Finite Group Databasebeta

← C22×C3×#1020⋊C2 #72274620

Subgroups — C22×C3×#1020⋊C2 #72274620

Order 2304.

6062 conjugacy classes of subgroups; only the 208 normal subgroups are shown.

normal subgroups only

This group has 6062 conjugacy classes of subgroups — too many to draw. Only its 208 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#1020C212yesmaximal, complemented
2C2×C3×#1020⋊C2C2112yesmaximal, direct product factor
3C22×C3×#120⋊C3C211yesmaximal, characteristic, complemented
4C22×#1020⋊C2C311yesmaximal, characteristic, direct product factor
5C22×C3×B(4,𝔽2)—31nomaximal
6C22×C3×C22⋊A4C2211yescharacteristic, complemented
7C2×C3×#1020C22112yescomplemented
8C23×C3×S4—41nomaximal
9C2×C3×#120⋊C3C2216yescomplemented
10C3×#1020⋊C2C22116yesdirect product factor
11C22×#1020C2×C312yescomplemented
12C22×C3×H(5,𝔽2)—31no—
13C22×C3×#120—32no—
14C22×C3×#120D311yescharacteristic, complemented
15C22×#120⋊C3—21no—
16C22×#120⋊C3C2×C311yescharacteristic, complemented
17C2×#1020⋊C2C2×C3112yesdirect product factor
18C2×C3×B(4,𝔽2)—312no—
19C2×C3×B(4,𝔽2)—312no—
20C22×C3×#99—32no—
21C22×C3×#99—31no—
22C2×C3×C22⋊A4C2313yescomplemented
23C23×C3×A4—41no—
24C3×#1020C2318yescomplemented
25C22×C3×S4—42no—
26C22×C3×S4—412no—
27C22×C3×S4—82no—
28C3×#120⋊C3C2314yescomplemented
29C22×B(4,𝔽2)—31noSylow
30C26×C3C2×D311yescharacteristic, complemented
31C22×C22⋊A4—21no—
32C22×C22⋊A4C22×C311yescharacteristic, complemented
33C2×#1020C22×C3112yescomplemented
34C23×S4—41no—
35C2×C3×H(5,𝔽2)—312no—
36C2×C3×H(5,𝔽2)—612no—
37C2×C3×H(5,𝔽2)—66no—
38C22×C3×#41—62no—
39C22×C3×#41—61no—
40C23×C3×D4—32no—
41C23×C3×D4—31no—
42C23×C3×D4—62no—
43C23×C3×D4—62no—
44C23×C3×D4—61no—
45C2×C3×#120—36no—
46C2×C3×#120—312no—
47C2×C3×#120—312no—
48C2×C3×#120—612no—
49C2×C3×#120C2×D316yescomplemented
50C22×C3×C22⋊C4—31no—
51C22×C3×C22⋊C4—32no—
52C22×C3×C22⋊C4—62no—
53C2×#120⋊C3—26no—
54C2×#120⋊C3C22×C316yescomplemented
55#1020⋊C2C22×C3116yesdirect product factor
56C3×B(4,𝔽2)—348no—
57C3×B(4,𝔽2)—316no—
58C2×C3×#99—324no—
59C2×C3×#99—312no—
60C23×C3×D3—161no—
61C3×C22⋊A4C2411yescharacteristic, complemented
62C22×C3×A4—41no—
63C22×C3×A4—46no—
64C22×C3×A4—82no—
65C2×C3×S4—416no—
66C2×C3×S4—412no—
67C2×C3×S4—812no—
68C22×H(5,𝔽2)—31no—
69C22×#120—32no—
70C22×#120C3×D311yescharacteristic, complemented
71C2×B(4,𝔽2)—312no—
72C2×B(4,𝔽2)—312no—
73C22×#99—32no—
74C22×#99—31no—
75C25×C3—31no—
76C25×C3—32no—
77C25×C3—33no—
78C25×C3—62no—
79C25×C3—66no—
80C25×C3—61no—
81C25×C3S411yescharacteristic, complemented
82C25×C3C22×D313yescomplemented
83C2×C22⋊A4—23no—
84C2×C22⋊A4C23×C313yescomplemented
85C23×A4—41no—
86C23×A4—81no—
87#1020C23×C318yescomplemented
88C22×S4—42no—
89C22×S4—412no—
90C22×S4—82no—
91C3×H(5,𝔽2)—316no—
92C3×H(5,𝔽2)—648no—
93C3×H(5,𝔽2)—624no—
94C3×H(5,𝔽2)—648no—
95C2×C3×#41—624no—
96C2×C3×#41—612no—
97C2×C3×#41—612no—
98C2×C3×#41—624no—
99C22×C3×Q8—61no—
100C22×C3×D4—312no—
101C22×C3×D4—312no—
102C22×C3×D4—36no—
103C22×C3×D4—36no—
104C22×C3×D4—62no—
105C22×C3×D4—62no—
106C22×C3×D4—66no—
107C22×C3×D4—66no—
108C22×C3×D4—66no—
109C22×C3×D4—624no—
110C22×C3×D4—62no—
111C22×C3×D4—61no—
112C22×C3×D4—612no—
113C22×C3×D4—624no—
114C22×C3×D4—62no—
115C22×C3×D4—612no—
116C22×C3×D4—612no—
117C22×C3×D4—612no—
118C22×C3×D4—122no—
119C22×C3×D4—126no—
120C22×C3×D4—126no—
121C22×C3×D4—122no—
122C23×C3×C4—32no—
123C23×C3×C4—31no—
124C23×C3×C4—61no—
125C23×C3×C4—62no—
126C23×C3×C4—62no—
127C3×#120—312no—
128C3×#120—324no—
129C3×#120—38no—
130C3×#120—64no—
131C3×#120—624no—
132C3×#120—624no—
133C3×#120C22×D314yescomplemented
134C2×C3×C22⋊C4—36no—
135C2×C3×C22⋊C4—312no—
136C2×C3×C22⋊C4—63no—
137C2×C3×C22⋊C4—612no—
138C2×C3×C22⋊C4—612no—
139C2×C3×C22⋊C4—66no—
140#120⋊C3—24no—
141#120⋊C3C23×C314yescomplemented
142C3×#99—332no—
143C3×#99—316no—
144C23×C32—161no—
145C22×C3×D3—162no—
146C22×C3×D3—1612no—
147C2×C3×A4—43no—
148C2×C3×A4—44no—
149C2×C3×A4—86no—
150C3×S4—48no—
151C3×S4—88no—
152C26C2×C3×D311yescharacteristic, complemented
153C2×H(5,𝔽2)—312no—
154C2×H(5,𝔽2)—612no—
155C2×H(5,𝔽2)—66no—
156C22×#41—62no—
157C22×#41—61no—
158C23×D4—32no—
159C23×D4—31no—
160C23×D4—62no—
161C23×D4—61no—
162C23×D4—62no—
163C2×#120—36no—
164C2×#120—312no—
165C2×#120—312no—
166C2×#120—612no—
167C2×#120C2×C3×D316yescomplemented
168C22×C22⋊C4—31no—
169C22×C22⋊C4—32no—
170C22×C22⋊C4—62no—
171B(4,𝔽2)—348no—
172B(4,𝔽2)—316no—
173C2×#99—324no—
174C2×#99—312no—
175C24×C3—22no—
176C24×C3—32no—
177C24×C3—36no—
178C24×C3—33no—
179C24×C3—33no—
180C24×C3—312no—
181C24×C3—33no—
182C24×C3—66no—
183C24×C3—66no—
184C24×C3—61no—
185C24×C3—61no—
186C24×C3—66no—
187C24×C3—62no—
188C24×C3—63no—
189C24×C3—66no—
190C24×C3—66no—
191C24×C3—66no—
192C24×C3—66no—
193C24×C3—62no—
194C24×C3—612no—
195C24×C3—121no—
196C24×C3—123no—
197C24×C3—126no—
198C24×C3—126no—
199C24×C3—122no—
200C24×C3—126no—
201C24×C3—126no—
202C24×C3—126no—
203C24×C3—122no—
204C24×C3—241no—
205C24×C3—243no—
206C24×C3C2×S411yescharacteristic
207C24×C3C2×S416yescomplemented
208C24×C3C23×D311yescharacteristic, complemented
209C23×D3—161no—
210C22⋊A4—21no—
211C22⋊A4C24×C311yescharacteristic, complemented
212C22×A4—41no—
213C22×A4—46no—
214C22×A4—82no—
215C22×A4—81no—
216C22×A4—86no—
217C22×A4—162no—
218C2×S4—416no—
219C2×S4—412no—
220C2×S4—812no—
221C3×#41—632no—
222C3×#41—616no—
223C3×#41—648no—
224C3×#41—696no—
225C2×C3×Q8—612no—
226C2×C3×D4—38no—
227C2×C3×D4—324no—
228C2×C3×D4—34no—
229C2×C3×D4—312no—
230C2×C3×D4—612no—
231C2×C3×D4—612no—
232C2×C3×D4—612no—
233C2×C3×D4—612no—
234C2×C3×D4—612no—
235C2×C3×D4—612no—
236C2×C3×D4—612no—
237C2×C3×D4—612no—
238C2×C3×D4—624no—
239C2×C3×D4—612no—
240C2×C3×D4—632no—
241C2×C3×D4—624no—
242C2×C3×D4—612no—
243C2×C3×D4—648no—
244C2×C3×D4—696no—
245C2×C3×D4—612no—
246C2×C3×D4—616no—
247C2×C3×D4—68no—
248C2×C3×D4—624no—
249C2×C3×D4—1212no—
250C2×C3×D4—1212no—
251C2×C3×D4—1212no—
252C2×C3×D4—1212no—
253C2×C3×D4—1212no—
254C2×C3×D4—1212no—
255C2×C3×D4—1224no—
256C22×C3×C4—312no—
257C22×C3×C4—36no—
258C22×C3×C4—62no—
259C22×C3×C4—61no—
260C22×C3×C4—63no—
261C22×C3×C4—66no—
262C22×C3×C4—612no—
263C22×C3×C4—624no—
264C22×C3×C4—612no—
265C22×C3×C4—126no—
266C22×C3×C4—122no—
267C3×C22⋊C4—34no—
268C3×C22⋊C4—38no—
269C3×C22⋊C4—66no—
270C3×C22⋊C4—68no—
271C3×C22⋊C4—624no—
272C3×C22⋊C4—612no—
273C22×C32—166no—
274C22×C32—161no—
275C2×C3×D3—1612no—
276C2×C3×D3—1616no—
277C3×A4—41no—
278C3×A4—82no—
279C25—31no—
280C25—32no—
281C25—33no—
282C25—62no—
283C25—66no—
284C25—61no—
285C25C3×S411yescharacteristic, complemented
286C25C22×C3×D313yescomplemented
287H(5,𝔽2)—316no—
288H(5,𝔽2)—648no—
289H(5,𝔽2)—624no—
290H(5,𝔽2)—648no—
291C2×#41—624no—
292C2×#41—612no—
293C2×#41—612no—
294C2×#41—624no—
295C22×Q8—61no—
296C22×D4—312no—
297C22×D4—312no—
298C22×D4—36no—
299C22×D4—36no—
300C22×D4—62no—
301C22×D4—62no—
302C22×D4—66no—
303C22×D4—66no—
304C22×D4—66no—
305C22×D4—624no—
306C22×D4—62no—
307C22×D4—61no—
308C22×D4—612no—
309C22×D4—624no—
310C22×D4—62no—
311C22×D4—612no—
312C22×D4—612no—
313C22×D4—612no—
314C22×D4—122no—
315C22×D4—126no—
316C22×D4—126no—
317C22×D4—122no—
318C23×C4—32no—
319C23×C4—31no—
320C23×C4—61no—
321C23×C4—62no—
322C23×C4—62no—
323#120—312no—
324#120—324no—
325#120—38no—
326#120—64no—
327#120—624no—
328#120—624no—
329#120C22×C3×D314yescomplemented
330C2×C22⋊C4—36no—
331C2×C22⋊C4—312no—
332C2×C22⋊C4—63no—
333C2×C22⋊C4—612no—
334C2×C22⋊C4—612no—
335C2×C22⋊C4—66no—
336#99—332no—
337#99—316no—
338C23×C3—26no—
339C23×C3—31no—
340C23×C3—33no—
341C23×C3—36no—
342C23×C3—36no—
343C23×C3—36no—
344C23×C3—36no—
345C23×C3—31no—
346C23×C3—33no—
347C23×C3—38no—
348C23×C3—41no—
349C23×C3—62no—
350C23×C3—66no—
351C23×C3—66no—
352C23×C3—66no—
353C23×C3—66no—
354C23×C3—66no—
355C23×C3—612no—
356C23×C3—612no—
357C23×C3—62no—
358C23×C3—66no—
359C23×C3—66no—
360C23×C3—66no—
361C23×C3—66no—
362C23×C3—612no—
363C23×C3—616no—
364C23×C3—123no—
365C23×C3—123no—
366C23×C3—123no—
367C23×C3—123no—
368C23×C3—123no—
369C23×C3—126no—
370C23×C3—126no—
371C23×C3—126no—
372C23×C3—126no—
373C23×C3—126no—
374C23×C3—126no—
375C23×C3—126no—
376C23×C3—1212no—
377C23×C3—1212no—
378C23×C3—126no—
379C23×C3—1212no—
380C23×C3—1212no—
381C23×C3—122no—
382C23×C3—161no—
383C23×C3—2412no—
384C23×C3—241no—
385C23×C3—246no—
386C23×C3—246no—
387C23×C3—2412no—
388C23×C3—321no—
389C23×C3C22×S413yes—
390C23×C3C22×S414yescomplemented
391C22×D3—162no—
392C22×D3—1612no—
393C2×A4—43no—
394C2×A4—44no—
395C2×A4—86no—
396C2×A4—83no—
397C2×A4—84no—
398C2×A4—166no—
399S4—48no—
400S4—88no—
401C3×Q8—616no—
402C3×D4—624no—
403C3×D4—68no—
404C3×D4—68no—
405C3×D4—624no—
406C3×D4—624no—
407C3×D4—632no—
408C3×D4—616no—
409C3×D4—68no—
410C3×D4—128no—
411C3×D4—1224no—
412C3×D4—128no—
413C3×D4—1224no—
414C2×C3×C4—38no—
415C2×C3×C4—34no—
416C2×C3×C4—612no—
417C2×C3×C4—66no—
418C2×C3×C4—66no—
419C2×C3×C4—612no—
420C2×C3×C4—616no—
421C2×C3×C4—632no—
422C2×C3×C4—68no—
423C2×C3×C4—1212no—
424C2×C3×C4—1212no—
425C2×C32—164no—
426C2×C32—163no—
427C3×D3—168no—
428C24—22no—
429C24—32no—
430C24—36no—
431C24—33no—
432C24—33no—
433C24—312no—
434C24—33no—
435C24—66no—
436C24—66no—
437C24—61no—
438C24—61no—
439C24—66no—
440C24—62no—
441C24—63no—
442C24—66no—
443C24—66no—
444C24—66no—
445C24—66no—
446C24—62no—
447C24—612no—
448C24—121no—
449C24—123no—
450C24—126no—
451C24—126no—
452C24—122no—
453C24—126no—
454C24—126no—
455C24—126no—
456C24—122no—
457C24—241no—
458C24—243no—
459C24C2×C3×S411yescharacteristic
460C24C2×C3×S416yescomplemented
461C24C23×C3×D311yescharacteristic, complemented
462#41—632no—
463#41—616no—
464#41—648no—
465#41—696no—
466C2×Q8—612no—
467C2×D4—38no—
468C2×D4—324no—
469C2×D4—34no—
470C2×D4—312no—
471C2×D4—612no—
472C2×D4—612no—
473C2×D4—612no—
474C2×D4—612no—
475C2×D4—612no—
476C2×D4—612no—
477C2×D4—612no—
478C2×D4—612no—
479C2×D4—624no—
480C2×D4—612no—
481C2×D4—632no—
482C2×D4—624no—
483C2×D4—612no—
484C2×D4—648no—
485C2×D4—696no—
486C2×D4—612no—
487C2×D4—616no—
488C2×D4—68no—
489C2×D4—624no—
490C2×D4—1212no—
491C2×D4—1212no—
492C2×D4—1212no—
493C2×D4—1212no—
494C2×D4—1212no—
495C2×D4—1212no—
496C2×D4—1224no—
497C22×C4—312no—
498C22×C4—36no—
499C22×C4—62no—
500C22×C4—61no—
501C22×C4—63no—
502C22×C4—66no—
503C22×C4—612no—
504C22×C4—624no—
505C22×C4—612no—
506C22×C4—126no—
507C22×C4—122no—
508C22⋊C4—34no—
509C22⋊C4—38no—
510C22⋊C4—66no—
511C22⋊C4—68no—
512C22⋊C4—624no—
513C22⋊C4—612no—
514C22×C3—22no—
515C22×C3—33no—
516C22×C3—33no—
517C22×C3—33no—
518C22×C3—32no—
519C22×C3—36no—
520C22×C3—46no—
521C22×C3—61no—
522C22×C3—64no—
523C22×C3—66no—
524C22×C3—66no—
525C22×C3—68no—
526C22×C3—66no—
527C22×C3—62no—
528C22×C3—66no—
529C22×C3—66no—
530C22×C3—66no—
531C22×C3—126no—
532C22×C3—121no—
533C22×C3—123no—
534C22×C3—126no—
535C22×C3—126no—
536C22×C3—1212no—
537C22×C3—126no—
538C22×C3—122no—
539C22×C3—126no—
540C22×C3—128no—
541C22×C3—1212no—
542C22×C3—166no—
543C22×C3—161no—
544C22×C3—2416no—
545C22×C3—2412no—
546C22×C3—326no—
547C22×C3—321no—
548C22×C3#1020⋊C211yescharacteristic, direct product factor
549C22×C3C23×S411yescharacteristic
550C2×D3—1612no—
551C2×D3—1616no—
552A4—41no—
553A4—82no—
554A4—81no—
555A4—162no—
556C3×C4—68no—
557C3×C4—64no—
558C3×C4—128no—
559C32—161noSylow
560C23—26no—
561C23—31no—
562C23—33no—
563C23—36no—
564C23—36no—
565C23—36no—
566C23—36no—
567C23—31no—
568C23—33no—
569C23—38no—
570C23—41no—
571C23—62no—
572C23—66no—
573C23—66no—
574C23—66no—
575C23—66no—
576C23—66no—
577C23—612no—
578C23—612no—
579C23—62no—
580C23—66no—
581C23—66no—
582C23—66no—
583C23—66no—
584C23—612no—
585C23—616no—
586C23—123no—
587C23—123no—
588C23—123no—
589C23—123no—
590C23—123no—
591C23—126no—
592C23—126no—
593C23—126no—
594C23—126no—
595C23—126no—
596C23—126no—
597C23—126no—
598C23—1212no—
599C23—1212no—
600C23—126no—
601C23—1212no—
602C23—1212no—
603C23—122no—
604C23—2412no—
605C23—241no—
606C23—246no—
607C23—246no—
608C23—2412no—
609C23C22×C3×S413yes—
610C23C22×C3×S414yescomplemented
611Q8—616no—
612D4—624no—
613D4—68no—
614D4—68no—
615D4—624no—
616D4—624no—
617D4—632no—
618D4—616no—
619D4—68no—
620D4—128no—
621D4—1224no—
622D4—128no—
623D4—1224no—
624C2×C4—38no—
625C2×C4—34no—
626C2×C4—612no—
627C2×C4—66no—
628C2×C4—66no—
629C2×C4—612no—
630C2×C4—616no—
631C2×C4—632no—
632C2×C4—68no—
633C2×C4—1212no—
634C2×C4—1212no—
635C2×C3—31no—
636C2×C3—33no—
637C2×C3—44no—
638C2×C3—62no—
639C2×C3—66no—
640C2×C3—128no—
641C2×C3—164no—
642C2×C3—163no—
643C2×C3—324no—
644C2×C3—323no—
645C2×C3C2×#1020⋊C213yesdirect product factor
646D3—168no—
647C22—22no—
648C22—33no—
649C22—33no—
650C22—33no—
651C22—32no—
652C22—36no—
653C22—46no—
654C22—61no—
655C22—64no—
656C22—66no—
657C22—66no—
658C22—68no—
659C22—66no—
660C22—62no—
661C22—66no—
662C22—66no—
663C22—66no—
664C22—126no—
665C22—121no—
666C22—123no—
667C22—126no—
668C22—126no—
669C22—1212no—
670C22—126no—
671C22—122no—
672C22—126no—
673C22—128no—
674C22—1212no—
675C22—2416no—
676C22—2412no—
677C22C3×#1020⋊C211yescharacteristic, direct product factor
678C22C23×C3×S411yescharacteristic
679C4—68no—
680C4—64no—
681C4—128no—
682C3—161no—
683C3—321no—
684C3C22×#1020⋊C211yescharacteristic, direct product factor
685C2—31no—
686C2—33no—
687C2—44no—
688C2—62no—
689C2—66no—
690C2—128no—
691C2C2×C3×#1020⋊C213yesdirect product factor
subgroupG/H
center, hypercenterC22×C3#1020⋊C2
commutator, cosocle, hypocenterC22⋊A4C24×C3
FrattiniC22C23×C3×S4
FittingC22×C3×#120D3
socleC24×C3C2×S4
LCSG ▷ C22⋊A4
UCSC1 ◁ C22×C3

Complements

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

NH ≅ G/Ntype
1. C22×C3×#1020C2semidirect
2. C2×C3×#1020⋊C2C2direct (N × H)
3. C22×C3×#120⋊C3C2semidirect
4. C22×#1020⋊C2C3direct (N × H)
6. C22×C3×C22⋊A4C22semidirect
7. C2×C3×#1020C22semidirect
9. C2×C3×#120⋊C3C22semidirect
10. C3×#1020⋊C2C22direct (N × H)
11. C22×#1020C2×C3semidirect
14. C22×C3×#120D3semidirect
16. C22×#120⋊C3C2×C3semidirect
17. C2×#1020⋊C2C2×C3direct (N × H)
22. C2×C3×C22⋊A4C23semidirect
24. C3×#1020C23semidirect
28. C3×#120⋊C3C23semidirect
30. C26×C3C2×D3semidirect
32. C22×C22⋊A4C22×C3semidirect
33. C2×#1020C22×C3semidirect
49. C2×C3×#120C2×D3semidirect
54. C2×#120⋊C3C22×C3semidirect
55. #1020⋊C2C22×C3direct (N × H)
61. C3×C22⋊A4C24semidirect
70. C22×#120C3×D3semidirect
81. C25×C3S4semidirect
82. C25×C3C22×D3semidirect
84. C2×C22⋊A4C23×C3semidirect
87. #1020C23×C3semidirect
133. C3×#120C22×D3semidirect
141. #120⋊C3C23×C3semidirect
152. C26C2×C3×D3semidirect
167. C2×#120C2×C3×D3semidirect
207. C24×C3C2×S4semidirect
208. C24×C3C23×D3semidirect
211. C22⋊A4C24×C3semidirect
285. C25C3×S4semidirect
286. C25C22×C3×D3semidirect
329. #120C22×C3×D3semidirect
390. C23×C3C22×S4semidirect
460. C24C2×C3×S4semidirect
461. C24C23×C3×D3semidirect
548. C22×C3#1020⋊C2direct (N × H)
610. C23C22×C3×S4semidirect
645. C2×C3C2×#1020⋊C2direct (N × H)
677. C22C3×#1020⋊C2direct (N × H)
684. C3C22×#1020⋊C2direct (N × H)
691. C2C2×C3×#1020⋊C2direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#10202
C2×C3×#1020⋊C22
C22×C3×#120⋊C32
C22×#1020⋊C23
C22×C3×B(4,𝔽2)3
C23×C3×S44