Finite Group Databasebeta

← #76861

Subgroups — #76861

Order 384.

396 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 125 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×#393C211yesmaximal, characteristic
2#5230C212yesmaximal, complemented
3#4949C211yesmaximal, characteristic
4#4948C212yesmaximal
5#4943C211yesmaximal, characteristic
6#4942C212yesmaximal
7#4659C212yesmaximal, complemented
8#4657C212yesmaximal, complemented
9#4657C212yesmaximal, complemented
10#2841—31nomaximal, Sylow
11C22×C3×Q8C2211yescharacteristic
12C23×Dic3C2211yescharacteristic
13C2×C3×C22⋊C4C2212yes—
14C2×C3×C22⋊C4C2211yescharacteristic
15C2×#217C2212yes—
16C2×#217C2211yescharacteristic
17C2×C4⋊Dic3C2211yescharacteristic
18C2×C4⋊Dic3C2212yes—
19C2×Dic3⋊C4C2212yes—
20C2×Dic3⋊C4C2212yes—
21C2×Dic3⋊C4C2212yes—
22C2×Dic3⋊C4C2212yes—
23C2×Dic3⋊C4C2212yes—
24C3×#95C2211yescharacteristic
25C3×#95C2212yes—
26C3⋊#95C2212yes—
27C3⋊#95C2212yes—
28C3⋊#95C2212yes—
29C3⋊#95C2211yescharacteristic
30C3⋊#95C2212yes—
31C3⋊#95C2212yes—
32#400—32no—
33#399—31no—
34#398—32no—
35#398—32no—
36#396—32no—
37#395—32no—
38#393D311yescharacteristic
39#381—31no—
40#380—32no—
41C24×C3C2311yescharacteristic
42C2×C3×Q8—41no—
43C2×C3×Q8—42no—
44C22×C3×C4C2311yescharacteristic
45C22×C3×C4C2311yescharacteristic
46C22×C3×C4C2312yes—
47C22×C3×C4C2312yes—
48C22×Dic3—41no—
49C22×Dic3—41no—
50C22×Dic3Q814yescomplemented
51C22×Dic3C2312yes—
52C22×Dic3C2312yes—
53C22×Dic3C2312yes—
54C22×Dic3C2312yes—
55C3×C22⋊C4—42no—
56C3×C22⋊C4—41no—
57#217—42no—
58#217—41no—
59C4⋊Dic3—41no—
60C4⋊Dic3—42no—
61Dic3⋊C4—42no—
62Dic3⋊C4—42no—
63Dic3⋊C4—42no—
64Dic3⋊C4—42no—
65Dic3⋊C4—42no—
66C22×Q8C2×D311yescharacteristic
67C23×C4—31no—
68C2×C4⋊C4—32no—
69C2×C4⋊C4—32no—
70C2×C4⋊C4—32no—
71C2×C4⋊C4—32no—
72C2×C4⋊C4—32no—
73C2×C4⋊C4—32no—
74C2×C4⋊C4—31no—
75C2×C22⋊C4—32no—
76C2×C22⋊C4—31no—
77C2×C22⋊C4C2×D312yes—
78C2×C22⋊C4C2×D311yescharacteristic
79#95—32no—
80#95—32no—
81#95—32no—
82#95—32no—
83#95—32no—
84#95—31no—
85#95C2×D311yescharacteristic
86#95C2×D312yes—
87C23×C3—22no—
88C23×C3—21no—
89C23×C3—21no—
90C23×C3—22no—
91C23×C3C2×Q812yes—
92C23×C3C2411yescharacteristic
93C3×Q8—44no—
94C2×C3×C4—22no—
95C2×C3×C4—22no—
96C2×C3×C4—22no—
97C2×C3×C4—22no—
98C2×C3×C4—22no—
99C2×C3×C4—21no—
100C2×C3×C4—21no—
101C2×C3×C4—22no—
102C2×C3×C4—22no—
103C2×C3×C4—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—22no—
114C2×Dic3—22no—
115C2×Dic3—44no—
116C2×Dic3C2×Q814yes—
117C24C22×D311yescharacteristic
118C2×Q8—41no—
119C2×Q8—42no—
120C22×C4—32no—
121C22×C4—34no—
122C22×C4—32no—
123C22×C4—32no—
124C22×C4—32no—
125C22×C4—121no—
126C22×C4—121no—
127C22×C4C22×D311yescharacteristic
128C22×C4C22×D312yes—
129C22×C4C22×D311yescharacteristic
130C22×C4C22×D312yes—
131C4⋊C4—122no—
132C4⋊C4—122no—
133C4⋊C4—122no—
134C4⋊C4—122no—
135C4⋊C4—122no—
136C4⋊C4—121no—
137C4⋊C4—122no—
138C22⋊C4—42no—
139C22⋊C4—41no—
140C22⋊C4—122no—
141C22⋊C4—121no—
142C22×C3—22no—
143C22×C3—24no—
144C22×C3—41no—
145C22×C3—42no—
146C22×C3—41no—
147C22×C3C22×Q811yescharacteristic
148C22×C3H(5,𝔽2)11yescharacteristic
149C22×C3H(5,𝔽2)12yes—
150C22×C3H(5,𝔽2)11yescharacteristic
151C22×C32−1+412yes—
152C3×C4—42no—
153C3×C4—41no—
154C3×C4—42no—
155C3×C4—41no—
156Dic3—24no—
157Dic3—42no—
158Dic3—42no—
159Dic3—42no—
160C23—22no—
161C23—21no—
162C23—21no—
163C23—22no—
164C23D3×Q812yes—
165C23C23×D311yescharacteristic
166Q8—44no—
167C2×C4—21no—
168C2×C4—22no—
169C2×C4—21no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—34no—
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×C4—62no—
189C2×C4—124no—
190C2×C3—42no—
191C2×C3#54311yescharacteristic
192C2×C3#54412yes—
193C2×C3#56011yescharacteristic
194C2×C3#56212yes—
195C2×C3#56311yescharacteristic
196C22—22no—
197C22—24no—
198C22—41no—
199C22—42no—
200C22—41no—
201C22#100411yescharacteristic
202C22#100412yes—
203C22C2×D3×Q811yescharacteristic
204C22#100911yescharacteristic
205C22#101012yes—
206C4—41no—
207C4—42no—
208C4—41no—
209C4—42no—
210C4—64no—
211C4—122no—
212C4—122no—
213C4—122no—
214C3#284111yescharacteristic, complemented, Sylow
215C2—42no—
216C2#559511yescharacteristic
217C2#562312yes—
218C2#563411yescharacteristic
219C2#567211yescharacteristic
220C2#568412yes—
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#393C2
prime radicalC24×C3C23
hypercenter#393D3
hypocenterC3#2841
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
2. #5230C2semidirect
7. #4659C2semidirect
8. #4657C2semidirect
9. #4657C2semidirect
50. C22×Dic3Q8semidirect
214. C3#2841semidirect

Maximal subgroups

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

subgroupindex
C3×#3932
#52302
#49492
#49482
#49432
#49422
#46592
#46572
#28413