Finite Group Databasebeta

← C2×#64094⋊C3 #13152357

Subgroups — C2×#64094⋊C3 #13152357

Order 1728.

1287 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#638⋊C3C211yesmaximal, characteristic
2C22×#634⋊C3C211yesmaximal, characteristic, complemented
3C2×C22⋊C4×H(3,𝔽3)C211yesmaximal, characteristic, complemented
4C2×#3757⋊C3C211yesmaximal, characteristic, complemented
5C2×#3752⋊C3C211yesmaximal, characteristic, complemented
6C2×#3752⋊C3C211yesmaximal, characteristic, complemented
7C2×C4×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic, complemented
8#64094⋊C3C218yesmaximal, direct product factor
9C2×#64094C311yesmaximal, characteristic, complemented
10C2×C3×#885—31nomaximal
11C24×H(3,𝔽3)C2211yescharacteristic
12C23×(C3⋊D3)⋊C3C2211yescharacteristic
13C22×C4×H(3,𝔽3)C2211yescharacteristic
14C22×C4×H(3,𝔽3)C2211yescharacteristic
15C2×#638⋊C3—24no—
16C2×#638⋊C3C2214yes—
17C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
18C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
19C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
20C2×#634⋊C3—24no—
21C2×#634⋊C3C2214yescomplemented
22C22⋊C4×H(3,𝔽3)C2214yescomplemented
23#3757⋊C3C2214yescomplemented
24#3752⋊C3C2214yescomplemented
25#3752⋊C3C2214yescomplemented
26C4×(C3⋊Dic3)⋊C3C2214yescomplemented
27C22×#638C2×C311yescharacteristic
28C22×#634C2×C311yescharacteristic, complemented
29C22×C3×#67—31no—
30C22×C3×Dic6—31no—
31C2×C32×C22⋊C4—21no—
32C2×C32×C22⋊C4D311yescharacteristic, complemented
33C2×C32×C22⋊C4C2×C311yescharacteristic, complemented
34C2×#3757C2×C311yescharacteristic, complemented
35C2×#3752C2×C311yescharacteristic, complemented
36C2×#3752C2×C311yescharacteristic, complemented
37C2×C4×C3⋊Dic3C2×C311yescharacteristic, complemented
38#64094C2×C318yescomplemented
39C2×C3×#217—31no—
40C2×C3×(C2×D3)⋊C4—31no—
41C2×C3×(C2×D3)⋊C4—31no—
42C2×C3×C4×Dic3—31no—
43C3×#885—38no—
44C23×H(3,𝔽3)—22no—
45C23×H(3,𝔽3)—22no—
46C23×H(3,𝔽3)—22no—
47C23×H(3,𝔽3)C2312yes—
48C23×H(3,𝔽3)C2311yescharacteristic
49C22×(C3⋊D3)⋊C3—22no—
50C22×(C3⋊D3)⋊C3—22no—
51C22×(C3⋊D3)⋊C3—22no—
52C22×(C3⋊D3)⋊C3C2312yes—
53C2×C4×H(3,𝔽3)—22no—
54C2×C4×H(3,𝔽3)—22no—
55C2×C4×H(3,𝔽3)C2312yes—
56C2×C4×H(3,𝔽3)C2312yes—
57#638⋊C3—28no—
58C2×(C3⋊Dic3)⋊C3—22no—
59C2×(C3⋊Dic3)⋊C3—22no—
60C2×(C3⋊Dic3)⋊C3D414yes—
61C2×(C3⋊Dic3)⋊C3C2312yes—
62C2×(C3⋊Dic3)⋊C3C2312yes—
63C2×(C3⋊Dic3)⋊C3C2312yes—
64#634⋊C3—28no—
65C2×C3×#124—91no—
66C2×#885—31no—
67C2×#885—91no—
68C24×C32—21no—
69C24×C32C2×D311yescharacteristic
70C24×C32C22×C311yescharacteristic
71C23×C3⋊D3C22×C311yescharacteristic
72C23×C3×D3—31no—
73C22×C32×C4—21no—
74C22×C32×C4—21no—
75C22×C32×C4C2×D311yescharacteristic
76C22×C32×C4C2×D311yescharacteristic
77C22×C32×C4C22×C311yescharacteristic
78C22×C32×C4C22×C311yescharacteristic
79C2×#638—24no—
80C2×#638C22×C314yes—
81C22×C3⋊Dic3C22×C311yescharacteristic
82C22×C3⋊Dic3C22×C311yescharacteristic
83C22×C3⋊Dic3C22×C311yescharacteristic
84C2×#634—24no—
85C2×#634C22×C314yescomplemented
86C2×C3×#67—34no—
87C2×C3×#67—64no—
88C22×C3×Dic3—31no—
89C22×C3×Dic3—31no—
90C22×C3×Dic3—31no—
91C2×C3×Dic6—34no—
92C2×C3×Dic6—64no—
93C32×C22⋊C4—24no—
94C32×C22⋊C4C2×D314yescomplemented
95C32×C22⋊C4C22×C314yescomplemented
96#3757C22×C314yescomplemented
97#3752C22×C314yescomplemented
98#3752C22×C314yescomplemented
99C4×C3⋊Dic3C22×C314yescomplemented
100C3×#217—34no—
101C3×(C2×D3)⋊C4—34no—
102C3×(C2×D3)⋊C4—34no—
103C3×C4×Dic3—34no—
104C22×H(3,𝔽3)—22no—
105C22×H(3,𝔽3)—22no—
106C22×H(3,𝔽3)—22no—
107C22×H(3,𝔽3)—44no—
108C22×H(3,𝔽3)C2×D412yes—
109C22×H(3,𝔽3)#4112yes—
110C22×H(3,𝔽3)#4112yes—
111C22×H(3,𝔽3)C2411yescharacteristic
112C2×(C3⋊D3)⋊C3—22no—
113C2×(C3⋊D3)⋊C3—22no—
114C2×(C3⋊D3)⋊C3—22no—
115C2×(C3⋊D3)⋊C3—44no—
116C4×H(3,𝔽3)—22no—
117C4×H(3,𝔽3)—22no—
118(C3⋊Dic3)⋊C3—22no—
119(C3⋊Dic3)⋊C3—22no—
120(C3⋊Dic3)⋊C3C2×D414yes—
121C22×C3×Q8—91no—
122C22×C3×D4—91no—
123C22×#67—31no—
124C22×#67—91no—
125C22×Dic6—31no—
126C22×Dic6—91no—
127C3×#124—98no—
128C2×C3×C22⋊C4—31no—
129C2×C3×C22⋊C4—31no—
130C2×C3×C22⋊C4—61no—
131C2×C3×C22⋊C4—91no—
132C2×C3×C22⋊C4—91no—
133C2×C3×C22⋊C4—91no—
134C2×C3×C22⋊C4C3×D311yescharacteristic
135C2×C3×C42—91no—
136C2×#217—31no—
137C2×#217—91no—
138C2×(C2×D3)⋊C4—31no—
139C2×(C2×D3)⋊C4—31no—
140C2×(C2×D3)⋊C4—91no—
141C2×(C2×D3)⋊C4—91no—
142C2×C4×Dic3—31no—
143C2×C4×Dic3—91no—
144#885—38no—
145#885—98no—
146C23×C32—24no—
147C23×C32—22no—
148C23×C32—22no—
149C23×C32—22no—
150C23×C32—22no—
151C23×C32—22no—
152C23×C32—22no—
153C23×C32—21no—
154C23×C32—22no—
155C23×C32—42no—
156C23×C32—42no—
157C23×C32C22×D312yes—
158C23×C32C22×D311yescharacteristic
159C23×C32C23×C312yes—
160C23×C32C23×C311yescharacteristic
161C22×C3⋊D3—22no—
162C22×C3⋊D3—22no—
163C22×C3⋊D3—22no—
164C22×C3⋊D3C23×C312yes—
165C22×C3×D3—32no—
166C22×C3×D3—62no—
167C22×C3×D3—62no—
168C22×C3×D3—62no—
169C2×C32×C4—22no—
170C2×C32×C4—22no—
171C2×C32×C4—22no—
172C2×C32×C4—22no—
173C2×C32×C4—22no—
174C2×C32×C4—22no—
175C2×C32×C4—42no—
176C2×C32×C4—42no—
177C2×C32×C4C22×D312yes—
178C2×C32×C4C22×D312yes—
179C2×C32×C4C23×C312yes—
180C2×C32×C4C23×C312yes—
181#638—28no—
182C2×C3⋊Dic3—22no—
183C2×C3⋊Dic3—22no—
184C2×C3⋊Dic3C3×D414yes—
185C2×C3⋊Dic3C23×C312yes—
186C2×C3⋊Dic3C23×C312yes—
187C2×C3⋊Dic3C23×C312yes—
188#634—28no—
189C3×#67—68no—
190C2×C3×Dic3—34no—
191C2×C3×Dic3—32no—
192C2×C3×Dic3—32no—
193C2×C3×Dic3—32no—
194C2×C3×Dic3—62no—
195C2×C3×Dic3—62no—
196C3×Dic6—68no—
197C2×#124—91noSylow
198C2×H(3,𝔽3)—42no—
199C2×H(3,𝔽3)#12414yescomplemented
200C2×H(3,𝔽3)C22×D411yescharacteristic
201C2×H(3,𝔽3)C2×#4111yescharacteristic
202C2×H(3,𝔽3)C2×#4111yescharacteristic
203(C3⋊D3)⋊C3—42no—
204C24×C3—31no—
205C24×C3—31no—
206C24×C3—61no—
207C24×C3—91no—
208C24×C3C2×C3×D311yescharacteristic
209C23×D3—31no—
210C23×D3—91no—
211C2×C3×Q8—94no—
212C2×C3×Q8—184no—
213C2×C3×D4—94no—
214C2×C3×D4—184no—
215C22×C3×C4—31no—
216C22×C3×C4—31no—
217C22×C3×C4—31no—
218C22×C3×C4—31no—
219C22×C3×C4—61no—
220C22×C3×C4—61no—
221C22×C3×C4—91no—
222C22×C3×C4—91no—
223C22×C3×C4—91no—
224C22×C3×C4C2×C3×D311yescharacteristic
225C22×C3×C4C2×C3×D311yescharacteristic
226C2×#67—34no—
227C2×#67—64no—
228C2×#67—94no—
229C2×#67—184no—
230C22×Dic3—31no—
231C22×Dic3—31no—
232C22×Dic3—31no—
233C22×Dic3—91no—
234C22×Dic3—91no—
235C22×Dic3—91no—
236C2×Dic6—34no—
237C2×Dic6—64no—
238C2×Dic6—94no—
239C2×Dic6—184no—
240C3×C22⋊C4—34no—
241C3×C22⋊C4—34no—
242C3×C22⋊C4—64no—
243C3×C22⋊C4—94no—
244C3×C22⋊C4—94no—
245C3×C22⋊C4—94no—
246C3×C22⋊C4C2×C3×D314yes—
247C3×C42—94no—
248#217—34no—
249#217—94no—
250(C2×D3)⋊C4—34no—
251(C2×D3)⋊C4—34no—
252(C2×D3)⋊C4—94no—
253(C2×D3)⋊C4—94no—
254C4×Dic3—34no—
255C4×Dic3—94no—
256C22×C32—24no—
257C22×C32—22no—
258C22×C32—22no—
259C22×C32—22no—
260C22×C32—22no—
261C22×C32—22no—
262C22×C32—22no—
263C22×C32—21no—
264C22×C32—22no—
265C22×C32—22no—
266C22×C32—22no—
267C22×C32—42no—
268C22×C32—42no—
269C22×C32—48no—
270C22×C32—44no—
271C22×C32—44no—
272C22×C32#23512yes—
273C22×C32D3×D412yes—
274C22×C32#23712yes—
275C22×C32C2×C3×D412yes—
276C22×C32C3×#4112yes—
277C22×C32C3×#4112yes—
278C22×C32C23×D311yescharacteristic
279C22×C32C24×C311yescharacteristic
280C2×C3⋊D3—22no—
281C2×C3⋊D3—22no—
282C2×C3⋊D3—22no—
283C2×C3⋊D3—44no—
284C2×C3×D3—62no—
285C2×C3×D3—62no—
286C2×C3×D3—62no—
287C2×C3×D3—124no—
288C32×C4—22no—
289C32×C4—22no—
290C32×C4—22no—
291C32×C4—22no—
292C32×C4—42no—
293C32×C4—42no—
294C3⋊Dic3—22no—
295C3⋊Dic3—22no—
296C3⋊Dic3C2×C3×D414yes—
297C3×Dic3—34no—
298C3×Dic3—62no—
299C3×Dic3—62no—
300C22×Q8—91no—
301C22×D4—91no—
302#124—98no—
303C2×C22⋊C4—91no—
304C2×C22⋊C4—91no—
305C2×C22⋊C4—91no—
306C2×C22⋊C4(C3⋊D3)⋊C311yescharacteristic, complemented
307C2×C42—91no—
308H(3,𝔽3)C2×#12411yescharacteristic, complemented, Sylow
309C23×C3—22no—
310C23×C3—22no—
311C23×C3—22no—
312C23×C3—32no—
313C23×C3—32no—
314C23×C3—31no—
315C23×C3—31no—
316C23×C3—62no—
317C23×C3—62no—
318C23×C3—62no—
319C23×C3—62no—
320C23×C3—62no—
321C23×C3—62no—
322C23×C3—64no—
323C23×C3—62no—
324C23×C3—61no—
325C23×C3—92no—
326C23×C3—122no—
327C23×C3—122no—
328C23×C3—182no—
329C23×C3—182no—
330C23×C3—182no—
331C23×C3C22×C3×D312yes—
332C23×C3C22×C3×D311yescharacteristic
333C22×D3—32no—
334C22×D3—62no—
335C22×D3—62no—
336C22×D3—62no—
337C22×D3—92no—
338C22×D3—182no—
339C22×D3—182no—
340C22×D3—182no—
341C3×Q8—188no—
342C3×D4—188no—
343C2×C3×C4—22no—
344C2×C3×C4—22no—
345C2×C3×C4—32no—
346C2×C3×C4—32no—
347C2×C3×C4—32no—
348C2×C3×C4—32no—
349C2×C3×C4—62no—
350C2×C3×C4—62no—
351C2×C3×C4—62no—
352C2×C3×C4—62no—
353C2×C3×C4—62no—
354C2×C3×C4—62no—
355C2×C3×C4—92no—
356C2×C3×C4—94no—
357C2×C3×C4—92no—
358C2×C3×C4—92no—
359C2×C3×C4—122no—
360C2×C3×C4—122no—
361C2×C3×C4—182no—
362C2×C3×C4—182no—
363C2×C3×C4C22×C3×D312yes—
364C2×C3×C4C22×C3×D312yes—
365#67—68no—
366#67—188no—
367C2×Dic3—32no—
368C2×Dic3—34no—
369C2×Dic3—32no—
370C2×Dic3—32no—
371C2×Dic3—62no—
372C2×Dic3—62no—
373C2×Dic3—94no—
374C2×Dic3—92no—
375C2×Dic3—92no—
376C2×Dic3—92no—
377C2×Dic3—182no—
378C2×Dic3—182no—
379Dic6—68no—
380Dic6—188no—
381C2×C32—24no—
382C2×C32—21no—
383C2×C32—21no—
384C2×C32—21no—
385C2×C32—44no—
386C2×C32—42no—
387C2×C32—42no—
388C2×C32#88514yescomplemented
389C2×C32C3×#12414yescomplemented
390C2×C32C2×#23511yescharacteristic
391C2×C32C2×D3×D411yescharacteristic
392C2×C32C2×#23711yescharacteristic
393C2×C32C22×C3×D411yescharacteristic
394C2×C32C2×C3×#4111yescharacteristic
395C2×C32C2×C3×#4111yescharacteristic
396C3⋊D3—42no—
397C3×D3—122no—
398C24—91no—
399C24C2×(C3⋊D3)⋊C311yescharacteristic
400C2×Q8—94no—
401C2×Q8—184no—
402C2×D4—94no—
403C2×D4—184no—
404C22×C4—91no—
405C22×C4—91no—
406C22×C4—91no—
407C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
408C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
409C22⋊C4—94no—
410C22⋊C4—94no—
411C22⋊C4—94no—
412C22⋊C4C2×(C3⋊D3)⋊C314yescomplemented
413C42—94no—
414C22×C3—22no—
415C22×C3—22no—
416C22×C3—22no—
417C22×C3—31no—
418C22×C3—31no—
419C22×C3—32no—
420C22×C3—32no—
421C22×C3—32no—
422C22×C3—32no—
423C22×C3—32no—
424C22×C3—32no—
425C22×C3—44no—
426C22×C3—62no—
427C22×C3—62no—
428C22×C3—62no—
429C22×C3—62no—
430C22×C3—62no—
431C22×C3—62no—
432C22×C3—64no—
433C22×C3—61no—
434C22×C3—62no—
435C22×C3—62no—
436C22×C3—62no—
437C22×C3—122no—
438C22×C3—122no—
439C22×C3—124no—
440C22×C3—124no—
441C22×C3—128no—
442C22×C3—182no—
443C22×C3—182no—
444C22×C3—182no—
445C22×C3—364no—
446C22×C3C3×#23512yes—
447C22×C3C3×D3×D412yes—
448C22×C3C3×#23712yes—
449C22×C3C23×C3×D311yescharacteristic
450C2×D3—62no—
451C2×D3—62no—
452C2×D3—62no—
453C2×D3—124no—
454C2×D3—182no—
455C2×D3—182no—
456C2×D3—182no—
457C2×D3—364no—
458C3×C4—22no—
459C3×C4—22no—
460C3×C4—62no—
461C3×C4—62no—
462C3×C4—62no—
463C3×C4—62no—
464C3×C4—94no—
465C3×C4—122no—
466C3×C4—122no—
467C3×C4—182no—
468C3×C4—182no—
469Dic3—34no—
470Dic3—62no—
471Dic3—62no—
472Dic3—94no—
473Dic3—182no—
474Dic3—182no—
475C32—21no—
476C32C2×#88511yescharacteristic, complemented
477C32C2×C3×#12411yescharacteristic, complemented
478C23—22no—
479C23—22no—
480C23—22no—
481C23—92no—
482C23—182no—
483C23—182no—
484C23—182no—
485C23C22×(C3⋊D3)⋊C312yes—
486C23C22×(C3⋊D3)⋊C311yescharacteristic
487Q8—188no—
488D4—188no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—92no—
492C2×C4—94no—
493C2×C4—92no—
494C2×C4—92no—
495C2×C4—182no—
496C2×C4—182no—
497C2×C4C22×(C3⋊D3)⋊C312yes—
498C2×C4C22×(C3⋊D3)⋊C312yes—
499C2×C3—31no—
500C2×C3—31no—
501C2×C3—31no—
502C2×C3—31no—
503C2×C3—31no—
504C2×C3—31no—
505C2×C3—34no—
506C2×C3—34no—
507C2×C3—42no—
508C2×C3—61no—
509C2×C3—61no—
510C2×C3—61no—
511C2×C3—64no—
512C2×C3—122no—
513C2×C3—122no—
514C2×C3—124no—
515C2×C3—362no—
516C2×C3C3×#88514yes—
517C2×C3C2×C3×#23511yescharacteristic
518C2×C3C2×C3×D3×D411yescharacteristic
519C2×C3C2×C3×#23711yescharacteristic
520D3—122no—
521D3—362no—
522C22—22no—
523C22—22no—
524C22—22no—
525C22—44no—
526C22—182no—
527C22—182no—
528C22—182no—
529C22—364no—
530C22#3828⋊C312yes—
531C22D4×(C3⋊D3)⋊C312yes—
532C22#3830⋊C312yes—
533C22C23×(C3⋊D3)⋊C311yescharacteristic
534C4—22no—
535C4—22no—
536C4—94no—
537C4—182no—
538C4—182no—
539C3—31no—
540C3—31no—
541C3—61no—
542C3C2×C3×#88511yescharacteristic
543C2—42no—
544C2—362no—
545C2#64094⋊C314yesdirect product factor
546C2C2×#3828⋊C311yescharacteristic
547C2C2×D4×(C3⋊D3)⋊C311yescharacteristic
548C2C2×#3830⋊C311yescharacteristic
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
prime radicalC22×#638⋊C3C2
hypercenterC2×C22⋊C4(C3⋊D3)⋊C3
hypocenterC32C2×C3×#124
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C22×#634⋊C3C2semidirect
3. C2×C22⋊C4×H(3,𝔽3)C2semidirect
4. C2×#3757⋊C3C2semidirect
5. C2×#3752⋊C3C2semidirect
6. C2×#3752⋊C3C2semidirect
7. C2×C4×(C3⋊Dic3)⋊C3C2semidirect
8. #64094⋊C3C2direct (N × H)
9. C2×#64094C3semidirect
21. C2×#634⋊C3C22semidirect
22. C22⋊C4×H(3,𝔽3)C22semidirect
23. #3757⋊C3C22semidirect
24. #3752⋊C3C22semidirect
25. #3752⋊C3C22semidirect
26. C4×(C3⋊Dic3)⋊C3C22semidirect
28. C22×#634C2×C3semidirect
32. C2×C32×C22⋊C4D3semidirect
33. C2×C32×C22⋊C4C2×C3semidirect
34. C2×#3757C2×C3semidirect
35. C2×#3752C2×C3semidirect
36. C2×#3752C2×C3semidirect
37. C2×C4×C3⋊Dic3C2×C3semidirect
38. #64094C2×C3semidirect
85. C2×#634C22×C3semidirect
94. C32×C22⋊C4C2×D3semidirect
95. C32×C22⋊C4C22×C3semidirect
96. #3757C22×C3semidirect
97. #3752C22×C3semidirect
98. #3752C22×C3semidirect
99. C4×C3⋊Dic3C22×C3semidirect
199. C2×H(3,𝔽3)#124semidirect
306. C2×C22⋊C4(C3⋊D3)⋊C3semidirect
308. H(3,𝔽3)C2×#124semidirect
388. C2×C32#885semidirect
389. C2×C32C3×#124semidirect
412. C22⋊C4C2×(C3⋊D3)⋊C3semidirect
476. C32C2×#885semidirect
477. C32C2×C3×#124semidirect
545. C2#64094⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#638⋊C32
C22×#634⋊C32
C2×C22⋊C4×H(3,𝔽3)2
C2×#3757⋊C32
C2×#3752⋊C32
C2×C4×(C3⋊Dic3)⋊C32
#64094⋊C32
C2×#640943
C2×C3×#8853