Finite Group Databasebeta

← #77646

Subgroups — #77646

Order 384.

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

normal subgroups only

This group has 444 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
1C2×#878C212yesmaximal
2C3×#397C211yesmaximal, characteristic
3#4979C212yesmaximal, complemented
4#4953C211yesmaximal, characteristic
5#4952C212yesmaximal
6#4943C211yesmaximal, characteristic
7#4660C212yesmaximal, complemented
8#4658C212yesmaximal, complemented
9#4656C212yesmaximal, complemented
10#2817—31nomaximal, Sylow
11C23×Dic3C2211yescharacteristic
12C22×Dic6C2212yes—
13C2×C3×C4⋊C4C2211yescharacteristic
14C2×C3×C22⋊C4C2212yes—
15C2×C3×C22⋊C4C2211yescharacteristic
16C2×#217C2212yes—
17C2×#217C2211yescharacteristic
18C2×C4⋊Dic3C2212yes—
19C2×C4⋊Dic3C2211yescharacteristic
20C2×Dic3⋊C4C2212yes—
21C2×Dic3⋊C4C2212yes—
22C2×Dic3⋊C4C2212yes—
23#878—28no—
24C3×#95C2212yes—
25C3×#95C2211yescharacteristic
26C3⋊#95C2212yes—
27C3⋊#95C2212yes—
28C3⋊#95C2211yescharacteristic
29C3⋊#95C2212yes—
30C3⋊#95C2212yes—
31C3⋊#95C2212yes—
32C3⋊#95C2212yes—
33C2×C22⋊Q8—32no—
34#400—32no—
35#400—32no—
36#397—32no—
37#397—31no—
38#397D311yescharacteristic
39#395—32no—
40#395—32no—
41#381—31no—
42C24×C3C2311yescharacteristic
43C22×C3×C4C2311yescharacteristic
44C22×C3×C4C2311yescharacteristic
45C22×C3×C4C2312yes—
46C22×C3×C4C2312yes—
47C22×Dic3—41no—
48C22×Dic3—41no—
49C22×Dic3D414yes—
50C22×Dic3C2312yes—
51C22×Dic3C2312yes—
52C22×Dic3C2312yes—
53C22×Dic3C2312yes—
54C2×Dic6—24no—
55C2×Dic6—42no—
56C2×Dic6—42no—
57C3×C4⋊C4—41no—
58C3×C22⋊C4—24no—
59C3×C22⋊C4—41no—
60#217—24no—
61#217—41no—
62C4⋊Dic3—24no—
63C4⋊Dic3—41no—
64Dic3⋊C4—24no—
65Dic3⋊C4—24no—
66Dic3⋊C4—42no—
67C22×Q8—32no—
68C23×C4—31no—
69C22⋊Q8—68no—
70C2×C4⋊C4—32no—
71C2×C4⋊C4—32no—
72C2×C4⋊C4—32no—
73C2×C4⋊C4—32no—
74C2×C4⋊C4—31no—
75C2×C4⋊C4C2×D311yescharacteristic
76C2×C22⋊C4—32no—
77C2×C22⋊C4—31no—
78C2×C22⋊C4C2×D312yes—
79C2×C22⋊C4C2×D311yescharacteristic
80#95—32no—
81#95—32no—
82#95—32no—
83#95—32no—
84#95—32no—
85#95—32no—
86#95—31no—
87#95C2×D312yes—
88#95C2×D311yescharacteristic
89C23×C3—21no—
90C23×C3—21no—
91C23×C3—22no—
92C23×C3—22no—
93C23×C3C2×D412yes—
94C23×C3C2411yescharacteristic
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—22no—
102C2×C3×C4—22no—
103C2×C3×C4—22no—
104C2×C3×C4—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—22no—
116C2×Dic3—44no—
117C2×Dic3C2×D414yes—
118Dic6—48no—
119C24C22×D311yescharacteristic
120C2×Q8—64no—
121C2×Q8—122no—
122C2×Q8—122no—
123C22×C4—32no—
124C22×C4—34no—
125C22×C4—32no—
126C22×C4—32no—
127C22×C4—32no—
128C22×C4—121no—
129C22×C4—121no—
130C22×C4C22×D311yescharacteristic
131C22×C4C22×D311yescharacteristic
132C22×C4C22×D312yes—
133C22×C4C22×D312yes—
134C4⋊C4—41no—
135C4⋊C4—64no—
136C4⋊C4—64no—
137C4⋊C4—64no—
138C4⋊C4—122no—
139C4⋊C4—121no—
140C22⋊C4—24no—
141C22⋊C4—41no—
142C22⋊C4—64no—
143C22⋊C4—121no—
144C22×C3—22no—
145C22×C3—24no—
146C22×C3—41no—
147C22×C3—41no—
148C22×C3—42no—
149C22×C3C22×D411yescharacteristic
150C22×C3H(5,𝔽2)11yescharacteristic
151C22×C3H(5,𝔽2)11yescharacteristic
152C22×C32−1+412yes—
153C22×C32−1+412yes—
154C3×C4—42no—
155C3×C4—42no—
156C3×C4—41no—
157C3×C4—41no—
158Dic3—24no—
159Dic3—42no—
160Dic3—42no—
161Dic3—42no—
162C23—21no—
163C23—21no—
164C23—22no—
165C23—22no—
166C23D3×D412yes—
167C23C23×D311yescharacteristic
168Q8—128no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—34no—
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—62no—
190C2×C4—62no—
191C2×C4—124no—
192C2×C3—42no—
193C2×C3#53411yescharacteristic
194C2×C3#53612yes—
195C2×C3#56311yescharacteristic
196C2×C3#56311yescharacteristic
197C2×C3#56312yes—
198C22—24no—
199C22—22no—
200C22—41no—
201C22—41no—
202C22—42no—
203C22C2×D3×D411yescharacteristic
204C22#100411yescharacteristic
205C22#100911yescharacteristic
206C22#101012yes—
207C22#101012yes—
208C4—42no—
209C4—41no—
210C4—41no—
211C4—42no—
212C4—64no—
213C4—122no—
214C4—122no—
215C4—122no—
216C3#281711yescharacteristic, complemented, Sylow
217C2—42no—
218C2#560511yescharacteristic
219C2#564812yes—
220C2#565111yescharacteristic
221C2#567211yescharacteristic
222C2#571212yes—
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#397C2
prime radicalC24×C3C23
hypercenter#397D3
hypocenterC3#2817
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
3. #4979C2semidirect
7. #4660C2semidirect
8. #4658C2semidirect
9. #4656C2semidirect
216. C3#2817semidirect

Maximal subgroups

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

subgroupindex
C2×#8782
C3×#3972
#49792
#49532
#49522
#49432
#46602
#46582
#46562
#28173