Finite Group Databasebeta

← #74167

Subgroups — #74167

Order 384.

512 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 149 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×#897C211yesmaximal, characteristic
2C2×D3×C4⋊C4C211yesmaximal, characteristic
3C3×#382C211yesmaximal, characteristic, complemented
4#4977C211yesmaximal, characteristic, complemented
5#4973C211yesmaximal, characteristic, complemented
6C4×(C2×D3)⋊C4C211yesmaximal, characteristic
7#4929C211yesmaximal, characteristic, complemented
8#4664C214yesmaximal
9#4660C214yesmaximal, complemented
10#2503—31nomaximal, Sylow
11C22×C4×D3C2212yes—
12C22×C4×D3C2211yescharacteristic
13C22×Dic6C2211yescharacteristic
14C2×C3×C4⋊C4C2211yescharacteristic
15C2×C3×C4⋊C4C2211yescharacteristic
16C2×C3×C42C2211yescharacteristic
17C2×(C2×D3)⋊C4C2212yes—
18C2×(C2×D3)⋊C4C2212yes—
19C2×C4⋊Dic3C2212yes—
20C2×C4⋊Dic3C2211yescharacteristic
21C2×Dic3⋊C4C2212yes—
22C2×C4×Dic3C2211yescharacteristic
23#897—24no—
24D3×C4⋊C4—24no—
25C3×#95C2214yes—
26C3⋊#95C2212yes—
27C3⋊#95C2214yes—
28C3⋊#95C2214yes—
29C3⋊#95C2214yes—
30C2×C22⋊Q8—31no—
31C22×C4⋊C4—31no—
32#400—34no—
33#391—31no—
34#391—31no—
35#382—31no—
36#382D311yescharacteristic, complemented
37#381—34no—
38C4×C22⋊C4—31no—
39C23×D3C2311yescharacteristic
40C22×C3×C4C2312yes—
41C22×C3×C4C2312yes—
42C22×C3×C4C2312yes—
43C22×C3×C4C2311yescharacteristic
44C22×Dic3C2312yes—
45C22×Dic3C2312yes—
46C22×Dic3C2312yes—
47C22×Dic3C2311yescharacteristic
48C2×C4×D3—22no—
49C2×C4×D3—22no—
50C2×C4×D3—24no—
51C2×C4×D3—28no—
52C2×C4×D3D414yes—
53C2×Dic6—22no—
54C2×Dic6—22no—
55C2×Dic6—22no—
56C3×C4⋊C4—22no—
57C3×C4⋊C4—22no—
58C3×C42—22no—
59(C2×D3)⋊C4—24no—
60(C2×D3)⋊C4—24no—
61C4⋊Dic3—22no—
62C4⋊Dic3—24no—
63Dic3⋊C4—24no—
64C4×Dic3—22no—
65C22×Q8—31no—
66C23×C4—31no—
67C23×C4—32no—
68C22⋊Q8—64no—
69C2×C4⋊C4—31no—
70C2×C4⋊C4—32no—
71C2×C4⋊C4—32no—
72C2×C4⋊C4—64no—
73C2×C4⋊C4C2×D311yescharacteristic
74C2×C4⋊C4C2×D311yescharacteristic
75C2×C22⋊C4—32no—
76C2×C22⋊C4—32no—
77C2×C42—31no—
78C2×C42C2×D311yescharacteristic
79#95—32no—
80#95—34no—
81#95—34no—
82#95—34no—
83#95C2×D314yes—
84C23×C3C2411yescharacteristic
85C22×D3—21no—
86C22×D3—21no—
87C22×D3—22no—
88C22×D3C2×D412yes—
89C22×D3#4114yes—
90C2×C3×C4—22no—
91C2×C3×C4—22no—
92C2×C3×C4—22no—
93C2×C3×C4—22no—
94C2×C3×C4—24no—
95C2×C3×C4—24no—
96C2×C3×C4—22no—
97C2×C3×C4C2×D412yes—
98C2×C3×C4#4112yes—
99C2×C3×C4#4112yes—
100C2×Dic3—21no—
101C2×Dic3—21no—
102C2×Dic3—22no—
103C2×Dic3—22no—
104C2×Dic3—22no—
105C2×Dic3—22no—
106C2×Dic3—22no—
107C2×Dic3—22no—
108C2×Dic3—24no—
109C2×Dic3—22no—
110C2×Dic3C2×D412yes—
111C4×D3—28no—
112C4×D3—48no—
113Dic6—44no—
114C24—31no—
115C2×Q8—62no—
116C2×Q8—62no—
117C2×Q8—62no—
118C22×C4—31no—
119C22×C4—34no—
120C22×C4—32no—
121C22×C4—32no—
122C22×C4—32no—
123C22×C4—64no—
124C22×C4—68no—
125C22×C4—62no—
126C22×C4—62no—
127C22×C4C22×D312yes—
128C22×C4C22×D312yes—
129C22×C4C22×D312yes—
130C22×C4C22×D311yescharacteristic
131C4⋊C4—22no—
132C4⋊C4—22no—
133C4⋊C4—64no—
134C4⋊C4—64no—
135C4⋊C4—62no—
136C22⋊C4—64no—
137C22⋊C4—64no—
138C42—22no—
139C42—62no—
140C22×C3C22×D411yescharacteristic
141C22×C3C2×#4111yescharacteristic
142C22×C3C2×#4111yescharacteristic
143C22×C3C2×#4112yes—
144C22×C32−1+412yes—
145C2×D3—22no—
146C2×D3—22no—
147C2×D3—24no—
148C2×D3—24no—
149C2×D3#12414yes—
150C3×C4—42no—
151C3×C4—42no—
152C3×C4—42no—
153C3×C4#12314yes—
154Dic3—22no—
155Dic3—42no—
156Dic3—42no—
157Dic3—42no—
158C23—32no—
159C23—34no—
160C23—61no—
161C23—61no—
162C23—62no—
163C23C23×D311yescharacteristic
164Q8—124no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—24no—
170C2×C4—24no—
171C2×C4—22no—
172C2×C4—32no—
173C2×C4—61no—
174C2×C4—61no—
175C2×C4—68no—
176C2×C4—62no—
177C2×C4—62no—
178C2×C4—62no—
179C2×C4—62no—
180C2×C4—62no—
181C2×C4—62no—
182C2×C4—64no—
183C2×C4—62no—
184C2×C4—128no—
185C2×C4#23512yes—
186C2×C4D3×D412yes—
187C2×C4#23912yes—
188C2×C3C2×#12311yescharacteristic
189C2×C3C2×#12411yescharacteristic
190C2×C3#53611yescharacteristic
191C2×C3#55212yes—
192C2×C3#55212yes—
193D3—24no—
194C22—34no—
195C22—62no—
196C22—64no—
197C22—62no—
198C22—64no—
199C22C2×#23511yescharacteristic
200C22C2×D3×D411yescharacteristic
201C22C2×#23911yescharacteristic
202C22D3×#4112yes—
203C22#101012yes—
204C4—42no—
205C4—42no—
206C4—42no—
207C4—62no—
208C4—122no—
209C4—122no—
210C4—122no—
211C4#89414yes—
212C3#250311yescharacteristic, complemented, Sylow
213C2—64no—
214C2C2×#89411yescharacteristic
215C2#552912yes—
216C2D3×#12411yescharacteristic
217C2#567511yescharacteristic
218C2#569412yes—
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#382C2
prime radicalC23×D3C23
hypercenter#382D3
hypocenterC3#2503
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #382

Complements

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

NH ≅ G/Ntype
3. C3×#382C2semidirect
4. #4977C2semidirect
5. #4973C2semidirect
7. #4929C2semidirect
9. #4660C2semidirect
36. #382D3semidirect
212. C3#2503semidirect

Maximal subgroups

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

subgroupindex
C2×#8972
C2×D3×C4⋊C42
C3×#3822
#49772
#49732
C4×(C2×D3)⋊C42
#49292
#46642
#46602
#25033