Finite Group Databasebeta

← #76414

Subgroups — #76414

Order 384.

472 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 137 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
1C3×#390C211yesmaximal, characteristic
2#5220C212yesmaximal
3#4986C211yesmaximal, characteristic, complemented
4#4979C211yesmaximal, characteristic, complemented
5#4948C212yesmaximal, complemented
6#4947C212yesmaximal, complemented
7#4946C212yesmaximal, complemented
8#4942C212yesmaximal, complemented
9#4933C212yesmaximal, complemented
10#2784—31nomaximal, Sylow
11C22×C3×D4C2211yescharacteristic
12C23×Dic3C2212yes—
13C22×Dic6C2211yescharacteristic
14C2×C3×C4⋊C4C2211yescharacteristic
15C2×C3×C22⋊C4C2212yes—
16C2×C3×C22⋊C4C2212yes—
17C2×C3×C42C2211yescharacteristic
18C2×#217C2212yes—
19C2×#217C2212yes—
20C2×#217C2212yes—
21C2×C4⋊Dic3C2211yescharacteristic
22C2×Dic3⋊C4C2212yes—
23C2×Dic3⋊C4C2212yes—
24C2×Dic3⋊C4C2212yes—
25C3⋊#95C2212yes—
26C3⋊#95C2212yes—
27C3⋊#95C2212yes—
28C3⋊#95C2212yes—
29C3⋊#95C2211yescharacteristic
30C3⋊#95C2211yescharacteristic
31C3⋊#95C2212yes—
32#400—31no—
33#396—32no—
34#395—31no—
35#393—32no—
36#390D311yescharacteristic
37#387—32no—
38#386—32no—
39#385—32no—
40#380—32no—
41C24×C3C2312yes—
42C2×C3×D4—41no—
43C2×C3×D4—41no—
44C2×C3×D4Q814yescomplemented
45C22×C3×C4C2311yescharacteristic
46C22×C3×C4C2312yes—
47C22×C3×C4C2311yescharacteristic
48C22×C3×C4C2311yescharacteristic
49C22×Dic3—24no—
50C22×Dic3—42no—
51C22×Dic3—42no—
52C22×Dic3C2312yes—
53C22×Dic3C2312yes—
54C22×Dic3C2312yes—
55C22×Dic3C2312yes—
56C2×Dic6—42no—
57C2×Dic6—41no—
58C3×C4⋊C4—41no—
59C3×C22⋊C4—42no—
60C3×C22⋊C4—42no—
61C3×C42—41no—
62#217—24no—
63#217—42no—
64#217—42no—
65C4⋊Dic3—41no—
66Dic3⋊C4—24no—
67Dic3⋊C4—42no—
68Dic3⋊C4—42no—
69C22×Q8—31no—
70C22×D4C2×D311yescharacteristic
71C23×C4—32no—
72C2×C4⋊C4—32no—
73C2×C4⋊C4—32no—
74C2×C4⋊C4—32no—
75C2×C4⋊C4—31no—
76C2×C4⋊C4C2×D311yescharacteristic
77C2×C22⋊C4—32no—
78C2×C22⋊C4—32no—
79C2×C22⋊C4—32no—
80C2×C22⋊C4C2×D312yes—
81C2×C22⋊C4C2×D312yes—
82C2×C42C2×D311yescharacteristic
83#95—31no—
84#95—32no—
85#95—31no—
86#95—32no—
87#95—32no—
88#95—32no—
89#95—32no—
90C23×C3—22no—
91C23×C3—22no—
92C23×C3—22no—
93C23×C3—22no—
94C23×C3—22no—
95C23×C3—22no—
96C23×C3C2×Q814yes—
97C23×C3C2411yescharacteristic
98C3×D4—44no—
99C2×C3×C4—21no—
100C2×C3×C4—21no—
101C2×C3×C4—22no—
102C2×C3×C4—22no—
103C2×C3×C4—22no—
104C2×C3×C4—21no—
105C2×C3×C4—22no—
106C2×C3×C4—22no—
107C2×C3×C4—21no—
108C2×C3×C4C2×Q812yes—
109C2×Dic3—22no—
110C2×Dic3—22no—
111C2×Dic3—22no—
112C2×Dic3—22no—
113C2×Dic3—22no—
114C2×Dic3—22no—
115C2×Dic3—22no—
116C2×Dic3—22no—
117C2×Dic3—22no—
118C2×Dic3—22no—
119C2×Dic3—22no—
120C2×Dic3—48no—
121C2×Dic3#4114yes—
122Dic6—44no—
123C24C22×D312yes—
124C2×Q8—122no—
125C2×Q8—121no—
126C2×D4—41no—
127C2×D4—41no—
128C2×D4Dic614yescomplemented
129C22×C4—32no—
130C22×C4—32no—
131C22×C4—32no—
132C22×C4—32no—
133C22×C4—64no—
134C22×C4—122no—
135C22×C4—122no—
136C22×C4C22×D311yescharacteristic
137C22×C4C22×D312yes—
138C22×C4C22×D311yescharacteristic
139C22×C4C22×D311yescharacteristic
140C4⋊C4—41no—
141C4⋊C4—64no—
142C4⋊C4—122no—
143C4⋊C4—122no—
144C4⋊C4—121no—
145C22⋊C4—42no—
146C22⋊C4—42no—
147C22⋊C4—64no—
148C22⋊C4—122no—
149C22⋊C4—122no—
150C42—41no—
151C22×C3—24no—
152C22×C3—24no—
153C22×C3—24no—
154C22×C3—42no—
155C22×C3—42no—
156C22×C3—42no—
157C22×C3—42no—
158C22×C3C22×Q811yescharacteristic
159C22×C3C2×#4112yes—
160C22×C3H(5,𝔽2)11yescharacteristic
161C22×C3H(5,𝔽2)12yes—
162C22×C3H(5,𝔽2)11yescharacteristic
163C3×C4—22no—
164C3×C4—42no—
165C3×C4—41no—
166C3×C4—41no—
167Dic3—24no—
168Dic3—42no—
169Dic3—42no—
170Dic3—42no—
171C23—22no—
172C23—22no—
173C23—22no—
174C23—22no—
175C23—22no—
176C23—22no—
177C23C2×Dic614yes—
178C23C23×D311yescharacteristic
179Q8—124no—
180D4—44no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—22no—
190C2×C4—34no—
191C2×C4—62no—
192C2×C4—62no—
193C2×C4—62no—
194C2×C4—62no—
195C2×C4—62no—
196C2×C4—62no—
197C2×C4—62no—
198C2×C4—62no—
199C2×C4—62no—
200C2×C4—62no—
201C2×C4—62no—
202C2×C4—128no—
203C2×C4C2×Dic612yes—
204C2×C3—44no—
205C2×C3#53812yes—
206C2×C3#54311yescharacteristic
207C2×C3#55412yes—
208C2×C3#55511yescharacteristic
209C2×C3#56011yescharacteristic
210C22—24no—
211C22—24no—
212C22—24no—
213C22—42no—
214C22—42no—
215C22—42no—
216C22—42no—
217C22C22×Dic611yescharacteristic
218C22#100411yescharacteristic
219C22#100412yes—
220C22#100411yescharacteristic
221C22D3×#4112yes—
222C4—22no—
223C4—41no—
224C4—41no—
225C4—42no—
226C4—64no—
227C4—122no—
228C4—122no—
229C4—122no—
230C3#278411yescharacteristic, complemented, Sylow
231C2—44no—
232C2#548311yescharacteristic
233C2#553912yes—
234C2#561312yes—
235C2#567011yescharacteristic
236C2#572011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#390C2
prime radicalC22×C3×D4C22
hypercenter#390D3
hypocenterC3#2784
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
3. #4986C2semidirect
4. #4979C2semidirect
5. #4948C2semidirect
6. #4947C2semidirect
7. #4946C2semidirect
8. #4942C2semidirect
9. #4933C2semidirect
44. C2×C3×D4Q8semidirect
128. C2×D4Dic6semidirect
230. C3#2784semidirect

Maximal subgroups

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

subgroupindex
C3×#3902
#52202
#49862
#49792
#49482
#49472
#49462
#49422
#49332
#27843