Finite Group Databasebeta

← #44560324

Subgroups — #44560324

Order 32128.

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

normal subgroups only

This group has 418 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
1#30632251C211yesmaximal, characteristic
2#30632170C211yesmaximal, characteristic, complemented
3#30632012C212yesmaximal, complemented
4#30632012C212yesmaximal, complemented
5#30632006C212yesmaximal, complemented
6#30632001C212yesmaximal, complemented
7#30631729C211yesmaximal, characteristic, complemented
8#30631725C211yesmaximal, characteristic, complemented
9#30631693C211yesmaximal, characteristic, complemented
10C2×#22548637C211yesmaximal, characteristic, complemented
11#408×C251C211yesmaximal, characteristic, complemented
12C22×D1004C2211yescharacteristic
13#22548711—21no—
14#22548704—21no—
15#22548689C2212yes—
16#22548689C2212yes—
17#22548683—22no—
18Q8×Dic251—21no—
19D4×Dic251—22no—
20D4×Dic251—21no—
21#22548674C2211yescharacteristic
22#22548674C2211yescharacteristic
23#22548674C2212yescomplemented
24C2×C4⋊Dic251C2211yescharacteristic
25C2×C4⋊Dic251C2212yescomplemented
26#22548649C2211yescharacteristic, complemented
27#22548649C2211yescharacteristic, complemented
28#22548649C2212yescomplemented
29#22548647C2211yescharacteristic, complemented
30#22548647C2211yescharacteristic, complemented
31#22548647C2212yescomplemented
32#22548637C2212yes—
33#22548637C2212yes—
34#22548637C2212yes—
35#22548637C2212yes—
36C2×#41×C251C2211yescharacteristic
37C2×M16×C251C2211yescharacteristic
38C22×C8×C251C2211yescharacteristic
39C22⋊C8×C251C2212yescomplemented
40C22⋊C8×C251C2212yescomplemented
41C23×D251—21no—
42C2×D1004—21no—
43C2×D1004—22no—
44C2×D1004—22no—
45C2×D1004C2312yes—
46C2×C22⋊D251—22no—
47C2×C22⋊D251—22no—
48C22×Dic251—21no—
49C22×Dic251—22no—
50C4⋊Dic251—21no—
51C4⋊Dic251—22no—
52C4⋊Dic251C2312yes—
53C4⋊Dic251C2311yescharacteristic
54C4⋊Dic251C2311yescharacteristic
55C4⋊Dic251C2311yescharacteristic
56C4⋊Dic251C2311yescharacteristic
57C22⋊Dic251—21no—
58C22⋊Dic251—21no—
59C22⋊Dic251—22no—
60(C2×D251)⋊C4—22no—
61(C2×D251)⋊C4—22no—
62C4×Dic251—21no—
63C4×Dic251—21no—
64C4×Dic251—22no—
65#41×C251—22no—
66#41×C251—22no—
67C2×Q8×C251D411yescharacteristic
68C2×D4×C251D412yes—
69C2×D4×C251D411yescharacteristic
70C22×C4×C251C2312yes—
71C22×C4×C251C2311yescharacteristic
72M16×C251—22no—
73C2×C8×C251—22no—
74C2×C8×C251C2311yescharacteristic
75C2×C8×C251C2311yescharacteristic
76C2×C8×C251C2312yes—
77C22×D251—22no—
78C22×D251—42no—
79C22×D251—81no—
80C22×D251—81no—
81D1004—22no—
82D1004—22no—
83D1004—42no—
84C22⋊D251—82no—
85C22⋊D251—82no—
86C2×Dic251—21no—
87C2×Dic251—21no—
88C2×Dic251—21no—
89C2×Dic251—21no—
90C2×Dic251—22no—
91C2×Dic251—81no—
92C2×Dic251—82no—
93C23×C251C2×D412yes—
94C23×C251C2×D411yescharacteristic
95Q8×C251—22no—
96D4×C251—24no—
97D4×C251—22no—
98C2×C4×C251—22no—
99C2×C4×C251—22no—
100C2×C4×C251C2×D412yes—
101C2×C4×C251C2×D411yescharacteristic
102C2×C4×C251#4112yes—
103C2×C4×C251#4112yes—
104C2×C4×C251C2411yescharacteristic
105C8×C251—21no—
106C8×C251—21no—
107C8×C251—22no—
108C22×C251—21no—
109C22×C251—21no—
110C22×C251—22no—
111C22×C251—22no—
112C22×C251—22no—
113C22×C251#12312yes—
114C22×C251C22×D411yescharacteristic
115C2×D251—42no—
116C2×D251—42no—
117C2×D251—42no—
118C2×D251—84no—
119C4×C251—22no—
120C4×C251#12312yes—
121C4×C251C2×#4111yescharacteristic
122C4×C251C2×#4111yescharacteristic
123Dic251—41no—
124Dic251—42no—
125Dic251—41no—
126Dic251—41no—
127Dic251—41no—
128C2×C251—22no—
129C2×C251—42no—
130C2×C251C2×#12311yescharacteristic
131C2×C251#57611yescharacteristic
132C2×C251#57711yescharacteristic
133D251—82no—
134C251#309611yescharacteristic, complemented, Sylow
135#3096—2511nomaximal, Sylow
136#535—2511no—
137#520—2511no—
138#482—2512no—
139#482—2512no—
140#481—2512no—
141#480—2512no—
142#428—2511no—
143#427—2511no—
144#418—2511no—
145C2×#102—2511no—
146#408D25111yescharacteristic, complemented
147C2×#41C2×D25111yescharacteristic
148C22×D4—2511no—
149C2×M16C2×D25111yescharacteristic
150C22×C8C2×D25111yescharacteristic
151#127—5021no—
152#124—5021no—
153#121—2512no—
154#121—2512no—
155#120—5022no—
156C4×Q8—5021no—
157C4×D4—5022no—
158C4×D4—5021no—
159#117—2511no—
160#117—2512no—
161#117—2511no—
162C2×C4⋊C4—2511no—
163C2×C4⋊C4—2512no—
164#107—2511no—
165#107—2511no—
166#107—2512no—
167#106—2511no—
168#106—2511no—
169#106—2512no—
170#102—2512no—
171#102—2512no—
172#102—2512no—
173#102—2512no—
174C22⋊C8C2×D25112yescomplemented
175C22⋊C8C2×D25112yescomplemented
176C24—5021no—
177#41—22no—
178#41—22no—
179C2×Q8D100411yescharacteristic
180C2×D4—5022no—
181C2×D4—5022no—
182C2×D4—5021no—
183C2×D4—5022no—
184C2×D4—5022no—
185C2×D4—2512no—
186C2×D4D100412yes—
187C2×D4D100411yescharacteristic
188C22×C4—5022no—
189C22×C4—5021no—
190C22×C4C22×D25112yes—
191C22×C4C22×D25111yescharacteristic
192M16—22no—
193C2×C8—22no—
194C2×C8C22×D25111yescharacteristic
195C2×C8C22×D25112yes—
196C2×C8C22×D25111yescharacteristic
197C4⋊C4—5022no—
198C4⋊C4—5021no—
199C4⋊C4—2512no—
200C4⋊C4—2511no—
201C4⋊C4—2511no—
202C4⋊C4—2511no—
203C4⋊C4—2511no—
204C22⋊C4—5022no—
205C22⋊C4—5021no—
206C22⋊C4—5021no—
207C22⋊C4—5022no—
208C22⋊C4—5022no—
209C42—5022no—
210C42—5021no—
211C42—5021no—
212C23—20081no—
213C23—20081no—
214C23—10042no—
215C23—5022no—
216C23C2×D100412yes—
217C23C2×D100411yescharacteristic
218Q8—22no—
219D4—24no—
220D4—22no—
221D4—20082no—
222D4—20082no—
223D4—10042no—
224D4—5022no—
225D4—5022no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—20081no—
229C2×C4—20082no—
230C2×C4—5021no—
231C2×C4—5021no—
232C2×C4—5021no—
233C2×C4—5022no—
234C2×C4—5021no—
235C2×C4C2×D100412yes—
236C2×C4C2×D100411yescharacteristic
237C2×C4#1686080912yes—
238C2×C4#1686080912yes—
239C2×C4C23×D25111yescharacteristic
240C8—21no—
241C8—21no—
242C8—22no—
243C22—21no—
244C22—21no—
245C22—22no—
246C22—22no—
247C22—22no—
248C22—20084no—
249C22—10042no—
250C22—10042no—
251C22—10042no—
252C22#2254869912yes—
253C22C22×D100411yescharacteristic
254C4—22no—
255C4—10042no—
256C4—10041no—
257C4—10041no—
258C4—10041no—
259C4—10041no—
260C4#2254869912yes—
261C4C2×#1686080911yescharacteristic
262C4C2×#1686080911yescharacteristic
263C2—22no—
264C2—42no—
265C2—20082no—
266C2C2×#2254869911yescharacteristic
267C2#3063249211yescharacteristic
268C2#3063249911yescharacteristic
subgroupG/H
centerC22C22×D1004
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#408×C251C2
socleC22×C251C22×D4
prime radical#30632251C2
hypercenter#408D251
hypocenterC251#3096
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C2×#41 ◁ #408

Complements

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

NH ≅ G/Ntype
2. #30632170C2semidirect
3. #30632012C2semidirect
4. #30632012C2semidirect
5. #30632006C2semidirect
6. #30632001C2semidirect
7. #30631729C2semidirect
8. #30631725C2semidirect
9. #30631693C2semidirect
10. C2×#22548637C2semidirect
11. #408×C251C2semidirect
23. #22548674C22semidirect
25. C2×C4⋊Dic251C22semidirect
26. #22548649C22semidirect
27. #22548649C22semidirect
28. #22548649C22semidirect
29. #22548647C22semidirect
30. #22548647C22semidirect
31. #22548647C22semidirect
39. C22⋊C8×C251C22semidirect
40. C22⋊C8×C251C22semidirect
134. C251#3096semidirect
146. #408D251semidirect
174. C22⋊C8C2×D251semidirect
175. C22⋊C8C2×D251semidirect

Maximal subgroups

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

subgroupindex
#306322512
#306321702
#306320122
#306320062
#306320012
#306317292
#306317252
#306316932
C2×#225486372
#408×C2512
#3096251