Finite Group Databasebeta

← #78152

Subgroups — #78152

Order 384.

512 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 141 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×#896C211yesmaximal, characteristic
2C2×D3×C4⋊C4C211yesmaximal, characteristic
3C3×#400C211yesmaximal, characteristic, complemented
4#4986C211yesmaximal, characteristic, complemented
5#4985C211yesmaximal, characteristic, complemented
6#4666C211yesmaximal, characteristic
7#4664C212yesmaximal
8#4664C212yesmaximal
9#4659C211yesmaximal, characteristic, complemented
10#4650C212yesmaximal, complemented
11#4646C212yesmaximal, complemented
12#2503—31nomaximal, Sylow
13C22×C4×D3C2212yes—
14C22×C4×D3C2211yescharacteristic
15C22×Dic6C2211yescharacteristic
16C2×C3×C4⋊C4C2211yescharacteristic
17C2×C3×C4⋊C4C2211yescharacteristic
18C2×(C2×D3)⋊C4C2212yes—
19C2×(C2×D3)⋊C4C2211yescharacteristic
20C2×(C2×D3)⋊C4C2211yescharacteristic
21C2×C4⋊Dic3C2211yescharacteristic
22C2×Dic3⋊C4C2212yes—
23C2×Dic3⋊C4C2211yescharacteristic
24C2×Dic3⋊C4C2211yescharacteristic
25C2×C4×Dic3C2212yes—
26#896—24no—
27D3×C4⋊C4—24no—
28C3×#95C2212yes—
29C3×#95C2212yes—
30C3×#95C2211yescharacteristic
31C3⋊#95C2212yes—
32C3⋊#95C2212yes—
33C3⋊#95C2212yes—
34C3⋊#95C2212yes—
35C3⋊#95C2212yes—
36C3⋊#95C2212yes—
37C3⋊#95C2211yescharacteristic
38C2×C22⋊Q8—31no—
39C22×C4⋊C4—31no—
40#400—31no—
41#400—31no—
42#400—31no—
43#400D311yescharacteristic, complemented
44#391—32no—
45#382—32no—
46#381—32no—
47#381—32no—
48C4×C22⋊C4—31no—
49C23×D3C2311yescharacteristic
50C22×C3×C4C2312yes—
51C22×C3×C4C2311yescharacteristic
52C22×C3×C4C2312yes—
53C22×C3×C4C2311yescharacteristic
54C22×C3×C4C2311yescharacteristic
55C22×Dic3C2312yes—
56C22×Dic3C2311yescharacteristic
57C22×Dic3C2312yes—
58C22×Dic3C2311yescharacteristic
59C22×Dic3C2311yescharacteristic
60C2×C4×D3—24no—
61C2×C4×D3—24no—
62C2×C4×D3—28no—
63C2×C4×D3D414yes—
64C2×Dic6—22no—
65C2×Dic6—24no—
66C3×C4⋊C4—22no—
67C3×C4⋊C4—22no—
68(C2×D3)⋊C4—22no—
69(C2×D3)⋊C4—22no—
70(C2×D3)⋊C4—24no—
71C4⋊Dic3—22no—
72Dic3⋊C4—22no—
73Dic3⋊C4—24no—
74Dic3⋊C4—22no—
75C4×Dic3—24no—
76C22×Q8—31no—
77C23×C4—31no—
78C23×C4—32no—
79C22⋊Q8—64no—
80C2×C4⋊C4—32no—
81C2×C4⋊C4—31no—
82C2×C4⋊C4—31no—
83C2×C4⋊C4—31no—
84C2×C4⋊C4—64no—
85C2×C4⋊C4C2×D311yescharacteristic
86C2×C4⋊C4C2×D311yescharacteristic
87C2×C22⋊C4—31no—
88C2×C22⋊C4—31no—
89C2×C22⋊C4—32no—
90C2×C42—32no—
91#95—31no—
92#95—32no—
93#95—32no—
94#95—32no—
95#95—32no—
96#95—32no—
97#95—32no—
98#95C2×D311yescharacteristic
99#95C2×D312yes—
100#95C2×D312yes—
101C23×C3C2411yescharacteristic
102C22×D3—22no—
103C22×D3—21no—
104C22×D3—21no—
105C22×D3C2×D412yes—
106C22×D3#4114yes—
107C2×C3×C4—22no—
108C2×C3×C4—21no—
109C2×C3×C4—22no—
110C2×C3×C4—21no—
111C2×C3×C4—22no—
112C2×C3×C4—22no—
113C2×C3×C4—22no—
114C2×C3×C4—24no—
115C2×C3×C4—21no—
116C2×C3×C4—21no—
117C2×C3×C4—22no—
118C2×C3×C4C2×D412yes—
119C2×Dic3—21no—
120C2×Dic3—22no—
121C2×Dic3—21no—
122C2×Dic3—22no—
123C2×Dic3—22no—
124C2×Dic3—22no—
125C2×Dic3—24no—
126C2×Dic3—21no—
127C2×Dic3—22no—
128C2×Dic3—21no—
129C2×Dic3C2×D412yes—
130C2×Dic3#4114yes—
131C4×D3—28no—
132C4×D3—48no—
133Dic6—44no—
134C24—31no—
135C2×Q8—62no—
136C2×Q8—64no—
137C22×C4—31no—
138C22×C4—34no—
139C22×C4—32no—
140C22×C4—31no—
141C22×C4—31no—
142C22×C4—32no—
143C22×C4—64no—
144C22×C4—64no—
145C22×C4—68no—
146C22×C4C22×D312yes—
147C22×C4C22×D311yescharacteristic
148C22×C4C22×D311yescharacteristic
149C22×C4C22×D312yes—
150C22×C4C22×D311yescharacteristic
151C4⋊C4—22no—
152C4⋊C4—22no—
153C4⋊C4—64no—
154C4⋊C4—62no—
155C4⋊C4—62no—
156C4⋊C4—62no—
157C22⋊C4—62no—
158C22⋊C4—62no—
159C22⋊C4—64no—
160C42—64no—
161C22×C3C22×D411yescharacteristic
162C22×C3C2×#4112yes—
163C22×C3C2×#4112yes—
164C22×C32−1+411yescharacteristic
165C22×C32−1+411yescharacteristic
166C2×D3—22no—
167C2×D3—22no—
168C2×D3—24no—
169C2×D3—24no—
170C2×D3#12414yes—
171C3×C4—22no—
172C3×C4—41no—
173C3×C4—42no—
174C3×C4—41no—
175C3×C4—42no—
176Dic3—41no—
177Dic3—42no—
178Dic3—41no—
179Dic3—42no—
180Dic3#12314yes—
181C23—32no—
182C23—34no—
183C23—61no—
184C23—62no—
185C23—61no—
186C23C23×D311yescharacteristic
187Q8—124no—
188C2×C4—22no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—22no—
192C2×C4—21no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—24no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—32no—
200C2×C4—34no—
201C2×C4—68no—
202C2×C4—61no—
203C2×C4—61no—
204C2×C4—62no—
205C2×C4—62no—
206C2×C4—61no—
207C2×C4—62no—
208C2×C4—62no—
209C2×C4—64no—
210C2×C4—62no—
211C2×C4—61no—
212C2×C4—128no—
213C2×C4D3×D412yes—
214C2×C3C2×#12311yescharacteristic
215C2×C3C2×#12411yescharacteristic
216C2×C3#53611yescharacteristic
217C2×C3#55212yes—
218C2×C3#55212yes—
219D3—24no—
220C22—34no—
221C22—62no—
222C22—64no—
223C22—64no—
224C22—62no—
225C22C2×D3×D411yescharacteristic
226C22#100711yescharacteristic
227C22D3×#4112yes—
228C22D3×#4112yes—
229C22#101011yescharacteristic
230C4—22no—
231C4—34no—
232C4—41no—
233C4—42no—
234C4—42no—
235C4—41no—
236C4—121no—
237C4—122no—
238C4—122no—
239C4—121no—
240C3#250311yescharacteristic, complemented, Sylow
241C2—64no—
242C2#562811yescharacteristic
243C2D3×#12311yescharacteristic
244C2D3×#12411yescharacteristic
245C2#569312yes—
246C2#570812yes—
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#400C2
prime radicalC23×D3C23
hypercenter#400D3
hypocenterC3#2503
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. #4986C2semidirect
5. #4985C2semidirect
9. #4659C2semidirect
10. #4650C2semidirect
11. #4646C2semidirect
43. #400D3semidirect
240. C3#2503semidirect

Maximal subgroups

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

subgroupindex
C2×#8962
C2×D3×C4⋊C42
C3×#4002
#49862
#49852
#46662
#46642
#46592
#46502
#46462
#25033