Finite Group Databasebeta

← #83001

Subgroups — #83001

Order 384.

612 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 157 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×#124C211yesmaximal, characteristic, complemented
2C2×#946C211yesmaximal, characteristic, complemented
3C2×#938C211yesmaximal, characteristic
4C2×#872C211yesmaximal, characteristic, complemented
5#5229C211yesmaximal, characteristic, complemented
6#5221C211yesmaximal, characteristic, complemented
7#4961C214yesmaximal, complemented
8#4948C214yesmaximal, complemented
9#4935C211yesmaximal, characteristic, complemented
10#2620—31nomaximal, Sylow
11C22×C3×Q8C2211yescharacteristic
12C22×C3×D4C2211yescharacteristic
13C22×#67C2212yes—
14C22×C4×D3C2211yescharacteristic
15C3×#124—24no—
16C2×C3×C22⋊C4C2214yes—
17C2×C3×C42C2211yescharacteristic
18C2×#217C2212yes—
19C2×#217C2214yes—
20#946—24no—
21#938—24no—
22C2×(C2×D3)⋊C4C2212yes—
23C2×(C2×D3)⋊C4C2212yes—
24C2×C4⋊Dic3C2211yescharacteristic
25C2×C4⋊Dic3C2211yescharacteristic
26C2×C4⋊Dic3C2214yes—
27C2×Dic3⋊C4C2212yes—
28C2×Dic3⋊C4C2212yes—
29C2×C4×Dic3C2211yescharacteristic
30#872—24no—
31C3⋊#95C2214yes—
32C2×#124D311yescharacteristic, complemented
33C2×C22⋊Q8—31no—
34C2×#121—31no—
35C2×#117—31no—
36#396—34no—
37#394—34no—
38#391—31no—
39#390—31no—
40#384—31no—
41C24×C3C2312yes—
42C23×D3C2311yescharacteristic
43C2×C3×Q8—22no—
44C2×C3×Q8—22no—
45C2×C3×Q8—22no—
46C2×C3×D4—22no—
47C2×C3×D4—22no—
48C2×C3×D4—22no—
49C22×C3×C4C2312yes—
50C22×C3×C4C2312yes—
51C22×C3×C4C2311yescharacteristic
52C2×#67—24no—
53C2×#67—44no—
54C22×Dic3C2312yes—
55C22×Dic3C2312yes—
56C22×Dic3C2312yes—
57C22×Dic3C2311yescharacteristic
58C2×C4×D3—22no—
59C2×C4×D3—22no—
60C2×C4×D3D414yes—
61C3×C22⋊C4—28no—
62C3×C42D414yescomplemented
63#217—24no—
64#217—44no—
65(C2×D3)⋊C4—24no—
66(C2×D3)⋊C4—24no—
67C4⋊Dic3—22no—
68C4⋊Dic3—22no—
69C4⋊Dic3—44no—
70Dic3⋊C4—24no—
71Dic3⋊C4—24no—
72C4×Dic3—22no—
73C22×Q8C2×D311yescharacteristic
74C22×D4—32no—
75C22×D4C2×D311yescharacteristic
76C23×C4—31no—
77#124—24no—
78C22⋊Q8—64no—
79#121—64no—
80#117—64no—
81C2×C4⋊C4—32no—
82C2×C4⋊C4—31no—
83C2×C4⋊C4—31no—
84C2×C4⋊C4—32no—
85C2×C4⋊C4—34no—
86C2×C22⋊C4—32no—
87C2×C22⋊C4—32no—
88C2×C22⋊C4—32no—
89C2×C22⋊C4—34no—
90C2×C22⋊C4C2×D314yes—
91C2×C42—31no—
92C2×C42C2×D311yescharacteristic
93#95—34no—
94C23×C3—22no—
95C23×C3—22no—
96C23×C3—22no—
97C23×C3—24no—
98C23×C3—24no—
99C23×C3C2411yescharacteristic
100C22×D3—21no—
101C22×D3—21no—
102C22×D3—22no—
103C22×D3—22no—
104C22×D3C2×D412yes—
105C3×Q8—44no—
106C3×D4—44no—
107C2×C3×C4—22no—
108C2×C3×C4—22no—
109C2×C3×C4—22no—
110C2×C3×C4—22no—
111C2×C3×C4—22no—
112C2×C3×C4C2×D412yes—
113C2×C3×C4C2×D414yes—
114C2×C3×C4C2×D412yes—
115C2×C3×C4#4112yes—
116#67—48no—
117C2×Dic3—21no—
118C2×Dic3—21no—
119C2×Dic3—22no—
120C2×Dic3—22no—
121C2×Dic3—22no—
122C2×Dic3—24no—
123C2×Dic3—24no—
124C2×Dic3—22no—
125C2×Dic3—22no—
126C2×Dic3C2×D412yes—
127C4×D3—28no—
128C24—31no—
129C24C22×D312yes—
130C2×Q8—22no—
131C2×Q8—22no—
132C2×Q8—22no—
133C2×D4—22no—
134C2×D4—22no—
135C2×D4—22no—
136C2×D4—64no—
137C2×D4—124no—
138C22×C4—31no—
139C22×C4—34no—
140C22×C4—32no—
141C22×C4—32no—
142C22×C4—32no—
143C22×C4—62no—
144C22×C4—62no—
145C22×C4C22×D312yes—
146C22×C4C22×D312yes—
147C22×C4C22×D311yescharacteristic
148C4⋊C4—64no—
149C4⋊C4—62no—
150C4⋊C4—62no—
151C4⋊C4—64no—
152C4⋊C4—124no—
153C22⋊C4—28no—
154C22⋊C4—64no—
155C22⋊C4—64no—
156C22⋊C4—64no—
157C22⋊C4—124no—
158C42—62no—
159C42#6714yescomplemented
160C22×C3—42no—
161C22×C3—42no—
162C22×C3—42no—
163C22×C3—44no—
164C22×C3—44no—
165C22×C3C22×D411yescharacteristic
166C22×C3C22×D411yescharacteristic
167C22×C3C2×#4111yescharacteristic
168C22×C3H(5,𝔽2)12yes—
169C22×C32−1+412yes—
170C2×D3—22no—
171C2×D3—22no—
172C2×D3—22no—
173C2×D3—42no—
174C2×D3—42no—
175C3×C4—24no—
176C3×C4—42no—
177C3×C4#12114yes—
178Dic3—22no—
179Dic3—42no—
180Dic3—42no—
181Dic3—42no—
182C23—22no—
183C23—22no—
184C23—24no—
185C23—24no—
186C23—22no—
187C23—32no—
188C23—61no—
189C23—62no—
190C23—62no—
191C23—61no—
192C23C23×D311yescharacteristic
193Q8—44no—
194D4—44no—
195D4—128no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—32no—
202C2×C4—61no—
203C2×C4—61no—
204C2×C4—68no—
205C2×C4—62no—
206C2×C4—62no—
207C2×C4—62no—
208C2×C4—64no—
209C2×C4—64no—
210C2×C4—62no—
211C2×C4—62no—
212C2×C4D3×D412yes—
213C2×C4#23712yes—
214C2×C4C2×#6714yes—
215C2×C4C2×#6712yes—
216C2×C3—82no—
217C2×C3C2×#12111yescharacteristic
218C2×C3#53511yescharacteristic
219C2×C3#53611yescharacteristic
220C2×C3#53712yes—
221C2×C3#54212yes—
222D3—42no—
223C22—42no—
224C22—42no—
225C22—44no—
226C22—42no—
227C22—44no—
228C22—62no—
229C22—62no—
230C22—62no—
231C22—122no—
232C22—122no—
233C22C2×D3×D411yescharacteristic
234C22C2×#23711yescharacteristic
235C22#100912yes—
236C22#101012yes—
237C22C22×#6711yescharacteristic
238C4—24no—
239C4—42no—
240C4—62no—
241C4—122no—
242C4—122no—
243C4—122no—
244C4#93814yes—
245C3#262011yescharacteristic, complemented, Sylow
246C2—82no—
247C2—122no—
248C2#567411yescharacteristic
249C2#567511yescharacteristic
250C2#568212yes—
251C2C2×#93811yescharacteristic
252C2#583312yes—
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC2×C3×#124C2
prime radicalC2×#938C2
hypercenterC2×#124D3
hypocenterC3#2620
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. C2×C3×#124C2semidirect
2. C2×#946C2semidirect
4. C2×#872C2semidirect
5. #5229C2semidirect
6. #5221C2semidirect
7. #4961C2semidirect
8. #4948C2semidirect
9. #4935C2semidirect
32. C2×#124D3semidirect
62. C3×C42D4semidirect
159. C42#67semidirect
245. C3#2620semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#1242
C2×#9462
C2×#9382
C2×#8722
#52292
#52212
#49612
#49482
#49352
#26203