Finite Group Databasebeta

← #44560812

Subgroups — #44560812

Order 32128.

656 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 145 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×#22548742C214yesmaximal, complemented
2#30632238C211yesmaximal, characteristic, complemented
3#127×D251C211yesmaximal, characteristic, complemented
4#30632073C214yesmaximal, complemented
5#30632064C211yesmaximal, characteristic, complemented
6#30632063C211yesmaximal, characteristic, complemented
7#30632061C211yesmaximal, characteristic, complemented
8C4×#16860793C211yesmaximal, characteristic, complemented
9#492×C251C211yesmaximal, characteristic, complemented
10C2×#16860811C2212yes—
11C2×D4×D251—22no—
12C2×D4×D251C2212yes—
13#22548742—216no—
14C2×#16860797C2214yescomplemented
15C2×D2008C2214yescomplemented
16C2×#16860795C2214yescomplemented
17C2×#16860793C2212yes—
18#22548712C2211yescharacteristic
19#22548712C2211yescharacteristic, complemented
20#22548711—22no—
21#22548711C2212yes—
22#22548704C2212yescomplemented
23#22548690—22no—
24#22548677—22no—
25C42×D251C2211yescharacteristic
26#22548625C2212yescomplemented
27C4×C251⋊C8C2211yescharacteristic, complemented
28C2×SD16×C251C2214yescomplemented
29C4⋊Q8×C251C2211yescharacteristic, complemented
30#127×C251C2211yescharacteristic
31C4×C8×C251C2211yescharacteristic, complemented
32C23×D251—22no—
33#16860811—28no—
34C2×D1004—22no—
35C2×D1004—22no—
36C2×D1004—22no—
37C2×D1004C2312yes—
38C2×D1004C2312yes—
39C2×C22⋊D251—24no—
40C2×C22⋊D251—22no—
41D4×D251—28no—
42D4×D251—48no—
43C2×C4×D251—22no—
44C2×C4×D251D412yes—
45C2×C4×D251C2311yescharacteristic
46#16860797—28no—
47D2008—28no—
48#16860795—28no—
49#16860793D418yescomplemented
50C2×C251⋊C8C2312yes—
51(C2×D251)⋊C4—24no—
52C4×Dic251D412yes—
53C4×Dic251C2311yescharacteristic
54C2×Q8×C251C2312yes—
55C2×D4×C251—22no—
56C2×D4×C251C2312yes—
57SD16×C251—28no—
58C2×C8×C251C2312yes—
59C4⋊C4×C251—22no—
60C42×C251C2311yescharacteristic
61C22×D251—22no—
62C22×D251—22no—
63C22×D251—48no—
64C22×D251—44no—
65C22×D251C2×D411yescharacteristic
66D1004—24no—
67D1004—24no—
68D1004—44no—
69D1004—44no—
70D1004—44no—
71C22⋊D251—48no—
72C22⋊D251—44no—
73C2×Dic251C2×D412yes—
74C2×Dic251C2×D411yescharacteristic
75C4×D251—24no—
76C4×D251—44no—
77C4×D251C2×D414yes—
78C251⋊C8C2×D414yescomplemented
79C23×C251—22no—
80Q8×C251—24no—
81D4×C251—24no—
82D4×C251—44no—
83C2×C4×C251—22no—
84C2×C4×C251C2×D412yes—
85C2×C4×C251C2411yescharacteristic
86C8×C251C2×D414yescomplemented
87C22×C251—42no—
88C22×C251—44no—
89C22×C251C22×D411yescharacteristic
90C2×D251—22no—
91C2×D251—42no—
92C2×D251—42no—
93C2×D251—44no—
94C2×D251—44no—
95C2×D251—88no—
96C2×D251#12712yes—
97C4×C251—42no—
98C4×C251AGL(1,ℤ/8)14yescomplemented
99C4×C251C22×D412yes—
100Dic251—22no—
101Dic251#12712yes—
102C2×C251—82no—
103C2×C251C2×#12711yescharacteristic
104C2×C251C2×AGL(1,ℤ/8)12yes—
105D251—22no—
106D251—82no—
107D251—82no—
108C251#315111yescharacteristic, complemented, Sylow
109#3151—2511nomaximal, Sylow
110C2×AGL(1,ℤ/8)—2514no—
111#532—2511no—
112C2×#127—2511no—
113#496—2514no—
114#493—2511no—
115#493—2511no—
116#492—2511no—
117#492D25111yescharacteristic, complemented
118C4×M16—2511no—
119C2×#41—2512no—
120C22×D4—5022no—
121C22×D4—2512no—
122AGL(1,ℤ/8)—50216no—
123C2×SD16—2514no—
124C2×SD16C2×D25114yescomplemented
125C2×D8—2514no—
126C2×D8—2514no—
127C2×M16—2512no—
128C4⋊Q8C2×D25111yescharacteristic, complemented
129#127—5022no—
130#127—2512no—
131#127—2511no—
132#127—2511no—
133#127C2×D25111yescharacteristic
134#124—2512no—
135#121—5022no—
136C4×D4—5022no—
137C2×C42—2511no—
138#97—2512no—
139C4×C8—2511no—
140C4×C8C2×D25111yescharacteristic, complemented
141C24—5022no—
142#41—5028no—
143C2×Q8C22×D25112yes—
144C2×D4—22no—
145C2×D4—10048no—
146C2×D4—5022no—
147C2×D4—5028no—
148C2×D4—5022no—
149C2×D4—5024no—
150C2×D4—5022no—
151C2×D4—5022no—
152C2×D4—2512no—
153C2×D4—2512no—
154C2×D4C22×D25112yes—
155C22×C4—5022no—
156C22×C4—2512no—
157C22×C4—2511no—
158SD16—28no—
159SD16—5028no—
160D8—5028no—
161D8—5028no—
162M16—2518no—
163C2×C8—2512no—
164C2×C8C22×D25112yes—
165C4⋊C4—22no—
166C22⋊C4—5024no—
167C42—2512no—
168C42—2511no—
169C42C22×D25111yescharacteristic
170C23—22no—
171C23—10048no—
172C23—10044no—
173C23—5022no—
174C23—5022no—
175C23—2511no—
176Q8—24no—
177D4—24no—
178D4—44no—
179D4—10044no—
180D4—10048no—
181D4—10044no—
182D4—10044no—
183D4—10044no—
184D4—5024no—
185D4—5024no—
186C2×C4—22no—
187C2×C4—10044no—
188C2×C4—5024no—
189C2×C4—2512no—
190C2×C4—2511no—
191C2×C4—2514no—
192C2×C4D4×D25112yes—
193C2×C4C23×D25111yescharacteristic
194C8—2514no—
195C8D4×D25114yescomplemented
196C22—44no—
197C22—42no—
198C22—20088no—
199C22—10044no—
200C22—10044no—
201C22—10042no—
202C22—10042no—
203C22—5022no—
204C22—2512no—
205C22C2×D4×D25111yescharacteristic
206C4—42no—
207C4—5022no—
208C4—2512no—
209C4#2254874214yescomplemented
210C4C2×D4×D25112yes—
211C2—82no—
212C2—20082no—
213C2—20082no—
214C2—5022no—
215C2#127×D25111yescharacteristic
216C2C2×#2254874212yes—
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#492×C251C2
socleC22×C251C22×D4
hypercenter#492D251
hypocenterC251#3151
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ #492

Complements

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

NH ≅ G/Ntype
1. C2×#22548742C2semidirect
2. #30632238C2semidirect
3. #127×D251C2semidirect
4. #30632073C2semidirect
5. #30632064C2semidirect
6. #30632063C2semidirect
7. #30632061C2semidirect
8. C4×#16860793C2semidirect
9. #492×C251C2semidirect
14. C2×#16860797C22semidirect
15. C2×D2008C22semidirect
16. C2×#16860795C22semidirect
19. #22548712C22semidirect
22. #22548704C22semidirect
26. #22548625C22semidirect
27. C4×C251⋊C8C22semidirect
28. C2×SD16×C251C22semidirect
29. C4⋊Q8×C251C22semidirect
31. C4×C8×C251C22semidirect
49. #16860793D4semidirect
78. C251⋊C8C2×D4semidirect
86. C8×C251C2×D4semidirect
98. C4×C251AGL(1,ℤ/8)semidirect
108. C251#3151semidirect
117. #492D251semidirect
124. C2×SD16C2×D251semidirect
128. C4⋊Q8C2×D251semidirect
140. C4×C8C2×D251semidirect
195. C8D4×D251semidirect
209. C4#22548742semidirect

Maximal subgroups

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

subgroupindex
C2×#225487422
#306322382
#127×D2512
#306320732
#306320642
#306320632
#306320612
C4×#168607932
#492×C2512
#3151251