Finite Group Databasebeta

← C2×#64133⋊C3 #13153551

Subgroups — C2×#64133⋊C3 #13153551

Order 1728.

1363 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C4×H(3,𝔽3)C211yesmaximal, characteristic
2C22×#634⋊C3C211yesmaximal, characteristic, complemented
3C2×#3757⋊C3C212yesmaximal
4C2×C4⋊((C3⋊Dic3)⋊C3)C211yesmaximal, characteristic, complemented
5#64133⋊C3C218yesmaximal, direct product factor
6C2×#3750⋊C3C212yesmaximal, complemented
7C2×#64133C311yesmaximal, characteristic, complemented
8C2×C3×#924—31nomaximal
9C24×H(3,𝔽3)C2211yescharacteristic
10C22×C4×H(3,𝔽3)—24no—
11C22×C4×H(3,𝔽3)C2211yescharacteristic
12C22×C4×H(3,𝔽3)C2214yes—
13C22×C4×H(3,𝔽3)C2211yescharacteristic
14C22×(C3⋊Dic3)⋊C3C2212yes—
15C22×(C3⋊Dic3)⋊C3C2212yes—
16C2×#634⋊C3—24no—
17C2×#634⋊C3C2214yescomplemented
18#3757⋊C3C2218yes—
19C4⋊((C3⋊Dic3)⋊C3)C2214yescomplemented
20#3750⋊C3C2218yescomplemented
21C23×C32×C4—21no—
22C23×C32×C4D311yescharacteristic
23C23×C32×C4C2×C311yescharacteristic
24C22×#634C2×C311yescharacteristic, complemented
25C22×C3×Dic6—31no—
26C2×#3757C2×C312yes—
27C2×C4⋊(C3⋊Dic3)C2×C311yescharacteristic, complemented
28#64133C2×C318yescomplemented
29C2×#3750C2×C312yescomplemented
30C2×C3×#217—32no—
31C2×C3×C4⋊Dic3—31no—
32C3×#924—38no—
33C2×C3×Dic3⋊C4—32no—
34C23×H(3,𝔽3)—22no—
35C23×H(3,𝔽3)—22no—
36C23×H(3,𝔽3)Q814yescomplemented
37C23×H(3,𝔽3)C2312yes—
38C23×H(3,𝔽3)C2311yescharacteristic
39C2×C4×H(3,𝔽3)—22no—
40C2×C4×H(3,𝔽3)—28no—
41C2×C4×H(3,𝔽3)D414yes—
42C2×C4×H(3,𝔽3)C2312yes—
43C2×C4×H(3,𝔽3)C2312yes—
44C2×(C3⋊Dic3)⋊C3—24no—
45C2×(C3⋊Dic3)⋊C3—24no—
46C2×(C3⋊Dic3)⋊C3C2314yes—
47C2×(C3⋊Dic3)⋊C3C2314yes—
48#634⋊C3—28no—
49C2×C3×C22⋊Q8—91no—
50C2×#924—31no—
51C2×#924—91no—
52C24×C32—21no—
53C24×C32C2×D311yescharacteristic
54C24×C32C22×C311yescharacteristic
55C22×C32×C4—28no—
56C22×C32×C4—21no—
57C22×C32×C4—24no—
58C22×C32×C4—21no—
59C22×C32×C4—24no—
60C22×C32×C4—24no—
61C22×C32×C4C2×D311yescharacteristic
62C22×C32×C4C2×D314yes—
63C22×C32×C4C2×D311yescharacteristic
64C22×C32×C4C22×C311yescharacteristic
65C22×C32×C4C22×C314yes—
66C22×C32×C4C22×C311yescharacteristic
67C22×C3⋊Dic3C22×C312yes—
68C22×C3⋊Dic3C22×C312yes—
69C2×#634—24no—
70C2×#634C22×C314yescomplemented
71C22×C3×Dic3—32no—
72C22×C3×Dic3—32no—
73C2×C3×Dic6—34no—
74C2×C3×Dic6—64no—
75#3757C22×C318yes—
76C4⋊(C3⋊Dic3)C22×C314yescomplemented
77#3750C22×C318yescomplemented
78C3×#217—38no—
79C3×C4⋊Dic3—34no—
80C3×Dic3⋊C4—38no—
81C22×H(3,𝔽3)—28no—
82C22×H(3,𝔽3)—22no—
83C22×H(3,𝔽3)—22no—
84C22×H(3,𝔽3)C2×D412yes—
85C22×H(3,𝔽3)C2×Q812yes—
86C22×H(3,𝔽3)C2×Q814yescomplemented
87C22×H(3,𝔽3)#4112yes—
88C22×H(3,𝔽3)C2411yescharacteristic
89C4×H(3,𝔽3)—22no—
90C4×H(3,𝔽3)C2×D414yes—
91(C3⋊Dic3)⋊C3—24no—
92(C3⋊Dic3)⋊C3—24no—
93C22×C3×Q8—91no—
94C23×C3×C4—31no—
95C23×C3×C4—31no—
96C23×C3×C4—61no—
97C23×C3×C4C3×D311yescharacteristic
98C22×Dic6—31no—
99C22×Dic6—91no—
100C3×C22⋊Q8—98no—
101C2×C3×C4⋊C4—92no—
102C2×C3×C4⋊C4—91no—
103C2×C3×C22⋊C4—92no—
104C2×#217—32no—
105C2×#217—92no—
106C2×C4⋊Dic3—31no—
107C2×C4⋊Dic3—91no—
108#924—38no—
109#924—98no—
110C2×Dic3⋊C4—32no—
111C2×Dic3⋊C4—92no—
112C23×C32—24no—
113C23×C32—24no—
114C23×C32—22no—
115C23×C32—22no—
116C23×C32—22no—
117C23×C32—22no—
118C23×C32—24no—
119C23×C32—22no—
120C23×C32—21no—
121C23×C32Dic614yescomplemented
122C23×C32C3×Q814yescomplemented
123C23×C32C22×D312yes—
124C23×C32C22×D311yescharacteristic
125C23×C32C23×C312yes—
126C23×C32C23×C311yescharacteristic
127C2×C32×C4—24no—
128C2×C32×C4—216no—
129C2×C32×C4—22no—
130C2×C32×C4—22no—
131C2×C32×C4—28no—
132C2×C32×C4—28no—
133C2×C32×C4—24no—
134C2×C32×C4—22no—
135C2×C32×C4—22no—
136C2×C32×C4#6714yes—
137C2×C32×C4C3×D414yes—
138C2×C32×C4C22×D312yes—
139C2×C32×C4C22×D312yes—
140C2×C32×C4C23×C312yes—
141C2×C32×C4C23×C312yes—
142C2×C3⋊Dic3—24no—
143C2×C3⋊Dic3—24no—
144C2×C3⋊Dic3C23×C314yes—
145C2×C3⋊Dic3C23×C314yes—
146#634—28no—
147C2×C3×Dic3—34no—
148C2×C3×Dic3—34no—
149C2×C3×Dic3—64no—
150C2×C3×Dic3—64no—
151C3×Dic6—68no—
152C2×C22⋊Q8—91noSylow
153C2×H(3,𝔽3)—24no—
154C2×H(3,𝔽3)C22⋊Q814yescomplemented
155C2×H(3,𝔽3)C22×D411yescharacteristic
156C2×H(3,𝔽3)C22×Q811yescharacteristic
157C2×H(3,𝔽3)C2×#4111yescharacteristic
158C24×C3—31no—
159C24×C3—31no—
160C24×C3—61no—
161C24×C3C2×C3×D311yescharacteristic
162C2×C3×Q8—94no—
163C2×C3×Q8—184no—
164C22×C3×C4—24no—
165C22×C3×C4—31no—
166C22×C3×C4—31no—
167C22×C3×C4—31no—
168C22×C3×C4—34no—
169C22×C3×C4—31no—
170C22×C3×C4—34no—
171C22×C3×C4—64no—
172C22×C3×C4—64no—
173C22×C3×C4—68no—
174C22×C3×C4—64no—
175C22×C3×C4—61no—
176C22×C3×C4—61no—
177C22×C3×C4—92no—
178C22×C3×C4—92no—
179C22×C3×C4C2×C3×D311yescharacteristic
180C22×C3×C4C2×C3×D314yes—
181C22×C3×C4C2×C3×D311yescharacteristic
182C22×Dic3—32no—
183C22×Dic3—32no—
184C22×Dic3—92no—
185C22×Dic3—92no—
186C2×Dic6—34no—
187C2×Dic6—64no—
188C2×Dic6—94no—
189C2×Dic6—184no—
190C3×C4⋊C4—98no—
191C3×C4⋊C4—94no—
192C3×C22⋊C4—98no—
193#217—38no—
194#217—98no—
195C4⋊Dic3—34no—
196C4⋊Dic3—94no—
197Dic3⋊C4—38no—
198Dic3⋊C4—98no—
199C22×C32—24no—
200C22×C32—216no—
201C22×C32—24no—
202C22×C32—28no—
203C22×C32—28no—
204C22×C32—22no—
205C22×C32—22no—
206C22×C32—22no—
207C22×C32—22no—
208C22×C32—21no—
209C22×C32—22no—
210C22×C32—24no—
211C22×C32—22no—
212C22×C32—22no—
213C22×C32C2×Dic612yes—
214C22×C32C2×Dic614yescomplemented
215C22×C32#23512yes—
216C22×C32C2×#6712yes—
217C22×C32C2×C3×D412yes—
218C22×C32C2×C3×Q812yes—
219C22×C32C2×C3×Q814yescomplemented
220C22×C32C3×#4112yes—
221C22×C32C23×D311yescharacteristic
222C22×C32C24×C311yescharacteristic
223C32×C4—24no—
224C32×C4—22no—
225C32×C4—22no—
226C32×C4—24no—
227C32×C4C2×#6714yes—
228C32×C4C2×C3×D414yes—
229C3⋊Dic3—24no—
230C3⋊Dic3—24no—
231C3×Dic3—64no—
232C3×Dic3—64no—
233C22×Q8—91no—
234C23×C4(C3⋊D3)⋊C311yescharacteristic
235C22⋊Q8—98no—
236C2×C4⋊C4—92no—
237C2×C4⋊C4—91no—
238C2×C22⋊C4—92no—
239H(3,𝔽3)C2×C22⋊Q811yescharacteristic, complemented, Sylow
240C23×C3—22no—
241C23×C3—22no—
242C23×C3—34no—
243C23×C3—34no—
244C23×C3—32no—
245C23×C3—32no—
246C23×C3—31no—
247C23×C3—31no—
248C23×C3—62no—
249C23×C3—62no—
250C23×C3—62no—
251C23×C3—62no—
252C23×C3—64no—
253C23×C3—64no—
254C23×C3—64no—
255C23×C3—62no—
256C23×C3—61no—
257C23×C3C3×Dic614yes—
258C23×C3C22×C3×D312yes—
259C23×C3C22×C3×D311yescharacteristic
260C3×Q8—188no—
261C2×C3×C4—22no—
262C2×C3×C4—28no—
263C2×C3×C4—34no—
264C2×C3×C4—34no—
265C2×C3×C4—32no—
266C2×C3×C4—32no—
267C2×C3×C4—32no—
268C2×C3×C4—32no—
269C2×C3×C4—68no—
270C2×C3×C4—68no—
271C2×C3×C4—62no—
272C2×C3×C4—62no—
273C2×C3×C4—64no—
274C2×C3×C4—616no—
275C2×C3×C4—64no—
276C2×C3×C4—62no—
277C2×C3×C4—62no—
278C2×C3×C4—94no—
279C2×C3×C4—94no—
280C2×C3×C4—184no—
281C2×C3×C4—184no—
282C2×C3×C4C3×#6714yes—
283C2×C3×C4C22×C3×D312yes—
284C2×C3×C4C22×C3×D312yes—
285C2×Dic3—34no—
286C2×Dic3—34no—
287C2×Dic3—64no—
288C2×Dic3—64no—
289C2×Dic3—94no—
290C2×Dic3—94no—
291C2×Dic3—184no—
292C2×Dic3—184no—
293Dic6—68no—
294Dic6—188no—
295C2×C32—28no—
296C2×C32—24no—
297C2×C32—24no—
298C2×C32—24no—
299C2×C32—21no—
300C2×C32—21no—
301C2×C32—21no—
302C2×C32#92414yescomplemented
303C2×C32C3×C22⋊Q814yescomplemented
304C2×C32C22×Dic611yescharacteristic
305C2×C32C2×#23511yescharacteristic
306C2×C32C22×#6711yescharacteristic
307C2×C32C22×C3×D411yescharacteristic
308C2×C32C22×C3×Q811yescharacteristic
309C2×C32C2×C3×#4111yescharacteristic
310C24C2×(C3⋊D3)⋊C311yescharacteristic
311C2×Q8—94no—
312C2×Q8—184no—
313C22×C4—24no—
314C22×C4—92no—
315C22×C4—92no—
316C22×C4C2×(C3⋊D3)⋊C314yes—
317C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
318C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
319C4⋊C4—98no—
320C4⋊C4—94no—
321C22⋊C4—98no—
322C22×C3—28no—
323C22×C3—22no—
324C22×C3—22no—
325C22×C3—31no—
326C22×C3—31no—
327C22×C3—32no—
328C22×C3—32no—
329C22×C3—34no—
330C22×C3—34no—
331C22×C3—32no—
332C22×C3—32no—
333C22×C3—32no—
334C22×C3—32no—
335C22×C3—62no—
336C22×C3—62no—
337C22×C3—62no—
338C22×C3—62no—
339C22×C3—68no—
340C22×C3—68no—
341C22×C3—64no—
342C22×C3—64no—
343C22×C3—616no—
344C22×C3—61no—
345C22×C3—62no—
346C22×C3—64no—
347C22×C3—62no—
348C22×C3—62no—
349C22×C3C2×C3×Dic612yes—
350C22×C3C2×C3×Dic614yes—
351C22×C3C3×#23512yes—
352C22×C3C2×C3×#6712yes—
353C22×C3C23×C3×D311yescharacteristic
354C3×C4—22no—
355C3×C4—34no—
356C3×C4—34no—
357C3×C4—62no—
358C3×C4—62no—
359C3×C4—64no—
360C3×C4—64no—
361C3×C4—184no—
362C3×C4—184no—
363C3×C4C2×C3×#6714yes—
364Dic3—64no—
365Dic3—64no—
366Dic3—184no—
367Dic3—184no—
368C32—21no—
369C32C2×#92411yescharacteristic, complemented
370C32C2×C3×C22⋊Q811yescharacteristic, complemented
371C23—22no—
372C23—22no—
373C23#634⋊C314yescomplemented
374C23C22×(C3⋊D3)⋊C312yes—
375C23C22×(C3⋊D3)⋊C311yescharacteristic
376Q8—188no—
377C2×C4—22no—
378C2×C4—28no—
379C2×C4—94no—
380C2×C4—94no—
381C2×C4—184no—
382C2×C4—184no—
383C2×C4#638⋊C314yes—
384C2×C4C22×(C3⋊D3)⋊C312yes—
385C2×C4C22×(C3⋊D3)⋊C312yes—
386C2×C3—24no—
387C2×C3—31no—
388C2×C3—31no—
389C2×C3—31no—
390C2×C3—31no—
391C2×C3—34no—
392C2×C3—34no—
393C2×C3—31no—
394C2×C3—31no—
395C2×C3—64no—
396C2×C3—64no—
397C2×C3—68no—
398C2×C3—61no—
399C2×C3—61no—
400C2×C3—64no—
401C2×C3—61no—
402C2×C3C3×#92414yes—
403C2×C3C22×C3×Dic611yescharacteristic
404C2×C3C2×C3×#23511yescharacteristic
405C2×C3C22×C3×#6711yescharacteristic
406C22—22no—
407C22—28no—
408C22—22no—
409C22C2×#634⋊C312yes—
410C22C2×#634⋊C314yescomplemented
411C22#3828⋊C312yes—
412C22C2×#638⋊C312yes—
413C22C23×(C3⋊D3)⋊C311yescharacteristic
414C4—22no—
415C4—184no—
416C4—184no—
417C4C2×#638⋊C314yes—
418C3—31no—
419C3—31no—
420C3—61no—
421C3C2×C3×#92411yescharacteristic
422C2—24no—
423C2#64133⋊C314yesdirect product factor
424C2C22×#634⋊C311yescharacteristic
425C2C2×#3828⋊C311yescharacteristic
426C2C22×#638⋊C311yescharacteristic
subgroupG/H
centerC23C22×(C3⋊D3)⋊C3
commutator, cosocleC22×C32C24×C3
FrattiniC22×C3C23×C3×D3
FittingC23×C4×H(3,𝔽3)C2
socleC23×C3C22×C3×D3
prime radicalC24×H(3,𝔽3)C22
hypercenterC23×C4(C3⋊D3)⋊C3
hypocenterC32C2×C3×C22⋊Q8
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23 ◁ C23×C4

Complements

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

NH ≅ G/Ntype
2. C22×#634⋊C3C2semidirect
4. C2×C4⋊((C3⋊Dic3)⋊C3)C2semidirect
5. #64133⋊C3C2direct (N × H)
6. C2×#3750⋊C3C2semidirect
7. C2×#64133C3semidirect
17. C2×#634⋊C3C22semidirect
19. C4⋊((C3⋊Dic3)⋊C3)C22semidirect
20. #3750⋊C3C22semidirect
24. C22×#634C2×C3semidirect
27. C2×C4⋊(C3⋊Dic3)C2×C3semidirect
28. #64133C2×C3semidirect
29. C2×#3750C2×C3semidirect
36. C23×H(3,𝔽3)Q8semidirect
70. C2×#634C22×C3semidirect
76. C4⋊(C3⋊Dic3)C22×C3semidirect
77. #3750C22×C3semidirect
86. C22×H(3,𝔽3)C2×Q8semidirect
121. C23×C32Dic6semidirect
122. C23×C32C3×Q8semidirect
154. C2×H(3,𝔽3)C22⋊Q8semidirect
214. C22×C32C2×Dic6semidirect
219. C22×C32C2×C3×Q8semidirect
239. H(3,𝔽3)C2×C22⋊Q8semidirect
302. C2×C32#924semidirect
303. C2×C32C3×C22⋊Q8semidirect
369. C32C2×#924semidirect
370. C32C2×C3×C22⋊Q8semidirect
373. C23#634⋊C3semidirect
410. C22C2×#634⋊C3semidirect
423. C2#64133⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C4×H(3,𝔽3)2
C22×#634⋊C32
C2×#3757⋊C32
C2×C4⋊((C3⋊Dic3)⋊C3)2
#64133⋊C32
C2×#3750⋊C32
C2×#641333
C2×C3×#9243