Finite Group Databasebeta

← #78181

Subgroups — #78181

Order 384.

468 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 151 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×#888C211yesmaximal, characteristic
2C2×#887C211yesmaximal, characteristic
3C3×#400C211yesmaximal, characteristic
4#4986C211yesmaximal, characteristic
5#4985C211yesmaximal, characteristic
6#4659C211yesmaximal, characteristic
7#4650C212yesmaximal
8#4649C211yesmaximal, characteristic
9#4646C212yesmaximal
10#4646C212yesmaximal
11#4646C212yesmaximal
12#2568—31nomaximal, Sylow
13C22×Dic6C2212yes—
14C22×Dic6C2211yescharacteristic
15C2×C3×C4⋊C4C2211yescharacteristic
16C2×C3×C4⋊C4C2211yescharacteristic
17C2×C4⋊Dic3C2211yescharacteristic
18C2×Dic3⋊C4C2212yes—
19C2×Dic3⋊C4C2212yes—
20C2×Dic3⋊C4C2211yescharacteristic
21C2×Dic3⋊C4C2211yescharacteristic
22C2×C4×Dic3C2211yescharacteristic
23C2×C4×Dic3C2211yescharacteristic
24C2×C4×Dic3C2212yes—
25C2×C4×Dic3C2211yescharacteristic
26#888—24no—
27#887—24no—
28C3×#95C2212yes—
29C3×#95C2212yes—
30C3×#95C2211yescharacteristic
31C3⋊#95C2212yes—
32C3⋊#95C2212yes—
33C3⋊#95C2212yes—
34C3⋊#95C2212yes—
35C3⋊#95C2212yes—
36C3⋊#95C2212yes—
37C3⋊#95C2211yescharacteristic
38C2×C4⋊Q8—31no—
39C2×C4×Q8—31no—
40#400—31no—
41#400—31no—
42#400—31no—
43#400D311yescharacteristic
44#391—32no—
45#391—32no—
46#391—32no—
47#382—32no—
48C4×C4⋊C4—31no—
49C22×C3×C4C2312yes—
50C22×C3×C4C2311yescharacteristic
51C22×C3×C4C2312yes—
52C22×C3×C4C2311yescharacteristic
53C22×C3×C4C2311yescharacteristic
54C22×Dic3C2312yes—
55C22×Dic3C2311yescharacteristic
56C22×Dic3C2312yes—
57C22×Dic3C2311yescharacteristic
58C22×Dic3C2311yescharacteristic
59C22×Dic3C2311yescharacteristic
60C2×Dic6—22no—
61C2×Dic6—22no—
62C2×Dic6—24no—
63C2×Dic6—28no—
64C2×Dic6Q814yescomplemented
65C3×C4⋊C4—22no—
66C3×C4⋊C4—22no—
67C4⋊Dic3—22no—
68Dic3⋊C4—22no—
69Dic3⋊C4—22no—
70Dic3⋊C4—24no—
71Dic3⋊C4—24no—
72C4×Dic3—22no—
73C4×Dic3—22no—
74C4×Dic3—24no—
75C4×Dic3D414yes—
76C22×Q8—31no—
77C22×Q8—32no—
78C4⋊Q8—64no—
79C4×Q8—64no—
80C2×C4⋊C4—31no—
81C2×C4⋊C4—31no—
82C2×C4⋊C4—32no—
83C2×C4⋊C4—31no—
84C2×C4⋊C4—32no—
85C2×C4⋊C4C2×D311yescharacteristic
86C2×C4⋊C4C2×D311yescharacteristic
87C2×C42—31no—
88C2×C42—31no—
89C2×C42—31no—
90C2×C42—32no—
91#95—31no—
92#95—32no—
93#95—32no—
94#95—32no—
95#95—32no—
96#95—32no—
97#95—32no—
98#95C2×D311yescharacteristic
99#95C2×D312yes—
100#95C2×D312yes—
101C23×C3C2411yescharacteristic
102C2×C3×C4—21no—
103C2×C3×C4—21no—
104C2×C3×C4—22no—
105C2×C3×C4—21no—
106C2×C3×C4—22no—
107C2×C3×C4—22no—
108C2×C3×C4—22no—
109C2×C3×C4—24no—
110C2×C3×C4—22no—
111C2×C3×C4—22no—
112C2×C3×C4C2×D412yes—
113C2×C3×C4C2×Q812yes—
114C2×Dic3—21no—
115C2×Dic3—21no—
116C2×Dic3—21no—
117C2×Dic3—22no—
118C2×Dic3—22no—
119C2×Dic3—22no—
120C2×Dic3—24no—
121C2×Dic3—22no—
122C2×Dic3—22no—
123C2×Dic3C2×D412yes—
124C2×Dic3C2×D412yes—
125C2×Dic3C2×Q812yes—
126C2×Dic3C2×Q812yes—
127C2×Dic3#4112yes—
128C2×Dic3#4114yes—
129Dic6—28no—
130Dic6—48no—
131C2×Q8—34no—
132C2×Q8—62no—
133C2×Q8—62no—
134C2×Q8—64no—
135C2×Q8—68no—
136C22×C4—31no—
137C22×C4—31no—
138C22×C4—31no—
139C22×C4—32no—
140C22×C4—31no—
141C22×C4—32no—
142C22×C4C22×D312yes—
143C22×C4C22×D311yescharacteristic
144C22×C4C22×D312yes—
145C22×C4C22×D311yescharacteristic
146C22×C4C22×D311yescharacteristic
147C4⋊C4—22no—
148C4⋊C4—22no—
149C4⋊C4—62no—
150C4⋊C4—64no—
151C4⋊C4—64no—
152C4⋊C4—62no—
153C4⋊C4—62no—
154C42—34no—
155C42—62no—
156C42—62no—
157C42—64no—
158C22×C3C22×D411yescharacteristic
159C22×C3C22×Q811yescharacteristic
160C22×C3C2×#4111yescharacteristic
161C22×C3C2×#4112yes—
162C22×C32−1+412yes—
163C3×C4—22no—
164C3×C4—22no—
165C3×C4—42no—
166C3×C4—41no—
167C3×C4—42no—
168Dic3—22no—
169Dic3—42no—
170Dic3—41no—
171Dic3—42no—
172Dic3C22⋊Q814yes—
173Dic3#12414yes—
174C23C23×D311yescharacteristic
175Q8—68no—
176Q8—128no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—22no—
180C2×C4—21no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—24no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—32no—
188C2×C4—32no—
189C2×C4—32no—
190C2×C4—32no—
191C2×C4—34no—
192C2×C4—32no—
193C2×C4—61no—
194C2×C4—61no—
195C2×C4—61no—
196C2×C4—62no—
197C2×C4—62no—
198C2×C4—62no—
199C2×C4—64no—
200C2×C4—62no—
201C2×C4—62no—
202C2×C4D3×D412yes—
203C2×C4D3×Q812yes—
204C2×C3C2×C22⋊Q811yescharacteristic
205C2×C3C2×#12411yescharacteristic
206C2×C3#53611yescharacteristic
207C2×C3#55612yes—
208C2×C3#55712yes—
209C22C2×D3×D411yescharacteristic
210C22C2×D3×Q811yescharacteristic
211C22D3×#4112yes—
212C22D3×#4111yescharacteristic
213C22#101012yes—
214C4—22no—
215C4—22no—
216C4—34no—
217C4—34no—
218C4—41no—
219C4—42no—
220C4—42no—
221C4—62no—
222C4—121no—
223C4—122no—
224C4—122no—
225C3#256811yescharacteristic, complemented, Sylow
226C2D3×C22⋊Q811yescharacteristic
227C2#564811yescharacteristic
228C2D3×#12411yescharacteristic
229C2#568512yes—
230C2#570212yes—
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, prime radical, cosocleC23×C3C24
FittingC3×#400C2
hypercenter#400D3
hypocenterC3#2568
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
64. C2×Dic6Q8semidirect
225. C3#2568semidirect

Maximal subgroups

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

subgroupindex
C2×#8882
C2×#8872
C3×#4002
#49862
#49852
#46592
#46502
#46492
#46462
#25683