Finite Group Databasebeta

← #78089

Subgroups — #78089

Order 384.

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

normal subgroups only

This group has 408 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×#399C211yesmaximal, characteristic
2#4982C212yesmaximal, complemented
3#4976C212yesmaximal, complemented
4#4976C212yesmaximal, complemented
5#4949C211yesmaximal, characteristic
6#4946C211yesmaximal, characteristic
7#4946C212yesmaximal
8#4944C212yesmaximal
9#4658C212yesmaximal, complemented
10#2807—31nomaximal, Sylow
11C23×Dic3C2211yescharacteristic
12C2×C3×C4⋊C4C2211yescharacteristic
13C2×C3×C4⋊C4C2212yes—
14C2×C3×C22⋊C4C2212yes—
15C2×C3×C22⋊C4C2211yescharacteristic
16C2×#217C2212yes—
17C2×#217C2211yescharacteristic
18C2×C4⋊Dic3C2211yescharacteristic
19C2×Dic3⋊C4C2212yes—
20C2×Dic3⋊C4C2212yes—
21C2×Dic3⋊C4C2212yes—
22C2×Dic3⋊C4C2212yes—
23C2×Dic3⋊C4C2212yes—
24C2×C4×Dic3C2212yes—
25C3×#95C2211yescharacteristic
26C3⋊#95C2212yes—
27C3⋊#95C2212yes—
28C3⋊#95C2212yes—
29C3⋊#95C2212yes—
30C3⋊#95C2212yes—
31C3⋊#95C2211yescharacteristic
32#400—32no—
33#399—31no—
34#399D311yescharacteristic
35#398—32no—
36#389—32no—
37#389—32no—
38#388—32no—
39#385—31no—
40#385—32no—
41C24×C3C2311yescharacteristic
42C22×C3×C4C2312yes—
43C22×C3×C4C2311yescharacteristic
44C22×C3×C4C2312yes—
45C22×C3×C4C2311yescharacteristic
46C22×Dic3—42no—
47C22×Dic3—41no—
48C22×Dic3C2312yes—
49C22×Dic3C2312yes—
50C22×Dic3C2312yes—
51C22×Dic3C2312yes—
52C3×C4⋊C4—41no—
53C3×C4⋊C4—42no—
54C3×C22⋊C4—42no—
55C3×C22⋊C4—41no—
56#217—24no—
57#217Q814yes—
58C4⋊Dic3—41no—
59Dic3⋊C4—24no—
60Dic3⋊C4—24no—
61Dic3⋊C4—42no—
62Dic3⋊C4—42no—
63Dic3⋊C4—42no—
64C4×Dic3—42no—
65C23×C4—31no—
66C2×C4⋊C4—32no—
67C2×C4⋊C4—32no—
68C2×C4⋊C4—32no—
69C2×C4⋊C4—32no—
70C2×C4⋊C4—32no—
71C2×C4⋊C4—31no—
72C2×C4⋊C4C2×D311yescharacteristic
73C2×C4⋊C4C2×D312yes—
74C2×C22⋊C4—31no—
75C2×C22⋊C4—32no—
76C2×C22⋊C4C2×D312yes—
77C2×C22⋊C4C2×D311yescharacteristic
78C2×C42—32no—
79#95—32no—
80#95—32no—
81#95—32no—
82#95—32no—
83#95—32no—
84#95—31no—
85#95C2×D311yescharacteristic
86C23×C3—22no—
87C23×C3—21no—
88C23×C3—22no—
89C23×C3—21no—
90C23×C3C2×Q812yes—
91C23×C3C2411yescharacteristic
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—21no—
99C2×C3×C4—22no—
100C2×C3×C4—21no—
101C2×C3×C4—22no—
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—44no—
113C2×Dic3C2×Q814yes—
114C2×Dic3#4114yes—
115C24C22×D311yescharacteristic
116C22×C4—32no—
117C22×C4—32no—
118C22×C4—32no—
119C22×C4—32no—
120C22×C4—122no—
121C22×C4—121no—
122C22×C4C22×D312yes—
123C22×C4C22×D311yescharacteristic
124C22×C4C22×D312yes—
125C22×C4C22×D311yescharacteristic
126C4⋊C4—41no—
127C4⋊C4—42no—
128C4⋊C4—64no—
129C4⋊C4—64no—
130C4⋊C4—122no—
131C4⋊C4—122no—
132C4⋊C4—122no—
133C4⋊C4—121no—
134C22⋊C4—34no—
135C22⋊C4—42no—
136C22⋊C4—41no—
137C22⋊C4—64no—
138C42—122no—
139C22×C3—22no—
140C22×C3—24no—
141C22×C3—42no—
142C22×C3—41no—
143C22×C3—41no—
144C22×C3C22×Q811yescharacteristic
145C22×C3C2×#4112yes—
146C22×C3H(5,𝔽2)12yes—
147C22×C3H(5,𝔽2)11yescharacteristic
148C22×C32−1+411yescharacteristic
149C3×C4—42no—
150C3×C4—41no—
151C3×C4—42no—
152C3×C4—41no—
153Dic3—24no—
154Dic3—24no—
155Dic3—42no—
156Dic3—42no—
157C23—22no—
158C23—21no—
159C23—22no—
160C23—21no—
161C23D3×Q812yes—
162C23C23×D311yescharacteristic
163C2×C4—22no—
164C2×C4—22no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—34no—
174C2×C4—34no—
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×C4—62no—
185C2×C4—124no—
186C2×C3—42no—
187C2×C3#53912yes—
188C2×C3#54411yescharacteristic
189C2×C3#55412yes—
190C2×C3#55911yescharacteristic
191C2×C3#56011yescharacteristic
192C22—24no—
193C22—22no—
194C22—42no—
195C22—41no—
196C22—41no—
197C22#100412yes—
198C22#100411yescharacteristic
199C22C2×D3×Q811yescharacteristic
200C22D3×#4112yes—
201C22#101011yescharacteristic
202C4—42no—
203C4—41no—
204C4—42no—
205C4—41no—
206C4—64no—
207C4—64no—
208C4—122no—
209C4—122no—
210C3#280711yescharacteristic, complemented, Sylow
211C2—42no—
212C2#562311yescharacteristic
213C2#563612yes—
214C2#566012yes—
215C2#566411yescharacteristic
216C2#572011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#399C2
prime radicalC24×C3C23
hypercenter#399D3
hypocenterC3#2807
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
2. #4982C2semidirect
3. #4976C2semidirect
4. #4976C2semidirect
9. #4658C2semidirect
210. C3#2807semidirect

Maximal subgroups

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

subgroupindex
C3×#3992
#49822
#49762
#49492
#49462
#49442
#46582
#28073