Finite Group Databasebeta

← C2×#5246 #84118

Subgroups — C2×#5246 #84118

Order 384.

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

normal subgroups only

This group has 708 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×C3×2−1+4C211yesmaximal, characteristic
2C2×#951C212yesmaximal, complemented
3C2×#944C211yesmaximal, characteristic, complemented
4#5246C218yesmaximal, direct product factor
5C2×#837C212yesmaximal, complemented
6C2×#836C211yesmaximal, characteristic, complemented
7C2×#456—31nomaximal, Sylow
8C3×2−1+4—24no—
9C3×2−1+4—22no—
10C3×2−1+4C2214yes—
11C2×C3×#41—22no—
12C2×C3×#41—22no—
13C2×C3×#41C2212yes—
14C22×C3×Q8—22no—
15C22×C3×Q8C2211yescharacteristic
16C22×Dic6C2211yescharacteristic
17#951—24no—
18#951C2218yescomplemented
19#944—22no—
20#944C2214yescomplemented
21C2×#216—22no—
22C2×#214—22no—
23C2×Dic3⋊C4—22no—
24C2×C4×Dic3C2211yescharacteristic
25C2×#208C2212yes—
26#837C2218yescomplemented
27#836C2214yescomplemented
28C2×2−1+4D311yescharacteristic
29C2×#137—32no—
30C2×C4⋊Q8—31no—
31#456—38no—
32C2×#104—32no—
33C2×#101—31no—
34C3×#41—28no—
35C3×#41—28no—
36C3×#41—28no—
37C3×#41—28no—
38C3×#41—22no—
39C3×#41—22no—
40C3×#41D414yes—
41C3×#41C2314yes—
42C2×C3×Q8—22no—
43C2×C3×Q8—28no—
44C2×C3×Q8—28no—
45C2×C3×Q8—22no—
46C2×C3×Q8—22no—
47C2×C3×Q8D412yes—
48C2×C3×Q8D412yes—
49C2×C3×Q8C2312yes—
50C2×C3×D4—22no—
51C2×C3×D4—22no—
52C2×C3×D4D412yes—
53C22×C3×C4—24no—
54C22×C3×C4—22no—
55C22×C3×C4D412yes—
56C22×C3×C4C2311yescharacteristic
57C22×Dic3—21no—
58C22×Dic3—21no—
59C2×Dic6—22no—
60C2×Dic6—22no—
61C2×Dic6—21no—
62C2×Dic6C2312yes—
63#216—28no—
64#214—28no—
65Dic3⋊C4—28no—
66C4×Dic3—21no—
67C4×Dic3C2312yes—
68#208—22no—
69#208C2314yes—
70C2×C3⋊C8—22no—
712−1+4—24no—
722−1+4—22no—
732−1+4C2×D314yes—
74C2×#41—22no—
75C2×#41—22no—
76C2×#41C2×D312yes—
77C22×Q8—22no—
78C22×Q8—31no—
79C22×Q8C2×D311yescharacteristic
80#137—38no—
81#137—64no—
82C2×Q16—62no—
83C2×SD16—62no—
84C2×M16—32no—
85C4⋊Q8—34no—
86C4⋊Q8—62no—
87C2×C4⋊C4—62no—
88C2×C42—31no—
89#104—38no—
90#101—34no—
91C23×C3—22no—
92C23×C3C2×D411yescharacteristic
93C3×Q8—28no—
94C3×Q8—22no—
95C3×Q8—24no—
96C3×Q8—24no—
97C3×Q8C2×D414yes—
98C3×D4—28no—
99C3×D4—24no—
100C3×D4—24no—
101C3×D4—22no—
102C3×D4C2×D414yes—
103C2×C3×C4—216no—
104C2×C3×C4—22no—
105C2×C3×C4—22no—
106C2×C3×C4—22no—
107C2×C3×C4—24no—
108C2×C3×C4—24no—
109C2×C3×C4C2×D414yes—
110C2×C3×C4C2×D411yescharacteristic
111C2×C3×C4C2×D412yes—
112C2×C3×C4C2×D412yes—
113C2×C3×C4C2411yescharacteristic
114C2×Dic3—24no—
115C2×Dic3—22no—
116C2×Dic3—22no—
117C2×Dic3—41no—
118C2×Dic3—41no—
119Dic6—24no—
120Dic6—42no—
121C3⋊C8—24no—
122#41—28no—
123#41—28no—
124#41—28no—
125#41—28no—
126#41—22no—
127#41—22no—
128#41#6714yes—
129#41C22×D314yes—
130C2×Q8—22no—
131C2×Q8—28no—
132C2×Q8—28no—
133C2×Q8—22no—
134C2×Q8—22no—
135C2×Q8—32no—
136C2×Q8—61no—
137C2×Q8—62no—
138C2×Q8—62no—
139C2×Q8#6712yes—
140C2×Q8#6712yes—
141C2×Q8C22×D312yes—
142C2×D4—22no—
143C2×D4—22no—
144C2×D4#6712yes—
145C22×C4—24no—
146C22×C4—22no—
147C22×C4—61no—
148C22×C4—61no—
149C22×C4#6712yes—
150C22×C4C22×D311yescharacteristic
151Q16—68no—
152SD16—68no—
153M16—34no—
154M16—62no—
155C2×C8—62no—
156C4⋊C4—68no—
157C42—32no—
158C42—61no—
159C22×C3—22no—
160C22×C3—24no—
161C22×C3—44no—
162C22×C3#12011yescharacteristic
163C22×C3#12011yescharacteristic
164C22×C3C22×D411yescharacteristic
165C3×C4—24no—
166C3×C4—24no—
167C3×C4#12012yes—
168C3×C4C22×D412yes—
169Dic3—24no—
170Dic3—42no—
171C23—22no—
172C23C2×#6711yescharacteristic
173Q8—28no—
174Q8—22no—
175Q8—24no—
176Q8—24no—
177Q8—64no—
178Q8—122no—
179Q8C2×#6714yes—
180D4—28no—
181D4—24no—
182D4—24no—
183D4—22no—
184D4C2×#6714yes—
185C2×C4—216no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—24no—
190C2×C4—24no—
191C2×C4—64no—
192C2×C4—62no—
193C2×C4—62no—
194C2×C4—121no—
195C2×C4—121no—
196C2×C4C2×#6714yes—
197C2×C4C2×#6711yescharacteristic
198C2×C4C2×#6712yes—
199C2×C4C2×#6712yes—
200C2×C4C23×D311yescharacteristic
201C8—64no—
202C2×C3—21no—
203C2×C3—21no—
204C2×C3—44no—
205C2×C3#45612yescomplemented
206C2×C3C2×#12011yescharacteristic
207C22—22no—
208C22—24no—
209C22—44no—
210C22#95311yescharacteristic
211C22#95311yescharacteristic
212C22C22×#6711yescharacteristic
213C4—24no—
214C4—24no—
215C4—64no—
216C4—122no—
217C4#95312yes—
218C4C22×#6712yes—
219C3C2×#45611yescharacteristic, complemented, Sylow
220C2—21no—
221C2—21no—
222C2—44no—
223C2#524612yesdirect product factor
224C2C2×#95311yescharacteristic
subgroupG/H
centerC22#953
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
Fitting, prime radicalC2×C3×2−1+4C2
socleC22×C3#120
hypercenterC2×2−1+4D3
hypocenterC3C2×#456
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×2−1+4

Complements

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

NH ≅ G/Ntype
2. C2×#951C2semidirect
3. C2×#944C2semidirect
4. #5246C2direct (N × H)
5. C2×#837C2semidirect
6. C2×#836C2semidirect
18. #951C22semidirect
20. #944C22semidirect
26. #837C22semidirect
27. #836C22semidirect
205. C2×C3#456semidirect
219. C3C2×#456semidirect
223. C2#5246direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×2−1+42
C2×#9512
C2×#9442
#52462
C2×#8372
C2×#8362
C2×#4563