Finite Group Databasebeta

← #84116

Subgroups — #84116

Order 384.

698 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 698 conjugacy classes of subgroups — too many to draw. Only its 161 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, complemented
2#5822C212yesmaximal, complemented
3#5816C211yesmaximal, characteristic, complemented
4#5246C214yesmaximal, complemented
5#5245C214yesmaximal, complemented
6#5236C212yesmaximal, complemented
7C2×#836C211yesmaximal, characteristic, complemented
8#3020—31nomaximal, Sylow
9C3×2−1+4—24no—
10C3×2−1+4—22no—
11C3×2−1+4C2214yes—
12C2×C3×#41—22no—
13C2×C3×#41—22no—
14C2×C3×#41C2212yes—
15C22×C3×Q8—22no—
16C22×C3×Q8C2211yescharacteristic
17C2×#235C2211yescharacteristic
18#951C2214yescomplemented
19#950—24no—
20#949C2214yescomplemented
21#947C2212yescomplemented
22#946—22no—
23#944C2212yescomplemented
24#942—22no—
25#932—22no—
26#929—22no—
27#926C2211yescharacteristic
28C2×#208C2212yes—
29#837C2218yescomplemented
30#836C2214yescomplemented
31C2×2−1+4D311yescharacteristic, complemented
32#575—32no—
33#536—31no—
34#456—34no—
35#454—34no—
36#421—32no—
37C2×#101—31no—
38C3×#41—28no—
39C3×#41—28no—
40C3×#41—28no—
41C3×#41—28no—
42C3×#41—22no—
43C3×#41—22no—
44C3×#41D414yes—
45C3×#41C2314yes—
46C2×C3×Q8—22no—
47C2×C3×Q8—28no—
48C2×C3×Q8—28no—
49C2×C3×Q8—22no—
50C2×C3×Q8—22no—
51C2×C3×Q8D412yes—
52C2×C3×Q8D412yes—
53C2×C3×Q8C2312yes—
54C2×C3×D4—22no—
55C2×C3×D4—22no—
56C2×C3×D4D412yes—
57C22×C3×C4—24no—
58C22×C3×C4—22no—
59C22×C3×C4D412yes—
60C22×C3×C4C2311yescharacteristic
61C2×#67—21no—
62#235—22no—
63#235—22no—
64C2×D12C2311yescharacteristic
65C2×C4×D3—21no—
66C2×Dic6C2311yescharacteristic
67#217—21no—
68#216—24no—
69#215—24no—
70#214—24no—
71#213—24no—
72(C2×D3)⋊C4—24no—
73C4⋊Dic3—21no—
74Dic3⋊C4—24no—
75C4×Dic3C2312yes—
76#208—22no—
77#208C2314yes—
78C2×C3⋊C8—22no—
792−1+4—24no—
802−1+4—22no—
812−1+4C2×D314yes—
82C2×#41—22no—
83C2×#41—22no—
84C2×#41—31no—
85C2×#41C2×D312yes—
86C22×Q8—22no—
87C22×Q8C2×D311yescharacteristic
88#137—34no—
89#137—62no—
90AGL(1,ℤ/8)—34no—
91AGL(1,ℤ/8)—62no—
92#135—64no—
93C2×M16—32no—
94C4⋊Q8—32no—
95#124—32no—
96#123—62no—
97C22⋊Q8—62no—
98#117—31no—
99#104—38no—
100#101—34no—
101C23×C3—22no—
102C23×C3C2×D411yescharacteristic
103C22×D3—21no—
104C3×Q8—28no—
105C3×Q8—22no—
106C3×Q8—24no—
107C3×Q8—24no—
108C3×Q8C2×D414yes—
109C3×D4—28no—
110C3×D4—24no—
111C3×D4—24no—
112C3×D4—22no—
113C3×D4C2×D414yes—
114C2×C3×C4—216no—
115C2×C3×C4—22no—
116C2×C3×C4—22no—
117C2×C3×C4—22no—
118C2×C3×C4—24no—
119C2×C3×C4—24no—
120C2×C3×C4C2×D414yes—
121C2×C3×C4C2×D411yescharacteristic
122C2×C3×C4C2×D412yes—
123C2×C3×C4C2×D412yes—
124C2×C3×C4C2411yescharacteristic
125#67—41no—
126#67—41no—
127C2×Dic3—22no—
128C2×Dic3—21no—
129D12—22no—
130C4×D3—42no—
131Dic6—22no—
132C3⋊C8—24no—
133#41—28no—
134#41—28no—
135#41—28no—
136#41—28no—
137#41—22no—
138#41—22no—
139#41—62no—
140#41—62no—
141#41#6714yes—
142#41C22×D314yes—
143C2×Q8—22no—
144C2×Q8—28no—
145C2×Q8—28no—
146C2×Q8—22no—
147C2×Q8—22no—
148C2×Q8—31no—
149C2×Q8#6712yes—
150C2×Q8#6712yes—
151C2×Q8C22×D312yes—
152C2×D4—22no—
153C2×D4—22no—
154C2×D4—31no—
155C2×D4—61no—
156C2×D4#6712yes—
157C22×C4—24no—
158C22×C4—22no—
159C22×C4—61no—
160C22×C4#6712yes—
161C22×C4C22×D311yescharacteristic
162Q16—64no—
163SD16—64no—
164SD16—64no—
165D8—64no—
166M16—34no—
167M16—62no—
168C2×C8—62no—
169C4⋊C4—61no—
170C4⋊C4—64no—
171C22⋊C4—61no—
172C22⋊C4—64no—
173C42—32no—
174C22×C3—24no—
175C22×C3—41no—
176C22×C3—44no—
177C22×C3#12011yescharacteristic
178C22×C3#12011yescharacteristic
179C22×C3C22×D411yescharacteristic
180C2×D3—42no—
181C2×D3—41no—
182C3×C4—24no—
183C3×C4—24no—
184C3×C4#12012yes—
185C3×C4C22×D412yes—
186Dic3—42no—
187Dic3—41no—
188C23—22no—
189C23—61no—
190C23C2×#6711yescharacteristic
191Q8—28no—
192Q8—22no—
193Q8—24no—
194Q8—24no—
195Q8—62no—
196Q8C2×#6714yes—
197D4—28no—
198D4—24no—
199D4—24no—
200D4—22no—
201D4—62no—
202D4—121no—
203D4—121no—
204D4C2×#6714yes—
205C2×C4—216no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—24no—
210C2×C4—24no—
211C2×C4—62no—
212C2×C4—61no—
213C2×C4—122no—
214C2×C4C2×#6714yes—
215C2×C4C2×#6711yescharacteristic
216C2×C4C2×#6712yes—
217C2×C4C2×#6712yes—
218C2×C4C23×D311yescharacteristic
219C8—64no—
220C2×C3—21no—
221C2×C3—21no—
222C2×C3—21no—
223C2×C3—44no—
224C2×C3C2×#12011yescharacteristic
225D3—81no—
226C22—24no—
227C22—41no—
228C22—44no—
229C22—122no—
230C22—121no—
231C22#95311yescharacteristic
232C22#95311yescharacteristic
233C22C22×#6711yescharacteristic
234C4—24no—
235C4—24no—
236C4—122no—
237C4—121no—
238C4#95312yes—
239C4C22×#6712yes—
240C3#302011yescharacteristic, complemented, Sylow
241C2—21no—
242C2—21no—
243C2—21no—
244C2—44no—
245C2—241no—
246C2C2×#95311yescharacteristic
subgroupG/H
centerC2C2×#953
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC2×C3×2−1+4C2
socleC2×C3C2×#120
hypercenterC2×2−1+4D3
hypocenterC3#3020
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×2−1+4

Complements

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

NH ≅ G/Ntype
1. C2×C3×2−1+4C2semidirect
2. #5822C2semidirect
3. #5816C2semidirect
4. #5246C2semidirect
5. #5245C2semidirect
6. #5236C2semidirect
7. C2×#836C2semidirect
18. #951C22semidirect
20. #949C22semidirect
21. #947C22semidirect
23. #944C22semidirect
29. #837C22semidirect
30. #836C22semidirect
31. C2×2−1+4D3semidirect
240. C3#3020semidirect

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
#58222
#58162
#52462
#52452
#52362
C2×#8362
#30203