Finite Group Databasebeta

← #379⋊H(3,𝔽3) #13141228

Subgroups — #379⋊H(3,𝔽3) #13141228

Order 1728.

623 conjugacy classes of subgroups; only the 36 normal subgroups are shown.

normal subgroups only

This group has 623 conjugacy classes of subgroups — too many to draw. Only its 36 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×C32⋊(C22⋊A4)C211yesmaximal, characteristic
2C32×#379C311yesmaximal, characteristic, complemented
3C3×#379⋊C3C313yesmaximal, complemented
4C32⋊(C22⋊A4)C411yescharacteristic, complemented
5C4×C22⋊H(3,𝔽3)—41nomaximal
6C25×C32C2×C311yescharacteristic
7C2×C3×C22⋊A4C2×C313yes—
8C2×C32×C22⋊C4—31no—
9C2×C32×C22⋊C4—31no—
10C2×C22⋊H(3,𝔽3)—41no—
11C2×C22⋊H(3,𝔽3)—82no—
12C3×#379—31no—
13C3×#379C3211yescharacteristic
14#379⋊C3—33no—
15C24×C32—31no—
16C24×C32—31no—
17C24×C32—62no—
18C24×C32—62no—
19C24×C32C3×C411yescharacteristic, complemented
20C3×C22⋊A4C3×C413yescomplemented
21C22×C32×C4—31no—
22C22×C32×C4A411yescharacteristic, complemented
23C3×C4×A4—43no—
24C32×C22⋊C4—62no—
25C32×C22⋊C4—62no—
26C22⋊H(3,𝔽3)—41no—
27C22⋊H(3,𝔽3)—82no—
28C4×H(3,𝔽3)—161no—
29C25×C3—31no—
30C25×C3C2×C3211yescharacteristic
31C2×C22⋊A4—33no—
32C2×C3×C22⋊C4—33no—
33C2×C3×C22⋊C4—33no—
34C2×C3×C22⋊C4—31no—
35C2×C3×C22⋊C4—31no—
36C23×C32—22no—
37C23×C32—32no—
38C23×C32—32no—
39C23×C32—31no—
40C23×C32—31no—
41C23×C32—62no—
42C23×C32—62no—
43C23×C32—62no—
44C23×C32—62no—
45C23×C32—62no—
46C23×C32—64no—
47C23×C32—62no—
48C23×C32—62no—
49C23×C32—64no—
50C23×C32C2×A411yescharacteristic
51C2×C3×A4—43no—
52C2×C3×A4—86no—
53C2×C32×C4—61no—
54C2×C32×C4—62no—
55C2×C32×C4—61no—
56#379H(3,𝔽3)11yescharacteristic, complemented, Sylow
57C2×H(3,𝔽3)—161no—
58C24×C3—33no—
59C24×C3—31no—
60C24×C3—33no—
61C24×C3—31no—
62C24×C3—31no—
63C24×C3—62no—
64C24×C3—62no—
65C24×C3—66no—
66C24×C3—66no—
67C24×C3C32×C411yescharacteristic
68C22⋊A4—33no—
69C22×C3×C4—33no—
70C22×C3×C4—31no—
71C22×C3×C4—31no—
72C22×C3×C4C3×A411yescharacteristic
73C4×A4—123no—
74C3×C22⋊C4—62no—
75C3×C22⋊C4—66no—
76C3×C22⋊C4—66no—
77C3×C22⋊C4—62no—
78C22×C32—22no—
79C22×C32—31no—
80C22×C32—31no—
81C22×C32—32no—
82C22×C32—32no—
83C22×C32—64no—
84C22×C32—62no—
85C22×C32—62no—
86C22×C32—62no—
87C22×C32—62no—
88C22×C32—62no—
89C22×C32—62no—
90C22×C32—64no—
91C22×C32—62no—
92C22×C32C4×A411yescharacteristic
93C3×A4—43no—
94C3×A4—86no—
95C32×C4—41no—
96C32×C4—121no—
97C32×C4—163no—
98C25C2×H(3,𝔽3)11yescharacteristic
99C2×C22⋊C4—31no—
100C2×C22⋊C4—31no—
101H(3,𝔽3)—161noSylow
102C23×C3—22no—
103C23×C3—32no—
104C23×C3—36no—
105C23×C3—36no—
106C23×C3—32no—
107C23×C3—33no—
108C23×C3—33no—
109C23×C3—31no—
110C23×C3—31no—
111C23×C3—31no—
112C23×C3—62no—
113C23×C3—62no—
114C23×C3—62no—
115C23×C3—66no—
116C23×C3—62no—
117C23×C3—66no—
118C23×C3—66no—
119C23×C3—62no—
120C23×C3—62no—
121C23×C3—64no—
122C23×C3—66no—
123C23×C3—612no—
124C23×C3—62no—
125C23×C3—62no—
126C23×C3—66no—
127C23×C3—66no—
128C23×C3—66no—
129C23×C3—64no—
130C23×C3—612no—
131C23×C3C2×C3×A411yescharacteristic
132C2×A4—123no—
133C2×A4—246no—
134C2×C3×C4—61no—
135C2×C3×C4—63no—
136C2×C3×C4—62no—
137C2×C3×C4—66no—
138C2×C3×C4—63no—
139C2×C3×C4—61no—
140C2×C32—31no—
141C2×C32—31no—
142C2×C32—62no—
143C2×C32—62no—
144C2×C32—163no—
145C2×C32#120⋊C311yescharacteristic
146C24—31no—
147C24—31no—
148C24—62no—
149C24—62no—
150C24C4×H(3,𝔽3)11yescharacteristic, complemented
151C22×C4—31no—
152C22×C4C22⋊H(3,𝔽3)11yescharacteristic, complemented
153C22⋊C4—62no—
154C22⋊C4—62no—
155C22×C3—22no—
156C22×C3—31no—
157C22×C3—33no—
158C22×C3—33no—
159C22×C3—31no—
160C22×C3—36no—
161C22×C3—36no—
162C22×C3—32no—
163C22×C3—32no—
164C22×C3—31no—
165C22×C3—62no—
166C22×C3—612no—
167C22×C3—66no—
168C22×C3—66no—
169C22×C3—66no—
170C22×C3—64no—
171C22×C3—62no—
172C22×C3—62no—
173C22×C3—62no—
174C22×C3—66no—
175C22×C3—62no—
176C22×C3—62no—
177C22×C3—62no—
178C22×C3—66no—
179C22×C3—66no—
180C22×C3—66no—
181C22×C3—612no—
182C22×C3—64no—
183C22×C3—62no—
184C22×C3C3×C4×A411yescharacteristic
185A4—123no—
186A4—246no—
187C3×C4—41no—
188C3×C4—121no—
189C3×C4—123no—
190C3×C4—121no—
191C3×C4—483no—
192C32—163no—
193C32#379⋊C311yescharacteristic, complemented
194C23—22no—
195C23—32no—
196C23—32no—
197C23—31no—
198C23—31no—
199C23—62no—
200C23—62no—
201C23—62no—
202C23—62no—
203C23—62no—
204C23—64no—
205C23—62no—
206C23—62no—
207C23—64no—
208C23C2×C22⋊H(3,𝔽3)11yescharacteristic
209C2×C4—61no—
210C2×C4—62no—
211C2×C4—61no—
212C2×C3—33no—
213C2×C3—33no—
214C2×C3—31no—
215C2×C3—31no—
216C2×C3—31no—
217C2×C3—66no—
218C2×C3—66no—
219C2×C3—62no—
220C2×C3—62no—
221C2×C3—483no—
222C2×C3C3×#120⋊C311yescharacteristic
223C22—22no—
224C22—31no—
225C22—31no—
226C22—32no—
227C22—32no—
228C22—64no—
229C22—62no—
230C22—62no—
231C22—62no—
232C22—62no—
233C22—62no—
234C22—62no—
235C22—64no—
236C22—62no—
237C22C4×C22⋊H(3,𝔽3)11yescharacteristic
238C4—41no—
239C4—121no—
240C3—31no—
241C3—483no—
242C3C3×#379⋊C311yescharacteristic
243C2—31no—
244C2—31no—
245C2—62no—
246C2—62no—
247C2#120⋊H(3,𝔽3)11yescharacteristic
subgroupG/H
centerC2×C3C3×#120⋊C3
commutatorC24×C3C32×C4
Frattini, socleC23×C3C2×C3×A4
FittingC32×#379C3
prime radicalC2×C32⋊(C22⋊A4)C2
cosocleC25×C3C2×C32
hypercenterC2×C32#120⋊C3
hypocenterC24C4×H(3,𝔽3)
LCSG ▷ C24×C3 ▷ C24
UCSC1 ◁ C2×C3 ◁ C2×C32

Complements

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

NH ≅ G/Ntype
2. C32×#379C3semidirect
3. C3×#379⋊C3C3semidirect
4. C32⋊(C22⋊A4)C4semidirect
19. C24×C32C3×C4semidirect
20. C3×C22⋊A4C3×C4semidirect
22. C22×C32×C4A4semidirect
56. #379H(3,𝔽3)semidirect
150. C24C4×H(3,𝔽3)semidirect
152. C22×C4C22⋊H(3,𝔽3)semidirect
193. C32#379⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×C32⋊(C22⋊A4)2
C32×#3793
C3×#379⋊C33
C4×C22⋊H(3,𝔽3)4