Finite Group Databasebeta

← #76956

Subgroups — #76956

Order 384.

476 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 129 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×#394C211yesmaximal, characteristic
2#5220C212yesmaximal
3#4986C211yesmaximal, characteristic, complemented
4#4973C211yesmaximal, characteristic, complemented
5#4951C212yesmaximal, complemented
6#4947C212yesmaximal, complemented
7#4943C212yesmaximal, complemented
8Dic3×C22⋊C4C212yesmaximal, complemented
9#4651C212yesmaximal, complemented
10#2572—31nomaximal, Sylow
11C22×C3×D4C2211yescharacteristic
12C23×Dic3C2212yes—
13C22×Dic6C2211yescharacteristic
14C2×C3×C4⋊C4C2211yescharacteristic
15C2×C3×C22⋊C4C2212yes—
16C2×C3×C22⋊C4C2212yes—
17C2×#217C2212yes—
18C2×#217C2212yes—
19C2×#217C2212yes—
20C2×C4⋊Dic3C2211yescharacteristic
21C2×C4×Dic3C2212yes—
22C2×C4×Dic3C2211yescharacteristic
23C3×#95C2211yescharacteristic
24C3⋊#95C2212yes—
25C3⋊#95C2212yes—
26C3⋊#95C2212yes—
27C3⋊#95C2212yes—
28C3⋊#95C2212yes—
29C3⋊#95C2212yes—
30C3⋊#95C2211yescharacteristic
31C3⋊#95C2212yes—
32#400—31no—
33#394D311yescharacteristic
34#393—32no—
35#391—31no—
36#388—32no—
37#386—32no—
38#383—32no—
39#381—32no—
40C4×C22⋊C4—32no—
41C24×C3C2312yes—
42C2×C3×D4—24no—
43C2×C3×D4—41no—
44C22×C3×C4C2311yescharacteristic
45C22×C3×C4C2312yes—
46C22×C3×C4C2311yescharacteristic
47C22×C3×C4C2311yescharacteristic
48C22×Dic3—24no—
49C22×Dic3—24no—
50C22×Dic3—42no—
51C22×Dic3C2312yes—
52C22×Dic3C2312yes—
53C22×Dic3C2312yes—
54C22×Dic3C2312yes—
55C2×Dic6—22no—
56C2×Dic6—41no—
57C2×Dic6—41no—
58C3×C4⋊C4—41no—
59C3×C22⋊C4—24no—
60C3×C22⋊C4—24no—
61#217—24no—
62#217—42no—
63#217—42no—
64C4⋊Dic3—41no—
65C4×Dic3—24no—
66C4×Dic3—41no—
67C22×Q8—31no—
68C22×D4C2×D311yescharacteristic
69C23×C4—32no—
70C2×C4⋊C4—31no—
71C2×C4⋊C4C2×D311yescharacteristic
72C2×C22⋊C4—32no—
73C2×C22⋊C4—32no—
74C2×C22⋊C4—32no—
75C2×C22⋊C4C2×D312yes—
76C2×C22⋊C4C2×D312yes—
77C2×C42—32no—
78C2×C42—31no—
79#95—32no—
80#95—32no—
81#95—31no—
82#95—32no—
83#95—32no—
84#95—32no—
85#95—32no—
86#95—32no—
87#95C2×D311yescharacteristic
88C23×C3—22no—
89C23×C3—22no—
90C23×C3—22no—
91C23×C3—22no—
92C23×C3—22no—
93C23×C3—22no—
94C23×C3#4114yes—
95C23×C3C2411yescharacteristic
96C3×D4—44no—
97C2×C3×C4—21no—
98C2×C3×C4—21no—
99C2×C3×C4—22no—
100C2×C3×C4—22no—
101C2×C3×C4—22no—
102C2×C3×C4—22no—
103C2×C3×C4—22no—
104C2×C3×C4—21no—
105C2×C3×C4—21no—
106C2×C3×C4#4112yes—
107C2×Dic3—22no—
108C2×Dic3—22no—
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—48no—
119C2×Dic3#4114yes—
120Dic6—44no—
121C24C22×D312yes—
122C2×Q8—62no—
123C2×Q8—121no—
124C2×Q8—121no—
125C2×D4—24no—
126C2×D4—41no—
127C22×C4—32no—
128C22×C4—32no—
129C22×C4—32no—
130C22×C4—32no—
131C22×C4—64no—
132C22×C4—64no—
133C22×C4—122no—
134C22×C4C22×D311yescharacteristic
135C22×C4C22×D312yes—
136C22×C4C22×D311yescharacteristic
137C22×C4C22×D311yescharacteristic
138C4⋊C4—41no—
139C4⋊C4—121no—
140C22⋊C4—24no—
141C22⋊C4—24no—
142C22⋊C4—64no—
143C22⋊C4—122no—
144C22⋊C4—122no—
145C42—64no—
146C42—121no—
147C22×C3—24no—
148C22×C3—24no—
149C22×C3—24no—
150C22×C3—42no—
151C22×C3—42no—
152C22×C3—42no—
153C22×C3—42no—
154C22×C3C2×#4111yescharacteristic
155C22×C3C2×#4112yes—
156C22×C3C2×#4112yes—
157C22×C3H(5,𝔽2)11yescharacteristic
158C22×C3H(5,𝔽2)11yescharacteristic
159C3×C4—22no—
160C3×C4—42no—
161C3×C4—41no—
162C3×C4—41no—
163Dic3—24no—
164Dic3—42no—
165Dic3—42no—
166Dic3—42no—
167C23—22no—
168C23—22no—
169C23—22no—
170C23—22no—
171C23—22no—
172C23—22no—
173C23#23714yes—
174C23C23×D311yescharacteristic
175Q8—124no—
176D4—44no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—22no—
186C2×C4—34no—
187C2×C4—62no—
188C2×C4—62no—
189C2×C4—62no—
190C2×C4—62no—
191C2×C4—62no—
192C2×C4—62no—
193C2×C4—62no—
194C2×C4—62no—
195C2×C4—62no—
196C2×C4—62no—
197C2×C4—62no—
198C2×C4—128no—
199C2×C4#23712yes—
200C2×C3—44no—
201C2×C3#52912yes—
202C2×C3#53811yescharacteristic
203C2×C3#55112yes—
204C2×C3#55111yescharacteristic
205C2×C3#55511yescharacteristic
206C22—24no—
207C22—24no—
208C22—24no—
209C22—42no—
210C22—42no—
211C22—42no—
212C22—42no—
213C22C2×#23712yes—
214C22C2×#23711yescharacteristic
215C22#100411yescharacteristic
216C22D3×#4112yes—
217C22#100911yescharacteristic
218C4—22no—
219C4—41no—
220C4—41no—
221C4—42no—
222C4—64no—
223C4—122no—
224C4—122no—
225C4—122no—
226C3#257211yescharacteristic, complemented, Sylow
227C2—44no—
228C2#558711yescharacteristic
229C2#560012yes—
230C2#561711yescharacteristic
231C2#565812yes—
232C2#567011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#394C2
prime radicalC22×C3×D4C22
hypercenter#394D3
hypocenterC3#2572
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
3. #4986C2semidirect
4. #4973C2semidirect
5. #4951C2semidirect
6. #4947C2semidirect
7. #4943C2semidirect
8. Dic3×C22⋊C4C2semidirect
9. #4651C2semidirect
226. C3#2572semidirect

Maximal subgroups

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

subgroupindex
C3×#3942
#52202
#49862
#49732
#49512
#49472
#49432
Dic3×C22⋊C42
#46512
#25723