Finite Group Databasebeta

← #44560707

Subgroups — #44560707

Order 32128.

392 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 117 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
1#30632242C211yesmaximal, characteristic, complemented
2#30632236C211yesmaximal, characteristic
3#30632054C212yesmaximal, complemented
4#30632050C212yesmaximal, complemented
5#30632049C211yesmaximal, characteristic, complemented
6#30632047C211yesmaximal, characteristic, complemented
7#30632046C211yesmaximal, characteristic, complemented
8#30631682C212yesmaximal, complemented
9#30631680C212yesmaximal, complemented
10#97×D251C211yesmaximal, characteristic, complemented
11#489×C251C211yesmaximal, characteristic, complemented
12C2×#16860811C2211yescharacteristic
13C2×#16860809C2211yescharacteristic
14C2×C8×D251C2212yes—
15#22548715C2211yescharacteristic
16#22548713C2211yescharacteristic
17#22548711C2211yescharacteristic
18#22548706C2212yes—
19#22548703C2211yescharacteristic
20#22548695—21no—
21#22548691—21no—
22#22548688—21no—
23#22548688—21no—
24Q8×Dic251—21no—
25#22548680—21no—
26(C22⋊D251)⋊C4—21no—
27(C22⋊D251)⋊C4—21no—
28#22548673C2212yes—
29C42×D251C2211yescharacteristic
30#22548640C2212yes—
31#22548639C2212yes—
32#22548638C2212yes—
33#22548637C2212yes—
34#22548636C2212yes—
35#22548635C2212yes—
36#22548626C2211yescharacteristic
37#22548625C2212yes—
38C4⋊Q8×C251C2211yescharacteristic
39#124×C251C2211yescharacteristic
40#103×C251C2212yes—
41#102×C251C2212yes—
42#97×C251C2211yescharacteristic
43#16860811—24no—
44#16860809—24no—
45C2×Dic502—21no—
46C2×Dic502C2311yescharacteristic
47C2×D1004—21no—
48C2×D1004C2311yescharacteristic
49C2×C22⋊D251—21no—
50C2×C22⋊D251—21no—
51C2×C22⋊D251—21no—
52C22×Dic251—21no—
53C2×C4×D251—21no—
54C2×C4×D251D411yescharacteristic
55C2×C4×D251D411yescharacteristic
56C2×C4×D251C2311yescharacteristic
57C2×C251⋊C8C2312yes—
58C8×D251—24no—
59C4⋊Dic251—22no—
60C4⋊Dic251—21no—
61C4⋊Dic251C2312yes—
62Dic251⋊C4—21no—
63Dic251⋊C4—21no—
64Dic251⋊C4—22no—
65(C2×D251)⋊C4—22no—
66(C2×D251)⋊C4—21no—
67(C2×D251)⋊C4—21no—
68C4×Dic251—22no—
69C4×Dic251D411yescharacteristic
70C4×Dic251D411yescharacteristic
71C4×Dic251C2311yescharacteristic
72C2×Q8×C251—21no—
73C2×Q8×C251C2311yescharacteristic
74C2×D4×C251C2311yescharacteristic
75C2×C8×C251C2312yes—
76C4⋊C4×C251—21no—
77C4⋊C4×C251C2312yes—
78C22⋊C4×C251—21no—
79C22⋊C4×C251—21no—
80C42×C251C2311yescharacteristic
81C22×D251—21no—
82C22×D251C2×D411yescharacteristic
83Dic502—22no—
84Dic502—42no—
85D1004—22no—
86D1004—42no—
87C22⋊D251—42no—
88C22⋊D251—42no—
89C22⋊D251—42no—
90C2×Dic251—22no—
91C2×Dic251—21no—
92C2×Dic251—42no—
93C2×Dic251C2×D411yescharacteristic
94C2×Dic251C2×D411yescharacteristic
95C2×Dic251C2×D411yescharacteristic
96C4×D251—41no—
97C4×D251—41no—
98C4×D251—42no—
99C4×D251#4114yes—
100C251⋊C8—22no—
101C23×C251—21no—
102Q8×C251—22no—
103Q8×C251—42no—
104D4×C251—22no—
105C2×C4×C251—21no—
106C2×C4×C251—22no—
107C2×C4×C251C2×D411yescharacteristic
108C2×C4×C251C2×D411yescharacteristic
109C2×C4×C251C2411yescharacteristic
110C8×C251—22no—
111C22×C251—41no—
112C22×C251—41no—
113C22×C251—41no—
114C22×C251C22×D411yescharacteristic
115C2×D251—21no—
116C2×D251—21no—
117C2×D251—41no—
118C2×D251—41no—
119C2×D251—41no—
120C2×D251#12412yes—
121C4×C251—21no—
122C4×C251—21no—
123C4×C251—41no—
124C4×C251—42no—
125C4×C251C2×#4112yes—
126Dic251—21no—
127Dic251—21no—
128Dic251—41no—
129Dic251—42no—
130Dic251#12412yes—
131C2×C251—81no—
132C2×C251C2×#12411yescharacteristic
133C2×C251#57511yescharacteristic
134C2×C251#57511yescharacteristic
135D251—22no—
136D251—81no—
137C251#313511yescharacteristic, complemented, Sylow
138#3135—2511nomaximal, Sylow
139#533—2511no—
140#532—2511no—
141#491—2512no—
142#490—2512no—
143#489—2511no—
144#489—2511no—
145#489—2511no—
146#489D25111yescharacteristic, complemented
147#416—2512no—
148#416—2512no—
149C2×#97—2511no—
150C2×#41—2511no—
151C2×#41—2511no—
152C22×C8—2512no—
153C4⋊Q8—2511no—
154C4⋊Q8—2511no—
155C4⋊Q8C2×D25111yescharacteristic
156#127—2511no—
157#125—2512no—
158#124—2511no—
159#124C2×D25111yescharacteristic
160C22⋊Q8—5021no—
161C22⋊Q8—5021no—
162#121—5021no—
163#121—5021no—
164C4×Q8—5021no—
165C4×Q8—5021no—
166C4×D4—5021no—
167C4×D4—5021no—
168#117—2512no—
169C2×C42—2511no—
170#103—2512no—
171#103—2512no—
172#103—2512no—
173#103C2×D25112yes—
174#102—2512no—
175#102—2512no—
176#102—2512no—
177#102C2×D25112yes—
178#97—2512no—
179#97—2511no—
180#97C2×D25111yescharacteristic
181#41—5024no—
182#41—5024no—
183C2×Q8—21no—
184C2×Q8—5021no—
185C2×Q8—2511no—
186C2×Q8C22×D25111yescharacteristic
187C2×D4—5021no—
188C2×D4—5021no—
189C2×D4—5021no—
190C2×D4—5021no—
191C2×D4—2511no—
192C2×D4C22×D25111yescharacteristic
193C22×C4—5021no—
194C22×C4—5021no—
195C22×C4—2511no—
196C22×C4—2511no—
197C22×C4—2511no—
198C2×C8—5024no—
199C2×C8—2512no—
200C2×C8C22×D25112yes—
201C4⋊C4—21no—
202C4⋊C4—5021no—
203C4⋊C4—5021no—
204C4⋊C4—5022no—
205C4⋊C4—5022no—
206C4⋊C4—5021no—
207C4⋊C4—2512no—
208C4⋊C4C22×D25112yes—
209C22⋊C4—21no—
210C22⋊C4—21no—
211C22⋊C4—5022no—
212C22⋊C4—5021no—
213C22⋊C4—5021no—
214C42—5022no—
215C42—2511no—
216C42—2511no—
217C42—2511no—
218C42C22×D25111yescharacteristic
219C23—21no—
220C23—5021no—
221C23—2511no—
222Q8—22no—
223Q8—42no—
224Q8—10042no—
225Q8—5022no—
226D4—22no—
227D4—10042no—
228D4—10042no—
229D4—10042no—
230D4—10042no—
231D4—5022no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—10041no—
235C2×C4—10041no—
236C2×C4—10042no—
237C2×C4—10042no—
238C2×C4—5022no—
239C2×C4—5021no—
240C2×C4—2511no—
241C2×C4—2511no—
242C2×C4—2511no—
243C2×C4—2514no—
244C2×C4D4×D25111yescharacteristic
245C2×C4D4×D25111yescharacteristic
246C2×C4C23×D25111yescharacteristic
247C8—22no—
248C8—5022no—
249C22—41no—
250C22—41no—
251C22—41no—
252C22—10041no—
253C22—10041no—
254C22—10041no—
255C22—5021no—
256C22—5021no—
257C22—2512no—
258C22C2×D4×D25111yescharacteristic
259C4—21no—
260C4—21no—
261C4—41no—
262C4—42no—
263C4—10041no—
264C4—10042no—
265C4—5021no—
266C4—5021no—
267C4—2512no—
268C4#41×D25112yes—
269C2—81no—
270C2—20081no—
271C2—5022no—
272C2#124×D25111yescharacteristic
273C2#3063248311yescharacteristic
274C2#3063248611yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#489×C251C2
socleC22×C251C22×D4
prime radical#30632236C2
hypercenter#489D251
hypocenterC251#3135
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ #489

Complements

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

NH ≅ G/Ntype
1. #30632242C2semidirect
3. #30632054C2semidirect
4. #30632050C2semidirect
5. #30632049C2semidirect
6. #30632047C2semidirect
7. #30632046C2semidirect
8. #30631682C2semidirect
9. #30631680C2semidirect
10. #97×D251C2semidirect
11. #489×C251C2semidirect
137. C251#3135semidirect
146. #489D251semidirect

Maximal subgroups

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

subgroupindex
#306322422
#306322362
#306320542
#306320502
#306320492
#306320472
#306320462
#306316822
#306316802
#97×D2512
#489×C2512
#3135251