Finite Group Databasebeta

← #77954

Subgroups — #77954

Order 384.

512 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 139 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×#892C211yesmaximal, characteristic
2C2×D3×C4⋊C4C211yesmaximal, characteristic
3C3×#398C211yesmaximal, characteristic, complemented
4#4989C212yesmaximal
5#4986C211yesmaximal, characteristic, complemented
6#4983C211yesmaximal, characteristic, complemented
7#4980C212yesmaximal, complemented
8#4664C211yesmaximal, characteristic
9#4664C212yesmaximal
10#4659C211yesmaximal, characteristic, complemented
11#4652C212yesmaximal, complemented
12#2484—31nomaximal, Sylow
13C22×C4×D3C2211yescharacteristic
14C22×C4×D3C2211yescharacteristic
15C22×C4×D3C2211yescharacteristic
16C2×C3×C4⋊C4C2212yes—
17C2×C3×C4⋊C4C2211yescharacteristic
18C2×C3×C4⋊C4C2211yescharacteristic
19C2×(C2×D3)⋊C4C2212yes—
20C2×(C2×D3)⋊C4C2212yes—
21C2×C4⋊Dic3C2211yescharacteristic
22C2×C4⋊Dic3C2211yescharacteristic
23C2×Dic3⋊C4C2211yescharacteristic
24C2×Dic3⋊C4C2212yes—
25C2×Dic3⋊C4C2211yescharacteristic
26C2×C4×Dic3C2212yes—
27#892—24no—
28D3×C4⋊C4—24no—
29C3×#95C2212yes—
30C3×#95C2211yescharacteristic
31C3⋊#95C2212yes—
32C3⋊#95C2212yes—
33C3⋊#95C2212yes—
34C3⋊#95C2212yes—
35C3⋊#95C2212yes—
36C3⋊#95C2211yescharacteristic
37C3⋊#95C2212yes—
38C2×#117—31no—
39C22×C4⋊C4—31no—
40#400—31no—
41#400—31no—
42#398—31no—
43#398D311yescharacteristic, complemented
44#387—32no—
45#387—32no—
46#385—32no—
47#381—31no—
48#381—32no—
49C23×D3C2311yescharacteristic
50C22×C3×C4C2312yes—
51C22×C3×C4C2311yescharacteristic
52C22×C3×C4C2312yes—
53C22×C3×C4C2311yescharacteristic
54C22×C3×C4C2311yescharacteristic
55C22×Dic3C2312yes—
56C22×Dic3C2312yes—
57C22×Dic3C2311yescharacteristic
58C22×Dic3C2311yescharacteristic
59C22×Dic3C2311yescharacteristic
60C2×C4×D3—22no—
61C2×C4×D3—24no—
62C2×C4×D3—22no—
63C2×C4×D3—22no—
64C2×C4×D3—22no—
65C2×C4×D3—24no—
66C2×C4×D3D414yes—
67C3×C4⋊C4—22no—
68C3×C4⋊C4—22no—
69C3×C4⋊C4—24no—
70(C2×D3)⋊C4—24no—
71(C2×D3)⋊C4—24no—
72C4⋊Dic3—22no—
73C4⋊Dic3—22no—
74Dic3⋊C4—22no—
75Dic3⋊C4—22no—
76Dic3⋊C4—24no—
77C4×Dic3—24no—
78C23×C4—31no—
79C23×C4—31no—
80C23×C4—31no—
81#117—64no—
82C2×C4⋊C4—31no—
83C2×C4⋊C4—31no—
84C2×C4⋊C4—31no—
85C2×C4⋊C4—31no—
86C2×C4⋊C4—32no—
87C2×C4⋊C4—64no—
88C2×C4⋊C4C2×D311yescharacteristic
89C2×C4⋊C4C2×D311yescharacteristic
90C2×C4⋊C4C2×D312yes—
91C2×C22⋊C4—32no—
92C2×C22⋊C4—32no—
93C2×C42—32no—
94#95—32no—
95#95—32no—
96#95—32no—
97#95—32no—
98#95—31no—
99#95—32no—
100#95—32no—
101#95C2×D312yes—
102#95C2×D311yescharacteristic
103C23×C3C2411yescharacteristic
104C22×D3—21no—
105C22×D3—21no—
106C22×D3—22no—
107C22×D3C2×D412yes—
108C22×D3#4114yes—
109C2×C3×C4—21no—
110C2×C3×C4—22no—
111C2×C3×C4—21no—
112C2×C3×C4—22no—
113C2×C3×C4—21no—
114C2×C3×C4—22no—
115C2×C3×C4—22no—
116C2×C3×C4—22no—
117C2×C3×C4—22no—
118C2×C3×C4—22no—
119C2×C3×C4—22no—
120C2×C3×C4C2×D412yes—
121C2×C3×C4#4112yes—
122C2×Dic3—21no—
123C2×Dic3—21no—
124C2×Dic3—22no—
125C2×Dic3—22no—
126C2×Dic3—21no—
127C2×Dic3—22no—
128C2×Dic3—22no—
129C2×Dic3—22no—
130C2×Dic3—24no—
131C2×Dic3—22no—
132C2×Dic3C2×D412yes—
133C2×Dic3#4112yes—
134C4×D3—28no—
135C4×D3—28no—
136C4×D3—44no—
137C24—31no—
138C22×C4—31no—
139C22×C4—31no—
140C22×C4—34no—
141C22×C4—32no—
142C22×C4—32no—
143C22×C4—31no—
144C22×C4—62no—
145C22×C4—62no—
146C22×C4—64no—
147C22×C4—62no—
148C22×C4—64no—
149C22×C4—62no—
150C22×C4C22×D312yes—
151C22×C4C22×D311yescharacteristic
152C22×C4C22×D312yes—
153C22×C4C22×D311yescharacteristic
154C22×C4C22×D311yescharacteristic
155C4⋊C4—22no—
156C4⋊C4—22no—
157C4⋊C4—24no—
158C4⋊C4—62no—
159C4⋊C4—62no—
160C4⋊C4—64no—
161C4⋊C4—62no—
162C4⋊C4—62no—
163C22⋊C4—64no—
164C22⋊C4—64no—
165C42—64no—
166C22×C3C22×D411yescharacteristic
167C22×C3C2×#4111yescharacteristic
168C22×C3C2×#4112yes—
169C22×C3C2×#4111yescharacteristic
170C22×C32−1+412yes—
171C2×D3—22no—
172C2×D3—22no—
173C2×D3—24no—
174C2×D3—24no—
175C2×D3#12314yes—
176C3×C4—22no—
177C3×C4—22no—
178C3×C4—41no—
179C3×C4—42no—
180C3×C4—42no—
181Dic3—22no—
182Dic3—22no—
183Dic3—41no—
184Dic3—42no—
185Dic3—42no—
186C23—32no—
187C23—34no—
188C23—61no—
189C23—62no—
190C23—61no—
191C23C23×D311yescharacteristic
192C2×C4—21no—
193C2×C4—22no—
194C2×C4—21no—
195C2×C4—22no—
196C2×C4—21no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—32no—
204C2×C4—32no—
205C2×C4—68no—
206C2×C4—68no—
207C2×C4—61no—
208C2×C4—61no—
209C2×C4—62no—
210C2×C4—62no—
211C2×C4—61no—
212C2×C4—62no—
213C2×C4—64no—
214C2×C4—62no—
215C2×C4—62no—
216C2×C4—62no—
217C2×C4—124no—
218C2×C4D3×D412yes—
219C2×C4#23712yes—
220C2×C3C2×#12311yescharacteristic
221C2×C3#52511yescharacteristic
222C2×C3#53611yescharacteristic
223C2×C3#55212yes—
224C2×C3#55212yes—
225D3—24no—
226C22—34no—
227C22—64no—
228C22—64no—
229C22—62no—
230C22—62no—
231C22C2×D3×D411yescharacteristic
232C22C2×#23711yescharacteristic
233C22#100712yes—
234C22D3×#4112yes—
235C22D3×#4111yescharacteristic
236C4—22no—
237C4—22no—
238C4—42no—
239C4—41no—
240C4—42no—
241C4—62no—
242C4—62no—
243C4—121no—
244C4—122no—
245C4—122no—
246C3#248411yescharacteristic, complemented, Sylow
247C2—64no—
248C2#560611yescharacteristic
249C2#563912yes—
250C2D3×#12311yescharacteristic
251C2#569312yes—
252C2#572411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#398C2
prime radicalC23×D3C23
hypercenter#398D3
hypocenterC3#2484
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
3. C3×#398C2semidirect
5. #4986C2semidirect
6. #4983C2semidirect
7. #4980C2semidirect
10. #4659C2semidirect
11. #4652C2semidirect
43. #398D3semidirect
246. C3#2484semidirect

Maximal subgroups

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

subgroupindex
C2×#8922
C2×D3×C4⋊C42
C3×#3982
#49892
#49862
#49832
#49802
#46642
#46592
#46522
#24843