Finite Group Databasebeta

← #74062

Subgroups — #74062

Order 384.

464 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 173 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×#924C211yesmaximal, characteristic
2C3×#381C211yesmaximal, characteristic
3#5212C211yesmaximal, characteristic
4#4945C212yesmaximal
5#4944C212yesmaximal
6#4655C212yesmaximal, complemented
7#4653C212yesmaximal, complemented
8#4647C212yesmaximal, complemented
9#4646C212yesmaximal, complemented
10#2378—31nomaximal, Sylow
11C23×C3×C4C2211yescharacteristic
12C22×Dic6C2211yescharacteristic
13C2×C3×C22⋊C4C2212yes—
14C2×#217C2211yescharacteristic
15C2×#217C2211yescharacteristic
16C2×#217C2212yes—
17C2×C4⋊Dic3C2212yes—
18C2×C4⋊Dic3C2211yescharacteristic
19#924C418yescomplemented
20C2×Dic3⋊C4C2212yes—
21C2×Dic3⋊C4C2212yes—
22C2×Dic3⋊C4C2212yes—
23C2×C4×Dic3C2212yes—
24C2×C4×Dic3C2212yes—
25C3×#95C2212yes—
26C3×#95C2211yescharacteristic
27C3×#95C2211yescharacteristic
28C3⋊#95C2212yes—
29C3⋊#95C2212yes—
30C3⋊#95C2212yes—
31C3⋊#95C2212yes—
32C3⋊#95C2212yes—
33C2×C22⋊Q8—31no—
34#391—32no—
35#389—32no—
36#388—32no—
37#388—32no—
38#387—32no—
39#387—32no—
40#381D311yescharacteristic
41#380—31no—
42C24×C3C2311yescharacteristic
43C22×C3×C4—42no—
44C22×C3×C4C2×C414yes—
45C22×C3×C4C2312yes—
46C22×C3×C4C2312yes—
47C22×C3×C4C2311yescharacteristic
48C22×C3×C4C2311yescharacteristic
49C22×Dic3C2312yes—
50C22×Dic3C2312yes—
51C22×Dic3C2312yes—
52C22×Dic3C2312yes—
53C2×Dic6—42no—
54C2×Dic6C2×C414yes—
55C3×C22⋊C4—42no—
56#217—41no—
57#217—41no—
58#217C2×C418yes—
59C4⋊Dic3—42no—
60C4⋊Dic3C2×C414yes—
61Dic3⋊C4—42no—
62Dic3⋊C4—42no—
63Dic3⋊C4C2×C418yes—
64C4×Dic3—42no—
65C4×Dic3—42no—
66C22×Q8—31no—
67C23×C4C2×D311yescharacteristic
68C22⋊Q8—38no—
69C2×C4⋊C4—31no—
70C2×C4⋊C4—32no—
71C2×C4⋊C4—32no—
72C2×C4⋊C4—32no—
73C2×C4⋊C4—32no—
74C2×C22⋊C4—32no—
75C2×C22⋊C4—31no—
76C2×C22⋊C4—31no—
77C2×C22⋊C4C2×D312yes—
78C2×C42—32no—
79C2×C42—32no—
80#95—32no—
81#95—32no—
82#95—32no—
83#95—32no—
84#95—32no—
85#95C2×D311yescharacteristic
86#95C2×D312yes—
87#95C2×D311yescharacteristic
88C23×C3—21no—
89C23×C3—22no—
90C23×C3—22no—
91C23×C3—21no—
92C23×C3C22×C412yes—
93C23×C3C2411yescharacteristic
94C2×C3×C4—22no—
95C2×C3×C4—22no—
96C2×C3×C4—22no—
97C2×C3×C4—22no—
98C2×C3×C4—22no—
99C2×C3×C4—22no—
100C2×C3×C4—22no—
101C2×C3×C4—22no—
102C2×C3×C4—44no—
103C2×C3×C4C22×C412yes—
104C2×C3×C4C22×C412yes—
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—22no—
114C2×Dic3—22no—
115C2×Dic3C22×C414yes—
116C2×Dic3C22×C414yes—
117Dic6—44no—
118C24C22×D311yescharacteristic
119C2×Q8—34no—
120C2×Q8—122no—
121C22×C4—32no—
122C22×C4—32no—
123C22×C4—32no—
124C22×C4—32no—
125C22×C4—42no—
126C22×C4C4×D314yes—
127C22×C4C22×D312yes—
128C22×C4C22×D312yes—
129C22×C4C22×D311yescharacteristic
130C22×C4C22×D311yescharacteristic
131C4⋊C4—38no—
132C4⋊C4—34no—
133C4⋊C4—122no—
134C4⋊C4—122no—
135C4⋊C4—122no—
136C22⋊C4—38no—
137C22⋊C4—42no—
138C22⋊C4—121no—
139C22⋊C4—121no—
140C42—122no—
141C42—122no—
142C22×C3—22no—
143C22×C3—22no—
144C22×C3—22no—
145C22×C3—42no—
146C22×C3—42no—
147C22×C3C23×C411yescharacteristic
148C22×C3H(5,𝔽2)12yes—
149C22×C3H(5,𝔽2)11yescharacteristic
150C22×C32−1+412yes—
151C22×C32−1+411yescharacteristic
152C3×C4—22no—
153C3×C4—22no—
154C3×C4—42no—
155C3×C4—42no—
156Dic3—24no—
157Dic3—24no—
158Dic3—42no—
159Dic3—42no—
160C23—22no—
161C23—21no—
162C23—22no—
163C23—21no—
164C23C2×C4×D312yes—
165C23C23×D311yescharacteristic
166Q8—124no—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—34no—
176C2×C4—34no—
177C2×C4—44no—
178C2×C4—62no—
179C2×C4—62no—
180C2×C4—62no—
181C2×C4—62no—
182C2×C4—62no—
183C2×C4—62no—
184C2×C4—62no—
185C2×C4—62no—
186C2×C4—62no—
187C2×C4—62no—
188C2×C4C2×C4×D312yes—
189C2×C4C2×C4×D312yes—
190C2×C3—42no—
191C2×C3#51811yescharacteristic
192C2×C3#51911yescharacteristic
193C2×C3#52011yescharacteristic
194C2×C3#56212yes—
195C2×C3#56312yes—
196C22—22no—
197C22—22no—
198C22—22no—
199C22—42no—
200C22—42no—
201C22C22×C4×D311yescharacteristic
202C22#100412yes—
203C22#100911yescharacteristic
204C22#101011yescharacteristic
205C22#101012yes—
206C4—22no—
207C4—22no—
208C4—42no—
209C4—42no—
210C4—64no—
211C4—64no—
212C4—122no—
213C4—122no—
214C3#237811yescharacteristic, complemented, Sylow
215C2—42no—
216C2#548611yescharacteristic
217C2#551611yescharacteristic
218C2#552311yescharacteristic
219C2#565112yes—
220C2#566712yes—
subgroupG/H
center, FrattiniC23C23×D3
commutatorC22×C3C23×C4
FittingC3×#381C2
socle, cosocleC23×C3C24
prime radicalC24×C3C23
hypercenter#381D3
hypocenterC3#2378
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ #381

Complements

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

NH ≅ G/Ntype
6. #4655C2semidirect
7. #4653C2semidirect
8. #4647C2semidirect
9. #4646C2semidirect
19. #924C4semidirect
214. C3#2378semidirect

Maximal subgroups

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

subgroupindex
C2×#9242
C3×#3812
#52122
#49452
#49442
#46552
#46532
#46472
#46462
#23783