Finite Group Databasebeta

← #74583

Subgroups — #74583

Order 384.

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

normal subgroups only

This group has 408 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×#385C211yesmaximal, characteristic
2#5210C212yesmaximal
3#4982C212yesmaximal, complemented
4#4972C212yesmaximal, complemented
5#4951C212yesmaximal
6#4949C211yesmaximal, characteristic
7#4949C211yesmaximal, characteristic
8#4660C212yesmaximal, complemented
9#4647C212yesmaximal, complemented
10#2807—31nomaximal, Sylow
11C23×C3×C4C2211yescharacteristic
12C2×C3×C4⋊C4C2212yes—
13C2×C3×C22⋊C4C2211yescharacteristic
14C2×C3×C22⋊C4C2211yescharacteristic
15C2×#217C2212yes—
16C2×#217C2212yes—
17C2×C4⋊Dic3C2212yes—
18C2×C4⋊Dic3C2211yescharacteristic
19C2×C4⋊Dic3C2211yescharacteristic
20C2×Dic3⋊C4C2212yes—
21C2×Dic3⋊C4C2212yes—
22C2×Dic3⋊C4C2212yes—
23C2×Dic3⋊C4C2212yes—
24C2×C4×Dic3C2212yes—
25C3×#95C2212yes—
26C3⋊#95C2212yes—
27C3⋊#95C2211yescharacteristic
28C3⋊#95C2212yes—
29C3⋊#95C2212yes—
30C3⋊#95C2212yes—
31C3⋊#95C2211yescharacteristic
32#400—32no—
33#399—31no—
34#399—31no—
35#398—32no—
36#389—32no—
37#389—32no—
38#388—32no—
39#385—32no—
40#385D311yescharacteristic
41C24×C3C2311yescharacteristic
42C22×C3×C4—41no—
43C22×C3×C4—42no—
44C22×C3×C4C2312yes—
45C22×C3×C4C2311yescharacteristic
46C22×C3×C4C2311yescharacteristic
47C22×C3×C4C2312yes—
48C22×Dic3C2312yes—
49C22×Dic3C2312yes—
50C22×Dic3C2312yes—
51C22×Dic3C2312yes—
52C3×C4⋊C4—42no—
53C3×C22⋊C4—41no—
54C3×C22⋊C4Q814yes—
55#217—24no—
56#217—42no—
57C4⋊Dic3—24no—
58C4⋊Dic3—41no—
59C4⋊Dic3—41no—
60Dic3⋊C4—24no—
61Dic3⋊C4—42no—
62Dic3⋊C4—42no—
63Dic3⋊C4—42no—
64C4×Dic3—42no—
65C23×C4C2×D311yescharacteristic
66C2×C4⋊C4—32no—
67C2×C4⋊C4—32no—
68C2×C4⋊C4—32no—
69C2×C4⋊C4—32no—
70C2×C4⋊C4—31no—
71C2×C4⋊C4—32no—
72C2×C4⋊C4—31no—
73C2×C4⋊C4C2×D312yes—
74C2×C22⋊C4—32no—
75C2×C22⋊C4—32no—
76C2×C22⋊C4C2×D311yescharacteristic
77C2×C22⋊C4C2×D311yescharacteristic
78C2×C42—32no—
79#95—32no—
80#95—32no—
81#95—32no—
82#95—32no—
83#95—31no—
84#95—31no—
85#95C2×D312yes—
86C23×C3—22no—
87C23×C3—22no—
88C23×C3—21no—
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—21no—
100C2×C3×C4—22no—
101C2×C3×C4—44no—
102C2×C3×C4C2×Q814yes—
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—22no—
114C2×Dic3#4114yes—
115C24C22×D311yescharacteristic
116C22×C4—32no—
117C22×C4—32no—
118C22×C4—32no—
119C22×C4—32no—
120C22×C4—42no—
121C22×C4—41no—
122C22×C4C22×D312yes—
123C22×C4C22×D311yescharacteristic
124C22×C4C22×D311yescharacteristic
125C22×C4C22×D312yes—
126C4⋊C4—42no—
127C4⋊C4—64no—
128C4⋊C4—64no—
129C4⋊C4—122no—
130C4⋊C4—122no—
131C4⋊C4—122no—
132C4⋊C4—121no—
133C4⋊C4—121no—
134C22⋊C4—41no—
135C22⋊C4—64no—
136C22⋊C4—122no—
137C22⋊C4Dic614yes—
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—24no—
150C3×C4—41no—
151C3×C4—41no—
152C3×C4—42no—
153Dic3—24no—
154Dic3—42no—
155Dic3—42no—
156Dic3—42no—
157C23—21no—
158C23—22no—
159C23—21no—
160C23—22no—
161C23C2×Dic612yes—
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—34no—
173C2×C4—44no—
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×C4—62no—
185C2×C4C2×Dic614yes—
186C2×C3—42no—
187C2×C3#53912yes—
188C2×C3#54411yescharacteristic
189C2×C3#55412yes—
190C2×C3#55911yescharacteristic
191C2×C3#56011yescharacteristic
192C22—22no—
193C22—24no—
194C22—42no—
195C22—41no—
196C22—41no—
197C22C22×Dic611yescharacteristic
198C22#100412yes—
199C22#100411yescharacteristic
200C22#100711yescharacteristic
201C22D3×#4112yes—
202C4—24no—
203C4—41no—
204C4—42no—
205C4—41no—
206C4—64no—
207C4—122no—
208C4—122no—
209C4—122no—
210C3#280711yescharacteristic, complemented, Sylow
211C2—42no—
212C2#550011yescharacteristic
213C2#553912yes—
214C2#559511yescharacteristic
215C2#563812yes—
216C2#566411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#385C2
prime radicalC24×C3C23
hypercenter#385D3
hypocenterC3#2807
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
3. #4982C2semidirect
4. #4972C2semidirect
8. #4660C2semidirect
9. #4647C2semidirect
210. C3#2807semidirect

Maximal subgroups

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

subgroupindex
C3×#3852
#52102
#49822
#49722
#49512
#49492
#46602
#46472
#28073