Finite Group Databasebeta

← #76806

Subgroups — #76806

Order 384.

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

normal subgroups only

This group has 470 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
1C2×Q8×Dic3C211yesmaximal, characteristic, complemented
2C2×#879C211yesmaximal, characteristic
3C3×#393C211yesmaximal, characteristic
4#5230C211yesmaximal, characteristic, complemented
5#4951C212yesmaximal
6#4949C211yesmaximal, characteristic
7#4945C212yesmaximal
8#4659C211yesmaximal, characteristic, complemented
9#4655C212yesmaximal, complemented
10#4648C211yesmaximal, characteristic, complemented
11#4647C212yesmaximal, complemented
12#2604—31nomaximal, Sylow
13C22×C3×Q8C2211yescharacteristic
14C2×C3×C22⋊C4C2212yes—
15C2×C3×C22⋊C4C2211yescharacteristic
16C2×#217C2211yescharacteristic
17C2×#217C2212yes—
18C2×#217C2211yescharacteristic
19Q8×Dic3—24no—
20C2×C4⋊Dic3C2212yes—
21C2×C4⋊Dic3C2211yescharacteristic
22C2×C4⋊Dic3C2211yescharacteristic
23C2×Dic3⋊C4C2211yescharacteristic
24C2×Dic3⋊C4C2212yes—
25C2×Dic3⋊C4C2212yes—
26C2×Dic3⋊C4C2211yescharacteristic
27C2×C4×Dic3C2212yes—
28C2×C4×Dic3C2212yes—
29C2×C4×Dic3C2211yescharacteristic
30#879—24no—
31C3×#95C2212yes—
32C3×#95C2211yescharacteristic
33C3⋊#95C2212yes—
34C3⋊#95C2212yes—
35C3⋊#95C2211yescharacteristic
36C3⋊#95C2212yes—
37C3⋊#95C2212yes—
38C2×#126—31no—
39C2×C4×Q8—31no—
40#400—31no—
41#399—31no—
42#395—31no—
43#393D311yescharacteristic
44#389—32no—
45#388—32no—
46#388—32no—
47#387—32no—
48#376—31no—
49C24×C3C2311yescharacteristic
50C2×C3×Q8—22no—
51C2×C3×Q8—24no—
52C22×C3×C4C2312yes—
53C22×C3×C4C2311yescharacteristic
54C22×C3×C4C2311yescharacteristic
55C22×C3×C4C2312yes—
56C22×Dic3C2311yescharacteristic
57C22×Dic3C2312yes—
58C22×Dic3C2311yescharacteristic
59C22×Dic3C2311yescharacteristic
60C22×Dic3C2311yescharacteristic
61C22×Dic3C2312yes—
62C3×C22⋊C4—22no—
63C3×C22⋊C4—24no—
64#217—22no—
65#217—22no—
66#217—24no—
67C4⋊Dic3—22no—
68C4⋊Dic3—22no—
69C4⋊Dic3—24no—
70Dic3⋊C4—22no—
71Dic3⋊C4—24no—
72Dic3⋊C4—22no—
73Dic3⋊C4—24no—
74C4×Dic3—24no—
75C4×Dic3—24no—
76C4×Dic3D414yes—
77C22×Q8C2×D311yescharacteristic
78#126—64no—
79C4×Q8—64no—
80C2×C4⋊C4—32no—
81C2×C4⋊C4—31no—
82C2×C4⋊C4—31no—
83C2×C4⋊C4—31no—
84C2×C4⋊C4—31no—
85C2×C4⋊C4—32no—
86C2×C4⋊C4—32no—
87C2×C22⋊C4—31no—
88C2×C22⋊C4—31no—
89C2×C22⋊C4—32no—
90C2×C22⋊C4C2×D311yescharacteristic
91C2×C22⋊C4C2×D312yes—
92C2×C42—31no—
93C2×C42—32no—
94C2×C42—32no—
95#95—32no—
96#95—31no—
97#95—32no—
98#95—32no—
99#95—32no—
100#95C2×D311yescharacteristic
101#95C2×D312yes—
102C23×C3—21no—
103C23×C3—22no—
104C23×C3—22no—
105C23×C3—21no—
106C23×C3—21no—
107C23×C3C2411yescharacteristic
108C3×Q8—28no—
109C2×C3×C4—22no—
110C2×C3×C4—22no—
111C2×C3×C4—22no—
112C2×C3×C4—21no—
113C2×C3×C4—22no—
114C2×C3×C4—22no—
115C2×C3×C4—22no—
116C2×C3×C4—22no—
117C2×C3×C4C2×D412yes—
118C2×C3×C4#4114yes—
119C2×Dic3—22no—
120C2×Dic3—22no—
121C2×Dic3—21no—
122C2×Dic3—22no—
123C2×Dic3—21no—
124C2×Dic3—22no—
125C2×Dic3—21no—
126C2×Dic3—21no—
127C2×Dic3—22no—
128C2×Dic3—22no—
129C2×Dic3—22no—
130C2×Dic3—21no—
131C2×Dic3—21no—
132C2×Dic3C2×D412yes—
133C2×Dic3C2×D412yes—
134C2×Dic3#4114yes—
135C24C22×D311yescharacteristic
136C2×Q8—22no—
137C2×Q8—24no—
138C22×C4—32no—
139C22×C4—31no—
140C22×C4—31no—
141C22×C4—31no—
142C22×C4—32no—
143C22×C4—31no—
144C22×C4C22×D312yes—
145C22×C4C22×D311yescharacteristic
146C22×C4C22×D311yescharacteristic
147C22×C4C22×D312yes—
148C4⋊C4—64no—
149C4⋊C4—62no—
150C4⋊C4—62no—
151C4⋊C4—64no—
152C4⋊C4—62no—
153C4⋊C4—62no—
154C4⋊C4—64no—
155C22⋊C4—22no—
156C22⋊C4—24no—
157C22⋊C4—62no—
158C22⋊C4—62no—
159C22⋊C4—64no—
160C42—34no—
161C42—64no—
162C42—64no—
163C22×C3—41no—
164C22×C3—41no—
165C22×C3—41no—
166C22×C3—42no—
167C22×C3—42no—
168C22×C3C22×D411yescharacteristic
169C22×C3C2×#4112yes—
170C22×C3C2×#4112yes—
171C22×C32−1+411yescharacteristic
172C22×C32−1+411yescharacteristic
173C3×C4—22no—
174C3×C4—24no—
175C3×C4—41no—
176C3×C4—42no—
177Dic3—22no—
178Dic3—22no—
179Dic3—24no—
180Dic3—41no—
181Dic3—42no—
182Dic3—41no—
183C23—21no—
184C23—22no—
185C23—21no—
186C23—22no—
187C23—21no—
188C23C23×D311yescharacteristic
189Q8—28no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—21no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—32no—
199C2×C4—32no—
200C2×C4—34no—
201C2×C4—62no—
202C2×C4—62no—
203C2×C4—61no—
204C2×C4—62no—
205C2×C4—61no—
206C2×C4—62no—
207C2×C4—61no—
208C2×C4—61no—
209C2×C4—62no—
210C2×C4—61no—
211C2×C4—62no—
212C2×C4—62no—
213C2×C4—61no—
214C2×C4D3×D412yes—
215C2×C4#23714yes—
216C2×C3—81no—
217C2×C3#52511yescharacteristic
218C2×C3#53211yescharacteristic
219C2×C3#53611yescharacteristic
220C2×C3#55212yes—
221C2×C3#55612yes—
222C22—42no—
223C22—41no—
224C22—42no—
225C22—41no—
226C22—41no—
227C22C2×D3×D411yescharacteristic
228C22C2×#23712yes—
229C22#100711yescharacteristic
230C22D3×#4112yes—
231C22#101011yescharacteristic
232C4—22no—
233C4—24no—
234C4—41no—
235C4—42no—
236C4—64no—
237C4—62no—
238C4—62no—
239C4—122no—
240C4—121no—
241C4—121no—
242C3#260411yescharacteristic, complemented, Sylow
243C2—81no—
244C2#558611yescharacteristic
245C2#562811yescharacteristic
246C2#563912yes—
247C2#566811yescharacteristic
248C2#567112yes—
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#393C2
prime radicalC24×C3C23
hypercenter#393D3
hypocenterC3#2604
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. C2×Q8×Dic3C2semidirect
4. #5230C2semidirect
8. #4659C2semidirect
9. #4655C2semidirect
10. #4648C2semidirect
11. #4647C2semidirect
242. C3#2604semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×Dic32
C2×#8792
C3×#3932
#52302
#49512
#49492
#49452
#46592
#46552
#46482
#46472
#26043