Finite Group Databasebeta

← C2×#64156⋊C3 #13153660

Subgroups — C2×#64156⋊C3 #13153660

Order 1728.

1311 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×Q8×H(3,𝔽3)C211yesmaximal, characteristic, complemented
2C22×C4⋊((C3⋊D3)⋊C3)C211yesmaximal, characteristic
3#64156⋊C3C218yesmaximal, direct product factor
4C2×#3752⋊C3C214yesmaximal, complemented
5C2×C4×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic, complemented
6C2×#64156C311yesmaximal, characteristic, complemented
7C2×C3×#947—31nomaximal
8C23×(C3⋊D3)⋊C3C2212yes—
9C2×Q8×H(3,𝔽3)—24no—
10C2×Q8×H(3,𝔽3)C2214yescomplemented
11C22×C4×H(3,𝔽3)C2212yes—
12C22×C4×H(3,𝔽3)C2211yescharacteristic
13C22×(C3⋊Dic3)⋊C3C2212yes—
14C2×C4⋊((C3⋊D3)⋊C3)—24no—
15C2×C4⋊((C3⋊D3)⋊C3)C2214yes—
16#3752⋊C3C22116yescomplemented
17C4×(C3⋊Dic3)⋊C3C2214yescomplemented
18C22×C32×Q8—21no—
19C22×C32×Q8D311yescharacteristic, complemented
20C22×C32×Q8C2×C311yescharacteristic, complemented
21C22×C4⋊(C3⋊D3)C2×C311yescharacteristic
22C22×C3×D12—31no—
23#64156C2×C318yescomplemented
24C2×#3752C2×C314yescomplemented
25C2×C4×C3⋊Dic3C2×C311yescharacteristic, complemented
26C3×#947—38no—
27C2×C3×(C2×D3)⋊C4—34no—
28C2×C3×C4×Dic3—31no—
29C23×H(3,𝔽3)C2311yescharacteristic
30C22×(C3⋊D3)⋊C3—24no—
31C22×(C3⋊D3)⋊C3—28no—
32C22×(C3⋊D3)⋊C3C2314yes—
33Q8×H(3,𝔽3)—28no—
34C2×C4×H(3,𝔽3)—24no—
35C2×C4×H(3,𝔽3)D414yes—
36C2×C4×H(3,𝔽3)C2314yes—
37C2×C4×H(3,𝔽3)C2312yes—
38C2×(C3⋊Dic3)⋊C3—24no—
39C2×(C3⋊Dic3)⋊C3C2314yes—
40C4⋊((C3⋊D3)⋊C3)—28no—
41C2×C3×#124—91no—
42C2×#947—31no—
43C2×#947—91no—
44C23×C3⋊D3C22×C312yes—
45C23×C3×D3—32no—
46C2×C32×Q8—28no—
47C2×C32×Q8—24no—
48C2×C32×Q8—24no—
49C2×C32×Q8—24no—
50C2×C32×Q8C2×D314yescomplemented
51C2×C32×Q8C22×C314yescomplemented
52C22×C32×C4—22no—
53C22×C32×C4—21no—
54C22×C32×C4C2×D312yes—
55C22×C32×C4C2×D311yescharacteristic
56C22×C32×C4C22×C312yes—
57C22×C32×C4C22×C311yescharacteristic
58C22×C3⋊Dic3C22×C312yes—
59C2×C4⋊(C3⋊D3)—24no—
60C2×C4⋊(C3⋊D3)C22×C314yes—
61C22×C3×Dic3—32no—
62C2×C3×D12—34no—
63C2×C3×D12—64no—
64#3752C22×C3116yescomplemented
65C4×C3⋊Dic3C22×C314yescomplemented
66C3×(C2×D3)⋊C4—316no—
67C3×C4×Dic3—34no—
68C22×H(3,𝔽3)C2×D412yes—
69C22×H(3,𝔽3)#4114yes—
70C22×H(3,𝔽3)C2411yescharacteristic
71C2×(C3⋊D3)⋊C3—24no—
72C2×(C3⋊D3)⋊C3—28no—
73C2×(C3⋊D3)⋊C3—48no—
74C4×H(3,𝔽3)—24no—
75C4×H(3,𝔽3)C2×D414yes—
76(C3⋊Dic3)⋊C3—24no—
77C22×C3×Q8—31no—
78C22×C3×Q8—31no—
79C22×C3×Q8—61no—
80C22×C3×Q8C3×D311yescharacteristic
81C22×C3×D4—91no—
82C22×D12—31no—
83C22×D12—91no—
84C3×#124—98no—
85C2×C3×C22⋊C4—94no—
86C2×C3×C42—91no—
87#947—38no—
88#947—98no—
89C2×(C2×D3)⋊C4—34no—
90C2×(C2×D3)⋊C4—94no—
91C2×C4×Dic3—31no—
92C2×C4×Dic3—91no—
93C23×C32—21no—
94C23×C32C22×D311yescharacteristic
95C23×C32C23×C311yescharacteristic
96C22×C3⋊D3—24no—
97C22×C3⋊D3—28no—
98C22×C3⋊D3C23×C314yes—
99C22×C3×D3—34no—
100C22×C3×D3—68no—
101C22×C3×D3—64no—
102C32×Q8—216no—
103C32×Q8—28no—
104C32×Q8—28no—
105C2×C32×C4—28no—
106C2×C32×C4—24no—
107C2×C32×C4—24no—
108C2×C32×C4—24no—
109C2×C32×C4—24no—
110C2×C32×C4—22no—
111C2×C32×C4#6714yes—
112C2×C32×C4C3×D414yes—
113C2×C32×C4C22×D314yes—
114C2×C32×C4C22×D312yes—
115C2×C32×C4C23×C314yes—
116C2×C32×C4C23×C312yes—
117C2×C3⋊Dic3—24no—
118C2×C3⋊Dic3C23×C314yes—
119C4⋊(C3⋊D3)—28no—
120C2×C3×Dic3—34no—
121C2×C3×Dic3—64no—
122C3×D12—68no—
123C2×#124—91noSylow
124C2×H(3,𝔽3)#12414yescomplemented
125C2×H(3,𝔽3)C22×D411yescharacteristic
126C2×H(3,𝔽3)C2×#4112yes—
127(C3⋊D3)⋊C3—44no—
128C24×C3—92no—
129C23×D3—32no—
130C23×D3—92no—
131C2×C3×Q8—24no—
132C2×C3×Q8—34no—
133C2×C3×Q8—34no—
134C2×C3×Q8—64no—
135C2×C3×Q8—64no—
136C2×C3×Q8—68no—
137C2×C3×Q8—64no—
138C2×C3×Q8C2×C3×D314yes—
139C2×C3×D4—94no—
140C2×C3×D4—184no—
141C22×C3×C4—31no—
142C22×C3×C4—31no—
143C22×C3×C4—32no—
144C22×C3×C4—32no—
145C22×C3×C4—62no—
146C22×C3×C4—61no—
147C22×C3×C4—92no—
148C22×C3×C4C2×C3×D312yes—
149C22×C3×C4C2×C3×D311yescharacteristic
150C22×Dic3—32no—
151C22×Dic3—92no—
152C2×D12—34no—
153C2×D12—64no—
154C2×D12—94no—
155C2×D12—184no—
156C3×C22⋊C4—916no—
157C3×C42—94no—
158(C2×D3)⋊C4—316no—
159(C2×D3)⋊C4—916no—
160C4×Dic3—34no—
161C4×Dic3—94no—
162C22×C32—21no—
163C22×C32—24no—
164C22×C32—22no—
165C22×C32#23914yes—
166C22×C32C2×#6712yes—
167C22×C32C2×C3×D412yes—
168C22×C32C3×#4114yes—
169C22×C32C23×D311yescharacteristic
170C22×C32C24×C311yescharacteristic
171C2×C3⋊D3—24no—
172C2×C3⋊D3—28no—
173C2×C3⋊D3—48no—
174C2×C3×D3—68no—
175C2×C3×D3—64no—
176C2×C3×D3—128no—
177C32×C4—28no—
178C32×C4—24no—
179C32×C4—24no—
180C32×C4—24no—
181C32×C4C2×#6714yes—
182C32×C4C2×C3×D414yes—
183C3⋊Dic3—24no—
184C3×Dic3—64no—
185C22×Q8(C3⋊D3)⋊C311yescharacteristic, complemented
186C22×D4—91no—
187#124—98no—
188C2×C22⋊C4—94no—
189C2×C42—91no—
190H(3,𝔽3)C2×#12411yescharacteristic, complemented, Sylow
191C23×C3—31no—
192C23×C3—31no—
193C23×C3—61no—
194C23×C3—94no—
195C23×C3—184no—
196C23×C3—188no—
197C23×C3C22×C3×D311yescharacteristic
198C22×D3—34no—
199C22×D3—68no—
200C22×D3—64no—
201C22×D3—94no—
202C22×D3—184no—
203C22×D3—188no—
204C3×Q8—28no—
205C3×Q8—68no—
206C3×Q8—68no—
207C3×Q8—616no—
208C3×D4—188no—
209C2×C3×C4—24no—
210C2×C3×C4—34no—
211C2×C3×C4—34no—
212C2×C3×C4—32no—
213C2×C3×C4—32no—
214C2×C3×C4—34no—
215C2×C3×C4—34no—
216C2×C3×C4—64no—
217C2×C3×C4—64no—
218C2×C3×C4—68no—
219C2×C3×C4—64no—
220C2×C3×C4—64no—
221C2×C3×C4—62no—
222C2×C3×C4—94no—
223C2×C3×C4—184no—
224C2×C3×C4C3×#6714yes—
225C2×C3×C4C22×C3×D314yes—
226C2×C3×C4C22×C3×D312yes—
227C2×Dic3—34no—
228C2×Dic3—64no—
229C2×Dic3—94no—
230C2×Dic3—184no—
231D12—68no—
232D12—188no—
233C2×C32—24no—
234C2×C32—22no—
235C2×C32—21no—
236C2×C32#94714yescomplemented
237C2×C32C3×#12414yescomplemented
238C2×C32C2×#23912yes—
239C2×C32C22×#6711yescharacteristic
240C2×C32C22×C3×D411yescharacteristic
241C2×C32C2×C3×#4112yes—
242C3⋊D3—44no—
243C3×D3—124no—
244C24—92no—
245C2×Q8—24no—
246C2×Q8C2×(C3⋊D3)⋊C314yescomplemented
247C2×D4—94no—
248C2×D4—184no—
249C22×C4—92no—
250C22×C4C2×(C3⋊D3)⋊C312yes—
251C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
252C22⋊C4—916no—
253C42—94no—
254C22×C3—31no—
255C22×C3—31no—
256C22×C3—34no—
257C22×C3—34no—
258C22×C3—32no—
259C22×C3—32no—
260C22×C3—61no—
261C22×C3—64no—
262C22×C3—62no—
263C22×C3—184no—
264C22×C3—188no—
265C22×C3—368no—
266C22×C3C3×#23914yes—
267C22×C3C2×C3×#6712yes—
268C22×C3C23×C3×D311yescharacteristic
269C2×D3—68no—
270C2×D3—64no—
271C2×D3—128no—
272C2×D3—184no—
273C2×D3—188no—
274C2×D3—368no—
275C3×C4—24no—
276C3×C4—34no—
277C3×C4—34no—
278C3×C4—64no—
279C3×C4—64no—
280C3×C4—68no—
281C3×C4—64no—
282C3×C4—184no—
283C3×C4C2×C3×#6714yes—
284Dic3—64no—
285Dic3—184no—
286C32—21no—
287C32C2×#94711yescharacteristic, complemented
288C32C2×C3×#12411yescharacteristic, complemented
289C23—94no—
290C23—188no—
291C23—184no—
292C23C22×(C3⋊D3)⋊C311yescharacteristic
293Q8—28no—
294D4—188no—
295C2×C4—24no—
296C2×C4—94no—
297C2×C4—184no—
298C2×C4#638⋊C314yes—
299C2×C4C22×(C3⋊D3)⋊C314yes—
300C2×C4C22×(C3⋊D3)⋊C312yes—
301C2×C3—32no—
302C2×C3—32no—
303C2×C3—31no—
304C2×C3—31no—
305C2×C3—34no—
306C2×C3—34no—
307C2×C3—62no—
308C2×C3—61no—
309C2×C3—64no—
310C2×C3—364no—
311C2×C3C3×#94714yes—
312C2×C3C2×C3×#23912yes—
313C2×C3C22×C3×#6711yescharacteristic
314D3—124no—
315D3—364no—
316C22—188no—
317C22—184no—
318C22—368no—
319C22#8872214yes—
320C22C2×#638⋊C312yes—
321C22C23×(C3⋊D3)⋊C311yescharacteristic
322C4—24no—
323C4—184no—
324C4C2×#638⋊C314yes—
325C3—31no—
326C3—31no—
327C3—61no—
328C3C2×C3×#94711yescharacteristic
329C2—364no—
330C2#64156⋊C314yesdirect product factor
331C2C2×#8872212yes—
332C2C22×#638⋊C311yescharacteristic
subgroupG/H
centerC23C22×(C3⋊D3)⋊C3
commutator, cosocleC22×C32C24×C3
FrattiniC22×C3C23×C3×D3
FittingC22×Q8×H(3,𝔽3)C2
socleC23×C3C22×C3×D3
prime radicalC22×C4⋊((C3⋊D3)⋊C3)C2
hypercenterC22×Q8(C3⋊D3)⋊C3
hypocenterC32C2×C3×#124
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
1. C22×Q8×H(3,𝔽3)C2semidirect
3. #64156⋊C3C2direct (N × H)
4. C2×#3752⋊C3C2semidirect
5. C2×C4×(C3⋊Dic3)⋊C3C2semidirect
6. C2×#64156C3semidirect
10. C2×Q8×H(3,𝔽3)C22semidirect
16. #3752⋊C3C22semidirect
17. C4×(C3⋊Dic3)⋊C3C22semidirect
19. C22×C32×Q8D3semidirect
20. C22×C32×Q8C2×C3semidirect
23. #64156C2×C3semidirect
24. C2×#3752C2×C3semidirect
25. C2×C4×C3⋊Dic3C2×C3semidirect
50. C2×C32×Q8C2×D3semidirect
51. C2×C32×Q8C22×C3semidirect
64. #3752C22×C3semidirect
65. C4×C3⋊Dic3C22×C3semidirect
124. C2×H(3,𝔽3)#124semidirect
185. C22×Q8(C3⋊D3)⋊C3semidirect
190. H(3,𝔽3)C2×#124semidirect
236. C2×C32#947semidirect
237. C2×C32C3×#124semidirect
246. C2×Q8C2×(C3⋊D3)⋊C3semidirect
287. C32C2×#947semidirect
288. C32C2×C3×#124semidirect
330. C2#64156⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C22×Q8×H(3,𝔽3)2
C22×C4⋊((C3⋊D3)⋊C3)2
#64156⋊C32
C2×#3752⋊C32
C2×C4×(C3⋊Dic3)⋊C32
C2×#641563
C2×C3×#9473