Finite Group Databasebeta

← #77043

Subgroups — #77043

Order 384.

512 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 133 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, complemented
2C3×#394C211yesmaximal, characteristic
3#5220C212yesmaximal
4#4981C211yesmaximal, characteristic, complemented
5#4973C211yesmaximal, characteristic, complemented
6#4952C212yesmaximal, complemented
7#4950C212yesmaximal, complemented
8#4945C212yesmaximal, complemented
9#4660C212yesmaximal, complemented
10#2728—31nomaximal, Sylow
11C22×C3×D4C2211yescharacteristic
12C23×Dic3C2212yes—
13C22×Dic6C2211yescharacteristic
14C2×C3×C4⋊C4C2211yescharacteristic
15C2×C3×C22⋊C4C2212yes—
16C2×C3×C22⋊C4C2212yes—
17C2×#217C2212yes—
18C2×#217C2212yes—
19C2×#217C2212yes—
20C2×C4⋊Dic3C2212yes—
21C2×C4⋊Dic3C2211yescharacteristic
22C2×Dic3⋊C4C2212yes—
23C2×Dic3⋊C4C2212yes—
24C2×C4×Dic3C2211yescharacteristic
25#878—28no—
26C3×#95C2211yescharacteristic
27C3⋊#95C2212yes—
28C3⋊#95C2212yes—
29C3⋊#95C2212yes—
30C3⋊#95C2212yes—
31C3⋊#95C2212yes—
32C3⋊#95C2211yescharacteristic
33C2×C22⋊Q8—32no—
34#400—32no—
35#398—31no—
36#397—32no—
37#394D311yescharacteristic
38#391—31no—
39#388—32no—
40#386—32no—
41#385—32no—
42C24×C3C2312yes—
43C2×C3×D4—24no—
44C2×C3×D4—41no—
45C22×C3×C4C2311yescharacteristic
46C22×C3×C4C2311yescharacteristic
47C22×C3×C4C2312yes—
48C22×C3×C4C2311yescharacteristic
49C22×Dic3—24no—
50C22×Dic3—42no—
51C22×Dic3—42no—
52C22×Dic3C2312yes—
53C22×Dic3C2312yes—
54C22×Dic3C2312yes—
55C22×Dic3C2312yes—
56C2×Dic6—41no—
57C2×Dic6—41no—
58C2×Dic6D414yescomplemented
59C3×C4⋊C4—41no—
60C3×C22⋊C4—24no—
61C3×C22⋊C4—24no—
62#217—24no—
63#217—24no—
64#217—42no—
65C4⋊Dic3—24no—
66C4⋊Dic3—41no—
67Dic3⋊C4—24no—
68Dic3⋊C4—24no—
69C4×Dic3—41no—
70C22×Q8—31no—
71C22×D4C2×D311yescharacteristic
72C23×C4—32no—
73C22⋊Q8—68no—
74C2×C4⋊C4—32no—
75C2×C4⋊C4—32no—
76C2×C4⋊C4—32no—
77C2×C4⋊C4—31no—
78C2×C4⋊C4C2×D311yescharacteristic
79C2×C22⋊C4—32no—
80C2×C22⋊C4—32no—
81C2×C22⋊C4—32no—
82C2×C22⋊C4C2×D312yes—
83C2×C22⋊C4C2×D312yes—
84C2×C42—31no—
85#95—32no—
86#95—32no—
87#95—32no—
88#95—32no—
89#95—32no—
90#95—31no—
91#95C2×D311yescharacteristic
92C23×C3—22no—
93C23×C3—22no—
94C23×C3—22no—
95C23×C3—22no—
96C23×C3—22no—
97C23×C3—22no—
98C23×C3#4114yes—
99C23×C3C2411yescharacteristic
100C3×D4—44no—
101C2×C3×C4—21no—
102C2×C3×C4—21no—
103C2×C3×C4—22no—
104C2×C3×C4—22no—
105C2×C3×C4—22no—
106C2×C3×C4—22no—
107C2×C3×C4—21no—
108C2×C3×C4—21no—
109C2×C3×C4—22no—
110C2×C3×C4C2×D412yes—
111C2×Dic3—22no—
112C2×Dic3—22no—
113C2×Dic3—22no—
114C2×Dic3—22no—
115C2×Dic3—22no—
116C2×Dic3—22no—
117C2×Dic3—22no—
118C2×Dic3—22no—
119C2×Dic3—22no—
120C2×Dic3—22no—
121C2×Dic3—22no—
122C2×Dic3—48no—
123C2×Dic3C2×D414yes—
124Dic6—44no—
125C24C22×D312yes—
126C2×Q8—34no—
127C2×Q8—121no—
128C2×Q8—121no—
129C2×D4—24no—
130C2×D4—41no—
131C22×C4—32no—
132C22×C4—32no—
133C22×C4—32no—
134C22×C4—32no—
135C22×C4—64no—
136C22×C4—122no—
137C22×C4—122no—
138C22×C4C22×D311yescharacteristic
139C22×C4C22×D311yescharacteristic
140C22×C4C22×D312yes—
141C22×C4C22×D311yescharacteristic
142C4⋊C4—41no—
143C4⋊C4—64no—
144C4⋊C4—64no—
145C4⋊C4—64no—
146C4⋊C4—121no—
147C22⋊C4—24no—
148C22⋊C4—24no—
149C22⋊C4—64no—
150C22⋊C4—64no—
151C22⋊C4—122no—
152C42—121no—
153C22×C3—24no—
154C22×C3—24no—
155C22×C3—24no—
156C22×C3—42no—
157C22×C3—42no—
158C22×C3—42no—
159C22×C3—42no—
160C22×C3C22×D411yescharacteristic
161C22×C3C2×#4112yes—
162C22×C3H(5,𝔽2)11yescharacteristic
163C22×C3H(5,𝔽2)11yescharacteristic
164C22×C32−1+412yes—
165C3×C4—22no—
166C3×C4—42no—
167C3×C4—41no—
168C3×C4—41no—
169Dic3—24no—
170Dic3—42no—
171Dic3—42no—
172Dic3—42no—
173C23—22no—
174C23—22no—
175C23—22no—
176C23—22no—
177C23—22no—
178C23—22no—
179C23#23714yes—
180C23C23×D311yescharacteristic
181Q8—124no—
182D4—44no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—22no—
192C2×C4—34no—
193C2×C4—62no—
194C2×C4—62no—
195C2×C4—62no—
196C2×C4—62no—
197C2×C4—62no—
198C2×C4—62no—
199C2×C4—62no—
200C2×C4—62no—
201C2×C4—62no—
202C2×C4—62no—
203C2×C4—62no—
204C2×C4—128no—
205C2×C4D3×D412yes—
206C2×C3—44no—
207C2×C3#53411yescharacteristic
208C2×C3#53912yes—
209C2×C3#54812yes—
210C2×C3#55111yescharacteristic
211C2×C3#56311yescharacteristic
212C22—24no—
213C22—24no—
214C22—24no—
215C22—42no—
216C22—42no—
217C22—42no—
218C22—42no—
219C22C2×D3×D411yescharacteristic
220C22C2×#23712yes—
221C22#100411yescharacteristic
222C22#100911yescharacteristic
223C22#101012yes—
224C4—22no—
225C4—41no—
226C4—41no—
227C4—42no—
228C4—64no—
229C4—122no—
230C4—122no—
231C4—122no—
232C3#272811yescharacteristic, complemented, Sylow
233C2—44no—
234C2#558711yescharacteristic
235C2#559812yes—
236C2#560511yescharacteristic
237C2#566512yes—
238C2#567211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#394C2
prime radicalC22×C3×D4C22
hypercenter#394D3
hypocenterC3#2728
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C2×#878C2semidirect
4. #4981C2semidirect
5. #4973C2semidirect
6. #4952C2semidirect
7. #4950C2semidirect
8. #4945C2semidirect
9. #4660C2semidirect
58. C2×Dic6D4semidirect
232. C3#2728semidirect

Maximal subgroups

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

subgroupindex
C2×#8782
C3×#3942
#52202
#49812
#49732
#49522
#49502
#49452
#46602
#27283