Finite Group Databasebeta

← C2×#5242 #84106

Subgroups — C2×#5242 #84106

Order 384.

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

normal subgroups only

This group has 760 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×C3×H(5,𝔽2)C211yesmaximal, characteristic
2C2×#951C212yesmaximal, complemented
3C2×#936C211yesmaximal, characteristic, complemented
4#5242C218yesmaximal, direct product factor
5C2×#837C212yesmaximal, complemented
6C2×#833C211yesmaximal, characteristic, complemented
7C2×#455—31nomaximal, Sylow
8C3×H(5,𝔽2)—24no—
9C3×H(5,𝔽2)—22no—
10C3×H(5,𝔽2)C2214yes—
11C2×C3×#41—22no—
12C2×C3×#41C2212yes—
13C22×C3×D4—22no—
14C22×C3×D4—22no—
15C22×C3×D4C2211yescharacteristic
16C22×Dic6C2211yescharacteristic, complemented
17C2×#217—22no—
18#951—24no—
19#951C2218yescomplemented
20C2×#216—22no—
21#936—22no—
22#936C2214yescomplemented
23C2×#214—22no—
24C2×C4×Dic3C2211yescharacteristic, complemented
25C2×#208C2212yescomplemented
26#837C2218yescomplemented
27#833C2214yescomplemented
28C2×H(5,𝔽2)D311yescharacteristic
29C2×#137—32no—
30C2×#124—31no—
31#455—38no—
32C2×#104—32no—
33C2×#100—31no—
34C24×C3—22no—
35C24×C3D412yes—
36C3×#41—28no—
37C3×#41—28no—
38C3×#41—22no—
39C3×#41—22no—
40C3×#41D414yes—
41C3×#41C2314yes—
42C2×C3×Q8D412yes—
43C2×C3×D4—24no—
44C2×C3×D4—28no—
45C2×C3×D4—28no—
46C2×C3×D4—22no—
47C2×C3×D4—28no—
48C2×C3×D4—28no—
49C2×C3×D4—22no—
50C2×C3×D4—22no—
51C2×C3×D4D412yes—
52C2×C3×D4D412yes—
53C2×C3×D4C2312yes—
54C22×C3×C4—22no—
55C22×C3×C4—22no—
56C22×C3×C4C2311yescharacteristic
57C22×Dic3—21no—
58C22×Dic3—21no—
59C2×Dic6—22no—
60C2×Dic6—22no—
61C2×Dic6—21no—
62C2×Dic6C2312yescomplemented
63#217—28no—
64#216—28no—
65#214—28no—
66C4×Dic3—21no—
67C4×Dic3C2312yescomplemented
68#208—22no—
69#208C2314yescomplemented
70C2×C3⋊C8—22no—
71H(5,𝔽2)—24no—
72H(5,𝔽2)—22no—
73H(5,𝔽2)C2×D314yes—
74C2×#41—22no—
75C2×#41C2×D312yes—
76C22×Q8—31no—
77C22×D4—22no—
78C22×D4—22no—
79C22×D4C2×D311yescharacteristic
80#137—38no—
81#137—64no—
82C2×Q16—62no—
83C2×SD16—62no—
84C2×M16—32no—
85#124—34no—
86#124—62no—
87C2×C22⋊C4—62no—
88C2×C42—31no—
89#104—38no—
90#100—34no—
91C23×C3—28no—
92C23×C3—22no—
93C23×C3—22no—
94C23×C3—22no—
95C23×C3—44no—
96C23×C3—84no—
97C23×C3C2×D411yescharacteristic
98C23×C3C2×D414yes—
99C3×Q8—22no—
100C3×Q8C2×D414yes—
101C3×D4—216no—
102C3×D4—24no—
103C3×D4—24no—
104C3×D4—24no—
105C3×D4—22no—
106C3×D4—24no—
107C3×D4C2×D414yes—
108C2×C3×C4—28no—
109C2×C3×C4—22no—
110C2×C3×C4—24no—
111C2×C3×C4—24no—
112C2×C3×C4C2×D411yescharacteristic
113C2×C3×C4C2×D412yes—
114C2×C3×C4C2×D412yes—
115C2×C3×C4C2411yescharacteristic
116C2×Dic3—22no—
117C2×Dic3—22no—
118C2×Dic3—42no—
119C2×Dic3—41no—
120C2×Dic3—41no—
121Dic6—24no—
122Dic6—42no—
123C3⋊C8—24no—
124C24—22no—
125C24#6712yes—
126#41—28no—
127#41—28no—
128#41—22no—
129#41—22no—
130#41#6714yes—
131#41C22×D314yes—
132C2×Q8—32no—
133C2×Q8—61no—
134C2×Q8—62no—
135C2×Q8—62no—
136C2×Q8#6712yes—
137C2×D4—24no—
138C2×D4—28no—
139C2×D4—28no—
140C2×D4—22no—
141C2×D4—28no—
142C2×D4—28no—
143C2×D4—22no—
144C2×D4—22no—
145C2×D4#6712yes—
146C2×D4#6712yes—
147C2×D4C22×D312yes—
148C22×C4—22no—
149C22×C4—22no—
150C22×C4—61no—
151C22×C4—61no—
152C22×C4C22×D311yescharacteristic
153Q16—68no—
154SD16—68no—
155M16—34no—
156M16—62no—
157C2×C8—62no—
158C22⋊C4—68no—
159C42—32no—
160C42—61no—
161C22×C3—22no—
162C22×C3—24no—
163C22×C3—24no—
164C22×C3—44no—
165C22×C3—44no—
166C22×C3—44no—
167C22×C3—82no—
168C22×C3—88no—
169C22×C3#12011yescharacteristic
170C22×C3#12011yescharacteristic
171C22×C3C22×D411yescharacteristic
172C3×C4—24no—
173C3×C4#12012yes—
174C3×C4C22×D412yes—
175Dic3—42no—
176Dic3—42no—
177C23—28no—
178C23—22no—
179C23—22no—
180C23—22no—
181C23—44no—
182C23—84no—
183C23C2×#6711yescharacteristic
184C23C2×#6714yes—
185Q8—22no—
186Q8—64no—
187Q8—122no—
188Q8C2×#6714yes—
189D4—216no—
190D4—24no—
191D4—24no—
192D4—24no—
193D4—22no—
194D4—24no—
195D4C2×#6714yes—
196C2×C4—28no—
197C2×C4—22no—
198C2×C4—24no—
199C2×C4—24no—
200C2×C4—62no—
201C2×C4—62no—
202C2×C4—122no—
203C2×C4—121no—
204C2×C4—121no—
205C2×C4C2×#6711yescharacteristic
206C2×C4C2×#6712yes—
207C2×C4C2×#6712yes—
208C2×C4C23×D311yescharacteristic
209C8—64no—
210C2×C3—21no—
211C2×C3—21no—
212C2×C3—44no—
213C2×C3—44no—
214C2×C3#45512yescomplemented
215C2×C3C2×#12011yescharacteristic
216C22—22no—
217C22—24no—
218C22—24no—
219C22—44no—
220C22—44no—
221C22—44no—
222C22—82no—
223C22—88no—
224C22#95311yescharacteristic
225C22#95311yescharacteristic
226C22C22×#6711yescharacteristic
227C4—24no—
228C4—122no—
229C4—122no—
230C4#95312yes—
231C4C22×#6712yes—
232C3C2×#45511yescharacteristic, complemented, Sylow
233C2—21no—
234C2—21no—
235C2—44no—
236C2—44no—
237C2#524212yesdirect product factor
238C2C2×#95311yescharacteristic
subgroupG/H
centerC22#953
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
Fitting, prime radicalC2×C3×H(5,𝔽2)C2
socleC22×C3#120
hypercenterC2×H(5,𝔽2)D3
hypocenterC3C2×#455
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
2. C2×#951C2semidirect
3. C2×#936C2semidirect
4. #5242C2direct (N × H)
5. C2×#837C2semidirect
6. C2×#833C2semidirect
16. C22×Dic6C22semidirect
19. #951C22semidirect
22. #936C22semidirect
24. C2×C4×Dic3C22semidirect
25. C2×#208C22semidirect
26. #837C22semidirect
27. #833C22semidirect
62. C2×Dic6C23semidirect
67. C4×Dic3C23semidirect
69. #208C23semidirect
214. C2×C3#455semidirect
232. C3C2×#455semidirect
237. C2#5242direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×H(5,𝔽2)2
C2×#9512
C2×#9362
#52422
C2×#8372
C2×#8332
C2×#4553