Finite Group Databasebeta

← #78046

Subgroups — #78046

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×#399C211yesmaximal, characteristic
2#4980C212yesmaximal, complemented
3#4978C212yesmaximal, complemented
4#4973C212yesmaximal, complemented
5#4949C211yesmaximal, characteristic
6#4947C212yesmaximal
7#4943C211yesmaximal, characteristic
8Dic3×C22⋊C4C212yesmaximal
9#4646C212yesmaximal, complemented
10#2607—31nomaximal, Sylow
11C23×Dic3C2211yescharacteristic
12C22×Dic6C2212yes—
13C2×C3×C4⋊C4C2212yes—
14C2×C3×C4⋊C4C2211yescharacteristic
15C2×C3×C22⋊C4C2212yes—
16C2×C3×C22⋊C4C2211yescharacteristic
17C2×#217C2212yes—
18C2×#217C2211yescharacteristic
19C2×C4⋊Dic3C2211yescharacteristic
20C2×Dic3⋊C4C2212yes—
21C2×C4×Dic3C2212yes—
22C2×C4×Dic3C2212yes—
23C3×#95C2211yescharacteristic
24C3⋊#95C2212yes—
25C3⋊#95C2212yes—
26C3⋊#95C2212yes—
27C3⋊#95C2212yes—
28C3⋊#95C2212yes—
29C3⋊#95C2212yes—
30C3⋊#95C2212yes—
31C3⋊#95C2211yescharacteristic
32#399—31no—
33#399D311yescharacteristic
34#393—32no—
35#391—32no—
36#391—32no—
37#387—32no—
38#387—32no—
39#381—31no—
40C4×C22⋊C4—32no—
41C24×C3C2311yescharacteristic
42C22×C3×C4C2312yes—
43C22×C3×C4C2311yescharacteristic
44C22×C3×C4C2311yescharacteristic
45C22×C3×C4C2312yes—
46C22×Dic3—41no—
47C22×Dic3—41no—
48C22×Dic3Q814yescomplemented
49C22×Dic3C2312yes—
50C22×Dic3C2312yes—
51C22×Dic3C2312yes—
52C22×Dic3C2312yes—
53C2×Dic6—24no—
54C2×Dic6—24no—
55C2×Dic6—42no—
56C3×C4⋊C4—24no—
57C3×C4⋊C4—41no—
58C3×C22⋊C4—24no—
59C3×C22⋊C4—41no—
60#217—24no—
61#217—41no—
62C4⋊Dic3—41no—
63Dic3⋊C4—24no—
64C4×Dic3—42no—
65C4×Dic3—42no—
66C22×Q8—32no—
67C23×C4—31no—
68C2×C4⋊C4—32no—
69C2×C4⋊C4—31no—
70C2×C4⋊C4C2×D312yes—
71C2×C4⋊C4C2×D311yescharacteristic
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—32no—
83#95—32no—
84#95—32no—
85#95—31no—
86#95C2×D311yescharacteristic
87C23×C3—22no—
88C23×C3—22no—
89C23×C3—21no—
90C23×C3—21no—
91C23×C3C2×Q812yes—
92C23×C3C2411yescharacteristic
93C2×C3×C4—22no—
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—21no—
101C2×C3×C4—21no—
102C2×C3×C4#4114yes—
103C2×Dic3—22no—
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—44no—
114C2×Dic3C2×Q814yes—
115C2×Dic3#4114yes—
116Dic6—48no—
117C24C22×D311yescharacteristic
118C2×Q8—64no—
119C2×Q8—64no—
120C2×Q8—122no—
121C22×C4—32no—
122C22×C4—32no—
123C22×C4—34no—
124C22×C4—32no—
125C22×C4—32no—
126C22×C4—121no—
127C22×C4—121no—
128C22×C4C22×D312yes—
129C22×C4C22×D311yescharacteristic
130C22×C4C22×D311yescharacteristic
131C22×C4C22×D312yes—
132C4⋊C4—24no—
133C4⋊C4—41no—
134C4⋊C4—64no—
135C4⋊C4—121no—
136C22⋊C4—24no—
137C22⋊C4—41no—
138C22⋊C4—64no—
139C22⋊C4—121no—
140C42—122no—
141C42—122no—
142C22×C3—24no—
143C22×C3—22no—
144C22×C3—42no—
145C22×C3—41no—
146C22×C3—41no—
147C22×C3C22×Q811yescharacteristic
148C22×C3C2×#4112yes—
149C22×C3C2×#4112yes—
150C22×C3H(5,𝔽2)11yescharacteristic
151C22×C3H(5,𝔽2)11yescharacteristic
152C3×C4—24no—
153C3×C4—41no—
154C3×C4—42no—
155C3×C4—41no—
156Dic3—24no—
157Dic3—24no—
158Dic3—42no—
159Dic3—42no—
160C23—22no—
161C23—21no—
162C23—22no—
163C23—21no—
164C23D3×Q812yes—
165C23C23×D311yescharacteristic
166Q8—128no—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—22no—
176C2×C4—34no—
177C2×C4—34no—
178C2×C4—62no—
179C2×C4—62no—
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—124no—
189C2×C4#23714yes—
190C2×C3—42no—
191C2×C3#53312yes—
192C2×C3#54311yescharacteristic
193C2×C3#55112yes—
194C2×C3#55911yescharacteristic
195C2×C3#55911yescharacteristic
196C22—22no—
197C22—24no—
198C22—42no—
199C22—41no—
200C22—41no—
201C22C2×#23712yes—
202C22#100411yescharacteristic
203C22C2×D3×Q811yescharacteristic
204C22D3×#4112yes—
205C22#100911yescharacteristic
206C4—24no—
207C4—41no—
208C4—42no—
209C4—41no—
210C4—64no—
211C4—64no—
212C4—122no—
213C4—122no—
214C3#260711yescharacteristic, complemented, Sylow
215C2—42no—
216C2#562212yes—
217C2#563411yescharacteristic
218C2#565812yes—
219C2#566411yescharacteristic
220C2#571311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#399C2
prime radicalC24×C3C23
hypercenter#399D3
hypocenterC3#2607
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
2. #4980C2semidirect
3. #4978C2semidirect
4. #4973C2semidirect
9. #4646C2semidirect
48. C22×Dic3Q8semidirect
214. C3#2607semidirect

Maximal subgroups

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

subgroupindex
C3×#3992
#49802
#49782
#49732
#49492
#49472
#49432
Dic3×C22⋊C42
#46462
#26073