Finite Group Databasebeta

← #10592530⋊C3 #13142047

Subgroups — #10592530⋊C3 #13142047

Order 1728.

367 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 367 conjugacy classes of subgroups — too many to draw. Only its 122 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
1C4×#634⋊C3C211yesmaximal, characteristic
2H(3,𝔽3)×C4⋊C8C211yesmaximal, characteristic
3#63642⋊C3C212yesmaximal
4#63641⋊C3C212yesmaximal
5C4×(C3⋊(C3⋊C8))⋊C3C211yesmaximal, characteristic
6#10592530C311yesmaximal, characteristic, complemented
7C3×#4831—31nomaximal
8C2×#634⋊C3C411yescharacteristic
9C2×C8×H(3,𝔽3)C2212yes—
10C42×H(3,𝔽3)C2211yescharacteristic
11C4⋊((C3⋊Dic3)⋊C3)C411yescharacteristic
12#3750⋊C3C412yes—
13C4×(C3⋊Dic3)⋊C3C2212yes—
14C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
15C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
16C4×#634C2×C311yescharacteristic
17C3×C4×Dic6—31no—
18C32×C4⋊C8—21no—
19C32×C4⋊C8D311yescharacteristic
20C32×C4⋊C8C2×C311yescharacteristic
21#63642C2×C312yes—
22#63641C2×C312yes—
23C4×C3⋊(C3⋊C8)C2×C311yescharacteristic
24C3×#815—32no—
25C3×Dic3⋊C8—32no—
26C3×C4×C3⋊C8—31no—
27C2×C4×H(3,𝔽3)C2×C411yescharacteristic
28C2×C4×H(3,𝔽3)C2×C411yescharacteristic
29C2×C4×H(3,𝔽3)C2311yescharacteristic
30C2×(C3⋊Dic3)⋊C3C2×C412yes—
31C2×(C3⋊Dic3)⋊C3C2×C412yes—
32#634⋊C3—22no—
33C8×H(3,𝔽3)—22no—
34(C3⋊(C3⋊C8))⋊C3—21no—
35(C3⋊(C3⋊C8))⋊C3Q812yescomplemented
36C3×#446—91no—
37#4831—31no—
38#4831—91no—
39C2×#634C3×C411yescharacteristic
40C2×C3×Dic6—31no—
41C2×C32×C8—22no—
42C2×C32×C8C2×D312yes—
43C2×C32×C8C22×C312yes—
44C32×C42—21no—
45C32×C42C2×D311yescharacteristic
46C32×C42C22×C311yescharacteristic
47C4⋊(C3⋊Dic3)C3×C411yescharacteristic
48#3750C3×C412yes—
49C4×C3⋊Dic3C22×C312yes—
50C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
51C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
52C3×C4⋊Dic3—31no—
53C3×Dic3⋊C4—32no—
54C3×C4×Dic3—32no—
55C2×C3×C3⋊C8—31no—
56C2×C3×C3⋊C8—31no—
57C22×H(3,𝔽3)C22×C411yescharacteristic
58C4×H(3,𝔽3)—21no—
59C4×H(3,𝔽3)M1612yes—
60C4×H(3,𝔽3)C2×Q811yescharacteristic
61C4×H(3,𝔽3)#4111yescharacteristic
62(C3⋊Dic3)⋊C3—22no—
63(C3⋊Dic3)⋊C3—22no—
64C3×C4×Q8—91no—
65C4×Dic6—31no—
66C4×Dic6—91no—
67C3×C4⋊C8—31no—
68C3×C4⋊C8—31no—
69C3×C4⋊C8—61no—
70C3×C4⋊C8—92no—
71C3×C4⋊C8C3×D311yescharacteristic
72C3×#97—92no—
73C3×C4×C8—91no—
74#815—32no—
75#815—92no—
76Dic3⋊C8—32no—
77Dic3⋊C8—92no—
78C4×C3⋊C8—31no—
79C4×C3⋊C8—91no—
80C2×C32×C4—21no—
81C2×C32×C4—21no—
82C2×C32×C4—21no—
83C2×C32×C4C4×D311yescharacteristic
84C2×C32×C4C4×D311yescharacteristic
85C2×C32×C4C2×C3×C411yescharacteristic
86C2×C32×C4C2×C3×C411yescharacteristic
87C2×C32×C4C22×D311yescharacteristic
88C2×C32×C4C23×C311yescharacteristic
89C2×C3⋊Dic3C2×C3×C412yes—
90C2×C3⋊Dic3C2×C3×C412yes—
91#634—22no—
92C2×C3×Dic3—32no—
93C2×C3×Dic3—32no—
94C3×Dic6—62no—
95C32×C8—22no—
96C32×C8—22no—
97C32×C8—42no—
98C3⋊(C3⋊C8)—21no—
99C3⋊(C3⋊C8)C3×Q812yescomplemented
100C3×C3⋊C8—32no—
101C3×C3⋊C8—61no—
102#446—91noSylow
103C2×H(3,𝔽3)C4×Q811yescharacteristic
104C2×H(3,𝔽3)C2×M1611yescharacteristic
105C2×H(3,𝔽3)#13111yescharacteristic
106C2×C3×Q8—91no—
107C2×Dic6—31no—
108C2×Dic6—91no—
109C2×C3×C8—32no—
110C2×C3×C8—32no—
111C2×C3×C8—62no—
112C2×C3×C8—91no—
113C2×C3×C8—91no—
114C2×C3×C8C2×C3×D312yes—
115C3×C4⋊C4—92no—
116C3×C4⋊C4—91no—
117C3×C42—31no—
118C3×C42—31no—
119C3×C42—61no—
120C3×C42—92no—
121C3×C42C2×C3×D311yescharacteristic
122C4⋊Dic3—31no—
123C4⋊Dic3—91no—
124Dic3⋊C4—32no—
125Dic3⋊C4—92no—
126C4×Dic3—32no—
127C4×Dic3—92no—
128C2×C3⋊C8—31no—
129C2×C3⋊C8—31no—
130C2×C3⋊C8—91no—
131C2×C3⋊C8—91no—
132C22×C32—21no—
133C22×C32C2×C4×D311yescharacteristic
134C22×C32C22×C3×C411yescharacteristic
135C32×C4—21no—
136C32×C4—21no—
137C32×C4—22no—
138C32×C4—21no—
139C32×C4—21no—
140C32×C4—41no—
141C32×C4ΓL(1,𝔽25)12yes—
142C32×C4C3×M1612yes—
143C32×C4#23711yescharacteristic
144C32×C4D3×Q811yescharacteristic
145C32×C4C2×C3×Q811yescharacteristic
146C32×C4C3×#4111yescharacteristic
147C3⋊Dic3—22no—
148C3⋊Dic3—22no—
149C3×Dic3—62no—
150C3×Dic3—62no—
151C4×Q8—91no—
152C4⋊C8—92no—
153C4⋊C8(C3⋊D3)⋊C311yescharacteristic
154#97—92no—
155C4×C8—91no—
156H(3,𝔽3)#44611yescharacteristic, complemented, Sylow
157C3×Q8—182no—
158C2×C3×C4—31no—
159C2×C3×C4—31no—
160C2×C3×C4—31no—
161C2×C3×C4—31no—
162C2×C3×C4—31no—
163C2×C3×C4—31no—
164C2×C3×C4—61no—
165C2×C3×C4—61no—
166C2×C3×C4—61no—
167C2×C3×C4—92no—
168C2×C3×C4—92no—
169C2×C3×C4C3×C4×D311yescharacteristic
170C2×C3×C4C3×C4×D311yescharacteristic
171C2×C3×C4C22×C3×D311yescharacteristic
172C2×Dic3—32no—
173C2×Dic3—32no—
174C2×Dic3—92no—
175C2×Dic3—92no—
176Dic6—62no—
177Dic6—182no—
178C3×C8—22no—
179C3×C8—62no—
180C3×C8—62no—
181C3×C8—92no—
182C3×C8—122no—
183C3×C8—181no—
184C3⋊C8—32no—
185C3⋊C8—61no—
186C3⋊C8—92no—
187C3⋊C8—181no—
188C2×C32—21no—
189C2×C32—21no—
190C2×C32—21no—
191C2×C32#88711yescharacteristic
192C2×C32C2×ΓL(1,𝔽25)11yescharacteristic
193C2×C32#90711yescharacteristic
194C2×C32C3×C4×Q811yescharacteristic
195C2×C32C2×C3×M1611yescharacteristic
196C2×C32C3×#13111yescharacteristic
197C2×Q8—91no—
198C2×C8—91no—
199C2×C8—91no—
200C2×C8C2×(C3⋊D3)⋊C312yes—
201C4⋊C4—92no—
202C4⋊C4—91no—
203C42—92no—
204C42C2×(C3⋊D3)⋊C311yescharacteristic
205C22×C3—31no—
206C22×C3—31no—
207C22×C3—61no—
208C22×C3C2×C3×C4×D311yescharacteristic
209C3×C4—21no—
210C3×C4—31no—
211C3×C4—31no—
212C3×C4—31no—
213C3×C4—31no—
214C3×C4—32no—
215C3×C4—32no—
216C3×C4—61no—
217C3×C4—61no—
218C3×C4—61no—
219C3×C4—61no—
220C3×C4—62no—
221C3×C4—121no—
222C3×C4—182no—
223C3×C4—182no—
224C3×C4C3×ΓL(1,𝔽25)12yes—
225C3×C4C3×#23711yescharacteristic
226C3×C4C3×D3×Q811yescharacteristic
227Dic3—62no—
228Dic3—62no—
229Dic3—182no—
230Dic3—182no—
231C32—21no—
232C32#483111yescharacteristic, complemented
233C32C3×#44611yescharacteristic, complemented
234Q8—182no—
235C2×C4—92no—
236C2×C4—92no—
237C2×C4C4×(C3⋊D3)⋊C311yescharacteristic
238C2×C4C4×(C3⋊D3)⋊C311yescharacteristic
239C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
240C8—22no—
241C8—92no—
242C8—181no—
243C2×C3—31no—
244C2×C3—31no—
245C2×C3—31no—
246C2×C3—31no—
247C2×C3—31no—
248C2×C3—31no—
249C2×C3—61no—
250C2×C3—61no—
251C2×C3—61no—
252C2×C3C3×#88711yescharacteristic
253C2×C3C2×C3×ΓL(1,𝔽25)11yescharacteristic
254C2×C3C3×#90711yescharacteristic
255C22C2×C4×(C3⋊D3)⋊C311yescharacteristic
256C4—21no—
257C4—182no—
258C4—182no—
259C4#3743⋊C312yes—
260C4#3830⋊C311yescharacteristic
261C4Q8×(C3⋊D3)⋊C311yescharacteristic
262C3—31no—
263C3—31no—
264C3—61no—
265C3C3×#483111yescharacteristic
266C2#64096⋊C311yescharacteristic
267C2C2×#3743⋊C311yescharacteristic
268C2#64116⋊C311yescharacteristic
subgroupG/H
centerC2×C4C22×(C3⋊D3)⋊C3
commutatorC22×C32C22×C3×C4
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×C4⋊C8C2
socleC22×C3C2×C3×C4×D3
prime radicalC22×H(3,𝔽3)C22×C4
cosocleC2×C32×C4C23×C3
hypercenterC4⋊C8(C3⋊D3)⋊C3
hypocenterC32C3×#446
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C2×C4 ◁ C4⋊C8

Complements

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

NH ≅ G/Ntype
6. #10592530C3semidirect
35. (C3⋊(C3⋊C8))⋊C3Q8semidirect
99. C3⋊(C3⋊C8)C3×Q8semidirect
156. H(3,𝔽3)#446semidirect
232. C32#4831semidirect
233. C32C3×#446semidirect

Maximal subgroups

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

subgroupindex
C4×#634⋊C32
H(3,𝔽3)×C4⋊C82
#63642⋊C32
#63641⋊C32
C4×(C3⋊(C3⋊C8))⋊C32
#105925303
C3×#48313