Finite Group Databasebeta

← #78360

Subgroups — #78360

Order 384.

508 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 508 conjugacy classes of subgroups — too many to draw. Only its 143 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×D3×C4⋊C4C211yesmaximal, characteristic
3C3×#400C211yesmaximal, characteristic, complemented
4#4985C211yesmaximal, characteristic, complemented
5#4979C211yesmaximal, characteristic, complemented
6#4668C212yesmaximal
7#4667C211yesmaximal, characteristic
8#4664C212yesmaximal
9#4661C212yesmaximal, complemented
10#4657C212yesmaximal, complemented
11#4656C211yesmaximal, characteristic, complemented
12#2627—31nomaximal, Sylow
13C22×C4×D3C2211yescharacteristic
14C22×C4×D3C2211yescharacteristic
15C22×C4×D3C2211yescharacteristic
16C22×Dic6C2211yescharacteristic
17C2×C3×C4⋊C4C2211yescharacteristic
18C2×C3×C4⋊C4C2211yescharacteristic
19C2×(C2×D3)⋊C4C2212yes—
20C2×(C2×D3)⋊C4C2212yes—
21C2×C4⋊Dic3C2212yes—
22C2×C4⋊Dic3C2211yescharacteristic
23C2×C4⋊Dic3C2212yes—
24C2×Dic3⋊C4C2211yescharacteristic
25C2×Dic3⋊C4C2211yescharacteristic
26C2×Dic3⋊C4C2212yes—
27C2×Dic3⋊C4C2212yes—
28#897—24no—
29D3×C4⋊C4—24no—
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#400D311yescharacteristic, complemented
42#398—32no—
43#398—32no—
44#395—31no—
45#395—31no—
46#385—32no—
47#381—32no—
48#380—31no—
49C23×D3C2311yescharacteristic
50C22×C3×C4C2312yes—
51C22×C3×C4C2312yes—
52C22×C3×C4C2311yescharacteristic
53C22×C3×C4C2311yescharacteristic
54C22×C3×C4C2311yescharacteristic
55C22×Dic3C2312yes—
56C22×Dic3C2311yescharacteristic
57C22×Dic3C2312yes—
58C22×Dic3C2311yescharacteristic
59C22×Dic3C2311yescharacteristic
60C2×C4×D3—22no—
61C2×C4×D3—22no—
62C2×C4×D3—22no—
63C2×C4×D3—42no—
64C2×C4×D3—42no—
65C2×C4×D3D414yes—
66C2×C4×D3Q814yescomplemented
67C2×Dic6—22no—
68C2×Dic6—42no—
69C3×C4⋊C4—22no—
70C3×C4⋊C4—22no—
71(C2×D3)⋊C4—24no—
72(C2×D3)⋊C4—24no—
73C4⋊Dic3—22no—
74C4⋊Dic3—24no—
75C4⋊Dic3—42no—
76Dic3⋊C4—22no—
77Dic3⋊C4—22no—
78Dic3⋊C4—42no—
79Dic3⋊C4—42no—
80C22×Q8—31no—
81C23×C4—31no—
82C23×C4—31no—
83C23×C4—31no—
84C22⋊Q8—64no—
85C2×C4⋊C4—31no—
86C2×C4⋊C4—31no—
87C2×C4⋊C4—31no—
88C2×C4⋊C4—32no—
89C2×C4⋊C4—32no—
90C2×C4⋊C4—32no—
91C2×C4⋊C4—32no—
92C2×C4⋊C4—64no—
93C2×C4⋊C4C2×D311yescharacteristic
94C2×C4⋊C4C2×D311yescharacteristic
95C2×C22⋊C4—32no—
96C2×C22⋊C4—32no—
97#95—31no—
98#95—32no—
99#95—32no—
100#95—32no—
101#95—32no—
102#95C2×D311yescharacteristic
103#95C2×D312yes—
104#95C2×D312yes—
105C23×C3C2411yescharacteristic
106C22×D3—22no—
107C22×D3—22no—
108C22×D3C2×D412yes—
109C22×D3C2×Q812yes—
110C22×D3#4112yes—
111C2×C3×C4—21no—
112C2×C3×C4—21no—
113C2×C3×C4—22no—
114C2×C3×C4—21no—
115C2×C3×C4—22no—
116C2×C3×C4—22no—
117C2×C3×C4—22no—
118C2×C3×C4—22no—
119C2×C3×C4—24no—
120C2×C3×C4—22no—
121C2×C3×C4C2×D412yes—
122C2×C3×C4C2×Q812yes—
123C2×Dic3—21no—
124C2×Dic3—21no—
125C2×Dic3—22no—
126C2×Dic3—21no—
127C2×Dic3—22no—
128C2×Dic3—22no—
129C2×Dic3—22no—
130C2×Dic3—22no—
131C2×Dic3—22no—
132C2×Dic3—22no—
133C2×Dic3—22no—
134C2×Dic3C2×D412yes—
135C2×Dic3C2×Q812yes—
136C4×D3—28no—
137C4×D3—44no—
138C4×D3—44no—
139Dic6—44no—
140C24—31no—
141C2×Q8—62no—
142C2×Q8—122no—
143C22×C4—31no—
144C22×C4—31no—
145C22×C4—34no—
146C22×C4—34no—
147C22×C4—32no—
148C22×C4—32no—
149C22×C4—31no—
150C22×C4—62no—
151C22×C4—62no—
152C22×C4—62no—
153C22×C4—122no—
154C22×C4—122no—
155C22×C4C22×D312yes—
156C22×C4C22×D312yes—
157C22×C4C22×D311yescharacteristic
158C22×C4C22×D311yescharacteristic
159C22×C4C22×D311yescharacteristic
160C4⋊C4—22no—
161C4⋊C4—22no—
162C4⋊C4—62no—
163C4⋊C4—62no—
164C4⋊C4—64no—
165C4⋊C4—62no—
166C4⋊C4—122no—
167C4⋊C4—122no—
168C4⋊C4—122no—
169C22⋊C4—64no—
170C22⋊C4—64no—
171C22×C3C22×D411yescharacteristic
172C22×C3C22×Q811yescharacteristic
173C22×C3C2×#4111yescharacteristic
174C22×C3H(5,𝔽2)12yes—
175C22×C32−1+412yes—
176C2×D3—22no—
177C2×D3—22no—
178C2×D3—24no—
179C2×D3—24no—
180C2×D3C22⋊Q814yes—
181C3×C4—22no—
182C3×C4—22no—
183C3×C4—41no—
184C3×C4—42no—
185C3×C4—42no—
186Dic3—22no—
187Dic3—22no—
188Dic3—42no—
189Dic3—41no—
190Dic3—42no—
191C23—32no—
192C23—32no—
193C23—32no—
194C23—62no—
195C23—62no—
196C23C23×D311yescharacteristic
197Q8—124no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—21no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—24no—
207C2×C4—22no—
208C2×C4—32no—
209C2×C4—32no—
210C2×C4—68no—
211C2×C4—61no—
212C2×C4—61no—
213C2×C4—62no—
214C2×C4—62no—
215C2×C4—61no—
216C2×C4—62no—
217C2×C4—62no—
218C2×C4—62no—
219C2×C4—62no—
220C2×C4—62no—
221C2×C4—62no—
222C2×C4—124no—
223C2×C4—124no—
224C2×C4D3×D412yes—
225C2×C4D3×Q812yes—
226C2×C3C2×C22⋊Q811yescharacteristic
227C2×C3#53411yescharacteristic
228C2×C3#53611yescharacteristic
229C2×C3#53912yes—
230C2×C3#54412yes—
231D3—24no—
232C22—34no—
233C22—62no—
234C22—62no—
235C22—64no—
236C22—64no—
237C22C2×D3×D411yescharacteristic
238C22C2×D3×Q811yescharacteristic
239C22D3×#4111yescharacteristic
240C22#100912yes—
241C22#101012yes—
242C4—22no—
243C4—22no—
244C4—41no—
245C4—42no—
246C4—42no—
247C4—62no—
248C4—62no—
249C4—121no—
250C4—122no—
251C4—122no—
252C3#262711yescharacteristic, complemented, Sylow
253C2—64no—
254C2D3×C22⋊Q811yescharacteristic
255C2#565311yescharacteristic
256C2#567511yescharacteristic
257C2#568912yes—
258C2#570712yes—
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#400C2
prime radicalC23×D3C23
hypercenter#400D3
hypocenterC3#2627
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. #4661C2semidirect
10. #4657C2semidirect
11. #4656C2semidirect
41. #400D3semidirect
66. C2×C4×D3Q8semidirect
252. C3#2627semidirect

Maximal subgroups

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

subgroupindex
C2×#8972
C2×D3×C4⋊C42
C3×#4002
#49852
#49792
#46682
#46672
#46642
#46612
#46572
#46562
#26273