Finite Group Databasebeta

← #78316

Subgroups — #78316

Order 384.

444 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 125 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×#897C211yesmaximal, characteristic
2C2×#896C211yesmaximal, characteristic
3C3×#400C211yesmaximal, characteristic, complemented
4#4985C211yesmaximal, characteristic, complemented
5#4979C211yesmaximal, characteristic, complemented
6#4675C212yesmaximal
7#4674C212yesmaximal
8#4664C211yesmaximal, characteristic
9#4660C212yesmaximal, complemented
10#4656C212yesmaximal, complemented
11#4656C211yesmaximal, characteristic, complemented
12#2817—31nomaximal, Sylow
13C22×C4×D3C2211yescharacteristic
14C22×Dic6C2211yescharacteristic
15C22×Dic6C2211yescharacteristic
16C2×C3×C4⋊C4C2211yescharacteristic
17C2×C3×C4⋊C4C2211yescharacteristic
18C2×(C2×D3)⋊C4C2211yescharacteristic
19C2×(C2×D3)⋊C4C2212yes—
20C2×(C2×D3)⋊C4C2211yescharacteristic
21C2×(C2×D3)⋊C4C2212yes—
22C2×C4⋊Dic3C2212yes—
23C2×C4⋊Dic3C2212yes—
24C2×Dic3⋊C4C2211yescharacteristic
25C2×Dic3⋊C4C2211yescharacteristic
26C2×Dic3⋊C4C2212yes—
27#897—24no—
28#896—24no—
29C3×#95C2211yescharacteristic
30C3×#95C2212yes—
31C3×#95C2212yes—
32C3⋊#95C2212yes—
33C3⋊#95C2212yes—
34C3⋊#95C2212yes—
35C3⋊#95C2212yes—
36C3⋊#95C2212yes—
37C3⋊#95C2211yescharacteristic
38C2×C22⋊Q8—31no—
39C2×C22⋊Q8—31no—
40#400—31no—
41#400—32no—
42#400D311yescharacteristic, complemented
43#397—32no—
44#397—32no—
45#395—31no—
46#395—32no—
47#395—31no—
48#381—31no—
49C23×D3C2311yescharacteristic
50C22×C3×C4C2312yes—
51C22×C3×C4C2311yescharacteristic
52C22×C3×C4C2312yes—
53C22×C3×C4C2311yescharacteristic
54C22×C3×C4C2311yescharacteristic
55C22×Dic3C2312yes—
56C22×Dic3C2311yescharacteristic
57C22×Dic3C2311yescharacteristic
58C22×Dic3C2312yes—
59C22×Dic3C2311yescharacteristic
60C2×C4×D3—42no—
61C2×C4×D3D414yes—
62C2×Dic6—22no—
63C2×Dic6—22no—
64C2×Dic6—42no—
65C2×Dic6—42no—
66C3×C4⋊C4—22no—
67C3×C4⋊C4—22no—
68(C2×D3)⋊C4—22no—
69(C2×D3)⋊C4—22no—
70(C2×D3)⋊C4—24no—
71(C2×D3)⋊C4—42no—
72C4⋊Dic3—24no—
73C4⋊Dic3—42no—
74Dic3⋊C4—22no—
75Dic3⋊C4—22no—
76Dic3⋊C4—42no—
77C22×Q8—31no—
78C22×Q8—31no—
79C23×C4—31no—
80C22⋊Q8—64no—
81C22⋊Q8—64no—
82C2×C4⋊C4—31no—
83C2×C4⋊C4—32no—
84C2×C4⋊C4—31no—
85C2×C4⋊C4—32no—
86C2×C4⋊C4—32no—
87C2×C4⋊C4C2×D311yescharacteristic
88C2×C4⋊C4C2×D311yescharacteristic
89C2×C22⋊C4—31no—
90C2×C22⋊C4—31no—
91C2×C22⋊C4—32no—
92C2×C22⋊C4—32no—
93#95—32no—
94#95—31no—
95#95—32no—
96#95—32no—
97#95—32no—
98#95—32no—
99#95C2×D311yescharacteristic
100#95C2×D312yes—
101#95C2×D312yes—
102C23×C3C2411yescharacteristic
103C22×D3—22no—
104C22×D3—21no—
105C22×D3—22no—
106C22×D3—21no—
107C22×D3C2×D412yes—
108C2×C3×C4—22no—
109C2×C3×C4—21no—
110C2×C3×C4—22no—
111C2×C3×C4—21no—
112C2×C3×C4—22no—
113C2×C3×C4—22no—
114C2×C3×C4—21no—
115C2×C3×C4—21no—
116C2×C3×C4—24no—
117C2×C3×C4—22no—
118C2×C3×C4—22no—
119C2×C3×C4C2×D412yes—
120C2×Dic3—22no—
121C2×Dic3—21no—
122C2×Dic3—21no—
123C2×Dic3—22no—
124C2×Dic3—22no—
125C2×Dic3—22no—
126C2×Dic3—22no—
127C2×Dic3—22no—
128C2×Dic3—21no—
129C2×Dic3—22no—
130C2×Dic3—21no—
131C2×Dic3—22no—
132C2×Dic3C2×D412yes—
133C4×D3—44no—
134Dic6—44no—
135Dic6—44no—
136C24—31no—
137C2×Q8—62no—
138C2×Q8—62no—
139C2×Q8—122no—
140C2×Q8—122no—
141C22×C4—31no—
142C22×C4—34no—
143C22×C4—32no—
144C22×C4—31no—
145C22×C4—31no—
146C22×C4—32no—
147C22×C4—122no—
148C22×C4C22×D312yes—
149C22×C4C22×D311yescharacteristic
150C22×C4C22×D312yes—
151C22×C4C22×D311yescharacteristic
152C22×C4C22×D311yescharacteristic
153C4⋊C4—22no—
154C4⋊C4—22no—
155C4⋊C4—62no—
156C4⋊C4—64no—
157C4⋊C4—62no—
158C4⋊C4—122no—
159C4⋊C4—122no—
160C22⋊C4—62no—
161C22⋊C4—64no—
162C22⋊C4—62no—
163C22⋊C4—122no—
164C22×C3C22×D411yescharacteristic
165C22×C3H(5,𝔽2)12yes—
166C22×C32−1+411yescharacteristic
167C22×C32−1+412yes—
168C22×C32−1+411yescharacteristic
169C2×D3—22no—
170C2×D3—22no—
171C2×D3—22no—
172C2×D3—42no—
173C2×D3—42no—
174C3×C4—22no—
175C3×C4—41no—
176C3×C4—42no—
177C3×C4—41no—
178C3×C4—42no—
179Dic3—22no—
180Dic3—41no—
181Dic3—42no—
182Dic3—41no—
183Dic3—42no—
184C23—32no—
185C23—61no—
186C23—62no—
187C23—62no—
188C23—61no—
189C23C23×D311yescharacteristic
190Q8—124no—
191Q8—124no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—24no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—21no—
203C2×C4—32no—
204C2×C4—62no—
205C2×C4—61no—
206C2×C4—62no—
207C2×C4—61no—
208C2×C4—62no—
209C2×C4—61no—
210C2×C4—62no—
211C2×C4—62no—
212C2×C4—62no—
213C2×C4—62no—
214C2×C4—62no—
215C2×C4—61no—
216C2×C4—124no—
217C2×C4D3×D412yes—
218C2×C3#53411yescharacteristic
219C2×C3#53611yescharacteristic
220C2×C3#53611yescharacteristic
221C2×C3#56312yes—
222C2×C3#56312yes—
223D3—42no—
224C22—62no—
225C22—62no—
226C22—62no—
227C22—122no—
228C22—122no—
229C22C2×D3×D411yescharacteristic
230C22#100711yescharacteristic
231C22#100912yes—
232C22#101012yes—
233C22#101011yescharacteristic
234C4—22no—
235C4—41no—
236C4—42no—
237C4—42no—
238C4—41no—
239C4—62no—
240C4—122no—
241C4—121no—
242C4—122no—
243C4—121no—
244C3#281711yescharacteristic, complemented, Sylow
245C2—122no—
246C2#562811yescharacteristic
247C2#565311yescharacteristic
248C2#567511yescharacteristic
249C2#569912yes—
250C2#571212yes—
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#400C2
prime radicalC23×D3C23
hypercenter#400D3
hypocenterC3#2817
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
3. C3×#400C2semidirect
4. #4985C2semidirect
5. #4979C2semidirect
9. #4660C2semidirect
10. #4656C2semidirect
11. #4656C2semidirect
42. #400D3semidirect
244. C3#2817semidirect

Maximal subgroups

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

subgroupindex
C2×#8972
C2×#8962
C3×#4002
#49852
#49792
#46752
#46742
#46642
#46602
#46562
#28173