Finite Group Databasebeta

← #74092

Subgroups — #74092

Order 384.

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

normal subgroups only

This group has 424 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×#381C211yesmaximal, characteristic
2#5212C211yesmaximal, characteristic
3C4×#217C211yesmaximal, characteristic
4#4948C212yesmaximal
5#4947C212yesmaximal
6#4660C212yesmaximal, complemented
7#4656C212yesmaximal, complemented
8#4654C212yesmaximal, complemented
9#4646C212yesmaximal, complemented
10#2665—31nomaximal, Sylow
11C23×C3×C4C2211yescharacteristic
12C22×Dic6C2212yes—
13C2×C3×C22⋊C4C2212yes—
14C2×#217C2212yes—
15C2×#217C2211yescharacteristic
16C2×#217C2211yescharacteristic
17C2×C4⋊Dic3C2212yes—
18C2×C4⋊Dic3C2212yes—
19C2×Dic3⋊C4C2212yes—
20C2×Dic3⋊C4C2212yes—
21C2×C4×Dic3C2212yes—
22C3×#95C2211yescharacteristic
23C3×#95C2211yescharacteristic
24C3×#95C2212yes—
25C3⋊#95C2212yes—
26C3⋊#95C2212yes—
27C3⋊#95C2212yes—
28C3⋊#95C2212yes—
29C3⋊#95C2212yes—
30C3⋊#95C2211yescharacteristic
31C3⋊#95C2211yescharacteristic
32#400—32no—
33#396—32no—
34#395—32no—
35#393—32no—
36#391—32no—
37#387—32no—
38#381D311yescharacteristic
39#380—31no—
40C4×C22⋊C4—31no—
41C24×C3C2311yescharacteristic
42C22×C3×C4—42no—
43C22×C3×C4Q814yescomplemented
44C22×C3×C4C2312yes—
45C22×C3×C4C2312yes—
46C22×C3×C4C2311yescharacteristic
47C22×C3×C4C2311yescharacteristic
48C22×Dic3C2312yes—
49C22×Dic3C2312yes—
50C22×Dic3C2312yes—
51C22×Dic3C2312yes—
52C2×Dic6—24no—
53C2×Dic6—42no—
54C2×Dic6—42no—
55C3×C22⋊C4—42no—
56#217—24no—
57#217—41no—
58#217—41no—
59C4⋊Dic3—24no—
60C4⋊Dic3—42no—
61Dic3⋊C4—42no—
62Dic3⋊C4—42no—
63C4×Dic3—42no—
64C22×Q8—32no—
65C23×C4C2×D311yescharacteristic
66C2×C4⋊C4—32no—
67C2×C4⋊C4—32no—
68C2×C4⋊C4—32no—
69C2×C4⋊C4—32no—
70C2×C22⋊C4—32no—
71C2×C22⋊C4—31no—
72C2×C22⋊C4—31no—
73C2×C22⋊C4C2×D312yes—
74C2×C42—32no—
75#95—31no—
76#95—31no—
77#95—32no—
78#95—32no—
79#95—32no—
80#95—32no—
81#95—32no—
82#95C2×D311yescharacteristic
83#95C2×D311yescharacteristic
84#95C2×D312yes—
85C23×C3—22no—
86C23×C3—21no—
87C23×C3—22no—
88C23×C3—21no—
89C23×C3C2×Q812yes—
90C23×C3C2411yescharacteristic
91C2×C3×C4—22no—
92C2×C3×C4—22no—
93C2×C3×C4—22no—
94C2×C3×C4—22no—
95C2×C3×C4—22no—
96C2×C3×C4—22no—
97C2×C3×C4—22no—
98C2×C3×C4—22no—
99C2×C3×C4—44no—
100C2×C3×C4C2×Q812yes—
101C2×C3×C4C2×Q812yes—
102C2×Dic3—22no—
103C2×Dic3—22no—
104C2×Dic3—22no—
105C2×Dic3—22no—
106C2×Dic3—22no—
107C2×Dic3—22no—
108C2×Dic3—22no—
109C2×Dic3—22no—
110C2×Dic3—22no—
111C2×Dic3—22no—
112C2×Dic3—22no—
113C2×Dic3#4114yes—
114Dic6—48no—
115C24C22×D311yescharacteristic
116C2×Q8—64no—
117C2×Q8—122no—
118C2×Q8—122no—
119C22×C4—32no—
120C22×C4—32no—
121C22×C4—32no—
122C22×C4—32no—
123C22×C4—42no—
124C22×C4Dic614yescomplemented
125C22×C4C22×D312yes—
126C22×C4C22×D312yes—
127C22×C4C22×D311yescharacteristic
128C22×C4C22×D311yescharacteristic
129C4⋊C4—64no—
130C4⋊C4—122no—
131C4⋊C4—122no—
132C4⋊C4—122no—
133C22⋊C4—42no—
134C22⋊C4—64no—
135C22⋊C4—121no—
136C22⋊C4—121no—
137C42—122no—
138C22×C3—22no—
139C22×C3—22no—
140C22×C3—22no—
141C22×C3—42no—
142C22×C3—42no—
143C22×C3C22×Q811yescharacteristic
144C22×C3C2×#4112yes—
145C22×C3H(5,𝔽2)12yes—
146C22×C3H(5,𝔽2)11yescharacteristic
147C22×C32−1+411yescharacteristic
148C3×C4—22no—
149C3×C4—22no—
150C3×C4—42no—
151C3×C4—42no—
152Dic3—24no—
153Dic3—42no—
154Dic3—42no—
155Dic3—42no—
156C23—22no—
157C23—22no—
158C23—21no—
159C23—21no—
160C23C2×Dic612yes—
161C23C23×D311yescharacteristic
162Q8—128no—
163C2×C4—22no—
164C2×C4—22no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—34no—
172C2×C4—44no—
173C2×C4—62no—
174C2×C4—62no—
175C2×C4—62no—
176C2×C4—62no—
177C2×C4—62no—
178C2×C4—62no—
179C2×C4—62no—
180C2×C4—62no—
181C2×C4—62no—
182C2×C4—62no—
183C2×C4—62no—
184C2×C4C2×Dic612yes—
185C2×C4C2×Dic612yes—
186C2×C3—42no—
187C2×C3#53311yescharacteristic
188C2×C3#53812yes—
189C2×C3#54212yes—
190C2×C3#54311yescharacteristic
191C2×C3#54411yescharacteristic
192C22—22no—
193C22—22no—
194C22—22no—
195C22—42no—
196C22—42no—
197C22C22×Dic611yescharacteristic
198C22#100412yes—
199C22D3×#4112yes—
200C22#100911yescharacteristic
201C22#101011yescharacteristic
202C4—22no—
203C4—22no—
204C4—42no—
205C4—42no—
206C4—64no—
207C4—122no—
208C4—122no—
209C4—122no—
210C3#266511yescharacteristic, complemented, Sylow
211C2—42no—
212C2#548311yescharacteristic
213C2#551711yescharacteristic
214C2#551911yescharacteristic
215C2#565712yes—
216C2#566212yes—
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#381C2
prime radicalC24×C3C23
hypercenter#381D3
hypocenterC3#2665
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #381

Complements

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

NH ≅ G/Ntype
6. #4660C2semidirect
7. #4656C2semidirect
8. #4654C2semidirect
9. #4646C2semidirect
43. C22×C3×C4Q8semidirect
124. C22×C4Dic6semidirect
210. C3#2665semidirect

Maximal subgroups

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

subgroupindex
C3×#3812
#52122
C4×#2172
#49482
#49472
#46602
#46562
#46542
#46462
#26653