Finite Group Databasebeta

← #74117

Subgroups — #74117

Order 384.

564 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 169 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
1C22×C4×Dic3C211yesmaximal, characteristic
2C2×#878C211yesmaximal, characteristic
3C3×#381C211yesmaximal, characteristic
4#5211C211yesmaximal, characteristic
5#4950C211yesmaximal, characteristic
6#4946C211yesmaximal, characteristic
7#4942C211yesmaximal, characteristic
8#4655C212yesmaximal, complemented
9#4654C212yesmaximal, complemented
10#4647C212yesmaximal, complemented
11#4646C212yesmaximal, complemented
12#2855—31nomaximal, Sylow
13C23×C3×C4C2211yescharacteristic
14C23×Dic3C2211yescharacteristic
15C23×Dic3C2211yescharacteristic
16C22×Dic6C2211yescharacteristic
17C2×C3×C22⋊C4C2211yescharacteristic
18C2×C3×C22⋊C4C2211yescharacteristic
19C2×#217C2211yescharacteristic
20C2×#217C2211yescharacteristic
21C2×C4⋊Dic3C2212yes—
22C2×C4⋊Dic3C2211yescharacteristic
23C2×Dic3⋊C4C2212yes—
24C2×Dic3⋊C4C2211yescharacteristic
25C2×Dic3⋊C4C2212yes—
26C2×Dic3⋊C4C2211yescharacteristic
27C2×C4×Dic3—24no—
28C2×C4×Dic3C2212yes—
29C2×C4×Dic3C2212yes—
30#878—24no—
31C3×#95C2212yes—
32C3×#95C2212yes—
33C3⋊#95C2212yes—
34C3⋊#95C2212yes—
35C3⋊#95C2212yes—
36C3⋊#95C2212yes—
37C3⋊#95C2212yes—
38C2×C22⋊Q8—31no—
39C22×C42—31no—
40#391—32no—
41#389—32no—
42#387—32no—
43#387—32no—
44#385—31no—
45#385—31no—
46#381—31no—
47#381D311yescharacteristic
48#380—31no—
49C24×C3C2311yescharacteristic
50C22×C3×C4—22no—
51C22×C3×C4—22no—
52C22×C3×C4Q814yescomplemented
53C22×C3×C4C2312yes—
54C22×C3×C4C2311yescharacteristic
55C22×C3×C4C2311yescharacteristic
56C22×C3×C4C2312yes—
57C22×Dic3—22no—
58C22×Dic3—24no—
59C22×Dic3—24no—
60C22×Dic3D414yes—
61C22×Dic3C2312yes—
62C22×Dic3C2311yescharacteristic
63C22×Dic3C2311yescharacteristic
64C22×Dic3C2311yescharacteristic
65C22×Dic3C2312yes—
66C22×Dic3C2311yescharacteristic
67C2×Dic6—22no—
68C2×Dic6—24no—
69C3×C22⋊C4—22no—
70C3×C22⋊C4—22no—
71#217—22no—
72#217—22no—
73C4⋊Dic3—22no—
74C4⋊Dic3—24no—
75Dic3⋊C4—22no—
76Dic3⋊C4—22no—
77Dic3⋊C4—24no—
78Dic3⋊C4—24no—
79C4×Dic3—24no—
80C4×Dic3—24no—
81C22×Q8—31no—
82C23×C4—31no—
83C23×C4—31no—
84C23×C4C2×D311yescharacteristic
85C22⋊Q8—64no—
86C2×C4⋊C4—31no—
87C2×C4⋊C4—31no—
88C2×C4⋊C4—31no—
89C2×C4⋊C4—32no—
90C2×C4⋊C4—32no—
91C2×C4⋊C4—32no—
92C2×C22⋊C4—31no—
93C2×C22⋊C4—31no—
94C2×C22⋊C4C2×D311yescharacteristic
95C2×C22⋊C4C2×D311yescharacteristic
96C2×C42—32no—
97C2×C42—32no—
98C2×C42—64no—
99#95—32no—
100#95—32no—
101#95—32no—
102#95—32no—
103#95—32no—
104#95C2×D312yes—
105#95C2×D312yes—
106C23×C3—22no—
107C23×C3—22no—
108C23×C3C2×D412yes—
109C23×C3C2×Q812yes—
110C23×C3#4112yes—
111C23×C3C2411yescharacteristic
112C2×C3×C4—28no—
113C2×C3×C4—22no—
114C2×C3×C4—22no—
115C2×C3×C4—21no—
116C2×C3×C4—21no—
117C2×C3×C4—22no—
118C2×C3×C4—22no—
119C2×C3×C4—22no—
120C2×C3×C4—22no—
121C2×C3×C4—22no—
122C2×C3×C4C2×Q814yes—
123C2×Dic3—28no—
124C2×Dic3—22no—
125C2×Dic3—22no—
126C2×Dic3—22no—
127C2×Dic3—28no—
128C2×Dic3—21no—
129C2×Dic3—21no—
130C2×Dic3—22no—
131C2×Dic3—22no—
132C2×Dic3—22no—
133C2×Dic3—22no—
134C2×Dic3—22no—
135C2×Dic3C2×D412yes—
136C2×Dic3C2×D412yes—
137C2×Dic3#4114yes—
138C2×Dic3#4114yes—
139Dic6—44no—
140C24C22×D311yescharacteristic
141C2×Q8—62no—
142C2×Q8—64no—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4—31no—
146C22×C4—34no—
147C22×C4—31no—
148C22×C4—32no—
149C22×C4—32no—
150C22×C4—31no—
151C22×C4—31no—
152C22×C4—64no—
153C22×C4—64no—
154C22×C4—62no—
155C22×C4Dic614yescomplemented
156C22×C4C22×D312yes—
157C22×C4C22×D311yescharacteristic
158C22×C4C22×D311yescharacteristic
159C22×C4C22×D312yes—
160C4⋊C4—62no—
161C4⋊C4—62no—
162C4⋊C4—62no—
163C4⋊C4—64no—
164C4⋊C4—64no—
165C4⋊C4—64no—
166C22⋊C4—22no—
167C22⋊C4—22no—
168C22⋊C4—62no—
169C22⋊C4—62no—
170C42—64no—
171C42—64no—
172C22×C3—22no—
173C22×C3—22no—
174C22×C3—24no—
175C22×C3—24no—
176C22×C3C22⋊Q814yes—
177C22×C3C22×D411yescharacteristic
178C22×C3C22×Q811yescharacteristic
179C22×C3C2×#4111yescharacteristic
180C22×C3C2×#4112yes—
181C22×C3C2×#4112yes—
182C3×C4—24no—
183C3×C4—41no—
184C3×C4—41no—
185C3×C4—42no—
186Dic3—22no—
187Dic3—24no—
188Dic3—41no—
189Dic3—41no—
190Dic3—42no—
191Dic3#12314yes—
192C23—22no—
193C23—22no—
194C23C2×Dic612yes—
195C23D3×D412yes—
196C23#23712yes—
197C23C23×D311yescharacteristic
198Q8—124no—
199C2×C4—28no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—32no—
210C2×C4—34no—
211C2×C4—32no—
212C2×C4—34no—
213C2×C4—68no—
214C2×C4—68no—
215C2×C4—62no—
216C2×C4—62no—
217C2×C4—62no—
218C2×C4—61no—
219C2×C4—61no—
220C2×C4—62no—
221C2×C4—62no—
222C2×C4—62no—
223C2×C4—62no—
224C2×C4—62no—
225C2×C4C2×Dic614yes—
226C2×C3—24no—
227C2×C3C2×C22⋊Q811yescharacteristic
228C2×C3C2×#12311yescharacteristic
229C2×C3#52511yescharacteristic
230C2×C3#52912yes—
231C2×C3#53312yes—
232C22—24no—
233C22—22no—
234C22—22no—
235C22—24no—
236C22#87814yes—
237C22C22×Dic611yescharacteristic
238C22C2×D3×D411yescharacteristic
239C22C2×#23711yescharacteristic
240C22D3×#4112yes—
241C22D3×#4112yes—
242C4—24no—
243C4—34no—
244C4—42no—
245C4—41no—
246C4—41no—
247C4—62no—
248C4—64no—
249C4—122no—
250C4—121no—
251C4—121no—
252C3#285511yescharacteristic, complemented, Sylow
253C2—24no—
254C2C2×#87811yescharacteristic
255C2#551712yes—
256C2#564912yes—
257C2D3×#12311yescharacteristic
258C2#565611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#381C2
prime radicalC24×C3C23
hypercenter#381D3
hypocenterC3#2855
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #381

Complements

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

NH ≅ G/Ntype
8. #4655C2semidirect
9. #4654C2semidirect
10. #4647C2semidirect
11. #4646C2semidirect
52. C22×C3×C4Q8semidirect
155. C22×C4Dic6semidirect
252. C3#2855semidirect

Maximal subgroups

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

subgroupindex
C22×C4×Dic32
C2×#8782
C3×#3812
#52112
#49502
#49462
#49422
#46552
#46542
#46472
#46462
#28553