Finite Group Databasebeta

← #64094⋊C3 #11721425

Subgroups — #64094⋊C3 #11721425

Order 864.

300 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 300 conjugacy classes of subgroups — too many to draw. Only its 71 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×#638⋊C3C211yesmaximal, characteristic
2C2×#634⋊C3C211yesmaximal, characteristic, complemented
3C22⋊C4×H(3,𝔽3)C211yesmaximal, characteristic, complemented
4#3757⋊C3C211yesmaximal, characteristic, complemented
5#3752⋊C3C211yesmaximal, characteristic, complemented
6#3752⋊C3C211yesmaximal, characteristic, complemented
7C4×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic, complemented
8#64094C311yesmaximal, characteristic, complemented
9C3×#885—31nomaximal
10C23×H(3,𝔽3)C2211yescharacteristic
11C22×(C3⋊D3)⋊C3C2211yescharacteristic
12C2×C4×H(3,𝔽3)C2211yescharacteristic
13C2×C4×H(3,𝔽3)C2211yescharacteristic
14#638⋊C3—22no—
15C2×(C3⋊Dic3)⋊C3C2211yescharacteristic
16C2×(C3⋊Dic3)⋊C3C2211yescharacteristic
17C2×(C3⋊Dic3)⋊C3C2211yescharacteristic
18#634⋊C3—22no—
19C2×#638C2×C311yescharacteristic
20C2×#634C2×C311yescharacteristic, complemented
21C2×C3×#67—31no—
22C2×C3×Dic6—31no—
23C32×C22⋊C4—21no—
24C32×C22⋊C4D311yescharacteristic, complemented
25C32×C22⋊C4C2×C311yescharacteristic, complemented
26#3757C2×C311yescharacteristic, complemented
27#3752C2×C311yescharacteristic, complemented
28#3752C2×C311yescharacteristic, complemented
29C4×C3⋊Dic3C2×C311yescharacteristic, complemented
30C3×#217—31no—
31C3×(C2×D3)⋊C4—31no—
32C3×(C2×D3)⋊C4—31no—
33C3×C4×Dic3—31no—
34C22×H(3,𝔽3)—21no—
35C22×H(3,𝔽3)—21no—
36C22×H(3,𝔽3)—21no—
37C22×H(3,𝔽3)C2311yescharacteristic
38C2×(C3⋊D3)⋊C3—21no—
39C2×(C3⋊D3)⋊C3—21no—
40C2×(C3⋊D3)⋊C3—21no—
41C4×H(3,𝔽3)—21no—
42C4×H(3,𝔽3)—21no—
43(C3⋊Dic3)⋊C3—21no—
44(C3⋊Dic3)⋊C3—21no—
45(C3⋊Dic3)⋊C3D412yes—
46C3×#124—91no—
47#885—31no—
48#885—91no—
49C23×C32—21no—
50C23×C32C2×D311yescharacteristic
51C23×C32C22×C311yescharacteristic
52C22×C3⋊D3C22×C311yescharacteristic
53C22×C3×D3—31no—
54C2×C32×C4—21no—
55C2×C32×C4—21no—
56C2×C32×C4C2×D311yescharacteristic
57C2×C32×C4C2×D311yescharacteristic
58C2×C32×C4C22×C311yescharacteristic
59C2×C32×C4C22×C311yescharacteristic
60#638—22no—
61C2×C3⋊Dic3C22×C311yescharacteristic
62C2×C3⋊Dic3C22×C311yescharacteristic
63C2×C3⋊Dic3C22×C311yescharacteristic
64#634—22no—
65C3×#67—62no—
66C2×C3×Dic3—31no—
67C2×C3×Dic3—31no—
68C2×C3×Dic3—31no—
69C3×Dic6—62no—
70C2×H(3,𝔽3)—41no—
71C2×H(3,𝔽3)C2×D411yescharacteristic
72C2×H(3,𝔽3)#4111yescharacteristic
73C2×H(3,𝔽3)#4111yescharacteristic
74(C3⋊D3)⋊C3—41no—
75C2×C3×Q8—91no—
76C2×C3×D4—91no—
77C2×#67—31no—
78C2×#67—91no—
79C2×Dic6—31no—
80C2×Dic6—91no—
81C3×C22⋊C4—31no—
82C3×C22⋊C4—31no—
83C3×C22⋊C4—61no—
84C3×C22⋊C4—91no—
85C3×C22⋊C4—91no—
86C3×C22⋊C4—91no—
87C3×C22⋊C4C3×D311yescharacteristic
88C3×C42—91no—
89#217—31no—
90#217—91no—
91(C2×D3)⋊C4—31no—
92(C2×D3)⋊C4—31no—
93(C2×D3)⋊C4—91no—
94(C2×D3)⋊C4—91no—
95C4×Dic3—31no—
96C4×Dic3—91no—
97C22×C32—22no—
98C22×C32—21no—
99C22×C32—21no—
100C22×C32—21no—
101C22×C32—21no—
102C22×C32—21no—
103C22×C32—21no—
104C22×C32—21no—
105C22×C32—41no—
106C22×C32—41no—
107C22×C32C22×D311yescharacteristic
108C22×C32C23×C311yescharacteristic
109C2×C3⋊D3—21no—
110C2×C3⋊D3—21no—
111C2×C3⋊D3—21no—
112C2×C3×D3—61no—
113C2×C3×D3—61no—
114C2×C3×D3—61no—
115C32×C4—21no—
116C32×C4—21no—
117C32×C4—21no—
118C32×C4—21no—
119C32×C4—41no—
120C32×C4—41no—
121C3⋊Dic3—21no—
122C3⋊Dic3—21no—
123C3⋊Dic3C3×D412yes—
124C3×Dic3—32no—
125C3×Dic3—61no—
126C3×Dic3—61no—
127#124—91noSylow
128H(3,𝔽3)#12411yescharacteristic, complemented, Sylow
129C23×C3—31no—
130C23×C3—31no—
131C23×C3—61no—
132C23×C3—91no—
133C23×C3C2×C3×D311yescharacteristic
134C22×D3—31no—
135C22×D3—91no—
136C3×Q8—182no—
137C3×D4—182no—
138C2×C3×C4—31no—
139C2×C3×C4—31no—
140C2×C3×C4—31no—
141C2×C3×C4—31no—
142C2×C3×C4—61no—
143C2×C3×C4—61no—
144C2×C3×C4—91no—
145C2×C3×C4—91no—
146C2×C3×C4—91no—
147C2×C3×C4C2×C3×D311yescharacteristic
148C2×C3×C4C2×C3×D311yescharacteristic
149#67—62no—
150#67—182no—
151C2×Dic3—31no—
152C2×Dic3—31no—
153C2×Dic3—31no—
154C2×Dic3—91no—
155C2×Dic3—91no—
156C2×Dic3—91no—
157Dic6—62no—
158Dic6—182no—
159C2×C32—21no—
160C2×C32—21no—
161C2×C32—21no—
162C2×C32—42no—
163C2×C32—41no—
164C2×C32—41no—
165C2×C32#23511yescharacteristic
166C2×C32D3×D411yescharacteristic
167C2×C32#23711yescharacteristic
168C2×C32C2×C3×D411yescharacteristic
169C2×C32C3×#4111yescharacteristic
170C2×C32C3×#4111yescharacteristic
171C3⋊D3—41no—
172C3×D3—121no—
173C2×Q8—91no—
174C2×D4—91no—
175C22⋊C4—91no—
176C22⋊C4—91no—
177C22⋊C4—91no—
178C22⋊C4(C3⋊D3)⋊C311yescharacteristic, complemented
179C42—91no—
180C22×C3—21no—
181C22×C3—21no—
182C22×C3—21no—
183C22×C3—31no—
184C22×C3—31no—
185C22×C3—61no—
186C22×C3—61no—
187C22×C3—61no—
188C22×C3—61no—
189C22×C3—61no—
190C22×C3—61no—
191C22×C3—62no—
192C22×C3—61no—
193C22×C3—121no—
194C22×C3—121no—
195C22×C3—181no—
196C22×C3—181no—
197C22×C3—181no—
198C22×C3C22×C3×D311yescharacteristic
199C2×D3—61no—
200C2×D3—61no—
201C2×D3—61no—
202C2×D3—181no—
203C2×D3—181no—
204C2×D3—181no—
205C3×C4—21no—
206C3×C4—21no—
207C3×C4—61no—
208C3×C4—61no—
209C3×C4—61no—
210C3×C4—61no—
211C3×C4—92no—
212C3×C4—121no—
213C3×C4—121no—
214C3×C4—181no—
215C3×C4—181no—
216Dic3—32no—
217Dic3—61no—
218Dic3—61no—
219Dic3—92no—
220Dic3—181no—
221Dic3—181no—
222C32—21no—
223C32#88511yescharacteristic, complemented
224C32C3×#12411yescharacteristic, complemented
225C23—91no—
226C23C2×(C3⋊D3)⋊C311yescharacteristic
227Q8—182no—
228D4—182no—
229C2×C4—91no—
230C2×C4—91no—
231C2×C4—91no—
232C2×C4C2×(C3⋊D3)⋊C311yescharacteristic
233C2×C4C2×(C3⋊D3)⋊C311yescharacteristic
234C2×C3—31no—
235C2×C3—31no—
236C2×C3—31no—
237C2×C3—31no—
238C2×C3—31no—
239C2×C3—31no—
240C2×C3—41no—
241C2×C3—61no—
242C2×C3—61no—
243C2×C3—61no—
244C2×C3—121no—
245C2×C3—121no—
246C2×C3—122no—
247C2×C3—361no—
248C2×C3C3×#23511yescharacteristic
249C2×C3C3×D3×D411yescharacteristic
250C2×C3C3×#23711yescharacteristic
251D3—121no—
252D3—361no—
253C22—21no—
254C22—21no—
255C22—21no—
256C22—181no—
257C22—181no—
258C22—181no—
259C22C22×(C3⋊D3)⋊C311yescharacteristic
260C4—21no—
261C4—21no—
262C4—92no—
263C4—181no—
264C4—181no—
265C3—31no—
266C3—31no—
267C3—61no—
268C3C3×#88511yescharacteristic
269C2—41no—
270C2—361no—
271C2#3828⋊C311yescharacteristic
272C2D4×(C3⋊D3)⋊C311yescharacteristic
273C2#3830⋊C311yescharacteristic
subgroupG/H
centerC22C22×(C3⋊D3)⋊C3
commutator, cosocleC22×C32C23×C3
Frattini, socleC22×C3C22×C3×D3
FittingC22⋊C4×H(3,𝔽3)C2
prime radicalC2×#638⋊C3C2
hypercenterC22⋊C4(C3⋊D3)⋊C3
hypocenterC32C3×#124
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#634⋊C3C2semidirect
3. C22⋊C4×H(3,𝔽3)C2semidirect
4. #3757⋊C3C2semidirect
5. #3752⋊C3C2semidirect
6. #3752⋊C3C2semidirect
7. C4×(C3⋊Dic3)⋊C3C2semidirect
8. #64094C3semidirect
20. C2×#634C2×C3semidirect
24. C32×C22⋊C4D3semidirect
25. C32×C22⋊C4C2×C3semidirect
26. #3757C2×C3semidirect
27. #3752C2×C3semidirect
28. #3752C2×C3semidirect
29. C4×C3⋊Dic3C2×C3semidirect
128. H(3,𝔽3)#124semidirect
178. C22⋊C4(C3⋊D3)⋊C3semidirect
223. C32#885semidirect
224. C32C3×#124semidirect

Maximal subgroups

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

subgroupindex
C2×#638⋊C32
C2×#634⋊C32
C22⋊C4×H(3,𝔽3)2
#3757⋊C32
#3752⋊C32
C4×(C3⋊Dic3)⋊C32
#640943
C3×#8853