Finite Group Databasebeta

← #76518

Subgroups — #76518

Order 384.

484 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 145 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×#946C211yesmaximal, characteristic
2C2×#892C211yesmaximal, characteristic
3C3×#391C211yesmaximal, characteristic, complemented
4#5229C211yesmaximal, characteristic, complemented
5#4975C211yesmaximal, characteristic, complemented
6C4×(C2×D3)⋊C4C211yesmaximal, characteristic
7#4936C211yesmaximal, characteristic, complemented
8#4675C214yesmaximal
9#4652C214yesmaximal, complemented
10#2499—31nomaximal, Sylow
11C22×C3×Q8C2211yescharacteristic
12C22×C4×D3C2211yescharacteristic
13C2×C3×C4⋊C4C2211yescharacteristic
14C2×C3×C42C2211yescharacteristic
15#946—24no—
16C2×(C2×D3)⋊C4C2212yes—
17C2×(C2×D3)⋊C4C2212yes—
18C2×(C2×D3)⋊C4C2212yes—
19C2×C4⋊Dic3C2211yescharacteristic
20C2×C4⋊Dic3C2211yescharacteristic
21C2×Dic3⋊C4C2212yes—
22C2×Dic3⋊C4C2214yes—
23C2×C4×Dic3C2212yes—
24C2×C4×Dic3C2211yescharacteristic
25#892—24no—
26C3×#95C2214yes—
27C3⋊#95C2212yes—
28C3⋊#95C2214yes—
29C3⋊#95C2214yes—
30C2×C22⋊Q8—31no—
31C2×#117—31no—
32#397—34no—
33#391—31no—
34#391D311yescharacteristic, complemented
35#387—34no—
36#384—31no—
37#382—31no—
38C4×C22⋊C4—31no—
39C23×D3C2311yescharacteristic
40C2×C3×Q8—22no—
41C2×C3×Q8—22no—
42C2×C3×Q8—22no—
43C22×C3×C4C2312yes—
44C22×C3×C4C2312yes—
45C22×C3×C4C2312yes—
46C22×C3×C4C2311yescharacteristic
47C22×Dic3C2312yes—
48C22×Dic3C2312yes—
49C22×Dic3C2311yescharacteristic
50C22×Dic3C2312yes—
51C2×C4×D3—22no—
52C2×C4×D3—22no—
53C2×C4×D3D414yes—
54C3×C4⋊C4—22no—
55C3×C42—22no—
56(C2×D3)⋊C4—24no—
57(C2×D3)⋊C4—24no—
58(C2×D3)⋊C4—24no—
59C4⋊Dic3—22no—
60C4⋊Dic3—22no—
61Dic3⋊C4—24no—
62Dic3⋊C4—28no—
63C4×Dic3—22no—
64C4×Dic3—24no—
65C22×Q8C2×D311yescharacteristic
66C23×C4—31no—
67C22⋊Q8—64no—
68#117—64no—
69C2×C4⋊C4—31no—
70C2×C4⋊C4—31no—
71C2×C4⋊C4—32no—
72C2×C4⋊C4—34no—
73C2×C4⋊C4C2×D311yescharacteristic
74C2×C22⋊C4—32no—
75C2×C22⋊C4—32no—
76C2×C22⋊C4—32no—
77C2×C42—32no—
78C2×C42—31no—
79C2×C42C2×D311yescharacteristic
80#95—32no—
81#95—34no—
82#95—34no—
83#95C2×D314yes—
84C23×C3C2411yescharacteristic
85C22×D3—22no—
86C22×D3—21no—
87C22×D3—21no—
88C22×D3—22no—
89C22×D3C2×D412yes—
90C3×Q8—44no—
91C2×C3×C4—22no—
92C2×C3×C4—22no—
93C2×C3×C4—24no—
94C2×C3×C4—22no—
95C2×C3×C4—22no—
96C2×C3×C4—22no—
97C2×C3×C4—22no—
98C2×C3×C4—22no—
99C2×C3×C4C2×D412yes—
100C2×C3×C4#4112yes—
101C2×C3×C4#4112yes—
102C2×Dic3—22no—
103C2×Dic3—21no—
104C2×Dic3—21no—
105C2×Dic3—22no—
106C2×Dic3—22no—
107C2×Dic3—22no—
108C2×Dic3—24no—
109C2×Dic3—24no—
110C2×Dic3C2×D412yes—
111C2×Dic3#4114yes—
112C4×D3—28no—
113C24—31no—
114C2×Q8—22no—
115C2×Q8—22no—
116C2×Q8—22no—
117C22×C4—32no—
118C22×C4—31no—
119C22×C4—34no—
120C22×C4—32no—
121C22×C4—32no—
122C22×C4—62no—
123C22×C4—62no—
124C22×C4C22×D312yes—
125C22×C4C22×D312yes—
126C22×C4C22×D312yes—
127C22×C4C22×D311yescharacteristic
128C4⋊C4—22no—
129C4⋊C4—64no—
130C4⋊C4—68no—
131C4⋊C4—62no—
132C4⋊C4—62no—
133C22⋊C4—64no—
134C22⋊C4—64no—
135C22⋊C4—64no—
136C42—22no—
137C42—64no—
138C42—62no—
139C22×C3C22×D411yescharacteristic
140C22×C3C2×#4111yescharacteristic
141C22×C3C2×#4111yescharacteristic
142C22×C3C2×#4112yes—
143C22×C32−1+412yes—
144C2×D3—22no—
145C2×D3—22no—
146C2×D3—22no—
147C2×D3—42no—
148C2×D3—42no—
149C3×C4—42no—
150C3×C4—42no—
151C3×C4—42no—
152C3×C4#12314yes—
153Dic3—22no—
154Dic3—24no—
155Dic3—42no—
156Dic3—42no—
157C23—32no—
158C23—61no—
159C23—61no—
160C23—62no—
161C23—62no—
162C23C23×D311yescharacteristic
163Q8—44no—
164C2×C4—22no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—24no—
172C2×C4—32no—
173C2×C4—34no—
174C2×C4—62no—
175C2×C4—61no—
176C2×C4—61no—
177C2×C4—62no—
178C2×C4—68no—
179C2×C4—62no—
180C2×C4—62no—
181C2×C4—64no—
182C2×C4—64no—
183C2×C4#23512yes—
184C2×C4D3×D412yes—
185C2×C4#23712yes—
186C2×C3C2×#12311yescharacteristic
187C2×C3#53211yescharacteristic
188C2×C3#53611yescharacteristic
189C2×C3#55212yes—
190C2×C3#55612yes—
191D3—42no—
192C22—62no—
193C22—62no—
194C22—62no—
195C22—122no—
196C22—122no—
197C22C2×#23511yescharacteristic
198C22C2×D3×D411yescharacteristic
199C22C2×#23711yescharacteristic
200C22#100712yes—
201C22D3×#4112yes—
202C4—42no—
203C4—42no—
204C4—42no—
205C4—62no—
206C4—64no—
207C4—122no—
208C4—122no—
209C4#88314yes—
210C3#249911yescharacteristic, complemented, Sylow
211C2—122no—
212C2C2×#88311yescharacteristic
213C2#558112yes—
214C2#564112yes—
215C2#566811yescharacteristic
216C2#572411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#391C2
prime radicalC23×D3C23
hypercenter#391D3
hypocenterC3#2499
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
3. C3×#391C2semidirect
4. #5229C2semidirect
5. #4975C2semidirect
7. #4936C2semidirect
9. #4652C2semidirect
34. #391D3semidirect
210. C3#2499semidirect

Maximal subgroups

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

subgroupindex
C2×#9462
C2×#8922
C3×#3912
#52292
#49752
C4×(C2×D3)⋊C42
#49362
#46752
#46522
#24993