Finite Group Databasebeta

← #76423

Subgroups — #76423

Order 384.

408 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 117 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#4981C211yesmaximal, characteristic, complemented
4#4979C211yesmaximal, characteristic, complemented
5#4952C212yesmaximal, complemented
6#4952C212yesmaximal, complemented
7#4948C212yesmaximal, complemented
8#4947C212yesmaximal, complemented
9#4933C212yesmaximal, complemented
10#2830—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×Dic3⋊C4C2212yes—
23C2×Dic3⋊C4C2212yes—
24C2×Dic3⋊C4C2212yes—
25C3⋊#95C2212yes—
26C3⋊#95C2212yes—
27C3⋊#95C2212yes—
28C3⋊#95C2212yes—
29C3⋊#95C2212yes—
30C3⋊#95C2211yescharacteristic
31C3⋊#95C2211yescharacteristic
32#398—31no—
33#397—32no—
34#397—32no—
35#396—32no—
36#395—31no—
37#394—32no—
38#393—32no—
39#390D311yescharacteristic
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—42no—
54C2×Dic6—41no—
55C3×C4⋊C4—41no—
56C3×C22⋊C4—42no—
57C3×C22⋊C4—42no—
58C3×C42—41no—
59#217—42no—
60#217—42no—
61#217—42no—
62#217—42no—
63C4⋊Dic3—41no—
64Dic3⋊C4—24no—
65Dic3⋊C4—42no—
66Dic3⋊C4—42no—
67C22×Q8—31no—
68C22×D4C2×D311yescharacteristic
69C2×C4⋊C4—32no—
70C2×C4⋊C4—32no—
71C2×C4⋊C4—32no—
72C2×C4⋊C4—31no—
73C2×C4⋊C4C2×D311yescharacteristic
74C2×C22⋊C4—32no—
75C2×C22⋊C4—32no—
76C2×C22⋊C4—32no—
77C2×C22⋊C4—32no—
78C2×C22⋊C4C2×D312yes—
79C2×C22⋊C4C2×D312yes—
80C2×C42C2×D311yescharacteristic
81#95—32no—
82#95—32no—
83#95—32no—
84#95—32no—
85#95—32no—
86#95—31no—
87#95—31no—
88C23×C3—22no—
89C23×C3—22no—
90C23×C3—22no—
91C23×C3—22no—
92C23×C3—22no—
93C23×C3—22no—
94C23×C3—22no—
95C23×C3C2411yescharacteristic
96C3×D4—44no—
97C2×C3×C4—21no—
98C2×C3×C4—21no—
99C2×C3×C4—22no—
100C2×C3×C4—22no—
101C2×C3×C4—21no—
102C2×C3×C4—22no—
103C2×C3×C4—22no—
104C2×C3×C4—21no—
105C2×C3×C4—22no—
106C2×C3×C4#4112yes—
107C2×Dic3—22no—
108C2×Dic3—22no—
109C2×Dic3—22no—
110C2×Dic3—22no—
111C2×Dic3—22no—
112C2×Dic3—22no—
113C2×Dic3—22no—
114C2×Dic3—22no—
115C2×Dic3—22no—
116C2×Dic3—22no—
117C2×Dic3—22no—
118C2×Dic3—22no—
119Dic6—44no—
120C24C22×D312yes—
121C2×Q8—122no—
122C2×Q8—121no—
123C2×D4—22no—
124C2×D4—41no—
125C2×D4—41no—
126C22×C4—32no—
127C22×C4—32no—
128C22×C4—32no—
129C22×C4—32no—
130C22×C4C22×D312yes—
131C22×C4C22×D311yescharacteristic
132C22×C4C22×D311yescharacteristic
133C22×C4C22×D311yescharacteristic
134C4⋊C4—41no—
135C4⋊C4—64no—
136C4⋊C4—122no—
137C4⋊C4—122no—
138C4⋊C4—121no—
139C22⋊C4—42no—
140C22⋊C4—42no—
141C22⋊C4—122no—
142C22⋊C4—122no—
143C22⋊C4—122no—
144C22⋊C4—122no—
145C42—41no—
146C22×C3—42no—
147C22×C3—42no—
148C22×C3—42no—
149C22×C3—42no—
150C22×C3—42no—
151C22×C3—42no—
152C22×C3—42no—
153C22×C3C2×#4111yescharacteristic
154C22×C3H(5,𝔽2)11yescharacteristic
155C22×C3H(5,𝔽2)11yescharacteristic
156C22×C3H(5,𝔽2)12yes—
157C22×C32−1+412yes—
158C3×C4—22no—
159C3×C4—41no—
160C3×C4—41no—
161C3×C4—42no—
162Dic3—42no—
163Dic3—42no—
164Dic3—42no—
165Dic3—42no—
166C23—22no—
167C23—22no—
168C23—22no—
169C23—22no—
170C23—22no—
171C23—22no—
172C23—22no—
173C23C23×D311yescharacteristic
174Q8—124no—
175D4—44no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—22no—
185C2×C4—62no—
186C2×C4—62no—
187C2×C4—62no—
188C2×C4—62no—
189C2×C4—62no—
190C2×C4—62no—
191C2×C4—62no—
192C2×C4—62no—
193C2×C4—62no—
194C2×C4—62no—
195C2×C4—62no—
196C2×C4—62no—
197C2×C4#23512yes—
198C2×C3—82no—
199C2×C3#53811yescharacteristic
200C2×C3#54212yes—
201C2×C3#56011yescharacteristic
202C2×C3#56212yes—
203C2×C3#56311yescharacteristic
204C22—42no—
205C22—42no—
206C22—42no—
207C22—42no—
208C22—42no—
209C22—42no—
210C22—42no—
211C22C2×#23511yescharacteristic
212C22#100411yescharacteristic
213C22#100412yes—
214C22#100411yescharacteristic
215C22#101012yes—
216C4—22no—
217C4—41no—
218C4—41no—
219C4—42no—
220C4—122no—
221C4—122no—
222C4—122no—
223C4—122no—
224C3#283011yescharacteristic, complemented, Sylow
225C2—82no—
226C2#549411yescharacteristic
227C2#556112yes—
228C2#562112yes—
229C2#567211yescharacteristic
230C2#572011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#390C2
prime radicalC22×C3×D4C22
hypercenter#390D3
hypocenterC3#2830
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
3. #4981C2semidirect
4. #4979C2semidirect
5. #4952C2semidirect
6. #4952C2semidirect
7. #4948C2semidirect
8. #4947C2semidirect
9. #4933C2semidirect
224. C3#2830semidirect

Maximal subgroups

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

subgroupindex
C3×#3902
#52222
#49812
#49792
#49522
#49482
#49472
#49332
#28303