Finite Group Databasebeta

← #10592532⋊C3 #13142055

Subgroups — #10592532⋊C3 #13142055

Order 1728.

451 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C42×(C3⋊D3)⋊C3C211yesmaximal, characteristic
2H(3,𝔽3)×C4⋊C8C211yesmaximal, characteristic, complemented
3#63647⋊C3C212yesmaximal
4C8×(C3⋊Dic3)⋊C3C212yesmaximal, complemented
5C4⋊((C3⋊(C3⋊C8))⋊C3)C211yesmaximal, characteristic, complemented
6#10592532C311yesmaximal, characteristic, complemented
7C3×#4833—31nomaximal
8C2×C4×(C3⋊D3)⋊C3C411yescharacteristic
9C2×C4×(C3⋊D3)⋊C3C411yescharacteristic
10C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
11C2×C8×H(3,𝔽3)C2212yes—
12C42×H(3,𝔽3)C2211yescharacteristic
13C4×(C3⋊Dic3)⋊C3C411yescharacteristic
14C4×(C3⋊Dic3)⋊C3C411yescharacteristic
15C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
16C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
17C42×C3⋊D3C2×C311yescharacteristic
18C3×C42×D3—31no—
19C32×C4⋊C8—21no—
20C32×C4⋊C8D311yescharacteristic, complemented
21C32×C4⋊C8C2×C311yescharacteristic, complemented
22#63647C2×C312yes—
23C8×C3⋊Dic3C2×C312yescomplemented
24C4⋊(C3⋊(C3⋊C8))C2×C311yescharacteristic, complemented
25C3×(C2×D3)⋊C8—32no—
26C3×C8×Dic3—32no—
27C3×C4⋊(C3⋊C8)—31no—
28C22×(C3⋊D3)⋊C3C2×C411yescharacteristic
29C2×C4×H(3,𝔽3)C2×C411yescharacteristic
30C2×C4×H(3,𝔽3)C2×C411yescharacteristic
31C2×C4×H(3,𝔽3)C2311yescharacteristic
32C2×(C3⋊Dic3)⋊C3C2×C411yescharacteristic
33C2×(C3⋊Dic3)⋊C3C2×C411yescharacteristic
34C2×(C3⋊Dic3)⋊C3C2×C411yescharacteristic
35C4×(C3⋊D3)⋊C3—21no—
36C4×(C3⋊D3)⋊C3—21no—
37C4×(C3⋊D3)⋊C3—22no—
38C4×(C3⋊D3)⋊C3C814yescomplemented
39C8×H(3,𝔽3)—22no—
40(C3⋊(C3⋊C8))⋊C3—22no—
41C3×#431—91no—
42#4833—31no—
43#4833—91no—
44C2×C4×C3⋊D3C3×C411yescharacteristic
45C2×C4×C3⋊D3C3×C411yescharacteristic
46C2×C4×C3⋊D3C22×C311yescharacteristic
47C2×C3×C4×D3—31no—
48C2×C3×C4×D3—31no—
49C2×C3×C4×D3—31no—
50C2×C32×C8—22no—
51C2×C32×C8C2×D312yes—
52C2×C32×C8C22×C312yes—
53C32×C42—21no—
54C32×C42C2×D311yescharacteristic
55C32×C42C22×C311yescharacteristic
56C4×C3⋊Dic3C3×C411yescharacteristic
57C4×C3⋊Dic3C3×C411yescharacteristic
58C4×C3⋊Dic3C22×C311yescharacteristic
59C2×C3⋊(C3⋊C8)C22×C312yes—
60C3×C4×Dic3—31no—
61C3×C4×Dic3—31no—
62C3×C4×Dic3—31no—
63C2×C3×C3⋊C8—32no—
64C22×H(3,𝔽3)C22×C411yescharacteristic
65C2×(C3⋊D3)⋊C3—21no—
66C2×(C3⋊D3)⋊C3—21no—
67C2×(C3⋊D3)⋊C3C2×C812yes—
68C4×H(3,𝔽3)—21no—
69C4×H(3,𝔽3)C2×C812yes—
70C4×H(3,𝔽3)#4111yescharacteristic
71C4×H(3,𝔽3)#4111yescharacteristic
72(C3⋊Dic3)⋊C3—21no—
73(C3⋊Dic3)⋊C3C2×C812yes—
74(C3⋊Dic3)⋊C3M1612yes—
75C2×C3×C42—91no—
76C42×D3—31no—
77C42×D3—91no—
78C3×C4⋊C8—31no—
79C3×C4⋊C8—31no—
80C3×C4⋊C8—61no—
81C3×C4⋊C8—91no—
82C3×C4⋊C8C3×D311yescharacteristic
83C3×C22⋊C8—92no—
84C3×C4×C8—92no—
85(C2×D3)⋊C8—32no—
86(C2×D3)⋊C8—92no—
87C8×Dic3—32no—
88C8×Dic3—92no—
89C4⋊(C3⋊C8)—31no—
90C4⋊(C3⋊C8)—91no—
91C22×C3⋊D3C2×C3×C411yescharacteristic
92C22×C3×D3—31no—
93C2×C32×C4—21no—
94C2×C32×C4—21no—
95C2×C32×C4—21no—
96C2×C32×C4C4×D311yescharacteristic
97C2×C32×C4C4×D311yescharacteristic
98C2×C32×C4C2×C3×C411yescharacteristic
99C2×C32×C4C2×C3×C411yescharacteristic
100C2×C32×C4C22×D311yescharacteristic
101C2×C32×C4C23×C311yescharacteristic
102C2×C3⋊Dic3C2×C3×C411yescharacteristic
103C2×C3⋊Dic3C2×C3×C411yescharacteristic
104C2×C3⋊Dic3C2×C3×C411yescharacteristic
105C4×C3⋊D3—21no—
106C4×C3⋊D3—21no—
107C4×C3⋊D3—22no—
108C4×C3⋊D3C3×C814yescomplemented
109C2×C3×Dic3—31no—
110C2×C3×Dic3—31no—
111C2×C3×Dic3—31no—
112C3×C4×D3—34no—
113C3×C4×D3—62no—
114C3×C4×D3—61no—
115C3×C4×D3—61no—
116C32×C8—22no—
117C32×C8—22no—
118C32×C8—42no—
119C3⋊(C3⋊C8)—22no—
120C3×C3⋊C8—62no—
121#431—91noSylow
122C2×H(3,𝔽3)#11711yescharacteristic
123C2×H(3,𝔽3)C22×C811yescharacteristic
124C2×H(3,𝔽3)C2×M1611yescharacteristic
125(C3⋊D3)⋊C3—22no—
126C22×C3×C4—91no—
127C22×C3×C4—91no—
128C22×C3×C4—91no—
129C2×C4×D3—31no—
130C2×C4×D3—31no—
131C2×C4×D3—31no—
132C2×C4×D3—91no—
133C2×C4×D3—91no—
134C2×C4×D3—91no—
135C2×C3×C8—32no—
136C2×C3×C8—32no—
137C2×C3×C8—62no—
138C2×C3×C8—92no—
139C2×C3×C8C2×C3×D312yes—
140C3×C42—31no—
141C3×C42—31no—
142C3×C42—61no—
143C3×C42—91no—
144C3×C42—91no—
145C3×C42—91no—
146C3×C42C2×C3×D311yescharacteristic
147C4×Dic3—31no—
148C4×Dic3—31no—
149C4×Dic3—31no—
150C4×Dic3—91no—
151C4×Dic3—91no—
152C4×Dic3—91no—
153C2×C3⋊C8—32no—
154C2×C3⋊C8—92no—
155C22×C32—21no—
156C22×C32C2×C4×D311yescharacteristic
157C22×C32C22×C3×C411yescharacteristic
158C2×C3⋊D3—21no—
159C2×C3⋊D3—21no—
160C2×C3⋊D3C2×C3×C812yes—
161C2×C3×D3—32no—
162C2×C3×D3—61no—
163C2×C3×D3—61no—
164C32×C4—21no—
165C32×C4—21no—
166C32×C4—22no—
167C32×C4—21no—
168C32×C4—21no—
169C32×C4—41no—
170C32×C4D3×C812yes—
171C32×C4C2×C3×C812yes—
172C32×C4#23711yescharacteristic
173C32×C4#23911yescharacteristic
174C32×C4C3×#4111yescharacteristic
175C32×C4C3×#4111yescharacteristic
176C3⋊Dic3—21no—
177C3⋊Dic3C2×C3×C812yes—
178C3⋊Dic3C3×M1612yes—
179C3×Dic3—32no—
180C3×Dic3—32no—
181C3×Dic3—61no—
182C2×C42—91no—
183C4⋊C8—91no—
184C4⋊C8(C3⋊D3)⋊C311yescharacteristic, complemented
185C22⋊C8—92no—
186C4×C8—92no—
187H(3,𝔽3)#43111yescharacteristic, complemented, Sylow
188C23×C3—91no—
189C22×D3—31no—
190C22×D3—91no—
191C2×C3×C4—31no—
192C2×C3×C4—31no—
193C2×C3×C4—31no—
194C2×C3×C4—31no—
195C2×C3×C4—31no—
196C2×C3×C4—31no—
197C2×C3×C4—61no—
198C2×C3×C4—61no—
199C2×C3×C4—61no—
200C2×C3×C4—94no—
201C2×C3×C4—91no—
202C2×C3×C4—91no—
203C2×C3×C4—91no—
204C2×C3×C4—181no—
205C2×C3×C4—181no—
206C2×C3×C4—182no—
207C2×C3×C4C3×C4×D311yescharacteristic
208C2×C3×C4C3×C4×D311yescharacteristic
209C2×C3×C4C22×C3×D311yescharacteristic
210C2×Dic3—31no—
211C2×Dic3—31no—
212C2×Dic3—31no—
213C2×Dic3—91no—
214C2×Dic3—91no—
215C2×Dic3—91no—
216C4×D3—34no—
217C4×D3—62no—
218C4×D3—61no—
219C4×D3—61no—
220C4×D3—94no—
221C4×D3—182no—
222C4×D3—181no—
223C4×D3—181no—
224C3×C8—22no—
225C3×C8—62no—
226C3×C8—62no—
227C3×C8—122no—
228C3×C8—182no—
229C3⋊C8—62no—
230C3⋊C8—182no—
231C2×C32—21no—
232C2×C32—21no—
233C2×C32—21no—
234C2×C32#89211yescharacteristic
235C2×C32C2×D3×C811yescharacteristic
236C2×C32D3×M1611yescharacteristic
237C2×C32C3×#11711yescharacteristic
238C2×C32C22×C3×C811yescharacteristic
239C2×C32C2×C3×M1611yescharacteristic
240C3⋊D3—22no—
241C3×D3—62no—
242C22×C4—91no—
243C22×C4—91no—
244C22×C4—91no—
245C2×C8—92no—
246C2×C8C2×(C3⋊D3)⋊C312yes—
247C42—91no—
248C42—91no—
249C42—91no—
250C42C2×(C3⋊D3)⋊C311yescharacteristic
251C22×C3—31no—
252C22×C3—31no—
253C22×C3—61no—
254C22×C3—92no—
255C22×C3—181no—
256C22×C3—181no—
257C22×C3C2×C3×C4×D311yescharacteristic
258C2×D3—32no—
259C2×D3—61no—
260C2×D3—61no—
261C2×D3—92no—
262C2×D3—181no—
263C2×D3—181no—
264C3×C4—21no—
265C3×C4—31no—
266C3×C4—31no—
267C3×C4—31no—
268C3×C4—31no—
269C3×C4—32no—
270C3×C4—32no—
271C3×C4—61no—
272C3×C4—61no—
273C3×C4—61no—
274C3×C4—61no—
275C3×C4—62no—
276C3×C4—92no—
277C3×C4—92no—
278C3×C4—121no—
279C3×C4—181no—
280C3×C4C3×D3×C812yes—
281C3×C4C3×#23711yescharacteristic
282C3×C4C3×#23911yescharacteristic
283Dic3—32no—
284Dic3—32no—
285Dic3—61no—
286Dic3—92no—
287Dic3—92no—
288Dic3—181no—
289C32—21no—
290C32#483311yescharacteristic, complemented
291C32C3×#43111yescharacteristic, complemented
292C23—91no—
293C2×C4—91no—
294C2×C4—91no—
295C2×C4—94no—
296C2×C4—91no—
297C2×C4—182no—
298C2×C4—181no—
299C2×C4—181no—
300C2×C4C4×(C3⋊D3)⋊C311yescharacteristic
301C2×C4C4×(C3⋊D3)⋊C311yescharacteristic
302C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
303C8—22no—
304C8—182no—
305C2×C3—31no—
306C2×C3—31no—
307C2×C3—31no—
308C2×C3—31no—
309C2×C3—31no—
310C2×C3—31no—
311C2×C3—61no—
312C2×C3—61no—
313C2×C3—61no—
314C2×C3—182no—
315C2×C3C3×#89211yescharacteristic
316C2×C3C2×C3×D3×C811yescharacteristic
317C2×C3C3×D3×M1611yescharacteristic
318D3—62no—
319D3—182no—
320C22—92no—
321C22—181no—
322C22—181no—
323C22C2×C4×(C3⋊D3)⋊C311yescharacteristic
324C4—21no—
325C4—92no—
326C4—92no—
327C4—181no—
328C4C8×(C3⋊D3)⋊C312yes—
329C4#3830⋊C311yescharacteristic
330C4#8872211yescharacteristic
331C3—31no—
332C3—31no—
333C3—61no—
334C3C3×#483311yescharacteristic
335C2—182no—
336C2#64101⋊C311yescharacteristic
337C2C2×C8×(C3⋊D3)⋊C311yescharacteristic
338C2M16×(C3⋊D3)⋊C311yescharacteristic
subgroupG/H
centerC2×C4C22×(C3⋊D3)⋊C3
commutatorC2×C32C22×C3×C8
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×C4⋊C8C2
socleC22×C3C2×C3×C4×D3
prime radicalC22×(C3⋊D3)⋊C3C2×C4
cosocleC2×C32×C4C23×C3
hypercenterC4⋊C8(C3⋊D3)⋊C3
hypocenterC32C3×#431
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C4 ◁ C4⋊C8

Complements

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

NH ≅ G/Ntype
2. H(3,𝔽3)×C4⋊C8C2semidirect
4. C8×(C3⋊Dic3)⋊C3C2semidirect
5. C4⋊((C3⋊(C3⋊C8))⋊C3)C2semidirect
6. #10592532C3semidirect
20. C32×C4⋊C8D3semidirect
21. C32×C4⋊C8C2×C3semidirect
23. C8×C3⋊Dic3C2×C3semidirect
24. C4⋊(C3⋊(C3⋊C8))C2×C3semidirect
38. C4×(C3⋊D3)⋊C3C8semidirect
108. C4×C3⋊D3C3×C8semidirect
184. C4⋊C8(C3⋊D3)⋊C3semidirect
187. H(3,𝔽3)#431semidirect
290. C32#4833semidirect
291. C32C3×#431semidirect

Maximal subgroups

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

subgroupindex
C42×(C3⋊D3)⋊C32
H(3,𝔽3)×C4⋊C82
#63647⋊C32
C8×(C3⋊Dic3)⋊C32
C4⋊((C3⋊(C3⋊C8))⋊C3)2
#105925323
C3×#48333