Finite Group Databasebeta

← #77072

Subgroups — #77072

Order 384.

420 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 121 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#5222C212yesmaximal
3#4986C211yesmaximal, characteristic, complemented
4#4975C211yesmaximal, characteristic, complemented
5#4951C212yesmaximal, complemented
6#4948C212yesmaximal, complemented
7#4944C212yesmaximal, complemented
8#4944C212yesmaximal, complemented
9#4652C212yesmaximal, complemented
10#2795—31nomaximal, Sylow
11C22×C3×D4C2211yescharacteristic
12C2×C3×C4⋊C4C2211yescharacteristic
13C2×C3×C22⋊C4C2212yes—
14C2×C3×C22⋊C4C2212yes—
15C2×#217C2212yes—
16C2×#217C2212yes—
17C2×#217C2212yes—
18C2×#217C2212yes—
19C2×C4⋊Dic3C2211yescharacteristic
20C2×C4⋊Dic3C2211yescharacteristic
21C2×Dic3⋊C4C2212yes—
22C2×Dic3⋊C4C2212yes—
23C2×Dic3⋊C4C2212yes—
24C2×C4×Dic3C2211yescharacteristic
25C2×C4×Dic3C2212yes—
26C3×#95C2211yescharacteristic
27C3⋊#95C2212yes—
28C3⋊#95C2212yes—
29C3⋊#95C2212yes—
30C3⋊#95C2211yescharacteristic
31C3⋊#95C2212yes—
32#400—31no—
33#396—32no—
34#394—32no—
35#394D311yescharacteristic
36#388—32no—
37#388—32no—
38#388—32no—
39#387—32no—
40#384—31no—
41C24×C3C2312yes—
42C2×C3×D4—42no—
43C2×C3×D4—41no—
44C22×C3×C4C2312yes—
45C22×C3×C4C2311yescharacteristic
46C22×C3×C4C2311yescharacteristic
47C22×C3×C4C2311yescharacteristic
48C22×Dic3C2312yes—
49C22×Dic3C2312yes—
50C22×Dic3C2312yes—
51C22×Dic3C2312yes—
52C3×C4⋊C4—41no—
53C3×C22⋊C4—24no—
54C3×C22⋊C4—42no—
55#217—24no—
56#217—24no—
57#217—42no—
58#217—42no—
59C4⋊Dic3—41no—
60C4⋊Dic3—41no—
61Dic3⋊C4—24no—
62Dic3⋊C4—42no—
63Dic3⋊C4—42no—
64C4×Dic3—22no—
65C4×Dic3—42no—
66C22×D4C2×D311yescharacteristic
67C2×C4⋊C4—32no—
68C2×C4⋊C4—32no—
69C2×C4⋊C4—31no—
70C2×C4⋊C4—32no—
71C2×C4⋊C4—31no—
72C2×C4⋊C4C2×D311yescharacteristic
73C2×C22⋊C4—32no—
74C2×C22⋊C4—32no—
75C2×C22⋊C4—32no—
76C2×C22⋊C4—32no—
77C2×C22⋊C4C2×D312yes—
78C2×C22⋊C4C2×D312yes—
79C2×C42—31no—
80C2×C42—32no—
81#95—32no—
82#95—32no—
83#95—32no—
84#95—31no—
85#95—32no—
86#95C2×D311yescharacteristic
87C23×C3—22no—
88C23×C3—22no—
89C23×C3—22no—
90C23×C3—22no—
91C23×C3—22no—
92C23×C3—22no—
93C23×C3—22no—
94C23×C3C2411yescharacteristic
95C3×D4—44no—
96C2×C3×C4—21no—
97C2×C3×C4—21no—
98C2×C3×C4—22no—
99C2×C3×C4—22no—
100C2×C3×C4—22no—
101C2×C3×C4—22no—
102C2×C3×C4—22no—
103C2×C3×C4—21no—
104C2×C3×C4—21no—
105C2×C3×C4#4112yes—
106C2×Dic3—22no—
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#4114yes—
118C24C22×D312yes—
119C2×D4—41no—
120C2×D4—42no—
121C22×C4—32no—
122C22×C4—32no—
123C22×C4—32no—
124C22×C4—32no—
125C22×C4C22×D312yes—
126C22×C4C22×D311yescharacteristic
127C22×C4C22×D311yescharacteristic
128C22×C4C22×D311yescharacteristic
129C4⋊C4—41no—
130C4⋊C4—64no—
131C4⋊C4—122no—
132C4⋊C4—122no—
133C4⋊C4—121no—
134C4⋊C4—121no—
135C22⋊C4—24no—
136C22⋊C4—42no—
137C22⋊C4—64no—
138C22⋊C4—64no—
139C22⋊C4—122no—
140C22⋊C4—122no—
141C42—62no—
142C42—122no—
143C22×C3—42no—
144C22×C3—42no—
145C22×C3—42no—
146C22×C3—42no—
147C22×C3—42no—
148C22×C3—42no—
149C22×C3—42no—
150C22×C3C2×#4111yescharacteristic
151C22×C3C2×#4112yes—
152C22×C3H(5,𝔽2)11yescharacteristic
153C22×C3H(5,𝔽2)12yes—
154C22×C32−1+411yescharacteristic
155C3×C4—22no—
156C3×C4—42no—
157C3×C4—41no—
158C3×C4—41no—
159Dic3—24no—
160Dic3—42no—
161Dic3—42no—
162Dic3—42no—
163C23—22no—
164C23—22no—
165C23—22no—
166C23—22no—
167C23—22no—
168C23—22no—
169C23—22no—
170C23C23×D311yescharacteristic
171D4—44no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—34no—
182C2×C4—62no—
183C2×C4—62no—
184C2×C4—62no—
185C2×C4—62no—
186C2×C4—62no—
187C2×C4—62no—
188C2×C4—62no—
189C2×C4—62no—
190C2×C4—62no—
191C2×C4—62no—
192C2×C4—62no—
193C2×C4#23712yes—
194C2×C3—82no—
195C2×C3#53911yescharacteristic
196C2×C3#54212yes—
197C2×C3#55312yes—
198C2×C3#55511yescharacteristic
199C2×C3#56011yescharacteristic
200C22—42no—
201C22—42no—
202C22—42no—
203C22—42no—
204C22—42no—
205C22—42no—
206C22—42no—
207C22C2×#23711yescharacteristic
208C22#100412yes—
209C22#100411yescharacteristic
210C22D3×#4112yes—
211C22#101011yescharacteristic
212C4—22no—
213C4—41no—
214C4—41no—
215C4—42no—
216C4—64no—
217C4—122no—
218C4—122no—
219C4—122no—
220C3#279511yescharacteristic, complemented, Sylow
221C2—82no—
222C2#559511yescharacteristic
223C2#560211yescharacteristic
224C2#561412yes—
225C2#566212yes—
226C2#567011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#394C2
prime radicalC22×C3×D4C22
hypercenter#394D3
hypocenterC3#2795
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
3. #4986C2semidirect
4. #4975C2semidirect
5. #4951C2semidirect
6. #4948C2semidirect
7. #4944C2semidirect
8. #4944C2semidirect
9. #4652C2semidirect
220. C3#2795semidirect

Maximal subgroups

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

subgroupindex
C3×#3942
#52222
#49862
#49752
#49512
#49482
#49442
#46522
#27953