Finite Group Databasebeta

← #76422

Subgroups — #76422

Order 384.

428 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 125 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×#390C211yesmaximal, characteristic
2#5222C212yesmaximal
3#4978C212yesmaximal, complemented
4#4953C212yesmaximal, complemented
5#4951C212yesmaximal, complemented
6#4951C212yesmaximal, complemented
7#4947C212yesmaximal, complemented
8#4929C211yesmaximal, characteristic, complemented
9#4928C211yesmaximal, characteristic, complemented
10#2808—31nomaximal, Sylow
11C22×C3×D4C2211yescharacteristic
12C22×Dic6C2211yescharacteristic
13C2×C3×C4⋊C4C2211yescharacteristic
14C2×C3×C22⋊C4C2212yes—
15C2×C3×C22⋊C4C2212yes—
16C2×C3×C42C2211yescharacteristic
17C2×#217C2212yes—
18C2×#217C2212yes—
19C2×#217C2212yes—
20C2×#217C2212yes—
21C2×C4⋊Dic3C2211yescharacteristic
22C2×C4⋊Dic3C2211yescharacteristic
23C2×C4⋊Dic3C2211yescharacteristic
24C2×Dic3⋊C4C2212yes—
25C2×C4×Dic3C2212yes—
26C3⋊#95C2212yes—
27C3⋊#95C2212yes—
28C3⋊#95C2212yes—
29C3⋊#95C2212yes—
30C3⋊#95C2212yes—
31C3⋊#95C2212yes—
32#397—32no—
33#394—32no—
34#393—32no—
35#391—31no—
36#390D311yescharacteristic
37#389—31no—
38#388—32no—
39#388—32no—
40#387—32no—
41C24×C3C2312yes—
42C2×C3×D4—22no—
43C2×C3×D4—41no—
44C2×C3×D4—41no—
45C22×C3×C4C2312yes—
46C22×C3×C4C2311yescharacteristic
47C22×C3×C4C2311yescharacteristic
48C22×C3×C4C2311yescharacteristic
49C22×Dic3C2312yes—
50C22×Dic3C2312yes—
51C22×Dic3C2312yes—
52C22×Dic3C2312yes—
53C2×Dic6—22no—
54C2×Dic6—41no—
55C2×Dic6—41no—
56C3×C4⋊C4—41no—
57C3×C22⋊C4—24no—
58C3×C22⋊C4—24no—
59C3×C42—41no—
60#217—42no—
61#217—42no—
62#217—42no—
63#217—42no—
64C4⋊Dic3—22no—
65C4⋊Dic3—41no—
66C4⋊Dic3—41no—
67Dic3⋊C4—24no—
68C4×Dic3—42no—
69C22×Q8—31no—
70C22×D4C2×D311yescharacteristic
71C2×C4⋊C4—32no—
72C2×C4⋊C4—31no—
73C2×C4⋊C4—31no—
74C2×C4⋊C4—31no—
75C2×C4⋊C4C2×D311yescharacteristic
76C2×C22⋊C4—32no—
77C2×C22⋊C4—32no—
78C2×C22⋊C4—32no—
79C2×C22⋊C4—32no—
80C2×C22⋊C4C2×D312yes—
81C2×C22⋊C4C2×D312yes—
82C2×C42—32no—
83C2×C42C2×D311yescharacteristic
84#95—32no—
85#95—32no—
86#95—32no—
87#95—32no—
88#95—32no—
89#95—32no—
90C23×C3—22no—
91C23×C3—22no—
92C23×C3—22no—
93C23×C3—22no—
94C23×C3—22no—
95C23×C3—22no—
96C23×C3—22no—
97C23×C3C2411yescharacteristic
98C3×D4—44no—
99C2×C3×C4—21no—
100C2×C3×C4—21no—
101C2×C3×C4—21no—
102C2×C3×C4—21no—
103C2×C3×C4—21no—
104C2×C3×C4—21no—
105C2×C3×C4—22no—
106C2×C3×C4—22no—
107C2×C3×C4—22no—
108C2×C3×C4#4112yes—
109C2×C3×C4#4112yes—
110C2×C3×C4#4112yes—
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×Dic3—22no—
120C2×Dic3—22no—
121C2×Dic3—22no—
122C2×Dic3—22no—
123Dic6—44no—
124C24C22×D312yes—
125C2×Q8—62no—
126C2×Q8—121no—
127C2×Q8—121no—
128C2×D4—22no—
129C2×D4—41no—
130C2×D4—41no—
131C22×C4—32no—
132C22×C4—32no—
133C22×C4—32no—
134C22×C4—32no—
135C22×C4C22×D312yes—
136C22×C4C22×D311yescharacteristic
137C22×C4C22×D311yescharacteristic
138C22×C4C22×D311yescharacteristic
139C4⋊C4—41no—
140C4⋊C4—64no—
141C4⋊C4—62no—
142C4⋊C4—121no—
143C4⋊C4—121no—
144C22⋊C4—24no—
145C22⋊C4—24no—
146C22⋊C4—122no—
147C22⋊C4—122no—
148C22⋊C4—122no—
149C22⋊C4—122no—
150C42—41no—
151C42—122no—
152C22×C3—42no—
153C22×C3—42no—
154C22×C3—42no—
155C22×C3—42no—
156C22×C3—42no—
157C22×C3—42no—
158C22×C3—42no—
159C22×C3C2×#4111yescharacteristic
160C22×C3C2×#4111yescharacteristic
161C22×C3C2×#4111yescharacteristic
162C22×C3H(5,𝔽2)11yescharacteristic
163C22×C3H(5,𝔽2)11yescharacteristic
164C22×C3H(5,𝔽2)11yescharacteristic
165C22×C32−1+411yescharacteristic
166C3×C4—22no—
167C3×C4—22no—
168C3×C4—22no—
169C3×C4—42no—
170Dic3—42no—
171Dic3—42no—
172Dic3—42no—
173Dic3—42no—
174C23—22no—
175C23—22no—
176C23—22no—
177C23—22no—
178C23—22no—
179C23—22no—
180C23—22no—
181C23C23×D311yescharacteristic
182Q8—124no—
183D4—44no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—62no—
194C2×C4—62no—
195C2×C4—62no—
196C2×C4—62no—
197C2×C4—62no—
198C2×C4—62no—
199C2×C4—62no—
200C2×C4—62no—
201C2×C4—62no—
202C2×C4—62no—
203C2×C4—62no—
204C2×C4—62no—
205C2×C4#23512yes—
206C2×C4#23712yes—
207C2×C4#23712yes—
208C2×C3—82no—
209C2×C3#53811yescharacteristic
210C2×C3#53811yescharacteristic
211C2×C3#53911yescharacteristic
212C2×C3#55511yescharacteristic
213C2×C3#55611yescharacteristic
214C2×C3#55911yescharacteristic
215C2×C3#56211yescharacteristic
216C22—42no—
217C22—42no—
218C22—42no—
219C22—42no—
220C22—42no—
221C22—42no—
222C22—42no—
223C22C2×#23511yescharacteristic
224C22C2×#23711yescharacteristic
225C22C2×#23711yescharacteristic
226C22#100411yescharacteristic
227C22#100411yescharacteristic
228C22#100911yescharacteristic
229C22#101011yescharacteristic
230C4—22no—
231C4—22no—
232C4—22no—
233C4—42no—
234C4—122no—
235C4—122no—
236C4—122no—
237C4—122no—
238C3#280811yescharacteristic, complemented, Sylow
239C2—82no—
240C2#549411yescharacteristic
241C2#554211yescharacteristic
242C2#556311yescharacteristic
243C2#560211yescharacteristic
244C2#561711yescharacteristic
245C2#568411yescharacteristic
246C2#571311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#390C2
prime radicalC22×C3×D4C22
hypercenter#390D3
hypocenterC3#2808
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
3. #4978C2semidirect
4. #4953C2semidirect
5. #4951C2semidirect
6. #4951C2semidirect
7. #4947C2semidirect
8. #4929C2semidirect
9. #4928C2semidirect
238. C3#2808semidirect

Maximal subgroups

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

subgroupindex
C3×#3902
#52222
#49782
#49532
#49512
#49472
#49292
#49282
#28083