Finite Group Databasebeta

← #44560012

Subgroups — #44560012

Order 32128.

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

normal subgroups only

This group has 524 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
1C22×#16860795C211yesmaximal, characteristic, complemented
2C2×#22548723C211yesmaximal, characteristic, complemented
3#30632256C211yesmaximal, characteristic, complemented
4#30632254C211yesmaximal, characteristic, complemented
5#30631939C214yesmaximal, complemented
6#30631922C214yesmaximal, complemented
7#30631723C211yesmaximal, characteristic, complemented
8#30631693C211yesmaximal, characteristic, complemented
9#418×C251C211yesmaximal, characteristic, complemented
10C2×#16860810C2211yescharacteristic
11C22×D1004C2211yescharacteristic
12C2×#16860795—24no—
13C2×#16860795C2212yes—
14C2×#16860795C2212yes—
15#22548723—24no—
16C2×#16860793C2212yescomplemented
17C22×C251⋊C8C2211yescharacteristic
18C2×C8×D251C2212yescomplemented
19#22548712—21no—
20#22548711—21no—
21#22548703—21no—
22#22548702—21no—
23#22548690C2214yescomplemented
24#22548686C2214yescomplemented
25#22548685—22no—
26#22548684—22no—
27#22548674C2211yescharacteristic
28#22548648C2212yescomplemented
29#22548646C2212yescomplemented
30#22548637C2214yescomplemented
31C22×D4×C251C2211yescharacteristic
32C2×M16×C251C2211yescharacteristic
33#117×C251C2211yescharacteristic
34#102×C251C2214yescomplemented
35C23×D251—21no—
36#16860810—22no—
37#16860810—22no—
38C2×Dic502D411yescharacteristic
39C2×D1004—24no—
40C2×D1004—22no—
41C2×D1004—21no—
42C2×D1004—22no—
43C2×D1004D411yescharacteristic
44C2×D1004C2312yes—
45C2×C22⋊D251—24no—
46C2×C22⋊D251D412yes—
47C2×C4×D251C2312yes—
48#16860795—24no—
49#16860795—24no—
50#16860793—24no—
51C2×C251⋊C8D414yescomplemented
52C2×C251⋊C8C2311yescharacteristic
53C2×C251⋊C8C2311yescharacteristic
54C8×D251—24no—
55C4⋊Dic251C2312yes—
56C22⋊Dic251—24no—
57C22⋊Dic251—21no—
58(C2×D251)⋊C4—24no—
59C4×Dic251—21no—
60C24×C251—21no—
61C2×D4×C251—22no—
62C2×D4×C251—21no—
63C2×D4×C251—22no—
64C2×D4×C251C2312yes—
65C22×C4×C251C2311yescharacteristic
66M16×C251—22no—
67C2×C8×C251C2312yes—
68C4⋊C4×C251C2312yes—
69C22⋊C4×C251—21no—
70C42×C251—21no—
71C22×D251—22no—
72C22×D251—42no—
73C22×D251—82no—
74C22×D251C2×D412yes—
75Dic502—22no—
76D1004—22no—
77D1004—22no—
78D1004—22no—
79D1004—42no—
80D1004—84no—
81C22⋊D251—24no—
82C22⋊D251—84no—
83C2×Dic251—22no—
84C2×Dic251C2×D412yes—
85C4×D251—22no—
86C4×D251—22no—
87C251⋊C8C2×D412yes—
88C251⋊C8C2×D412yes—
89C23×C251—22no—
90C23×C251—42no—
91C23×C251—82no—
92C23×C251C2×D411yescharacteristic
93D4×C251—22no—
94D4×C251—22no—
95D4×C251—42no—
96C2×C4×C251—22no—
97C2×C4×C251C2×D411yescharacteristic
98C2×C4×C251C2×D412yes—
99C2×C4×C251#4112yes—
100C2×C4×C251C2411yescharacteristic
101C8×C251—22no—
102C22×C251—22no—
103C22×C251—42no—
104C22×C251—44no—
105C22×C251—84no—
106C22×C251#12112yes—
107C22×C251C22×D411yescharacteristic
108C2×D251—22no—
109C2×D251—24no—
110C2×D251—42no—
111C2×D251—44no—
112C2×D251—84no—
113C4×C251—42no—
114C4×C251#12112yes—
115C4×C251C22×D411yescharacteristic
116C4×C251C2×#4111yescharacteristic
117Dic251—22no—
118Dic251—42no—
119C2×C251—22no—
120C2×C251—82no—
121C2×C251C2×#12111yescharacteristic
122C2×C251#57612yes—
123D251—42no—
124D251—82no—
125C251#305811yescharacteristic, complemented, Sylow
126#3058—2511nomaximal, Sylow
127C22×D8—2511no—
128C2×#131—2511no—
129#535—2511no—
130#535—2511no—
131#469—2514no—
132#466—2514no—
133#427—2511no—
134#418—2511no—
135#418D25111yescharacteristic, complemented
136C2×#41—2511no—
137C22×D4—2511no—
138C22×D4C2×D25111yescharacteristic
139C2×D8—5024no—
140C2×D8—2512no—
141C2×D8—2512no—
142#131—5024no—
143C2×M16—2512no—
144C2×M16C2×D25111yescharacteristic
145C22×C8—2512no—
146C22×C8—2511no—
147#127—5021no—
148#127—5021no—
149#124—5021no—
150#124—5021no—
151#121—2514no—
152#121—2514no—
153#120—5022no—
154#120—5022no—
155#117—2511no—
156#117C2×D25111yescharacteristic
157#107—2512no—
158#106—2512no—
159#102—2514no—
160#102C2×D25114yescomplemented
161C24—21no—
162C24—5021no—
163#41—5022no—
164#41—5022no—
165C2×Q8—2511no—
166C2×D4—22no—
167C2×D4—21no—
168C2×D4—22no—
169C2×D4—5024no—
170C2×D4—5024no—
171C2×D4—5021no—
172C2×D4—5022no—
173C2×D4—5022no—
174C2×D4—2512no—
175C2×D4—2511no—
176C2×D4—2512no—
177C2×D4C22×D25112yes—
178C22×C4—2512no—
179C22×C4C22×D25111yescharacteristic
180D8—5024no—
181D8—5024no—
182M16—22no—
183M16—5024no—
184C2×C8—5024no—
185C2×C8—2514no—
186C2×C8—2511no—
187C2×C8—2511no—
188C2×C8C22×D25112yes—
189C4⋊C4—2512no—
190C4⋊C4C22×D25112yes—
191C22⋊C4—21no—
192C22⋊C4—5024no—
193C22⋊C4—5021no—
194C22⋊C4—5024no—
195C42—21no—
196C42—5021no—
197C23—22no—
198C23—42no—
199C23—82no—
200C23—20082no—
201C23—10042no—
202C23—5022no—
203C23—2512no—
204C23D4×D25111yescharacteristic
205Q8—5022no—
206D4—22no—
207D4—22no—
208D4—42no—
209D4—20084no—
210D4—20084no—
211D4—10042no—
212D4—5024no—
213D4—5022no—
214D4—5022no—
215D4—5022no—
216C2×C4—22no—
217C2×C4—5022no—
218C2×C4—5022no—
219C2×C4—5022no—
220C2×C4—2512no—
221C2×C4D4×D25112yes—
222C2×C4D4×D25111yescharacteristic
223C2×C4#1686081012yes—
224C2×C4C23×D25111yescharacteristic
225C8—22no—
226C8—2512no—
227C8—2512no—
228C22—22no—
229C22—44no—
230C22—42no—
231C22—84no—
232C22—20084no—
233C22—10044no—
234C22—10042no—
235C22—5024no—
236C22—5022no—
237C22#2254868812yes—
238C22C2×D4×D25111yescharacteristic
239C4—42no—
240C4—10042no—
241C4—5022no—
242C4#2254868812yes—
243C4C2×D4×D25111yescharacteristic
244C4C2×#1686081011yescharacteristic
245C2—22no—
246C2—82no—
247C2—20082no—
248C2—10042no—
249C2C2×#2254868811yescharacteristic
250C2#3063249112yes—
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#418×C251C2
socleC22×C251C22×D4
hypercenter#418D251
hypocenterC251#3058
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
1. C22×#16860795C2semidirect
2. C2×#22548723C2semidirect
3. #30632256C2semidirect
4. #30632254C2semidirect
5. #30631939C2semidirect
6. #30631922C2semidirect
7. #30631723C2semidirect
8. #30631693C2semidirect
9. #418×C251C2semidirect
16. C2×#16860793C22semidirect
18. C2×C8×D251C22semidirect
23. #22548690C22semidirect
24. #22548686C22semidirect
28. #22548648C22semidirect
29. #22548646C22semidirect
30. #22548637C22semidirect
34. #102×C251C22semidirect
51. C2×C251⋊C8D4semidirect
125. C251#3058semidirect
135. #418D251semidirect
160. #102C2×D251semidirect

Maximal subgroups

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

subgroupindex
C22×#168607952
C2×#225487232
#306322562
#306322542
#306319392
#306319222
#306317232
#306316932
#418×C2512
#3058251