Finite Group Databasebeta

← C2×C3×#4754 #11867862

Subgroups — C2×C3×#4754 #11867862

Order 1152.

1009 conjugacy classes of subgroups.

The subgroup lattice has 605 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#942C211yesmaximal, characteristic, complemented
2C22×C3×#215C211yesmaximal, characteristic, complemented
3C2×C3×#908C211yesmaximal, characteristic
4C2×C3×#907C211yesmaximal, characteristic, complemented
5C2×C32×#101C211yesmaximal, characteristic, complemented
6C2×C3×#823C211yesmaximal, characteristic, complemented
7C2×C3×#822C211yesmaximal, characteristic, complemented
8C3×#4754C218yesmaximal, direct product factor
9C2×C3×#471—31nomaximal
10C2×#4754C311yesmaximal, characteristic, direct product factor
11C22×C32×Q8C2211yescharacteristic
12C2×C3×#235C2211yescharacteristic
13C22×C3×D12C2211yescharacteristic
14C2×C32×M16C2211yescharacteristic
15C2×C32×M16C2211yescharacteristic
16C2×C3×#216—21no—
17C3×#942—22no—
18C3×#942C2214yescomplemented
19C2×C3×#215—21no—
20C2×C3×#215—21no—
21C2×C3×#215—21no—
22C2×C3×#215—22no—
23C2×C3×#215C2214yescomplemented
24C2×C3×#208C2211yescharacteristic
25C22×C3×C3⋊C8C2211yescharacteristic
26C3×#908—22no—
27C3×#908C2214yes—
28C3×#907—22no—
29C3×#907C2214yescomplemented
30C2×C3×D24—21no—
31C2×C3×#204—21no—
32C2×C3×ΓL(1,𝔽25)—21no—
33C2×C3×D3×C8—21no—
34C32×#101C2214yescomplemented
35C3×#823C2214yescomplemented
36C3×#822C2214yescomplemented
37C2×C3×#137—31no—
38C2×C3×AGL(1,ℤ/8)—31no—
39C22×C3×SD16—31no—
40C2×C3×#131—31no—
41C2×#942C2×C311yescharacteristic, complemented
42C22×#215C2×C311yescharacteristic, complemented
43C2×#908C2×C311yescharacteristic
44C2×#907C2×C311yescharacteristic, complemented
45C3×#471—38no—
46C2×C3×#108—31no—
47C2×C3×#101—21no—
48C2×C3×#101D311yescharacteristic, complemented
49C2×C3×#101C2×C311yescharacteristic, complemented
50C2×C3×#100—31no—
51C2×#823C2×C311yescharacteristic, complemented
52C2×#822C2×C311yescharacteristic, complemented
53#4754C2×C318yesdirect product factor
54C23×C3×D3—21no—
55C2×C32×Q8—21no—
56C2×C32×Q8—21no—
57C2×C32×Q8—21no—
58C2×C32×Q8—21no—
59C2×C32×Q8—21no—
60C2×C32×Q8C2312yes—
61C22×C32×C4—21no—
62C22×C32×C4C2311yescharacteristic
63C2×C3×#67—21no—
64C3×#235—21no—
65C3×#235—21no—
66C3×#235D412yes—
67C3×#235C2312yes—
68C2×C3×D12—21no—
69C2×C3×D12—21no—
70C2×C3×D12—21no—
71C2×C3×D12—21no—
72C2×C3×D12—21no—
73C2×C3×D12D411yescharacteristic
74C2×C3×D12C2312yes—
75C2×C3×C4×D3—21no—
76C2×C3×Dic6D411yescharacteristic
77C32×M16—21no—
78C32×M16—21no—
79C32×M16C2312yes—
80C32×M16C2312yes—
81C2×C32×C8—21no—
82C2×C32×C8—21no—
83C3×#216—24no—
84C3×#215—24no—
85C3×#215—24no—
86C3×#215—24no—
87C3×#208—21no—
88C3×#208C2312yes—
89C2×C3×C3⋊C8—21no—
90C2×C3×C3⋊C8D411yescharacteristic
91C2×C3×C3⋊C8D411yescharacteristic
92C2×C3×C3⋊C8D412yes—
93C2×C3×C3⋊C8C2312yes—
94C3×D24—24no—
95C3×#204—24no—
96C3×ΓL(1,𝔽25)—24no—
97C3×D3×C8—24no—
98C2×#471—31noSylow
99C2×C3×#41—31no—
100C22×C3×Q8—21no—
101C22×C3×Q8C2×D311yescharacteristic
102C22×C3×Q8C22×C311yescharacteristic
103C22×C3×D4—31no—
104C2×#235C22×C311yescharacteristic
105C22×D12C22×C311yescharacteristic
106C3×#137—34no—
107C3×#137—62no—
108C3×AGL(1,ℤ/8)—34no—
109C3×AGL(1,ℤ/8)—62no—
110C2×C3×Q16—61no—
111C2×C3×SD16—34no—
112C2×C3×SD16—61no—
113C2×C3×SD16—61no—
114C2×C3×SD16—61no—
115C2×C3×SD16—62no—
116C2×C3×SD16—61no—
117C2×C3×D8—61no—
118C3×#131—34no—
119C3×#131—62no—
120C2×C3×M16—21no—
121C2×C3×M16—21no—
122C2×C3×M16—31no—
123C2×C3×M16—61no—
124C2×C3×M16C2×D311yescharacteristic
125C2×C3×M16C2×D311yescharacteristic
126C2×C3×M16C22×C311yescharacteristic
127C2×C3×M16C22×C311yescharacteristic
128C22×C3×C8—31no—
129C22×C3×C8—61no—
130C2×#216—21no—
131#942—22no—
132#942C22×C314yescomplemented
133C2×#215—21no—
134C2×#215—21no—
135C2×#215—21no—
136C2×#215—22no—
137C2×#215C22×C314yescomplemented
138C2×#208C22×C311yescharacteristic
139C22×C3⋊C8C22×C311yescharacteristic
140#908—22no—
141#908C22×C314yes—
142#907—22no—
143#907C22×C314yescomplemented
144C2×D24—21no—
145C2×#204—21no—
146C2×ΓL(1,𝔽25)—21no—
147C2×D3×C8—21no—
148C3×#108—34no—
149C3×#101—24no—
150C3×#101C2×D314yescomplemented
151C3×#101C22×C314yescomplemented
152C3×#100—34no—
153#823C22×C314yescomplemented
154#822C22×C314yescomplemented
155C23×C32#4111yescharacteristic
156C22×C3×D3—21no—
157C22×C3×D3—22no—
158C22×C3×D3—41no—
159C22×C3×D3—41no—
160C22×C3×D3—82no—
161C32×Q8—22no—
162C32×Q8—22no—
163C32×Q8—41no—
164C32×Q8—41no—
165C2×C32×C4—22no—
166C2×C32×C4—41no—
167C2×C32×C4—41no—
168C2×C32×C4C2×D411yescharacteristic
169C2×C32×C4C2×D411yescharacteristic
170C2×C32×C4C2×D411yescharacteristic
171C2×C32×C4C2×D411yescharacteristic
172C2×C32×C4#4111yescharacteristic
173C2×C32×C4C2411yescharacteristic
174C3×#67—22no—
175C3×#67—22no—
176C2×C3×Dic3—21no—
177C3×D12—21no—
178C3×D12—22no—
179C3×D12—22no—
180C3×D12—41no—
181C3×D12—41no—
182C3×D12C2×D412yes—
183C3×C4×D3—22no—
184C3×C4×D3—22no—
185C3×Dic6—21no—
186C3×Dic6C2×D412yes—
187C32×C8—22no—
188C32×C8—22no—
189C3×C3⋊C8—22no—
190C3×C3⋊C8C2×D412yes—
191C3×C3⋊C8C2×D412yes—
192C2×#137—31no—
193C2×AGL(1,ℤ/8)—31no—
194C22×SD16—31no—
195C2×#131—31no—
196#471—38no—
197C2×#108—31no—
198C2×#101C3×D311yescharacteristic, complemented
199C2×#100—31no—
200C24×C3—61no—
201C23×D3—21no—
202C3×#41—32no—
203C3×#41—32no—
204C3×#41—61no—
205C3×#41—61no—
206C2×C3×Q8—22no—
207C2×C3×Q8—21no—
208C2×C3×Q8—21no—
209C2×C3×Q8—21no—
210C2×C3×Q8—21no—
211C2×C3×Q8—21no—
212C2×C3×Q8—21no—
213C2×C3×Q8—21no—
214C2×C3×Q8—21no—
215C2×C3×Q8—21no—
216C2×C3×Q8—21no—
217C2×C3×Q8—22no—
218C2×C3×Q8—31no—
219C2×C3×Q8—41no—
220C2×C3×Q8—41no—
221C2×C3×Q8—41no—
222C2×C3×Q8—41no—
223C2×C3×Q8C22×D312yes—
224C2×C3×Q8C23×C312yes—
225C2×C3×D4—31no—
226C2×C3×D4—32no—
227C2×C3×D4—61no—
228C2×C3×D4—61no—
229C2×C3×D4—61no—
230C2×C3×D4—61no—
231C2×C3×D4—61no—
232C2×C3×D4—61no—
233C22×C3×C4—22no—
234C22×C3×C4—21no—
235C22×C3×C4—21no—
236C22×C3×C4—21no—
237C22×C3×C4—61no—
238C22×C3×C4C22×D311yescharacteristic
239C22×C3×C4C23×C311yescharacteristic
240C2×#67—21no—
241#235—21no—
242#235—21no—
243#235C3×D412yes—
244#235C23×C312yes—
245C2×D12—21no—
246C2×D12—21no—
247C2×D12—21no—
248C2×D12—21no—
249C2×D12—21no—
250C2×D12C3×D411yescharacteristic
251C2×D12C23×C312yes—
252C2×C4×D3—21no—
253C2×Dic6C3×D411yescharacteristic
254C3×Q16—64no—
255C3×SD16—64no—
256C3×SD16—64no—
257C3×SD16—64no—
258C3×SD16—64no—
259C3×D8—64no—
260C3×M16—21no—
261C3×M16—21no—
262C3×M16—21no—
263C3×M16—21no—
264C3×M16—22no—
265C3×M16—22no—
266C3×M16—32no—
267C3×M16—41no—
268C3×M16—41no—
269C3×M16—64no—
270C3×M16—61no—
271C3×M16C22×D312yes—
272C3×M16C22×D312yes—
273C3×M16C23×C312yes—
274C3×M16C23×C312yes—
275C2×C3×C8—21no—
276C2×C3×C8—21no—
277C2×C3×C8—21no—
278C2×C3×C8—21no—
279C2×C3×C8—31no—
280C2×C3×C8—31no—
281C2×C3×C8—32no—
282C2×C3×C8—32no—
283C2×C3×C8—41no—
284C2×C3×C8—41no—
285C2×C3×C8—64no—
286C2×C3×C8—61no—
287#216—24no—
288#215—24no—
289#215—24no—
290#215—24no—
291#208—21no—
292#208C23×C312yes—
293C2×C3⋊C8—21no—
294C2×C3⋊C8C3×D411yescharacteristic
295C2×C3⋊C8C3×D411yescharacteristic
296C2×C3⋊C8C3×D412yes—
297C2×C3⋊C8C23×C312yes—
298D24—24no—
299#204—24no—
300ΓL(1,𝔽25)—24no—
301D3×C8—24no—
302C22×C32—22no—
303C22×C32#12111yescharacteristic
304C22×C32#12111yescharacteristic
305C22×C32C2×#4111yescharacteristic
306C2×C3×D3—22no—
307C2×C3×D3—42no—
308C2×C3×D3—42no—
309C2×C3×D3—82no—
310C2×C3×D3—82no—
311C2×C3×D3—82no—
312C32×C4—42no—
313C32×C4#12111yescharacteristic
314C32×C4#12111yescharacteristic
315C32×C4C22×D411yescharacteristic
316C32×C4C22×D411yescharacteristic
317C3×Dic3—22no—
318C2×#41—31no—
319C22×Q8C2×C3×D311yescharacteristic
320C22×D4—31no—
321#137—34no—
322#137—62no—
323AGL(1,ℤ/8)—34no—
324AGL(1,ℤ/8)—62no—
325C2×Q16—61no—
326C2×SD16—34no—
327C2×SD16—61no—
328C2×SD16—61no—
329C2×SD16—61no—
330C2×SD16—62no—
331C2×SD16—61no—
332C2×D8—61no—
333#131—34no—
334#131—62no—
335C2×M16—31no—
336C2×M16—61no—
337C2×M16C2×C3×D311yescharacteristic
338C2×M16C2×C3×D311yescharacteristic
339C22×C8—31no—
340C22×C8—61no—
341#108—34no—
342#101C2×C3×D314yescomplemented
343#100—34no—
344C23×C3—21no—
345C23×C3—61no—
346C23×C3—62no—
347C23×C3—121no—
348C23×C3—121no—
349C23×C3—242no—
350C23×C3#23511yescharacteristic
351C23×C3C3×#4111yescharacteristic
352C22×D3—21no—
353C22×D3—22no—
354C22×D3—41no—
355C22×D3—41no—
356C22×D3—82no—
357C3×Q8—22no—
358C3×Q8—22no—
359C3×Q8—22no—
360C3×Q8—22no—
361C3×Q8—32no—
362C3×Q8—42no—
363C3×Q8—42no—
364C3×Q8—42no—
365C3×Q8—42no—
366C3×Q8—41no—
367C3×Q8—41no—
368C3×Q8—41no—
369C3×Q8—41no—
370C3×Q8—61no—
371C3×D4—32no—
372C3×D4—62no—
373C3×D4—62no—
374C3×D4—61no—
375C3×D4—62no—
376C3×D4—62no—
377C3×D4—121no—
378C3×D4—121no—
379C2×C3×C4—24no—
380C2×C3×C4—22no—
381C2×C3×C4—22no—
382C2×C3×C4—21no—
383C2×C3×C4—21no—
384C2×C3×C4—21no—
385C2×C3×C4—21no—
386C2×C3×C4—21no—
387C2×C3×C4—21no—
388C2×C3×C4—42no—
389C2×C3×C4—42no—
390C2×C3×C4—41no—
391C2×C3×C4—41no—
392C2×C3×C4—41no—
393C2×C3×C4—41no—
394C2×C3×C4—61no—
395C2×C3×C4—62no—
396C2×C3×C4—62no—
397C2×C3×C4#23511yescharacteristic
398C2×C3×C4D3×D411yescharacteristic
399C2×C3×C4D3×D411yescharacteristic
400C2×C3×C4D3×D411yescharacteristic
401C2×C3×C4D3×D411yescharacteristic
402C2×C3×C4C2×C3×D411yescharacteristic
403C2×C3×C4C2×C3×D411yescharacteristic
404C2×C3×C4C2×C3×D411yescharacteristic
405C2×C3×C4C2×C3×D411yescharacteristic
406C2×C3×C4C3×#4111yescharacteristic
407C2×C3×C4C23×D311yescharacteristic
408C2×C3×C4C24×C311yescharacteristic
409#67—22no—
410#67—22no—
411C2×Dic3—21no—
412D12—21no—
413D12—22no—
414D12—22no—
415D12—41no—
416D12—41no—
417D12C2×C3×D412yes—
418C4×D3—22no—
419C4×D3—22no—
420Dic6—21no—
421Dic6C2×C3×D412yes—
422C3×C8—22no—
423C3×C8—22no—
424C3×C8—22no—
425C3×C8—22no—
426C3×C8—32no—
427C3×C8—32no—
428C3×C8—42no—
429C3×C8—42no—
430C3×C8—62no—
431C3⋊C8—22no—
432C3⋊C8C2×C3×D412yes—
433C3⋊C8C2×C3×D412yes—
434C2×C32—21no—
435C2×C32—21no—
436C2×C32#47112yescomplemented
437C2×C32C2×#12111yescharacteristic
438C3×D3—42no—
439C3×D3—82no—
440C24—61no—
441#41—32no—
442#41—32no—
443#41—61no—
444#41—61no—
445C2×Q8—21no—
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8—31no—
451C2×Q8C22×C3×D312yes—
452C2×D4—32no—
453C2×D4—31no—
454C2×D4—61no—
455C2×D4—61no—
456C2×D4—61no—
457C2×D4—61no—
458C2×D4—61no—
459C2×D4—61no—
460C22×C4—21no—
461C22×C4—61no—
462C22×C4C22×C3×D311yescharacteristic
463Q16—64no—
464SD16—64no—
465SD16—64no—
466SD16—64no—
467SD16—64no—
468D8—64no—
469M16—21no—
470M16—21no—
471M16—32no—
472M16—64no—
473M16—61no—
474M16C22×C3×D312yes—
475M16C22×C3×D312yes—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—32no—
479C2×C8—31no—
480C2×C8—31no—
481C2×C8—32no—
482C2×C8—64no—
483C2×C8—61no—
484C22×C3—22no—
485C22×C3—22no—
486C22×C3—21no—
487C22×C3—21no—
488C22×C3—21no—
489C22×C3—42no—
490C22×C3—62no—
491C22×C3—122no—
492C22×C3—122no—
493C22×C3—242no—
494C22×C3—242no—
495C22×C3—242no—
496C22×C3#88411yescharacteristic
497C22×C3#88411yescharacteristic
498C22×C3C3×#12111yescharacteristic
499C22×C3C3×#12111yescharacteristic
500C22×C3C2×#23511yescharacteristic
501C22×C3C2×C3×#4111yescharacteristic
502C2×D3—22no—
503C2×D3—42no—
504C2×D3—42no—
505C2×D3—82no—
506C2×D3—82no—
507C2×D3—82no—
508C3×C4—21no—
509C3×C4—21no—
510C3×C4—21no—
511C3×C4—21no—
512C3×C4—44no—
513C3×C4—42no—
514C3×C4—42no—
515C3×C4—62no—
516C3×C4#88411yescharacteristic
517C3×C4#88411yescharacteristic
518C3×C4C3×#12111yescharacteristic
519C3×C4C3×#12111yescharacteristic
520C3×C4C2×D3×D411yescharacteristic
521C3×C4C2×D3×D411yescharacteristic
522C3×C4C22×C3×D411yescharacteristic
523C3×C4C22×C3×D411yescharacteristic
524Dic3—22no—
525C32C2×#47111yescharacteristic, complemented, Sylow
526C23—61no—
527C23—62no—
528C23—121no—
529C23—121no—
530C23—242no—
531C23C3×#23511yescharacteristic
532Q8—22no—
533Q8—22no—
534Q8—32no—
535Q8—41no—
536Q8—41no—
537Q8—61no—
538D4—32no—
539D4—62no—
540D4—62no—
541D4—61no—
542D4—62no—
543D4—62no—
544D4—121no—
545D4—121no—
546C2×C4—22no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—61no—
550C2×C4—62no—
551C2×C4—62no—
552C2×C4C3×#23511yescharacteristic
553C2×C4C3×D3×D411yescharacteristic
554C2×C4C3×D3×D411yescharacteristic
555C2×C4C3×D3×D411yescharacteristic
556C2×C4C3×D3×D411yescharacteristic
557C2×C4C23×C3×D311yescharacteristic
558C8—22no—
559C8—22no—
560C8—32no—
561C8—32no—
562C8—62no—
563C2×C3—21no—
564C2×C3—21no—
565C2×C3—21no—
566C2×C3—21no—
567C2×C3—21no—
568C2×C3—22no—
569C2×C3—41no—
570C2×C3—41no—
571C2×C3—122no—
572C2×C3—242no—
573C2×C3#475412yesdirect product factor
574C2×C3C3×#47112yescomplemented
575C2×C3C2×#88411yescharacteristic
576C2×C3C2×C3×#12111yescharacteristic
577D3—42no—
578D3—82no—
579C22—22no—
580C22—62no—
581C22—122no—
582C22—122no—
583C22—242no—
584C22—242no—
585C22—242no—
586C22C3×#88411yescharacteristic
587C22C3×#88411yescharacteristic
588C22C2×C3×#23511yescharacteristic
589C4—42no—
590C4—62no—
591C4C3×#88411yescharacteristic
592C4C3×#88411yescharacteristic
593C4C2×C3×D3×D411yescharacteristic
594C4C2×C3×D3×D411yescharacteristic
595C3—21no—
596C3C2×#475411yescharacteristic, direct product factor
597C3C2×C3×#47111yescharacteristic, complemented
598C2—21no—
599C2—21no—
600C2—122no—
601C2—242no—
602C2C3×#475412yesdirect product factor
603C2C2×C3×#88411yescharacteristic
subgroupG/H
centerC22×C3#884
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC2×C32×#101C2
socleC22×C32#121
prime radicalC2×C3×#908C2
hypercenterC2×C3×#101D3
hypocenterC3C2×C3×#471
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C2×C3×#101

Complements

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

NH ≅ G/Ntype
1. C2×C3×#942C2semidirect
2. C22×C3×#215C2semidirect
4. C2×C3×#907C2semidirect
5. C2×C32×#101C2semidirect
6. C2×C3×#823C2semidirect
7. C2×C3×#822C2semidirect
8. C3×#4754C2direct (N × H)
10. C2×#4754C3direct (N × H)
18. C3×#942C22semidirect
23. C2×C3×#215C22semidirect
29. C3×#907C22semidirect
34. C32×#101C22semidirect
35. C3×#823C22semidirect
36. C3×#822C22semidirect
41. C2×#942C2×C3semidirect
42. C22×#215C2×C3semidirect
44. C2×#907C2×C3semidirect
48. C2×C3×#101D3semidirect
49. C2×C3×#101C2×C3semidirect
51. C2×#823C2×C3semidirect
52. C2×#822C2×C3semidirect
53. #4754C2×C3direct (N × H)
132. #942C22×C3semidirect
137. C2×#215C22×C3semidirect
143. #907C22×C3semidirect
150. C3×#101C2×D3semidirect
151. C3×#101C22×C3semidirect
153. #823C22×C3semidirect
154. #822C22×C3semidirect
198. C2×#101C3×D3semidirect
342. #101C2×C3×D3semidirect
436. C2×C32#471semidirect
525. C32C2×#471semidirect
573. C2×C3#4754direct (N × H)
574. C2×C3C3×#471semidirect
596. C3C2×#4754direct (N × H)
597. C3C2×C3×#471semidirect
602. C2C3×#4754direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#9422
C22×C3×#2152
C2×C3×#9082
C2×C3×#9072
C2×C32×#1012
C2×C3×#8232
C2×C3×#8222
C3×#47542
C2×C3×#4713
C2×#47543