Finite Group Databasebeta

← C2×#64139⋊C3 #13153557

Subgroups — C2×#64139⋊C3 #13153557

Order 1728.

1727 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C4×H(3,𝔽3)C211yesmaximal, characteristic, complemented
2C22×#638⋊C3C212yesmaximal, complemented
3C22×C4⋊((C3⋊D3)⋊C3)C211yesmaximal, characteristic, complemented
4#64139⋊C3C218yesmaximal, direct product factor
5C2×#3752⋊C3C212yesmaximal, complemented
6C2×C4⋊((C3⋊Dic3)⋊C3)C211yesmaximal, characteristic, complemented
7C2×#64139C311yesmaximal, characteristic, complemented
8C2×C3×#930—31nomaximal
9C24×H(3,𝔽3)C2211yescharacteristic
10C23×(C3⋊D3)⋊C3C2212yes—
11C22×C4×H(3,𝔽3)—24no—
12C22×C4×H(3,𝔽3)C2211yescharacteristic
13C22×C4×H(3,𝔽3)C2211yescharacteristic
14C22×C4×H(3,𝔽3)C2214yescomplemented
15C2×#638⋊C3—28no—
16C2×#638⋊C3C2218yescomplemented
17C22×(C3⋊Dic3)⋊C3C2212yes—
18C2×C4⋊((C3⋊D3)⋊C3)—24no—
19C2×C4⋊((C3⋊D3)⋊C3)C2214yescomplemented
20#3752⋊C3C2218yescomplemented
21C4⋊((C3⋊Dic3)⋊C3)C2214yescomplemented
22C23×C32×C4—21no—
23C23×C32×C4D311yescharacteristic, complemented
24C23×C32×C4C2×C311yescharacteristic, complemented
25C22×#638C2×C312yescomplemented
26C22×C4⋊(C3⋊D3)C2×C311yescharacteristic, complemented
27C22×C3×#67—32no—
28C22×C3×D12—31no—
29#64139C2×C318yescomplemented
30C2×#3752C2×C312yescomplemented
31C2×C4⋊(C3⋊Dic3)C2×C311yescharacteristic, complemented
32C3×#930—38no—
33C2×C3×(C2×D3)⋊C4—32no—
34C2×C3×C4⋊Dic3—31no—
35C23×H(3,𝔽3)—22no—
36C23×H(3,𝔽3)—22no—
37C23×H(3,𝔽3)D414yescomplemented
38C23×H(3,𝔽3)C2312yes—
39C23×H(3,𝔽3)C2311yescharacteristic
40C22×(C3⋊D3)⋊C3—24no—
41C22×(C3⋊D3)⋊C3—24no—
42C22×(C3⋊D3)⋊C3—24no—
43C22×(C3⋊D3)⋊C3C2314yes—
44C2×C4×H(3,𝔽3)—22no—
45C2×C4×H(3,𝔽3)—28no—
46C2×C4×H(3,𝔽3)D414yescomplemented
47C2×C4×H(3,𝔽3)C2312yes—
48C2×C4×H(3,𝔽3)C2312yes—
49#638⋊C3—216no—
50C2×(C3⋊Dic3)⋊C3—24no—
51C2×(C3⋊Dic3)⋊C3C2314yes—
52C4⋊((C3⋊D3)⋊C3)—28no—
53C2×C3×#121—91no—
54C2×#930—31no—
55C2×#930—91no—
56C24×C32—21no—
57C24×C32C2×D311yescharacteristic
58C24×C32C22×C311yescharacteristic
59C23×C3⋊D3C22×C312yes—
60C23×C3×D3—32no—
61C22×C32×C4—28no—
62C22×C32×C4—21no—
63C22×C32×C4—21no—
64C22×C32×C4—24no—
65C22×C32×C4—24no—
66C22×C32×C4—24no—
67C22×C32×C4C2×D311yescharacteristic
68C22×C32×C4C2×D311yescharacteristic
69C22×C32×C4C2×D314yescomplemented
70C22×C32×C4C22×C311yescharacteristic
71C22×C32×C4C22×C311yescharacteristic
72C22×C32×C4C22×C314yescomplemented
73C2×#638—28no—
74C2×#638C22×C318yescomplemented
75C22×C3⋊Dic3C22×C312yes—
76C2×C4⋊(C3⋊D3)—24no—
77C2×C4⋊(C3⋊D3)C22×C314yescomplemented
78C2×C3×#67—38no—
79C2×C3×#67—68no—
80C22×C3×Dic3—32no—
81C2×C3×D12—34no—
82C2×C3×D12—64no—
83#3752C22×C318yescomplemented
84C4⋊(C3⋊Dic3)C22×C314yescomplemented
85C3×(C2×D3)⋊C4—38no—
86C3×C4⋊Dic3—34no—
87C22×H(3,𝔽3)—28no—
88C22×H(3,𝔽3)—22no—
89C22×H(3,𝔽3)—22no—
90C22×H(3,𝔽3)C2×D412yes—
91C22×H(3,𝔽3)C2×D412yes—
92C22×H(3,𝔽3)C2×D414yescomplemented
93C22×H(3,𝔽3)#4112yes—
94C22×H(3,𝔽3)C2411yescharacteristic
95C2×(C3⋊D3)⋊C3—24no—
96C2×(C3⋊D3)⋊C3—24no—
97C2×(C3⋊D3)⋊C3—24no—
98C2×(C3⋊D3)⋊C3—48no—
99C4×H(3,𝔽3)—22no—
100C4×H(3,𝔽3)C2×D414yescomplemented
101(C3⋊Dic3)⋊C3—24no—
102C22×C3×D4—91no—
103C22×C3×D4—92no—
104C23×C3×C4—31no—
105C23×C3×C4—31no—
106C23×C3×C4—61no—
107C23×C3×C4C3×D311yescharacteristic
108C22×#67—32no—
109C22×#67—92no—
110C22×D12—31no—
111C22×D12—91no—
112C3×#121—98no—
113C2×C3×C4⋊C4—91no—
114C2×C3×C22⋊C4—92no—
115#930—38no—
116#930—98no—
117C2×(C2×D3)⋊C4—32no—
118C2×(C2×D3)⋊C4—92no—
119C2×C4⋊Dic3—31no—
120C2×C4⋊Dic3—91no—
121C23×C32—24no—
122C23×C32—24no—
123C23×C32—22no—
124C23×C32—22no—
125C23×C32—22no—
126C23×C32—22no—
127C23×C32—24no—
128C23×C32—22no—
129C23×C32—21no—
130C23×C32D1214yescomplemented
131C23×C32C3×D414yescomplemented
132C23×C32C22×D312yes—
133C23×C32C22×D311yescharacteristic
134C23×C32C23×C312yes—
135C23×C32C23×C311yescharacteristic
136C22×C3⋊D3—24no—
137C22×C3⋊D3—24no—
138C22×C3⋊D3—24no—
139C22×C3⋊D3C23×C314yes—
140C22×C3×D3—34no—
141C22×C3×D3—64no—
142C22×C3×D3—64no—
143C22×C3×D3—64no—
144C2×C32×C4—24no—
145C2×C32×C4—216no—
146C2×C32×C4—22no—
147C2×C32×C4—22no—
148C2×C32×C4—28no—
149C2×C32×C4—28no—
150C2×C32×C4—24no—
151C2×C32×C4—22no—
152C2×C32×C4—22no—
153C2×C32×C4#6714yescomplemented
154C2×C32×C4C3×D414yescomplemented
155C2×C32×C4C22×D312yes—
156C2×C32×C4C22×D312yes—
157C2×C32×C4C23×C312yes—
158C2×C32×C4C23×C312yes—
159#638—216no—
160C2×C3⋊Dic3—24no—
161C2×C3⋊Dic3C23×C314yes—
162C4⋊(C3⋊D3)—28no—
163C3×#67—616no—
164C2×C3×Dic3—34no—
165C2×C3×Dic3—64no—
166C3×D12—68no—
167C2×#121—91noSylow
168C2×H(3,𝔽3)—24no—
169C2×H(3,𝔽3)#12114yescomplemented
170C2×H(3,𝔽3)C22×D411yescharacteristic
171C2×H(3,𝔽3)C22×D411yescharacteristic
172C2×H(3,𝔽3)C2×#4111yescharacteristic
173(C3⋊D3)⋊C3—44no—
174C24×C3—31no—
175C24×C3—31no—
176C24×C3—61no—
177C24×C3—92no—
178C24×C3C2×C3×D311yescharacteristic
179C23×D3—32no—
180C23×D3—92no—
181C2×C3×D4—94no—
182C2×C3×D4—98no—
183C2×C3×D4—184no—
184C2×C3×D4—188no—
185C22×C3×C4—24no—
186C22×C3×C4—31no—
187C22×C3×C4—31no—
188C22×C3×C4—31no—
189C22×C3×C4—31no—
190C22×C3×C4—34no—
191C22×C3×C4—34no—
192C22×C3×C4—64no—
193C22×C3×C4—64no—
194C22×C3×C4—68no—
195C22×C3×C4—61no—
196C22×C3×C4—61no—
197C22×C3×C4—64no—
198C22×C3×C4—92no—
199C22×C3×C4C2×C3×D311yescharacteristic
200C22×C3×C4C2×C3×D311yescharacteristic
201C22×C3×C4C2×C3×D314yes—
202C2×#67—38no—
203C2×#67—68no—
204C2×#67—98no—
205C2×#67—188no—
206C22×Dic3—32no—
207C22×Dic3—92no—
208C2×D12—34no—
209C2×D12—64no—
210C2×D12—94no—
211C2×D12—184no—
212C3×C4⋊C4—94no—
213C3×C22⋊C4—98no—
214(C2×D3)⋊C4—38no—
215(C2×D3)⋊C4—98no—
216C4⋊Dic3—34no—
217C4⋊Dic3—94no—
218C22×C32—24no—
219C22×C32—216no—
220C22×C32—24no—
221C22×C32—28no—
222C22×C32—28no—
223C22×C32—22no—
224C22×C32—22no—
225C22×C32—22no—
226C22×C32—22no—
227C22×C32—21no—
228C22×C32—22no—
229C22×C32—24no—
230C22×C32—22no—
231C22×C32—22no—
232C22×C32C2×D1212yes—
233C22×C32C2×D1214yescomplemented
234C22×C32#23512yes—
235C22×C32C2×#6712yes—
236C22×C32C2×C3×D412yes—
237C22×C32C2×C3×D412yes—
238C22×C32C2×C3×D414yescomplemented
239C22×C32C3×#4112yes—
240C22×C32C23×D311yescharacteristic
241C22×C32C24×C311yescharacteristic
242C2×C3⋊D3—24no—
243C2×C3⋊D3—24no—
244C2×C3⋊D3—24no—
245C2×C3⋊D3—48no—
246C2×C3×D3—64no—
247C2×C3×D3—64no—
248C2×C3×D3—64no—
249C2×C3×D3—128no—
250C32×C4—24no—
251C32×C4—22no—
252C32×C4—22no—
253C32×C4—24no—
254C32×C4C2×#6714yescomplemented
255C32×C4C2×C3×D414yescomplemented
256C3⋊Dic3—24no—
257C3×Dic3—64no—
258C22×D4—92no—
259C22×D4—91no—
260C23×C4(C3⋊D3)⋊C311yescharacteristic, complemented
261#121—98no—
262C2×C4⋊C4—91no—
263C2×C22⋊C4—92no—
264H(3,𝔽3)C2×#12111yescharacteristic, complemented, Sylow
265C23×C3—22no—
266C23×C3—22no—
267C23×C3—34no—
268C23×C3—34no—
269C23×C3—32no—
270C23×C3—32no—
271C23×C3—31no—
272C23×C3—31no—
273C23×C3—62no—
274C23×C3—62no—
275C23×C3—62no—
276C23×C3—62no—
277C23×C3—64no—
278C23×C3—64no—
279C23×C3—64no—
280C23×C3—62no—
281C23×C3—61no—
282C23×C3—94no—
283C23×C3—184no—
284C23×C3—184no—
285C23×C3—184no—
286C23×C3C3×D1214yes—
287C23×C3C22×C3×D312yes—
288C23×C3C22×C3×D311yescharacteristic
289C22×D3—34no—
290C22×D3—64no—
291C22×D3—64no—
292C22×D3—64no—
293C22×D3—94no—
294C22×D3—184no—
295C22×D3—184no—
296C22×D3—184no—
297C3×D4—188no—
298C3×D4—1816no—
299C2×C3×C4—22no—
300C2×C3×C4—28no—
301C2×C3×C4—34no—
302C2×C3×C4—34no—
303C2×C3×C4—32no—
304C2×C3×C4—32no—
305C2×C3×C4—32no—
306C2×C3×C4—32no—
307C2×C3×C4—68no—
308C2×C3×C4—68no—
309C2×C3×C4—62no—
310C2×C3×C4—62no—
311C2×C3×C4—64no—
312C2×C3×C4—616no—
313C2×C3×C4—64no—
314C2×C3×C4—62no—
315C2×C3×C4—62no—
316C2×C3×C4—94no—
317C2×C3×C4—184no—
318C2×C3×C4C3×#6714yes—
319C2×C3×C4C22×C3×D312yes—
320C2×C3×C4C22×C3×D312yes—
321#67—616no—
322#67—1816no—
323C2×Dic3—34no—
324C2×Dic3—64no—
325C2×Dic3—94no—
326C2×Dic3—184no—
327D12—68no—
328D12—188no—
329C2×C32—28no—
330C2×C32—24no—
331C2×C32—24no—
332C2×C32—24no—
333C2×C32—21no—
334C2×C32—21no—
335C2×C32—21no—
336C2×C32#93014yescomplemented
337C2×C32C3×#12114yescomplemented
338C2×C32C22×D1211yescharacteristic
339C2×C32C2×#23511yescharacteristic
340C2×C32C22×#6711yescharacteristic
341C2×C32C22×C3×D411yescharacteristic
342C2×C32C22×C3×D411yescharacteristic
343C2×C32C2×C3×#4111yescharacteristic
344C3⋊D3—44no—
345C3×D3—124no—
346C24—92no—
347C24C2×(C3⋊D3)⋊C311yescharacteristic
348C2×D4—98no—
349C2×D4—94no—
350C2×D4—188no—
351C2×D4—184no—
352C22×C4—24no—
353C22×C4—92no—
354C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
355C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
356C22×C4C2×(C3⋊D3)⋊C314yescomplemented
357C4⋊C4—94no—
358C22⋊C4—98no—
359C22×C3—28no—
360C22×C3—22no—
361C22×C3—22no—
362C22×C3—31no—
363C22×C3—31no—
364C22×C3—32no—
365C22×C3—32no—
366C22×C3—34no—
367C22×C3—34no—
368C22×C3—32no—
369C22×C3—32no—
370C22×C3—32no—
371C22×C3—32no—
372C22×C3—62no—
373C22×C3—62no—
374C22×C3—62no—
375C22×C3—62no—
376C22×C3—68no—
377C22×C3—68no—
378C22×C3—64no—
379C22×C3—64no—
380C22×C3—616no—
381C22×C3—61no—
382C22×C3—62no—
383C22×C3—64no—
384C22×C3—62no—
385C22×C3—62no—
386C22×C3—184no—
387C22×C3—184no—
388C22×C3—184no—
389C22×C3—368no—
390C22×C3C2×C3×D1212yes—
391C22×C3C2×C3×D1214yes—
392C22×C3C3×#23512yes—
393C22×C3C2×C3×#6712yes—
394C22×C3C23×C3×D311yescharacteristic
395C2×D3—64no—
396C2×D3—64no—
397C2×D3—64no—
398C2×D3—128no—
399C2×D3—184no—
400C2×D3—184no—
401C2×D3—184no—
402C2×D3—368no—
403C3×C4—22no—
404C3×C4—34no—
405C3×C4—34no—
406C3×C4—62no—
407C3×C4—62no—
408C3×C4—64no—
409C3×C4—64no—
410C3×C4—184no—
411C3×C4C2×C3×#6714yes—
412Dic3—64no—
413Dic3—184no—
414C32—21no—
415C32C2×#93011yescharacteristic, complemented
416C32C2×C3×#12111yescharacteristic, complemented
417C23—22no—
418C23—22no—
419C23—94no—
420C23—184no—
421C23—184no—
422C23—184no—
423C23C4⋊((C3⋊D3)⋊C3)14yescomplemented
424C23C22×(C3⋊D3)⋊C312yes—
425C23C22×(C3⋊D3)⋊C311yescharacteristic
426D4—1816no—
427D4—188no—
428C2×C4—22no—
429C2×C4—28no—
430C2×C4—94no—
431C2×C4—184no—
432C2×C4#638⋊C314yescomplemented
433C2×C4C22×(C3⋊D3)⋊C312yes—
434C2×C4C22×(C3⋊D3)⋊C312yes—
435C2×C3—24no—
436C2×C3—31no—
437C2×C3—31no—
438C2×C3—31no—
439C2×C3—31no—
440C2×C3—34no—
441C2×C3—34no—
442C2×C3—31no—
443C2×C3—31no—
444C2×C3—64no—
445C2×C3—64no—
446C2×C3—68no—
447C2×C3—61no—
448C2×C3—61no—
449C2×C3—64no—
450C2×C3—61no—
451C2×C3—364no—
452C2×C3C3×#93014yes—
453C2×C3C22×C3×D1211yescharacteristic
454C2×C3C2×C3×#23511yescharacteristic
455C2×C3C22×C3×#6711yescharacteristic
456D3—124no—
457D3—364no—
458C22—22no—
459C22—28no—
460C22—22no—
461C22—184no—
462C22—184no—
463C22—184no—
464C22—368no—
465C22C2×C4⋊((C3⋊D3)⋊C3)12yes—
466C22C2×C4⋊((C3⋊D3)⋊C3)14yescomplemented
467C22#3828⋊C312yes—
468C22C2×#638⋊C312yes—
469C22C23×(C3⋊D3)⋊C311yescharacteristic
470C4—22no—
471C4—184no—
472C4C2×#638⋊C314yescomplemented
473C3—31no—
474C3—31no—
475C3—61no—
476C3C2×C3×#93011yescharacteristic
477C2—24no—
478C2—364no—
479C2#64139⋊C314yesdirect product factor
480C2C22×C4⋊((C3⋊D3)⋊C3)11yescharacteristic
481C2C2×#3828⋊C311yescharacteristic
482C2C22×#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
hypercenterC23×C4(C3⋊D3)⋊C3
hypocenterC32C2×C3×#121
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23 ◁ C23×C4

Complements

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

NH ≅ G/Ntype
1. C23×C4×H(3,𝔽3)C2semidirect
2. C22×#638⋊C3C2semidirect
3. C22×C4⋊((C3⋊D3)⋊C3)C2semidirect
4. #64139⋊C3C2direct (N × H)
5. C2×#3752⋊C3C2semidirect
6. C2×C4⋊((C3⋊Dic3)⋊C3)C2semidirect
7. C2×#64139C3semidirect
14. C22×C4×H(3,𝔽3)C22semidirect
16. C2×#638⋊C3C22semidirect
19. C2×C4⋊((C3⋊D3)⋊C3)C22semidirect
20. #3752⋊C3C22semidirect
21. C4⋊((C3⋊Dic3)⋊C3)C22semidirect
23. C23×C32×C4D3semidirect
24. C23×C32×C4C2×C3semidirect
25. C22×#638C2×C3semidirect
26. C22×C4⋊(C3⋊D3)C2×C3semidirect
29. #64139C2×C3semidirect
30. C2×#3752C2×C3semidirect
31. C2×C4⋊(C3⋊Dic3)C2×C3semidirect
37. C23×H(3,𝔽3)D4semidirect
46. C2×C4×H(3,𝔽3)D4semidirect
69. C22×C32×C4C2×D3semidirect
72. C22×C32×C4C22×C3semidirect
74. C2×#638C22×C3semidirect
77. C2×C4⋊(C3⋊D3)C22×C3semidirect
83. #3752C22×C3semidirect
84. C4⋊(C3⋊Dic3)C22×C3semidirect
92. C22×H(3,𝔽3)C2×D4semidirect
100. C4×H(3,𝔽3)C2×D4semidirect
130. C23×C32D12semidirect
131. C23×C32C3×D4semidirect
153. C2×C32×C4#67semidirect
154. C2×C32×C4C3×D4semidirect
169. C2×H(3,𝔽3)#121semidirect
233. C22×C32C2×D12semidirect
238. C22×C32C2×C3×D4semidirect
254. C32×C4C2×#67semidirect
255. C32×C4C2×C3×D4semidirect
260. C23×C4(C3⋊D3)⋊C3semidirect
264. H(3,𝔽3)C2×#121semidirect
336. C2×C32#930semidirect
337. C2×C32C3×#121semidirect
356. C22×C4C2×(C3⋊D3)⋊C3semidirect
415. C32C2×#930semidirect
416. C32C2×C3×#121semidirect
423. C23C4⋊((C3⋊D3)⋊C3)semidirect
432. C2×C4#638⋊C3semidirect
466. C22C2×C4⋊((C3⋊D3)⋊C3)semidirect
472. C4C2×#638⋊C3semidirect
479. C2#64139⋊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×#638⋊C32
C22×C4⋊((C3⋊D3)⋊C3)2
#64139⋊C32
C2×#3752⋊C32
C2×C4⋊((C3⋊Dic3)⋊C3)2
C2×#641393
C2×C3×#9303