Finite Group Databasebeta

← #73950

Subgroups — #73950

Order 384.

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

normal subgroups only

This group has 816 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
1C22×#235C211yesmaximal, characteristic
2C2×#924C211yesmaximal, characteristic, complemented
3C2×#897C212yesmaximal, complemented
4C3×#380C211yesmaximal, characteristic, complemented
5#4987C212yesmaximal, complemented
6#4961C214yesmaximal, complemented
7#4672C212yesmaximal, complemented
8#4656C212yesmaximal, complemented
9#2616—31nomaximal, Sylow
10C23×C3×C4C2211yescharacteristic
11C22×#67C2212yes—
12C2×#235—24no—
13C2×#235—28no—
14C22×D12C2211yescharacteristic
15C22×C4×D3C2212yes—
16C22×Dic6C2211yescharacteristic
17C22×Dic6C2211yescharacteristic
18C2×C3×C4⋊C4C2212yes—
19C2×C3×C22⋊C4C2212yes—
20C2×#217C2212yes—
21C2×(C2×D3)⋊C4C2214yes—
22C2×(C2×D3)⋊C4C2214yes—
23C2×C4⋊Dic3C2214yes—
24C2×C4⋊Dic3C2211yescharacteristic
25#924—24no—
26C2×Dic3⋊C4C2212yes—
27#897—28no—
28C3×#95C2212yes—
29C3⋊#95C2214yes—
30C22×#41—31no—
31C2×C22⋊Q8—31no—
32C2×C22⋊Q8—32no—
33#395—32no—
34#394—34no—
35#394—32no—
36#380—32no—
37#380D311yescharacteristic, complemented
38C24×C3C2311yescharacteristic
39C23×D3C2312yes—
40C22×C3×C4—24no—
41C22×C3×C4D414yes—
42C22×C3×C4C2314yes—
43C22×C3×C4C2311yescharacteristic
44C22×C3×C4C2311yescharacteristic
45C2×#67—24no—
46C2×#67—24no—
47C2×#67—24no—
48C22×Dic3C2312yes—
49C22×Dic3C2312yes—
50C22×Dic3C2312yes—
51#235—232no—
52C2×D12—24no—
53C2×D12—22no—
54C2×C4×D3—24no—
55C2×C4×D3—24no—
56C2×C4×D3D418yes—
57C2×Dic6—22no—
58C2×Dic6—24no—
59C2×Dic6—24no—
60C2×Dic6—22no—
61C3×C4⋊C4—24no—
62C3×C22⋊C4—42no—
63#217—24no—
64(C2×D3)⋊C4—28no—
65(C2×D3)⋊C4—44no—
66C4⋊Dic3—22no—
67C4⋊Dic3—28no—
68Dic3⋊C4—24no—
69C2×#41—68no—
70C2×#41—64no—
71C22×Q8—31no—
72C22×Q8—31no—
73C22×D4—32no—
74C22×D4—31no—
75C23×C4—32no—
76C23×C4C2×D311yescharacteristic
77C22⋊Q8—68no—
78C22⋊Q8—64no—
79C2×C4⋊C4—31no—
80C2×C4⋊C4—32no—
81C2×C4⋊C4—34no—
82C2×C4⋊C4C2×D312yes—
83C2×C22⋊C4—32no—
84C2×C22⋊C4—34no—
85C2×C22⋊C4—34no—
86C2×C22⋊C4C2×D312yes—
87#95—34no—
88#95C2×D312yes—
89C23×C3—22no—
90C23×C3—21no—
91C23×C3—21no—
92C23×C3—22no—
93C23×C3C2×D412yes—
94C23×C3C2411yescharacteristic
95C22×D3—22no—
96C22×D3—22no—
97C22×D3—22no—
98C22×D3—22no—
99C22×D3—22no—
100C22×D3—22no—
101C22×D3C2×D414yes—
102C2×C3×C4—22no—
103C2×C3×C4—28no—
104C2×C3×C4—24no—
105C2×C3×C4—24no—
106C2×C3×C4—24no—
107C2×C3×C4C2×D412yes—
108C2×C3×C4C2×D412yes—
109C2×C3×C4C2×D414yes—
110#67—216no—
111C2×Dic3—22no—
112C2×Dic3—22no—
113C2×Dic3—22no—
114C2×Dic3—22no—
115C2×Dic3—24no—
116C2×Dic3—22no—
117C2×Dic3—22no—
118C2×Dic3C2×D414yes—
119D12—28no—
120C4×D3—216no—
121Dic6—28no—
122Dic6—44no—
123C24—32no—
124C24C22×D311yescharacteristic
125#41—632no—
126C2×Q8—64no—
127C2×Q8—62no—
128C2×Q8—62no—
129C2×Q8—64no—
130C2×D4—64no—
131C2×D4—64no—
132C2×D4—64no—
133C2×D4—64no—
134C2×D4—62no—
135C22×C4—24no—
136C22×C4—32no—
137C22×C4—38no—
138C22×C4—32no—
139C22×C4—32no—
140C22×C4—64no—
141C22×C4—64no—
142C22×C4D1214yes—
143C22×C4C22×D314yes—
144C22×C4C22×D311yescharacteristic
145C22×C4C22×D311yescharacteristic
146C4⋊C4—24no—
147C4⋊C4—68no—
148C4⋊C4—64no—
149C4⋊C4—62no—
150C22⋊C4—42no—
151C22⋊C4—68no—
152C22⋊C4—64no—
153C22⋊C4—124no—
154C22×C3—22no—
155C22×C3—22no—
156C22×C3—22no—
157C22×C3—42no—
158C22×C3—42no—
159C22×C3C22×D411yescharacteristic
160C22×C3C22×D412yes—
161C22×C3H(5,𝔽2)11yescharacteristic
162C22×C32−1+411yescharacteristic
163C22×C32−1+412yes—
164C2×D3—24no—
165C2×D3—24no—
166C2×D3—24no—
167C2×D3—42no—
168C2×D3—42no—
169C2×D3—42no—
170C2×D3—42no—
171C3×C4—22no—
172C3×C4—44no—
173C3×C4#12014yes—
174Dic3—24no—
175Dic3—42no—
176Dic3—42no—
177C23—22no—
178C23—21no—
179C23—21no—
180C23—22no—
181C23—34no—
182C23—62no—
183C23—62no—
184C23—62no—
185C23—62no—
186C23—62no—
187C23—62no—
188C23C2×D1212yes—
189C23C23×D311yescharacteristic
190Q8—68no—
191Q8—124no—
192D4—68no—
193D4—616no—
194C2×C4—22no—
195C2×C4—28no—
196C2×C4—24no—
197C2×C4—24no—
198C2×C4—24no—
199C2×C4—34no—
200C2×C4—62no—
201C2×C4—62no—
202C2×C4—616no—
203C2×C4—62no—
204C2×C4—62no—
205C2×C4—64no—
206C2×C4—62no—
207C2×C4—62no—
208C2×C4C2×D1212yes—
209C2×C4C2×D1212yes—
210C2×C4D3×D414yes—
211C2×C3—42no—
212C2×C3C2×#12011yescharacteristic
213C2×C3#53611yescharacteristic
214C2×C3#53612yes—
215C2×C3#53711yescharacteristic
216C2×C3#53712yes—
217D3—44no—
218C22—22no—
219C22—22no—
220C22—22no—
221C22—42no—
222C22—42no—
223C22—64no—
224C22—64no—
225C22—64no—
226C22—122no—
227C22—122no—
228C22—122no—
229C22—122no—
230C22C22×D1211yescharacteristic
231C22C2×D3×D412yes—
232C22#100411yescharacteristic
233C22#100711yescharacteristic
234C22#101012yes—
235C4—22no—
236C4—44no—
237C4—64no—
238C4—122no—
239C4—122no—
240C4#88214yes—
241C3#261611yescharacteristic, complemented, Sylow
242C2—42no—
243C2—124no—
244C2C2×#88211yescharacteristic
245C2#551011yescharacteristic
246C2#552611yescharacteristic
247C2#560812yes—
248C2#562812yes—
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#380C2
prime radicalC22×#235C2
hypercenter#380D3
hypocenterC3#2616
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #380

Complements

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

NH ≅ G/Ntype
2. C2×#924C2semidirect
3. C2×#897C2semidirect
4. C3×#380C2semidirect
5. #4987C2semidirect
6. #4961C2semidirect
7. #4672C2semidirect
8. #4656C2semidirect
37. #380D3semidirect
241. C3#2616semidirect

Maximal subgroups

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

subgroupindex
C22×#2352
C2×#9242
C2×#8972
C3×#3802
#49872
#49612
#46722
#46562
#26163