Finite Group Databasebeta

← #73357

Subgroups — #73357

Order 384.

544 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 173 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×#924C212yesmaximal
2C2×C3×#95C211yesmaximal, characteristic
3#5212C211yesmaximal, characteristic
4#5210C212yesmaximal
5C4×#217C211yesmaximal, characteristic
6#4657C212yesmaximal, complemented
7#4656C212yesmaximal, complemented
8#4647C212yesmaximal, complemented
9#4646C212yesmaximal, complemented
10#2549—31nomaximal, Sylow
11C23×C3×C4C2212yes—
12C23×C3×C4C2211yescharacteristic
13C22×Dic6C2212yes—
14C2×#217C2212yes—
15C2×#217C2211yescharacteristic
16C2×#217C2211yescharacteristic
17C2×C4⋊Dic3C2212yes—
18C2×C4⋊Dic3C2212yes—
19#924—28no—
20C2×Dic3⋊C4C2212yes—
21C2×Dic3⋊C4C2212yes—
22C2×Dic3⋊C4C2212yes—
23C2×Dic3⋊C4C2212yes—
24C2×C4×Dic3C2212yes—
25C3×#95C2212yes—
26C3×#95C2211yescharacteristic
27C3×#95C2211yescharacteristic
28C3⋊#95C2211yescharacteristic
29C3⋊#95C2212yes—
30C3⋊#95C2212yes—
31C3⋊#95C2211yescharacteristic
32C3⋊#95C2212yes—
33C2×C22⋊Q8—32no—
34#398—32no—
35#395—32no—
36#391—32no—
37#389—32no—
38#385—32no—
39#380—31no—
40C4×C22⋊C4—31no—
41C2×#95D311yescharacteristic
42C24×C3C2311yescharacteristic
43C22×C3×C4—24no—
44C22×C3×C4—24no—
45C22×C3×C4—42no—
46C22×C3×C4—42no—
47C22×C3×C4Q814yescomplemented
48C22×C3×C4C2312yes—
49C22×C3×C4C2311yescharacteristic
50C22×C3×C4C2311yescharacteristic
51C22×C3×C4C2312yes—
52C22×Dic3C2312yes—
53C22×Dic3C2312yes—
54C22×Dic3C2312yes—
55C22×Dic3C2312yes—
56C2×Dic6—24no—
57C2×Dic6—24no—
58C2×Dic6—42no—
59#217—24no—
60#217D414yes—
61#217Q814yescomplemented
62C4⋊Dic3—24no—
63C4⋊Dic3—42no—
64Dic3⋊C4—24no—
65Dic3⋊C4—24no—
66Dic3⋊C4—24no—
67Dic3⋊C4—42no—
68C4×Dic3—42no—
69C22×Q8—32no—
70C23×C4C2×D312yes—
71C23×C4C2×D311yescharacteristic
72C22⋊Q8—68no—
73C2×C4⋊C4—32no—
74C2×C4⋊C4—32no—
75C2×C4⋊C4—32no—
76C2×C4⋊C4—32no—
77C2×C4⋊C4—32no—
78C2×C4⋊C4—32no—
79C2×C22⋊C4—31no—
80C2×C22⋊C4—31no—
81C2×C22⋊C4—32no—
82C2×C42—32no—
83#95—32no—
84#95—31no—
85#95—31no—
86#95—32no—
87#95—32no—
88#95C2×D312yes—
89#95C2×D311yescharacteristic
90#95C2×D311yescharacteristic
91C23×C3—22no—
92C23×C3—22no—
93C23×C3C2×D412yes—
94C23×C3C2×Q812yes—
95C23×C3C2×Q812yes—
96C23×C3C2411yescharacteristic
97C2×C3×C4—22no—
98C2×C3×C4—22no—
99C2×C3×C4—22no—
100C2×C3×C4—22no—
101C2×C3×C4—22no—
102C2×C3×C4—22no—
103C2×C3×C4—22no—
104C2×C3×C4—44no—
105C2×C3×C4—48no—
106C2×C3×C4C2×Q812yes—
107C2×C3×C4C2×Q812yes—
108C2×C3×C4#4114yes—
109C2×Dic3—22no—
110C2×Dic3—22no—
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×Dic3C2×D414yes—
120C2×Dic3C2×Q814yes—
121Dic6—48no—
122C24C22×D311yescharacteristic
123C2×Q8—64no—
124C2×Q8—64no—
125C2×Q8—122no—
126C22×C4—24no—
127C22×C4—24no—
128C22×C4—32no—
129C22×C4—32no—
130C22×C4—32no—
131C22×C4—32no—
132C22×C4—42no—
133C22×C4—42no—
134C22×C4Dic614yescomplemented
135C22×C4C22×D312yes—
136C22×C4C22×D311yescharacteristic
137C22×C4C22×D311yescharacteristic
138C22×C4C22×D312yes—
139C4⋊C4—64no—
140C4⋊C4—64no—
141C4⋊C4—64no—
142C4⋊C4—64no—
143C4⋊C4—122no—
144C4⋊C4—122no—
145C22⋊C4—34no—
146C22⋊C4—34no—
147C22⋊C4—64no—
148C42—122no—
149C22×C3—22no—
150C22×C3—22no—
151C22×C3—24no—
152C22×C3—24no—
153C22×C3C4⋊Q814yes—
154C22×C3C22×D411yescharacteristic
155C22×C3C22×Q811yescharacteristic
156C22×C3C22×Q811yescharacteristic
157C22×C3C2×#4112yes—
158C22×C3H(5,𝔽2)12yes—
159C3×C4—22no—
160C3×C4—22no—
161C3×C4—24no—
162C3×C4—42no—
163Dic3—24no—
164Dic3—24no—
165Dic3—42no—
166Dic3—42no—
167C23—22no—
168C23—22no—
169C23C2×Dic612yes—
170C23D3×D412yes—
171C23D3×Q812yes—
172C23C23×D311yescharacteristic
173Q8—128no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—34no—
182C2×C4—34no—
183C2×C4—44no—
184C2×C4—48no—
185C2×C4—62no—
186C2×C4—62no—
187C2×C4—62no—
188C2×C4—62no—
189C2×C4—62no—
190C2×C4—62no—
191C2×C4—62no—
192C2×C4—62no—
193C2×C4—62no—
194C2×C4—62no—
195C2×C4C2×Dic612yes—
196C2×C4C2×Dic612yes—
197C2×C4#23514yes—
198C2×C3—24no—
199C2×C3C2×C4⋊Q811yescharacteristic
200C2×C3#53311yescharacteristic
201C2×C3#54311yescharacteristic
202C2×C3#54612yes—
203C2×C3#55412yes—
204C22—22no—
205C22—22no—
206C22—24no—
207C22—24no—
208C22#88814yes—
209C22C22×Dic611yescharacteristic
210C22C2×#23512yes—
211C22C2×D3×D411yescharacteristic
212C22#100412yes—
213C22C2×D3×Q811yescharacteristic
214C4—22no—
215C4—22no—
216C4—24no—
217C4—42no—
218C4—64no—
219C4—64no—
220C4—122no—
221C4—122no—
222C3#254911yescharacteristic, complemented, Sylow
223C2—24no—
224C2#547011yescharacteristic
225C2#548311yescharacteristic
226C2#549012yes—
227C2C2×#88811yescharacteristic
228C2#551112yes—
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC2×C3×#95C2
prime radicalC24×C3C23
hypercenterC2×#95D3
hypocenterC3#2549
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#95

Complements

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

NH ≅ G/Ntype
6. #4657C2semidirect
7. #4656C2semidirect
8. #4647C2semidirect
9. #4646C2semidirect
47. C22×C3×C4Q8semidirect
61. #217Q8semidirect
134. C22×C4Dic6semidirect
222. C3#2549semidirect

Maximal subgroups

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

subgroupindex
C2×#9242
C2×C3×#952
#52122
#52102
C4×#2172
#46572
#46562
#46472
#46462
#25493