidirect
11. #484×C251C2semidirect
30. #22548648C22semidirect
31. #22548647C22semidirect
32. #22548646C22semidirect
33. #22548640C22semidirect
34. #22548639C22semidirect
35. #22548638C22semidirect
40. #107×C251C22semidirect
41. #103×C251C22semidirect
139. C251#3095semidirect
146. #484D251semidirect
173. #107C2×D251semidirect
179. #103C2×D251semidirect

Maximal subgroups

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

subgroupindex
C22⋊Q8×D2512
C2×#225486732
#306320302
#306320262
#306320222
#306320182
#306320152
#306317282
#306316942
#306316492
#484×C2512
#3095251