Finite Group Databasebeta

← C2×#64091⋊C3 #13152363

Subgroups — C2×#64091⋊C3 #13152363

Order 1728.

2449 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C24×(C3⋊D3)⋊C3C211yesmaximal, characteristic, complemented
2C22×#638⋊C3C211yesmaximal, characteristic, complemented
3C22×C4⋊((C3⋊D3)⋊C3)C212yesmaximal, complemented
4C2×C22⋊C4×H(3,𝔽3)C211yesmaximal, characteristic, complemented
5C2×#3752⋊C3C212yesmaximal, complemented
6#64091⋊C3C218yesmaximal, direct product factor
7C2×#64091C311yesmaximal, characteristic, complemented
8C2×C3×#882—31nomaximal
9C24×H(3,𝔽3)C2211yescharacteristic
10C23×(C3⋊D3)⋊C3—24no—
11C23×(C3⋊D3)⋊C3—28no—
12C23×(C3⋊D3)⋊C3C2212yes—
13C23×(C3⋊D3)⋊C3C2214yescomplemented
14C23×(C3⋊D3)⋊C3C2211yescharacteristic, complemented
15C22×C4×H(3,𝔽3)C2212yescomplemented
16C2×#638⋊C3—24no—
17C2×#638⋊C3C2214yescomplemented
18C22×(C3⋊Dic3)⋊C3C2211yescharacteristic, complemented
19C2×C4⋊((C3⋊D3)⋊C3)—28no—
20C2×C4⋊((C3⋊D3)⋊C3)C2218yescomplemented
21C22⋊C4×H(3,𝔽3)C2214yescomplemented
22#3752⋊C3C2218yescomplemented
23C24×C3⋊D3C2×C311yescharacteristic, complemented
24C24×C3×D3—31no—
25C22×#638C2×C311yescharacteristic, complemented
26C22×C4⋊(C3⋊D3)C2×C312yescomplemented
27C22×C3×#67—31no—
28C22×C3×D12—32no—
29C2×C32×C22⋊C4—21no—
30C2×C32×C22⋊C4D311yescharacteristic, complemented
31C2×C32×C22⋊C4C2×C311yescharacteristic, complemented
32C2×#3752C2×C312yescomplemented
33#64091C2×C318yescomplemented
34C2×C3×(C2×D3)⋊C4—32no—
35C3×#882—38no—
36C23×H(3,𝔽3)—24no—
37C23×H(3,𝔽3)D414yescomplemented
38C23×H(3,𝔽3)C2312yes—
39C23×H(3,𝔽3)C2311yescharacteristic
40C22×(C3⋊D3)⋊C3—232no—
41C22×(C3⋊D3)⋊C3—24no—
42C22×(C3⋊D3)⋊C3—24no—
43C22×(C3⋊D3)⋊C3—28no—
44C22×(C3⋊D3)⋊C3—216no—
45C22×(C3⋊D3)⋊C3—24no—
46C22×(C3⋊D3)⋊C3—22no—
47C22×(C3⋊D3)⋊C3D418yescomplemented
48C22×(C3⋊D3)⋊C3C2314yes—
49C22×(C3⋊D3)⋊C3C2312yescomplemented
50C2×C4×H(3,𝔽3)—24no—
51C2×C4×H(3,𝔽3)C2314yescomplemented
52#638⋊C3—28no—
53C2×(C3⋊Dic3)⋊C3—22no—
54C2×(C3⋊Dic3)⋊C3C2312yescomplemented
55C4⋊((C3⋊D3)⋊C3)—216no—
56C2×C3×#120—91no—
57C2×#882—31no—
58C2×#882—91no—
59C24×C32—21no—
60C24×C32C2×D311yescharacteristic
61C24×C32C22×C311yescharacteristic
62C23×C3⋊D3—24no—
63C23×C3⋊D3—28no—
64C23×C3⋊D3C22×C312yes—
65C23×C3⋊D3C22×C314yescomplemented
66C23×C3⋊D3C22×C311yescharacteristic, complemented
67C23×C3×D3—32no—
68C23×C3×D3—34no—
69C23×C3×D3—31no—
70C23×C3×D3—68no—
71C23×C3×D3—64no—
72C22×C32×C4—22no—
73C22×C32×C4C2×D312yescomplemented
74C22×C32×C4C22×C312yescomplemented
75C2×#638—24no—
76C2×#638C22×C314yescomplemented
77C22×C3⋊Dic3C22×C311yescharacteristic, complemented
78C2×C4⋊(C3⋊D3)—28no—
79C2×C4⋊(C3⋊D3)C22×C318yescomplemented
80C2×C3×#67—34no—
81C2×C3×#67—64no—
82C22×C3×Dic3—31no—
83C2×C3×D12—38no—
84C2×C3×D12—68no—
85C32×C22⋊C4—24no—
86C32×C22⋊C4C2×D314yescomplemented
87C32×C22⋊C4C22×C314yescomplemented
88#3752C22×C318yescomplemented
89C3×(C2×D3)⋊C4—38no—
90C22×H(3,𝔽3)—28no—
91C22×H(3,𝔽3)—24no—
92C22×H(3,𝔽3)C2×D412yes—
93C22×H(3,𝔽3)C2×D414yes—
94C22×H(3,𝔽3)C2×D414yescomplemented
95C22×H(3,𝔽3)C2411yescharacteristic
96C2×(C3⋊D3)⋊C3—232no—
97C2×(C3⋊D3)⋊C3—216no—
98C2×(C3⋊D3)⋊C3—24no—
99C2×(C3⋊D3)⋊C3—24no—
100C2×(C3⋊D3)⋊C3—24no—
101C2×(C3⋊D3)⋊C3—22no—
102C2×(C3⋊D3)⋊C3—44no—
103C2×(C3⋊D3)⋊C3C2×D418yescomplemented
104C4×H(3,𝔽3)—24no—
105(C3⋊Dic3)⋊C3—22no—
106C25×C3—91no—
107C24×D3—31no—
108C24×D3—91no—
109C22×C3×D4—92no—
110C22×C3×D4—91no—
111C22×#67—31no—
112C22×#67—91no—
113C22×D12—32no—
114C22×D12—92no—
115C3×#120—98no—
116C2×C3×C22⋊C4—31no—
117C2×C3×C22⋊C4—31no—
118C2×C3×C22⋊C4—61no—
119C2×C3×C22⋊C4—92no—
120C2×C3×C22⋊C4C3×D311yescharacteristic
121C2×(C2×D3)⋊C4—32no—
122C2×(C2×D3)⋊C4—92no—
123#882—38no—
124#882—98no—
125C23×C32—24no—
126C23×C32—24no—
127C23×C32—24no—
128C23×C32—21no—
129C23×C32—22no—
130C23×C32—44no—
131C23×C32D1214yescomplemented
132C23×C32C3×D414yescomplemented
133C23×C32C22×D312yes—
134C23×C32C22×D311yescharacteristic
135C23×C32C23×C312yes—
136C23×C32C23×C311yescharacteristic
137C22×C3⋊D3—232no—
138C22×C3⋊D3—24no—
139C22×C3⋊D3—24no—
140C22×C3⋊D3—28no—
141C22×C3⋊D3—216no—
142C22×C3⋊D3—24no—
143C22×C3⋊D3—22no—
144C22×C3⋊D3C3×D418yescomplemented
145C22×C3⋊D3C23×C314yes—
146C22×C3⋊D3C23×C312yescomplemented
147C22×C3×D3—38no—
148C22×C3×D3—34no—
149C22×C3×D3—32no—
150C22×C3×D3—616no—
151C22×C3×D3—632no—
152C22×C3×D3—64no—
153C22×C3×D3—68no—
154C22×C3×D3—64no—
155C22×C3×D3—62no—
156C22×C3×D3—64no—
157C2×C32×C4—24no—
158C2×C32×C4—24no—
159C2×C32×C4—24no—
160C2×C32×C4—44no—
161C2×C32×C4C22×D314yescomplemented
162C2×C32×C4C23×C314yescomplemented
163#638—28no—
164C2×C3⋊Dic3—22no—
165C2×C3⋊Dic3C23×C312yescomplemented
166C4⋊(C3⋊D3)—216no—
167C3×#67—68no—
168C2×C3×Dic3—32no—
169C2×C3×Dic3—62no—
170C3×D12—616no—
171C2×#120—91noSylow
172C2×H(3,𝔽3)—24no—
173C2×H(3,𝔽3)#12014yescomplemented
174C2×H(3,𝔽3)C22×D411yescharacteristic
175C2×H(3,𝔽3)C22×D412yes—
176(C3⋊D3)⋊C3—28no—
177(C3⋊D3)⋊C3—42no—
178C24×C3—31no—
179C24×C3—31no—
180C24×C3—61no—
181C24×C3—92no—
182C24×C3—94no—
183C24×C3—91no—
184C24×C3—184no—
185C24×C3—188no—
186C24×C3C2×C3×D311yescharacteristic
187C23×D3—32no—
188C23×D3—34no—
189C23×D3—31no—
190C23×D3—68no—
191C23×D3—64no—
192C23×D3—92no—
193C23×D3—94no—
194C23×D3—91no—
195C23×D3—188no—
196C23×D3—184no—
197C2×C3×D4—98no—
198C2×C3×D4—94no—
199C2×C3×D4—188no—
200C2×C3×D4—184no—
201C22×C3×C4—32no—
202C22×C3×C4—32no—
203C22×C3×C4—62no—
204C22×C3×C4—91no—
205C22×C3×C4C2×C3×D312yes—
206C2×#67—34no—
207C2×#67—64no—
208C2×#67—94no—
209C2×#67—184no—
210C22×Dic3—31no—
211C22×Dic3—91no—
212C2×D12—38no—
213C2×D12—68no—
214C2×D12—98no—
215C2×D12—188no—
216C3×C22⋊C4—34no—
217C3×C22⋊C4—34no—
218C3×C22⋊C4—64no—
219C3×C22⋊C4—98no—
220C3×C22⋊C4C2×C3×D314yes—
221(C2×D3)⋊C4—38no—
222(C2×D3)⋊C4—98no—
223C22×C32—28no—
224C22×C32—28no—
225C22×C32—24no—
226C22×C32—24no—
227C22×C32—21no—
228C22×C32—22no—
229C22×C32—24no—
230C22×C32—24no—
231C22×C32—48no—
232C22×C32—44no—
233C22×C32C2×D1212yes—
234C22×C32C2×D1214yescomplemented
235C22×C32D3×D414yes—
236C22×C32C2×C3×D412yes—
237C22×C32C2×C3×D414yes—
238C22×C32C2×C3×D414yescomplemented
239C22×C32C23×D311yescharacteristic
240C22×C32C24×C311yescharacteristic
241C2×C3⋊D3—232no—
242C2×C3⋊D3—216no—
243C2×C3⋊D3—24no—
244C2×C3⋊D3—24no—
245C2×C3⋊D3—24no—
246C2×C3⋊D3—22no—
247C2×C3⋊D3—44no—
248C2×C3⋊D3C2×C3×D418yescomplemented
249C2×C3×D3—38no—
250C2×C3×D3—632no—
251C2×C3×D3—616no—
252C2×C3×D3—64no—
253C2×C3×D3—64no—
254C2×C3×D3—62no—
255C2×C3×D3—64no—
256C2×C3×D3—124no—
257C32×C4—24no—
258C32×C4—24no—
259C32×C4—44no—
260C3⋊Dic3—22no—
261C3×Dic3—62no—
262C25—91no—
263C22×D4—91no—
264C22×D4—92no—
265#120—98no—
266C2×C22⋊C4—92no—
267C2×C22⋊C4(C3⋊D3)⋊C311yescharacteristic, complemented
268H(3,𝔽3)C2×#12011yescharacteristic, complemented, Sylow
269C23×C3—24no—
270C23×C3—34no—
271C23×C3—34no—
272C23×C3—32no—
273C23×C3—32no—
274C23×C3—31no—
275C23×C3—31no—
276C23×C3—64no—
277C23×C3—64no—
278C23×C3—64no—
279C23×C3—62no—
280C23×C3—61no—
281C23×C3—94no—
282C23×C3—98no—
283C23×C3—92no—
284C23×C3—124no—
285C23×C3—184no—
286C23×C3—188no—
287C23×C3—184no—
288C23×C3—182no—
289C23×C3—184no—
290C23×C3—1816no—
291C23×C3—1832no—
292C23×C3C3×D1214yes—
293C23×C3C22×C3×D312yes—
294C23×C3C22×C3×D311yescharacteristic
295C22×D3—34no—
296C22×D3—38no—
297C22×D3—32no—
298C22×D3—616no—
299C22×D3—632no—
300C22×D3—64no—
301C22×D3—68no—
302C22×D3—64no—
303C22×D3—62no—
304C22×D3—64no—
305C22×D3—98no—
306C22×D3—94no—
307C22×D3—92no—
308C22×D3—1816no—
309C22×D3—184no—
310C22×D3—188no—
311C22×D3—184no—
312C22×D3—1832no—
313C22×D3—184no—
314C22×D3—182no—
315C3×D4—1816no—
316C3×D4—188no—
317C2×C3×C4—24no—
318C2×C3×C4—34no—
319C2×C3×C4—34no—
320C2×C3×C4—64no—
321C2×C3×C4—64no—
322C2×C3×C4—64no—
323C2×C3×C4—92no—
324C2×C3×C4—124no—
325C2×C3×C4—182no—
326C2×C3×C4C22×C3×D314yes—
327#67—68no—
328#67—188no—
329C2×Dic3—32no—
330C2×Dic3—62no—
331C2×Dic3—92no—
332C2×Dic3—182no—
333D12—616no—
334D12—1816no—
335C2×C32—24no—
336C2×C32—24no—
337C2×C32—24no—
338C2×C32—21no—
339C2×C32—22no—
340C2×C32—44no—
341C2×C32#88214yescomplemented
342C2×C32C3×#12014yescomplemented
343C2×C32C22×D1211yescharacteristic
344C2×C32C2×D3×D412yes—
345C2×C32C22×C3×D411yescharacteristic
346C2×C32C22×C3×D412yes—
347C3⋊D3—28no—
348C3⋊D3—42no—
349C3×D3—68no—
350C3×D3—122no—
351C24—92no—
352C24—94no—
353C24—91no—
354C24—188no—
355C24—184no—
356C24C2×(C3⋊D3)⋊C311yescharacteristic
357C2×D4—94no—
358C2×D4—98no—
359C2×D4—188no—
360C2×D4—184no—
361C22×C4—91no—
362C22×C4C2×(C3⋊D3)⋊C312yescomplemented
363C22⋊C4—98no—
364C22⋊C4C2×(C3⋊D3)⋊C314yescomplemented
365C22×C3—28no—
366C22×C3—24no—
367C22×C3—31no—
368C22×C3—31no—
369C22×C3—32no—
370C22×C3—32no—
371C22×C3—34no—
372C22×C3—34no—
373C22×C3—34no—
374C22×C3—34no—
375C22×C3—64no—
376C22×C3—68no—
377C22×C3—64no—
378C22×C3—68no—
379C22×C3—61no—
380C22×C3—62no—
381C22×C3—64no—
382C22×C3—64no—
383C22×C3—98no—
384C22×C3—124no—
385C22×C3—128no—
386C22×C3—184no—
387C22×C3—184no—
388C22×C3—1816no—
389C22×C3—182no—
390C22×C3—184no—
391C22×C3—1832no—
392C22×C3—364no—
393C22×C3C2×C3×D1212yes—
394C22×C3C2×C3×D1214yes—
395C22×C3C3×D3×D414yes—
396C22×C3C23×C3×D311yescharacteristic
397C2×D3—38no—
398C2×D3—632no—
399C2×D3—616no—
400C2×D3—64no—
401C2×D3—64no—
402C2×D3—62no—
403C2×D3—64no—
404C2×D3—98no—
405C2×D3—124no—
406C2×D3—184no—
407C2×D3—184no—
408C2×D3—1816no—
409C2×D3—1832no—
410C2×D3—184no—
411C2×D3—182no—
412C2×D3—364no—
413C3×C4—24no—
414C3×C4—64no—
415C3×C4—64no—
416C3×C4—124no—
417C3×C4—182no—
418Dic3—62no—
419Dic3—182no—
420C32—21no—
421C32C2×#88211yescharacteristic, complemented
422C32C2×C3×#12011yescharacteristic, complemented
423C23—24no—
424C23—94no—
425C23—98no—
426C23—92no—
427C23—1816no—
428C23—1832no—
429C23—184no—
430C23—188no—
431C23—184no—
432C23—182no—
433C23—184no—
434C23C4⋊((C3⋊D3)⋊C3)14yescomplemented
435C23C22×(C3⋊D3)⋊C312yes—
436C23C22×(C3⋊D3)⋊C311yescharacteristic
437D4—1816no—
438D4—188no—
439C2×C4—24no—
440C2×C4—92no—
441C2×C4—182no—
442C2×C4C22×(C3⋊D3)⋊C314yescomplemented
443C2×C3—24no—
444C2×C3—31no—
445C2×C3—31no—
446C2×C3—32no—
447C2×C3—32no—
448C2×C3—34no—
449C2×C3—34no—
450C2×C3—64no—
451C2×C3—64no—
452C2×C3—61no—
453C2×C3—62no—
454C2×C3—64no—
455C2×C3—124no—
456C2×C3—188no—
457C2×C3—362no—
458C2×C3C3×#88214yes—
459C2×C3C22×C3×D1211yescharacteristic
460C2×C3C2×C3×D3×D412yes—
461D3—68no—
462D3—122no—
463D3—188no—
464D3—362no—
465C22—24no—
466C22—28no—
467C22—98no—
468C22—1832no—
469C22—184no—
470C22—184no—
471C22—1816no—
472C22—182no—
473C22—184no—
474C22—364no—
475C22C2×C4⋊((C3⋊D3)⋊C3)12yes—
476C22C2×C4⋊((C3⋊D3)⋊C3)14yescomplemented
477C22D4×(C3⋊D3)⋊C314yes—
478C22C23×(C3⋊D3)⋊C311yescharacteristic
479C4—24no—
480C4—182no—
481C3—31no—
482C3—31no—
483C3—61no—
484C3C2×C3×#88211yescharacteristic
485C2—24no—
486C2—188no—
487C2—362no—
488C2#64091⋊C314yesdirect product factor
489C2C22×C4⋊((C3⋊D3)⋊C3)11yescharacteristic
490C2C2×D4×(C3⋊D3)⋊C312yes—
subgroupG/H
centerC23C22×(C3⋊D3)⋊C3
commutator, cosocleC22×C32C24×C3
FrattiniC22×C3C23×C3×D3
FittingC2×C22⋊C4×H(3,𝔽3)C2
socleC23×C3C22×C3×D3
hypercenterC2×C22⋊C4(C3⋊D3)⋊C3
hypocenterC32C2×C3×#120
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C24×(C3⋊D3)⋊C3C2semidirect
2. C22×#638⋊C3C2semidirect
3. C22×C4⋊((C3⋊D3)⋊C3)C2semidirect
4. C2×C22⋊C4×H(3,𝔽3)C2semidirect
5. C2×#3752⋊C3C2semidirect
6. #64091⋊C3C2direct (N × H)
7. C2×#64091C3semidirect
13. C23×(C3⋊D3)⋊C3C22semidirect
14. C23×(C3⋊D3)⋊C3C22semidirect
15. C22×C4×H(3,𝔽3)C22semidirect
17. C2×#638⋊C3C22semidirect
18. C22×(C3⋊Dic3)⋊C3C22semidirect
20. C2×C4⋊((C3⋊D3)⋊C3)C22semidirect
21. C22⋊C4×H(3,𝔽3)C22semidirect
22. #3752⋊C3C22semidirect
23. C24×C3⋊D3C2×C3semidirect
25. C22×#638C2×C3semidirect
26. C22×C4⋊(C3⋊D3)C2×C3semidirect
30. C2×C32×C22⋊C4D3semidirect
31. C2×C32×C22⋊C4C2×C3semidirect
32. C2×#3752C2×C3semidirect
33. #64091C2×C3semidirect
37. C23×H(3,𝔽3)D4semidirect
47. C22×(C3⋊D3)⋊C3D4semidirect
49. C22×(C3⋊D3)⋊C3C23semidirect
51. C2×C4×H(3,𝔽3)C23semidirect
54. C2×(C3⋊Dic3)⋊C3C23semidirect
65. C23×C3⋊D3C22×C3semidirect
66. C23×C3⋊D3C22×C3semidirect
73. C22×C32×C4C2×D3semidirect
74. C22×C32×C4C22×C3semidirect
76. C2×#638C22×C3semidirect
77. C22×C3⋊Dic3C22×C3semidirect
79. C2×C4⋊(C3⋊D3)C22×C3semidirect
86. C32×C22⋊C4C2×D3semidirect
87. C32×C22⋊C4C22×C3semidirect
88. #3752C22×C3semidirect
94. C22×H(3,𝔽3)C2×D4semidirect
103. C2×(C3⋊D3)⋊C3C2×D4semidirect
131. C23×C32D12semidirect
132. C23×C32C3×D4semidirect
144. C22×C3⋊D3C3×D4semidirect
146. C22×C3⋊D3C23×C3semidirect
161. C2×C32×C4C22×D3semidirect
162. C2×C32×C4C23×C3semidirect
165. C2×C3⋊Dic3C23×C3semidirect
173. C2×H(3,𝔽3)#120semidirect
234. C22×C32C2×D12semidirect
238. C22×C32C2×C3×D4semidirect
248. C2×C3⋊D3C2×C3×D4semidirect
267. C2×C22⋊C4(C3⋊D3)⋊C3semidirect
268. H(3,𝔽3)C2×#120semidirect
341. C2×C32#882semidirect
342. C2×C32C3×#120semidirect
362. C22×C4C2×(C3⋊D3)⋊C3semidirect
364. C22⋊C4C2×(C3⋊D3)⋊C3semidirect
421. C32C2×#882semidirect
422. C32C2×C3×#120semidirect
434. C23C4⋊((C3⋊D3)⋊C3)semidirect
442. C2×C4C22×(C3⋊D3)⋊C3semidirect
476. C22C2×C4⋊((C3⋊D3)⋊C3)semidirect
488. C2#64091⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C24×(C3⋊D3)⋊C32
C22×#638⋊C32
C22×C4⋊((C3⋊D3)⋊C3)2
C2×C22⋊C4×H(3,𝔽3)2
C2×#3752⋊C32
#64091⋊C32
C2×#640913
C2×C3×#8823