Finite Group Databasebeta

← C2×#30631840 #44559549

Subgroups — C2×#30631840 #44559549

Order 32128.

996 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 996 conjugacy classes of subgroups — too many to draw. Only its 181 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
1C23×D1004C211yesmaximal, characteristic
2C22×#16860799C212yesmaximal, complemented
3C2×#22548694C211yesmaximal, characteristic, complemented
4#30631840C218yesmaximal, direct product factor
5C2×#22548637C212yesmaximal, complemented
6C2×C22⋊C8×C251C211yesmaximal, characteristic, complemented
7C24×D251—21no—
8C22×Dic502C2211yescharacteristic, complemented
9C22×D1004—28no—
10C22×D1004—21no—
11C22×D1004—22no—
12C22×D1004C2214yes—
13C22×D1004C2212yes—
14C2×#16860799—28no—
15C2×#16860799C2218yescomplemented
16#22548694—22no—
17#22548694C2214yescomplemented
18C2×C4⋊Dic251C2211yescharacteristic, complemented
19C2×Dic251⋊C4—21no—
20C2×C22⋊Dic251—21no—
21#22548637C2218yescomplemented
22C23×C4×C251C2211yescharacteristic
23C22×C8×C251C2212yescomplemented
24C22⋊C8×C251C2214yescomplemented
25C23×D251—28no—
26C23×D251—24no—
27C23×D251—22no—
28C23×D251—44no—
29C23×D251—44no—
30C2×Dic502—21no—
31C2×Dic502—24no—
32C2×Dic502C2312yescomplemented
33C2×D1004—28no—
34C2×D1004—216no—
35C2×D1004—216no—
36C2×D1004—24no—
37C2×D1004—24no—
38C2×D1004—22no—
39C2×D1004D418yes—
40C2×D1004C2314yes—
41C22×Dic251—21no—
42C22×Dic251—21no—
43#16860799—216no—
44C4⋊Dic251—21no—
45C4⋊Dic251C2312yescomplemented
46Dic251⋊C4—24no—
47C22⋊Dic251—24no—
48C24×C251D411yescharacteristic
49C22×C4×C251—24no—
50C22×C4×C251D411yescharacteristic
51C22×C4×C251D412yes—
52C22×C4×C251C2312yes—
53C22×C4×C251C2311yescharacteristic
54C2×C8×C251—24no—
55C2×C8×C251C2314yescomplemented
56C22×D251—216no—
57C22×D251—28no—
58C22×D251—24no—
59C22×D251—48no—
60C22×D251—432no—
61C22×D251—44no—
62C22×D251—48no—
63C22×D251—44no—
64Dic502—24no—
65Dic502—42no—
66D1004—216no—
67D1004—28no—
68D1004—24no—
69D1004C2×D418yes—
70C2×Dic251—22no—
71C2×Dic251—22no—
72C2×Dic251—42no—
73C2×Dic251—42no—
74C23×C251—22no—
75C23×C251—22no—
76C23×C251SD1614yescomplemented
77C23×C251C2×D412yes—
78C23×C251C2×D411yescharacteristic
79C2×C4×C251—22no—
80C2×C4×C251—24no—
81C2×C4×C251—24no—
82C2×C4×C251C2×D414yes—
83C2×C4×C251C2×D412yes—
84C2×C4×C251C2×D411yescharacteristic
85C2×C4×C251C2411yescharacteristic
86C8×C251—24no—
87C22×C251—22no—
88C22×C251—22no—
89C22×C251—28no—
90C22×C251#12012yes—
91C22×C251C2×SD1612yes—
92C22×C251C2×SD1614yescomplemented
93C22×C251AGL(1,ℤ/8)12yes—
94C22×C251C22×D411yescharacteristic
95C2×D251—28no—
96C2×D251—432no—
97C2×D251—44no—
98C2×D251—44no—
99C2×D251—416no—
100C4×C251—22no—
101C4×C251#12012yes—
102C4×C251C22×D412yes—
103Dic251—42no—
104Dic251—42no—
105C2×C251—24no—
106C2×C251#45014yescomplemented
107C2×C251C2×#12011yescharacteristic
108C2×C251C22×SD1611yescharacteristic
109C2×C251C2×AGL(1,ℤ/8)11yescharacteristic
110D251—48no—
111C251C2×#45011yescharacteristic, complemented, Sylow
112C2×#450—2511nomaximal, Sylow
113C23×D4—2511no—
114C22×SD16—2512no—
115C2×C22⋊Q8—2511no—
116#450—2518no—
117C2×#102—2512no—
118C2×C22⋊C8D25111yescharacteristic, complemented
119C25—5021no—
120C22×Q8—2511no—
121C22×D4—5028no—
122C22×D4—5022no—
123C22×D4—5021no—
124C22×D4—2514no—
125C22×D4—2512no—
126C23×C4C2×D25111yescharacteristic
127C2×SD16—5028no—
128C2×SD16—2518no—
129C22×C8C2×D25112yescomplemented
130C22⋊Q8—5022no—
131C22⋊Q8—2514no—
132C2×C4⋊C4—5021no—
133C2×C4⋊C4—2511no—
134C2×C22⋊C4—5021no—
135#102—2518no—
136C22⋊C8C2×D25114yescomplemented
137C24—10044no—
138C24—10044no—
139C24—5028no—
140C24—5024no—
141C24—5022no—
142C24D100411yescharacteristic
143C2×Q8—5024no—
144C2×Q8—5021no—
145C2×Q8—2512no—
146C2×D4—5028no—
147C2×D4—50216no—
148C2×D4—50216no—
149C2×D4—5024no—
150C2×D4—5024no—
151C2×D4—5022no—
152C2×D4—2518no—
153C2×D4—2514no—
154C22×C4—24no—
155C22×C4—5021no—
156C22×C4—5021no—
157C22×C4D100412yes—
158C22×C4D100411yescharacteristic
159C22×C4C22×D25112yes—
160C22×C4C22×D25111yescharacteristic
161SD16—50216no—
162C2×C8—24no—
163C2×C8C22×D25114yescomplemented
164C4⋊C4—5024no—
165C4⋊C4—5021no—
166C4⋊C4—2512no—
167C22⋊C4—5024no—
168C23—22no—
169C23—22no—
170C23—10048no—
171C23—100432no—
172C23—10044no—
173C23—10044no—
174C23—10048no—
175C23—50216no—
176C23—5024no—
177C23—5028no—
178C23#1686079914yescomplemented
179C23C2×D100412yes—
180C23C2×D100411yescharacteristic
181Q8—10042no—
182Q8—5024no—
183D4—50216no—
184D4—5024no—
185D4—5028no—
186D4—2518no—
187C2×C4—22no—
188C2×C4—24no—
189C2×C4—24no—
190C2×C4—10042no—
191C2×C4—10042no—
192C2×C4—5022no—
193C2×C4—5022no—
194C2×C4D4×D25114yes—
195C2×C4C2×D100412yes—
196C2×C4C2×D100411yescharacteristic
197C2×C4C23×D25111yescharacteristic
198C8—24no—
199C22—22no—
200C22—28no—
201C22—22no—
202C22—100432no—
203C22—10044no—
204C22—10044no—
205C22—100416no—
206C22—5028no—
207C22#2254868512yes—
208C22C2×#1686079912yes—
209C22C2×#1686079914yescomplemented
210C22#2254874312yes—
211C22C22×D100411yescharacteristic
212C4—22no—
213C4—10042no—
214C4—10042no—
215C4#2254868512yes—
216C4C2×D4×D25112yes—
217C2—24no—
218C2—10048no—
219C2#3063184014yesdirect product factor
220C2C2×#2254868511yescharacteristic
221C2C22×#1686079911yescharacteristic
222C2C2×#2254874311yescharacteristic
subgroupG/H
centerC23C2×D1004
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
FittingC2×C22⋊C8×C251C2
socleC23×C251C2×D4
prime radicalC23×D1004C2
hypercenterC2×C22⋊C8D251
hypocenterC251C2×#450
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C22⋊C8

Complements

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

NH ≅ G/Ntype
2. C22×#16860799C2semidirect
3. C2×#22548694C2semidirect
4. #30631840C2direct (N × H)
5. C2×#22548637C2semidirect
6. C2×C22⋊C8×C251C2semidirect
8. C22×Dic502C22semidirect
15. C2×#16860799C22semidirect
17. #22548694C22semidirect
18. C2×C4⋊Dic251C22semidirect
21. #22548637C22semidirect
23. C22×C8×C251C22semidirect
24. C22⋊C8×C251C22semidirect
32. C2×Dic502C23semidirect
45. C4⋊Dic251C23semidirect
55. C2×C8×C251C23semidirect
76. C23×C251SD16semidirect
92. C22×C251C2×SD16semidirect
106. C2×C251#450semidirect
111. C251C2×#450semidirect
118. C2×C22⋊C8D251semidirect
129. C22×C8C2×D251semidirect
136. C22⋊C8C2×D251semidirect
163. C2×C8C22×D251semidirect
178. C23#16860799semidirect
209. C22C2×#16860799semidirect
219. C2#30631840direct (N × H)

Maximal subgroups

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

subgroupindex
C23×D10042
C22×#168607992
C2×#225486942
#306318402
C2×#225486372
C2×C22⋊C8×C2512
C2×#450251