Finite Group Databasebeta

← #74173

Subgroups — #74173

Order 384.

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

normal subgroups only

This group has 372 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×#382C211yesmaximal, characteristic
2#4984C211yesmaximal, characteristic
3Dic3×C4⋊C4C211yesmaximal, characteristic
4#4973C211yesmaximal, characteristic
5#4973C211yesmaximal, characteristic
6C4×Dic3⋊C4C211yesmaximal, characteristic
7#4929C211yesmaximal, characteristic
8C3⋊#401C212yesmaximal
9#4660C212yesmaximal
10#4654C212yesmaximal
11#4651C212yesmaximal
12#2514—31nomaximal, Sylow
13C22×Dic6C2211yescharacteristic
14C2×C3×C4⋊C4C2211yescharacteristic
15C2×C3×C4⋊C4C2211yescharacteristic
16C2×C3×C42C2211yescharacteristic
17C2×C4⋊Dic3C2212yes—
18C2×C4⋊Dic3C2211yescharacteristic
19C2×Dic3⋊C4C2211yescharacteristic
20C2×Dic3⋊C4C2211yescharacteristic
21C2×C4×Dic3C2212yes—
22C2×C4×Dic3C2211yescharacteristic
23C2×C4×Dic3C2211yescharacteristic
24C3×#95C2212yes—
25C3×#95C2212yes—
26C3⋊#95C2211yescharacteristic
27C3⋊#95C2212yes—
28C3⋊#95C2212yes—
29C3⋊#95C2212yes—
30C3⋊#95C2212yes—
31C3⋊#95C2212yes—
32C3⋊#95C2212yes—
33C3⋊#95C2212yes—
34C3⋊#95C2212yes—
35C3⋊#95C2211yescharacteristic
36#401—32no—
37#400—32no—
38#391—31no—
39#391—31no—
40#391—31no—
41#389—31no—
42#387—32no—
43#383—32no—
44#382D311yescharacteristic
45C4×C4⋊C4—31no—
46C4×C4⋊C4—31no—
47C22×C3×C4C2312yes—
48C22×C3×C4C2311yescharacteristic
49C22×C3×C4C2311yescharacteristic
50C22×C3×C4C2312yes—
51C22×C3×C4C2311yescharacteristic
52C22×Dic3C2311yescharacteristic
53C22×Dic3C2312yes—
54C22×Dic3C2311yescharacteristic
55C22×Dic3C2312yes—
56C22×Dic3C2311yescharacteristic
57C22×Dic3C2311yescharacteristic
58C2×Dic6—22no—
59C2×Dic6—22no—
60C2×Dic6—22no—
61C3×C4⋊C4—22no—
62C3×C4⋊C4—22no—
63C3×C42—22no—
64C4⋊Dic3—22no—
65C4⋊Dic3—24no—
66Dic3⋊C4—22no—
67Dic3⋊C4—22no—
68C4×Dic3—22no—
69C4×Dic3—22no—
70C4×Dic3—24no—
71C22×Q8—31no—
72C2×C4⋊C4—31no—
73C2×C4⋊C4—31no—
74C2×C4⋊C4—31no—
75C2×C4⋊C4—32no—
76C2×C4⋊C4C2×D311yescharacteristic
77C2×C4⋊C4C2×D311yescharacteristic
78C2×C42—31no—
79C2×C42—32no—
80C2×C42—31no—
81C2×C42C2×D311yescharacteristic
82#95—31no—
83#95—32no—
84#95—31no—
85#95—32no—
86#95—32no—
87#95—32no—
88#95—32no—
89#95—32no—
90#95—32no—
91#95—32no—
92#95C2×D312yes—
93#95C2×D312yes—
94C23×C3C2411yescharacteristic
95C2×C3×C4—22no—
96C2×C3×C4—21no—
97C2×C3×C4—21no—
98C2×C3×C4—22no—
99C2×C3×C4—21no—
100C2×C3×C4—21no—
101C2×C3×C4—22no—
102C2×C3×C4—22no—
103C2×C3×C4—21no—
104C2×C3×C4—22no—
105C2×C3×C4—22no—
106C2×C3×C4—21no—
107C2×C3×C4#4112yes—
108C2×C3×C4#4112yes—
109C2×C3×C4#4112yes—
110C2×Dic3—21no—
111C2×Dic3—21no—
112C2×Dic3—21no—
113C2×Dic3—21no—
114C2×Dic3—21no—
115C2×Dic3—21no—
116C2×Dic3—22no—
117C2×Dic3—21no—
118C2×Dic3—22no—
119C2×Dic3—22no—
120C2×Dic3—22no—
121C2×Dic3—22no—
122C2×Dic3—22no—
123C2×Dic3—21no—
124C2×Dic3—21no—
125C2×Dic3—21no—
126C2×Dic3#4112yes—
127C2×Dic3#4112yes—
128Dic6—44no—
129C2×Q8—62no—
130C2×Q8—62no—
131C2×Q8—62no—
132C22×C4—31no—
133C22×C4—31no—
134C22×C4—31no—
135C22×C4—32no—
136C22×C4—31no—
137C22×C4—32no—
138C22×C4C22×D312yes—
139C22×C4C22×D311yescharacteristic
140C22×C4C22×D311yescharacteristic
141C22×C4C22×D312yes—
142C22×C4C22×D311yescharacteristic
143C4⋊C4—22no—
144C4⋊C4—22no—
145C4⋊C4—62no—
146C4⋊C4—62no—
147C4⋊C4—64no—
148C4⋊C4—62no—
149C42—22no—
150C42—64no—
151C42—62no—
152C42—62no—
153C22×C3C2×#4111yescharacteristic
154C22×C3C2×#4111yescharacteristic
155C22×C3C2×#4111yescharacteristic
156C22×C3C2×#4111yescharacteristic
157C22×C3C2×#4111yescharacteristic
158C22×C32−1+411yescharacteristic
159C22×C32−1+411yescharacteristic
160C3×C4—41no—
161C3×C4—42no—
162C3×C4—42no—
163C3×C4—41no—
164C3×C4#12614yes—
165Dic3—22no—
166Dic3—22no—
167Dic3—42no—
168Dic3—42no—
169Dic3—41no—
170Dic3—41no—
171C23C23×D311yescharacteristic
172Q8—124no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—32no—
186C2×C4—32no—
187C2×C4—61no—
188C2×C4—61no—
189C2×C4—61no—
190C2×C4—61no—
191C2×C4—61no—
192C2×C4—62no—
193C2×C4—62no—
194C2×C4—61no—
195C2×C4—62no—
196C2×C4—62no—
197C2×C4—62no—
198C2×C4—62no—
199C2×C4—61no—
200C2×C4—61no—
201C2×C4—61no—
202C2×C4—61no—
203C2×C4#23512yes—
204C2×C4#23712yes—
205C2×C4#23912yes—
206C2×C3C2×#12611yescharacteristic
207C2×C3#52911yescharacteristic
208C2×C3#54111yescharacteristic
209C2×C3#55211yescharacteristic
210C2×C3#55611yescharacteristic
211C2×C3#55611yescharacteristic
212C2×C3#55611yescharacteristic
213C22C2×#23511yescharacteristic
214C22C2×#23711yescharacteristic
215C22C2×#23911yescharacteristic
216C22D3×#4111yescharacteristic
217C22D3×#4111yescharacteristic
218C22#101011yescharacteristic
219C22#101011yescharacteristic
220C4—42no—
221C4—42no—
222C4—41no—
223C4—41no—
224C4—62no—
225C4—62no—
226C4—122no—
227C4—121no—
228C4—122no—
229C4—121no—
230C4#89814yes—
231C3#251411yescharacteristic, complemented, Sylow
232C2C2×#89811yescharacteristic
233C2#552911yescharacteristic
234C2#553311yescharacteristic
235C2#567111yescharacteristic
236C2#567111yescharacteristic
237C2#569111yescharacteristic
238C2#569811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, prime radical, cosocleC23×C3C24
FittingC3×#382C2
hypercenter#382D3
hypocenterC3#2514
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #382

Complements

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

NH ≅ G/Ntype
231. C3#2514semidirect

Maximal subgroups

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

subgroupindex
C3×#3822
#49842
Dic3×C4⋊C42
#49732
C4×Dic3⋊C42
#49292
C3⋊#4012
#46602
#46542
#46512
#25143