Finite Group Databasebeta

← #76807

Subgroups — #76807

Order 384.

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

normal subgroups only

This group has 444 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×#393C211yesmaximal, characteristic
2#5229C212yesmaximal, complemented
3#4949C212yesmaximal
4#4947C211yesmaximal, characteristic
5#4943C211yesmaximal, characteristic
6Dic3×C22⋊C4C212yesmaximal
7#4653C212yesmaximal, complemented
8#4652C212yesmaximal, complemented
9#4646C212yesmaximal, complemented
10#2607—31nomaximal, Sylow
11C22×C3×Q8C2211yescharacteristic
12C23×Dic3C2211yescharacteristic
13C22×Dic6C2211yescharacteristic
14C2×C3×C22⋊C4C2212yes—
15C2×C3×C22⋊C4C2211yescharacteristic
16C2×#217C2212yes—
17C2×#217C2211yescharacteristic
18C2×C4⋊Dic3C2212yes—
19C2×Dic3⋊C4C2212yes—
20C2×Dic3⋊C4C2212yes—
21C2×C4×Dic3C2212yes—
22C2×C4×Dic3C2212yes—
23C3×#95C2212yes—
24C3×#95C2211yescharacteristic
25C3⋊#95C2212yes—
26C3⋊#95C2212yes—
27C3⋊#95C2212yes—
28C3⋊#95C2212yes—
29C3⋊#95C2212yes—
30C3⋊#95C2211yescharacteristic
31C3⋊#95C2212yes—
32#399—32no—
33#393—31no—
34#393D311yescharacteristic
35#391—32no—
36#391—32no—
37#387—32no—
38#387—32no—
39#381—31no—
40C4×C22⋊C4—32no—
41C24×C3C2311yescharacteristic
42C2×C3×Q8—24no—
43C2×C3×Q8—41no—
44C22×C3×C4C2311yescharacteristic
45C22×C3×C4C2311yescharacteristic
46C22×C3×C4C2312yes—
47C22×C3×C4C2312yes—
48C22×Dic3—41no—
49C22×Dic3—41no—
50C22×Dic3Q814yescomplemented
51C22×Dic3C2312yes—
52C22×Dic3C2312yes—
53C22×Dic3C2312yes—
54C22×Dic3C2312yes—
55C2×Dic6—24no—
56C2×Dic6—41no—
57C3×C22⋊C4—24no—
58C3×C22⋊C4—41no—
59#217—24no—
60#217—41no—
61C4⋊Dic3—24no—
62Dic3⋊C4—24no—
63Dic3⋊C4—42no—
64C4×Dic3—42no—
65C4×Dic3—42no—
66C22×Q8—31no—
67C22×Q8C2×D311yescharacteristic
68C23×C4—31no—
69C2×C4⋊C4—32no—
70C2×C4⋊C4—32no—
71C2×C4⋊C4—32no—
72C2×C22⋊C4—32no—
73C2×C22⋊C4—31no—
74C2×C22⋊C4C2×D312yes—
75C2×C22⋊C4C2×D311yescharacteristic
76C2×C42—32no—
77C2×C42—32no—
78#95—32no—
79#95—32no—
80#95—32no—
81#95—32no—
82#95—31no—
83#95—32no—
84#95—32no—
85#95C2×D311yescharacteristic
86#95C2×D312yes—
87C23×C3—22no—
88C23×C3—22no—
89C23×C3—21no—
90C23×C3—21no—
91C23×C3C2×Q812yes—
92C23×C3C2411yescharacteristic
93C3×Q8—44no—
94C2×C3×C4—22no—
95C2×C3×C4—22no—
96C2×C3×C4—22no—
97C2×C3×C4—22no—
98C2×C3×C4—22no—
99C2×C3×C4—22no—
100C2×C3×C4—22no—
101C2×C3×C4—21no—
102C2×C3×C4—21no—
103C2×C3×C4#4114yes—
104C2×Dic3—22no—
105C2×Dic3—22no—
106C2×Dic3—22no—
107C2×Dic3—22no—
108C2×Dic3—22no—
109C2×Dic3—22no—
110C2×Dic3—22no—
111C2×Dic3—22no—
112C2×Dic3—22no—
113C2×Dic3—22no—
114C2×Dic3—44no—
115C2×Dic3C2×Q814yes—
116C2×Dic3#4114yes—
117Dic6—44no—
118C24C22×D311yescharacteristic
119C2×Q8—24no—
120C2×Q8—41no—
121C2×Q8—64no—
122C2×Q8—121no—
123C22×C4—32no—
124C22×C4—32no—
125C22×C4—34no—
126C22×C4—32no—
127C22×C4—32no—
128C22×C4—121no—
129C22×C4—121no—
130C22×C4C22×D311yescharacteristic
131C22×C4C22×D311yescharacteristic
132C22×C4C22×D312yes—
133C22×C4C22×D312yes—
134C4⋊C4—64no—
135C4⋊C4—64no—
136C4⋊C4—122no—
137C22⋊C4—24no—
138C22⋊C4—41no—
139C22⋊C4—64no—
140C22⋊C4—121no—
141C42—122no—
142C42—122no—
143C22×C3—24no—
144C22×C3—22no—
145C22×C3—41no—
146C22×C3—41no—
147C22×C3—42no—
148C22×C3C22×Q811yescharacteristic
149C22×C3C2×#4112yes—
150C22×C3C2×#4112yes—
151C22×C3H(5,𝔽2)11yescharacteristic
152C22×C3H(5,𝔽2)11yescharacteristic
153C3×C4—24no—
154C3×C4—41no—
155C3×C4—41no—
156C3×C4—42no—
157Dic3—24no—
158Dic3—24no—
159Dic3—42no—
160Dic3—42no—
161C23—22no—
162C23—21no—
163C23—22no—
164C23—21no—
165C23D3×Q812yes—
166C23C23×D311yescharacteristic
167Q8—44no—
168Q8—124no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—34no—
179C2×C4—34no—
180C2×C4—62no—
181C2×C4—62no—
182C2×C4—62no—
183C2×C4—62no—
184C2×C4—62no—
185C2×C4—62no—
186C2×C4—62no—
187C2×C4—62no—
188C2×C4—62no—
189C2×C4—62no—
190C2×C4—124no—
191C2×C4#23714yes—
192C2×C3—42no—
193C2×C3#53312yes—
194C2×C3#54311yescharacteristic
195C2×C3#55111yescharacteristic
196C2×C3#55111yescharacteristic
197C2×C3#55912yes—
198C22—22no—
199C22—24no—
200C22—41no—
201C22—42no—
202C22—41no—
203C22C2×#23712yes—
204C22#100411yescharacteristic
205C22C2×D3×Q811yescharacteristic
206C22D3×#4112yes—
207C22#100911yescharacteristic
208C4—24no—
209C4—41no—
210C4—42no—
211C4—41no—
212C4—64no—
213C4—64no—
214C4—122no—
215C4—122no—
216C3#260711yescharacteristic, complemented, Sylow
217C2—42no—
218C2#558711yescharacteristic
219C2#562212yes—
220C2#563411yescharacteristic
221C2#567911yescharacteristic
222C2#568112yes—
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#393C2
prime radicalC24×C3C23
hypercenter#393D3
hypocenterC3#2607
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
2. #5229C2semidirect
7. #4653C2semidirect
8. #4652C2semidirect
9. #4646C2semidirect
50. C22×Dic3Q8semidirect
216. C3#2607semidirect

Maximal subgroups

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

subgroupindex
C3×#3932
#52292
#49492
#49472
#49432
Dic3×C22⋊C42
#46532
#46522
#46462
#26073