Finite Group Databasebeta

← C2×#14521222 #16855047

Subgroups — C2×#14521222 #16855047

Order 3968.

728 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 728 conjugacy classes of subgroups — too many to draw. Only its 165 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×#11758526C212yesmaximal, complemented
2C2×#11758524C211yesmaximal, characteristic, complemented
3#14521222C218yesmaximal, direct product factor
4C2×#11758425C212yesmaximal, complemented
5C2×#11758424C211yesmaximal, characteristic, complemented
6C2×C31×2−1+4C211yesmaximal, characteristic, complemented
7C31×2−1+4—24no—
8C31×2−1+4—22no—
9C31×2−1+4C2214yescomplemented
10C2×#41×C31—22no—
11C2×#41×C31—22no—
12C2×#41×C31C2212yes—
13C22×Q8×C31—22no—
14C22×Q8×C31C2211yescharacteristic
15C22×D124C2211yescharacteristic
16#11758526—24no—
17#11758526C2218yescomplemented
18#11758524—22no—
19#11758524C2214yescomplemented
20C2×#92481—22no—
21C2×#92479—22no—
22C2×(C2×D31)⋊C4—22no—
23C2×C4×Dic31C2211yescharacteristic
24C2×#92474C2212yes—
25#11758425C2218yescomplemented
26#11758424C2214yescomplemented
27C23×D31—21no—
28#41×C31—28no—
29#41×C31—28no—
30#41×C31—28no—
31#41×C31—28no—
32#41×C31—22no—
33#41×C31—22no—
34#41×C31D414yes—
35#41×C31C2314yes—
36C2×Q8×C31—22no—
37C2×Q8×C31—28no—
38C2×Q8×C31—28no—
39C2×Q8×C31—22no—
40C2×Q8×C31—22no—
41C2×Q8×C31D412yes—
42C2×Q8×C31D412yes—
43C2×Q8×C31C2312yes—
44C2×D4×C31—22no—
45C2×D4×C31—22no—
46C2×D4×C31D412yes—
47C22×C4×C31—24no—
48C22×C4×C31—22no—
49C22×C4×C31D412yes—
50C22×C4×C31C2311yescharacteristic
51C22×Dic31—21no—
52C2×D124—22no—
53C2×D124—22no—
54C2×D124—21no—
55C2×D124C2312yes—
56#92481—28no—
57#92479—28no—
58(C2×D31)⋊C4—28no—
59C4×Dic31—21no—
60C4×Dic31C2312yes—
61#92474—22no—
62#92474C2314yes—
63C2×C31⋊C8—22no—
64C23×C31—22no—
65C23×C31C2×D411yescharacteristic
66C22×D31—22no—
67C22×D31—44no—
68C22×D31—41no—
69C22×D31—41no—
70Q8×C31—28no—
71Q8×C31—22no—
72Q8×C31—24no—
73Q8×C31—24no—
74Q8×C31C2×D414yes—
75D4×C31—28no—
76D4×C31—24no—
77D4×C31—24no—
78D4×C31—22no—
79D4×C31C2×D414yes—
80C2×C4×C31—216no—
81C2×C4×C31—22no—
82C2×C4×C31—22no—
83C2×C4×C31—22no—
84C2×C4×C31—24no—
85C2×C4×C31—24no—
86C2×C4×C31C2×D414yes—
87C2×C4×C31C2×D411yescharacteristic
88C2×C4×C31C2×D412yes—
89C2×C4×C31C2×D412yes—
90C2×C4×C31C2411yescharacteristic
91C2×Dic31—22no—
92C2×Dic31—42no—
93D124—24no—
94D124—42no—
95C31⋊C8—24no—
96C2×#454—311nomaximal, Sylow
97C22×C31—22no—
98C22×C31—24no—
99C22×C31—44no—
100C22×C31#12011yescharacteristic
101C22×C31#12011yescharacteristic
102C22×C31C22×D411yescharacteristic
103C2×D31—44no—
104C2×D31—42no—
105C2×D31—82no—
106C2×D31—82no—
107C4×C31—24no—
108C4×C31—24no—
109C4×C31#12012yes—
110C4×C31C22×D412yes—
111Dic31—42no—
112C2×2−1+4D3111yescharacteristic, complemented
113C2×AGL(1,ℤ/8)—312no—
114C2×#124—311no—
115#454—318no—
116C2×#104—312no—
117C2×#101—311no—
118C2×C31—21no—
119C2×C31—21no—
120C2×C31—44no—
121C2×C31#45412yescomplemented
122C2×C31C2×#12011yescharacteristic
123D31—82no—
1242−1+4—24no—
1252−1+4—22no—
1262−1+4C2×D3114yescomplemented
127C2×#41—22no—
128C2×#41—22no—
129C2×#41C2×D3112yes—
130C22×Q8—22no—
131C22×Q8C2×D3111yescharacteristic
132C22×D4—311no—
133AGL(1,ℤ/8)—318no—
134AGL(1,ℤ/8)—624no—
135C2×SD16—622no—
136C2×D8—622no—
137C2×M16—312no—
138#124—314no—
139#124—622no—
140C2×C22⋊C4—622no—
141C2×C42—311no—
142#104—318no—
143#101—314no—
144C31C2×#45411yescharacteristic, complemented, Sylow
145C24—621no—
146#41—28no—
147#41—28no—
148#41—28no—
149#41—28no—
150#41—22no—
151#41—22no—
152#41C22⋊D3114yes—
153#41C22×D3114yes—
154C2×Q8—22no—
155C2×Q8—28no—
156C2×Q8—28no—
157C2×Q8—22no—
158C2×Q8—22no—
159C2×Q8C22⋊D3112yes—
160C2×Q8C22⋊D3112yes—
161C2×Q8C22×D3112yes—
162C2×D4—22no—
163C2×D4—22no—
164C2×D4—312no—
165C2×D4—622no—
166C2×D4—622no—
167C2×D4—621no—
168C2×D4C22⋊D3112yes—
169C22×C4—24no—
170C22×C4—22no—
171C22×C4—621no—
172C22×C4C22⋊D3112yes—
173C22×C4C22×D3111yescharacteristic
174SD16—628no—
175D8—628no—
176M16—314no—
177M16—622no—
178C2×C8—622no—
179C22⋊C4—628no—
180C42—312no—
181C42—621no—
182C23—22no—
183C23—622no—
184C23—1244no—
185C23—1241no—
186C23—1241no—
187C23C2×C22⋊D3111yescharacteristic
188Q8—28no—
189Q8—22no—
190Q8—24no—
191Q8—24no—
192Q8C2×C22⋊D3114yes—
193D4—28no—
194D4—24no—
195D4—24no—
196D4—22no—
197D4—624no—
198D4—1242no—
199D4C2×C22⋊D3114yes—
200C2×C4—216no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—24no—
205C2×C4—24no—
206C2×C4—622no—
207C2×C4—1242no—
208C2×C4C2×C22⋊D3114yes—
209C2×C4C2×C22⋊D3111yescharacteristic
210C2×C4C2×C22⋊D3112yes—
211C2×C4C2×C22⋊D3112yes—
212C2×C4C23×D3111yescharacteristic
213C8—624no—
214C22—22no—
215C22—24no—
216C22—44no—
217C22—1244no—
218C22—1242no—
219C22—2482no—
220C22—2482no—
221C22#1175853011yescharacteristic
222C22#1175853011yescharacteristic
223C22C22×C22⋊D3111yescharacteristic
224C4—24no—
225C4—24no—
226C4—1242no—
227C4#1175853012yes—
228C4C22×C22⋊D3112yes—
229C2—21no—
230C2—21no—
231C2—44no—
232C2—2482no—
233C2#1452122212yesdirect product factor
234C2C2×#1175853011yescharacteristic
subgroupG/H
centerC22#11758530
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC2×C31×2−1+4C2
socleC22×C31#120
hypercenterC2×2−1+4D31
hypocenterC31C2×#454
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×2−1+4

Complements

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

NH ≅ G/Ntype
1. C2×#11758526C2semidirect
2. C2×#11758524C2semidirect
3. #14521222C2direct (N × H)
4. C2×#11758425C2semidirect
5. C2×#11758424C2semidirect
6. C2×C31×2−1+4C2semidirect
9. C31×2−1+4C22semidirect
17. #11758526C22semidirect
19. #11758524C22semidirect
25. #11758425C22semidirect
26. #11758424C22semidirect
112. C2×2−1+4D31semidirect
121. C2×C31#454semidirect
126. 2−1+4C2×D31semidirect
144. C31C2×#454semidirect
233. C2#14521222direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#117585262
C2×#117585242
#145212222
C2×#117584252
C2×#117584242
C2×C31×2−1+42
C2×#45431