Finite Group Databasebeta

← C3×#11722273 #72222940

Subgroups — C3×#11722273 #72222940

Order 2592.

640 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 640 conjugacy classes of subgroups — too many to draw. Only its 46 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
1C2×C3×#6258C211yesmaximal, characteristic, complemented
2C2×C3×(C3⋊S4)⋊C3C211yesmaximal, characteristic, complemented
3C3×(C3⋊Dic3)⋊A4C211yesmaximal, characteristic, complemented
4C3×#64207—31nomaximal
5C3×#63980—31nomaximal
6#11722273C311yesmaximal, characteristic, direct product factor
7C2×C3×C22⋊H(3,𝔽3)C2211yescharacteristic
8C3×#6258—21no—
9C3×(C3⋊S4)⋊C3—21no—
10C3×#6200—41nomaximal
11C2×C3×C3⋊S4D311yescharacteristic, complemented
12C2×C32×S4—31no—
13C2×C32×S4—61no—
14C22×C3×C3⋊Dic3D311yescharacteristic, complemented
15C2×C32×#67—31no—
16C2×C32×#67—31no—
17C2×C3×#625—31no—
18C3×Dic3×A4—31no—
19C2×#6258C2×C311yescharacteristic, complemented
20C2×(C3⋊S4)⋊C3—21no—
21C2×(C3⋊S4)⋊C3C2×C311yescharacteristic, complemented
22C3×#3724—31no—
23C3×(C2×D3)⋊Dic3—31no—
24C3×(C2×D3)⋊Dic3—31no—
25(C3⋊Dic3)⋊A4—21no—
26(C3⋊Dic3)⋊A4C2×C311yescharacteristic, complemented
27C2×C3×#272—41no—
28C2×C3×(C3⋊D3)⋊C3—41no—
29C3×C22⋊H(3,𝔽3)D411yescharacteristic, complemented
30C3×(C3⋊Dic3)⋊C3—41no—
31C3×C4⋊S4—91no—
32#64207—31no—
33C3×#939—91no—
34C3×#939—91no—
35#63980—31no—
36C23×C33C2×D311yescharacteristic
37C22×C32×D3—31no—
38C22×C32×D3—31no—
39C2×C32×A4—21no—
40C2×C32×A4C2×D311yescharacteristic
41C3×C3⋊S4—21no—
42C32×S4—62no—
43C32×S4—61no—
44C2×C3×C3⋊Dic3—31no—
45C2×C3×C3⋊Dic3—31no—
46C32×#67—61no—
47C32×#67—61no—
48C32×#67—61no—
49C32×#67—61no—
50C2×C32×Dic3—31no—
51C2×C32×Dic3—31no—
52C3×#626—121no—
53C3×#625—61no—
54C3×#625—61no—
55C2×C22⋊H(3,𝔽3)—21no—
56C2×C22⋊H(3,𝔽3)—21no—
57C2×C22⋊H(3,𝔽3)—41no—
58C2×C22⋊H(3,𝔽3)C22×C311yescharacteristic
59#6258—21no—
60(C3⋊S4)⋊C3—22no—
61(C3⋊S4)⋊C3—21no—
62#6200—41no—
63C2×C3×H(3,𝔽3)—41no—
64C3×#272—81no—
65C3×(C3⋊D3)⋊C3—81no—
66C2×C3⋊S4C3×D311yescharacteristic, complemented
67C2×C3×S4—31no—
68C2×C3×S4—31no—
69C2×C3×S4—61no—
70C2×C3×S4—61no—
71C2×C3×S4—61no—
72C2×C3×S4—62no—
73C2×C3×S4—91no—
74C2×C32×D4—91no—
75C2×C32×D4—91no—
76C22×C3⋊Dic3C3×D311yescharacteristic, complemented
77C2×C3×#67—31no—
78C2×C3×#67—31no—
79C2×C3×#67—31no—
80C2×C3×#67—31no—
81C2×C3×#67—61no—
82C2×C3×#67—61no—
83C2×C3×#67—91no—
84C2×C3×#67—91no—
85C22×C3×Dic3—31no—
86C22×C3×Dic3—91no—
87C3×C4×A4—91no—
88C2×#625—31no—
89Dic3×A4—31no—
90Dic3×A4—61no—
91C3×#217—91no—
92C3×#217—91no—
93C3×(C2×D3)⋊C4—91no—
94C3×(C2×D3)⋊C4—91no—
95C3×Dic3⋊C4—91no—
96C3×Dic3⋊C4—91no—
97#3724—31no—
98(C2×D3)⋊Dic3—31no—
99(C2×D3)⋊Dic3—31no—
100C22×C33—31no—
101C22×C33—31no—
102C22×C33D1211yescharacteristic, complemented
103C2×C3×C3⋊D3—41no—
104C2×C32×D3—61no—
105C2×C32×D3—61no—
106C2×C32×D3—61no—
107C2×C32×D3—61no—
108C2×C32×D3—61no—
109C2×C32×D3—61no—
110C2×C32×D3—121no—
111C2×C32×D3—241no—
112C32×A4—21no—
113C32×A4#6711yescharacteristic, complemented
114C3×C3⋊Dic3—31no—
115C3×C3⋊Dic3S411yescharacteristic, complemented
116C32×Dic3—61no—
117C32×Dic3—61no—
118C32×Dic3—121no—
119C2×#272—41no—
120C2×(C3⋊D3)⋊C3—41no—
121C2×(C3⋊D3)⋊C3—81no—
122C22⋊H(3,𝔽3)—21no—
123C22⋊H(3,𝔽3)—21no—
124C22⋊H(3,𝔽3)—41no—
125C22⋊H(3,𝔽3)C3×D411yescharacteristic, complemented
126(C3⋊Dic3)⋊C3—41no—
127(C3⋊Dic3)⋊C3—81no—
128C4⋊S4—91no—
129C3×#121—271no—
130#939—91no—
131#939—91no—
132C3×H(3,𝔽3)—41noSylow
133C23×C32—21no—
134C23×C32—31no—
135C23×C32—61no—
136C23×C32—91no—
137C23×C32—91no—
138C23×C32D3211yescharacteristic
139C23×C32C2×C3×D311yescharacteristic
140C22×C3×D3—31no—
141C22×C3×D3—31no—
142C22×C3×D3—31no—
143C22×C3×D3—31no—
144C22×C3×D3—61no—
145C22×C3×D3—61no—
146C2×C3×A4—21no—
147C2×C3×A4—21no—
148C2×C3×A4—31no—
149C2×C3×A4—41no—
150C2×C3×A4—61no—
151C2×C3×A4—62no—
152C2×C3×A4—61no—
153C2×C3×A4C2×C3×D311yescharacteristic
154C3⋊S4—21no—
155C3×S4—62no—
156C3×S4—61no—
157C3×S4—62no—
158C3×S4—64no—
159C3×S4—62no—
160C3×S4—61no—
161C3×S4—181no—
162C32×D4—181no—
163C32×D4—181no—
164C32×D4—181no—
165C32×D4—181no—
166C2×C32×C4—91no—
167C2×C32×C4—91no—
168C2×C3⋊Dic3—31no—
169C2×C3⋊Dic3—31no—
170C3×#67—62no—
171C3×#67—61no—
172C3×#67—61no—
173C3×#67—61no—
174C3×#67—61no—
175C3×#67—61no—
176C3×#67—62no—
177C3×#67—62no—
178C3×#67—62no—
179C3×#67—61no—
180C3×#67—61no—
181C3×#67—61no—
182C3×#67—181no—
183C3×#67—181no—
184C3×#67—181no—
185C3×#67—181no—
186C2×C3×Dic3—31no—
187C2×C3×Dic3—31no—
188C2×C3×Dic3—31no—
189C2×C3×Dic3—31no—
190C2×C3×Dic3—61no—
191C2×C3×Dic3—61no—
192C2×C3×Dic3—91no—
193C2×C3×Dic3—91no—
194C2×C3×Dic3—91no—
195C2×C3×Dic3—91no—
196C2×C3×Dic3—181no—
197C2×C3×Dic3—181no—
198C3×D12—361no—
199#626—121no—
200#625—61no—
201#625—61no—
202C2×C33—31no—
203C2×C33—31no—
204C2×C33—41no—
205C2×C33—81no—
206C2×C33C2×S411yescharacteristic
207C3×C3⋊D3—81no—
208C32×D3—121no—
209C32×D3—121no—
210C32×D3—241no—
211C32×D3—242no—
212C2×H(3,𝔽3)—41no—
213C2×H(3,𝔽3)—81no—
214C2×H(3,𝔽3)—81no—
215C2×H(3,𝔽3)—161no—
216#272—81no—
217(C3⋊D3)⋊C3—81no—
218(C3⋊D3)⋊C3—82no—
219C2×S4—31no—
220C2×S4—61no—
221C2×S4—91no—
222C2×C3×D4—91no—
223C2×C3×D4—91no—
224C2×C3×D4—91no—
225C2×C3×D4—91no—
226C2×C3×D4—181no—
227C2×C3×D4—181no—
228C2×C3×D4—271no—
229C22×C3×C4—91no—
230C2×#67—31no—
231C2×#67—31no—
232C2×#67—91no—
233C2×#67—91no—
234C22×Dic3—31no—
235C22×Dic3—91no—
236C4×A4—91no—
237C4×A4—181no—
238C3×C4⋊C4—271no—
239C3×C22⋊C4—271no—
240C3×C22⋊C4—271no—
241#217—91no—
242#217—91no—
243(C2×D3)⋊C4—91no—
244(C2×D3)⋊C4—91no—
245Dic3⋊C4—91no—
246Dic3⋊C4—91no—
247C22×C32—21no—
248C22×C32—31no—
249C22×C32—31no—
250C22×C32—31no—
251C22×C32—31no—
252C22×C32—31no—
253C22×C32—31no—
254C22×C32—31no—
255C22×C32—61no—
256C22×C32—61no—
257C22×C32—61no—
258C22×C32—61no—
259C22×C32—61no—
260C22×C32—61no—
261C22×C32—61no—
262C22×C32—121no—
263C22×C32—121no—
264C22×C32—181no—
265C22×C32—181no—
266C22×C32—181no—
267C22×C32—181no—
268C22×C32—181no—
269C22×C32—181no—
270C22×C32#62611yescharacteristic
271C22×C32C3×D1211yescharacteristic, complemented
272C2×C3⋊D3—41no—
273C2×C3×D3—61no—
274C2×C3×D3—61no—
275C2×C3×D3—61no—
276C2×C3×D3—61no—
277C2×C3×D3—61no—
278C2×C3×D3—61no—
279C2×C3×D3—61no—
280C2×C3×D3—61no—
281C2×C3×D3—61no—
282C2×C3×D3—62no—
283C2×C3×D3—62no—
284C2×C3×D3—61no—
285C2×C3×D3—61no—
286C2×C3×D3—61no—
287C2×C3×D3—121no—
288C2×C3×D3—121no—
289C2×C3×D3—121no—
290C2×C3×D3—121no—
291C2×C3×D3—121no—
292C2×C3×D3—121no—
293C2×C3×D3—241no—
294C2×C3×D3—241no—
295C2×C3×D3—241no—
296C2×C3×D3—242no—
297C2×C3×D3—361no—
298C3×A4—21no—
299C3×A4—21no—
300C3×A4—31no—
301C3×A4—41no—
302C3×A4—61no—
303C3×A4—62no—
304C3×A4—61no—
305C3×A4C3×#6711yescharacteristic, complemented
306C32×C4—181no—
307C32×C4—181no—
308C32×C4—361no—
309C3⋊Dic3—31no—
310C3⋊Dic3C3×S411yescharacteristic, complemented
311C3×Dic3—31no—
312C3×Dic3—61no—
313C3×Dic3—61no—
314C3×Dic3—61no—
315C3×Dic3—61no—
316C3×Dic3—62no—
317C3×Dic3—62no—
318C3×Dic3—91no—
319C3×Dic3—91no—
320C3×Dic3—91no—
321C3×Dic3—121no—
322C3×Dic3—181no—
323C3×Dic3—241no—
324#121—271noSylow
325C33—41no—
326C33—81no—
327C33C4⋊S411yescharacteristic, complemented
328H(3,𝔽3)—41no—
329H(3,𝔽3)—81no—
330H(3,𝔽3)—81no—
331H(3,𝔽3)—161no—
332C23×C3—21no—
333C23×C3—31no—
334C23×C3—61no—
335C23×C3—91no—
336C23×C3—91no—
337C23×C3—91no—
338C23×C3—91no—
339C23×C3—181no—
340C23×C3—181no—
341C23×C3((C3⋊D3)⋊C3)⋊C211yescharacteristic
342C23×C3C3×D3211yescharacteristic
343C22×D3—31no—
344C22×D3—31no—
345C2×A4—31no—
346C2×A4—61no—
347C2×A4—61no—
348C2×A4—121no—
349S4—61no—
350S4—62no—
351S4—181no—
352C3×D4—182no—
353C3×D4—181no—
354C3×D4—181no—
355C3×D4—181no—
356C3×D4—181no—
357C3×D4—181no—
358C3×D4—182no—
359C3×D4—182no—
360C3×D4—182no—
361C3×D4—181no—
362C3×D4—181no—
363C3×D4—181no—
364C3×D4—541no—
365C3×D4—541no—
366C2×C3×C4—91no—
367C2×C3×C4—91no—
368C2×C3×C4—91no—
369C2×C3×C4—91no—
370C2×C3×C4—181no—
371C2×C3×C4—181no—
372C2×C3×C4—271no—
373C2×C3×C4—271no—
374#67—61no—
375#67—61no—
376#67—61no—
377#67—61no—
378#67—181no—
379#67—181no—
380#67—181no—
381#67—181no—
382C2×Dic3—31no—
383C2×Dic3—31no—
384C2×Dic3—91no—
385C2×Dic3—91no—
386C2×Dic3—91no—
387C2×Dic3—91no—
388C2×Dic3—181no—
389C2×Dic3—181no—
390D12—361no—
391C2×C32—21no—
392C2×C32—31no—
393C2×C32—31no—
394C2×C32—31no—
395C2×C32—31no—
396C2×C32—31no—
397C2×C32—31no—
398C2×C32—31no—
399C2×C32—41no—
400C2×C32—61no—
401C2×C32—61no—
402C2×C32—61no—
403C2×C32—61no—
404C2×C32—61no—
405C2×C32—61no—
406C2×C32—61no—
407C2×C32—81no—
408C2×C32—81no—
409C2×C32—121no—
410C2×C32—121no—
411C2×C32—121no—
412C2×C32—161no—
413C2×C32—241no—
414C2×C32—242no—
415C2×C32—241no—
416C2×C32—361no—
417C2×C32—361no—
418C2×C32D3×S411yescharacteristic
419C2×C32C2×C3×S411yescharacteristic
420C3⋊D3—81no—
421C3×D3—122no—
422C3×D3—121no—
423C3×D3—122no—
424C3×D3—121no—
425C3×D3—121no—
426C3×D3—121no—
427C3×D3—241no—
428C3×D3—242no—
429C3×D3—241no—
430C3×D3—242no—
431C3×D3—244no—
432C3×D3—242no—
433C3×D3—721no—
434C2×D4—91no—
435C2×D4—91no—
436C2×D4—271no—
437C22×C4—91no—
438C4⋊C4—271no—
439C22⋊C4—271no—
440C22⋊C4—271no—
441C22×C3—21no—
442C22×C3—31no—
443C22×C3—31no—
444C22×C3—31no—
445C22×C3—31no—
446C22×C3—31no—
447C22×C3—31no—
448C22×C3—31no—
449C22×C3—61no—
450C22×C3—61no—
451C22×C3—61no—
452C22×C3—61no—
453C22×C3—61no—
454C22×C3—61no—
455C22×C3—61no—
456C22×C3—121no—
457C22×C3—121no—
458C22×C3—181no—
459C22×C3—181no—
460C22×C3—181no—
461C22×C3—181no—
462C22×C3—181no—
463C22×C3—181no—
464C22×C3—181no—
465C22×C3—181no—
466C22×C3—181no—
467C22×C3—182no—
468C22×C3—182no—
469C22×C3—181no—
470C22×C3—181no—
471C22×C3—181no—
472C22×C3—361no—
473C22×C3—361no—
474C22×C3—361no—
475C22×C3—361no—
476C22×C3#620011yescharacteristic, complemented
477C22×C3C3×#62611yescharacteristic
478C2×D3—61no—
479C2×D3—61no—
480C2×D3—61no—
481C2×D3—61no—
482C2×D3—61no—
483C2×D3—61no—
484C2×D3—121no—
485C2×D3—241no—
486C2×D3—361no—
487A4—31no—
488A4—61no—
489A4—61no—
490A4—121no—
491C3×C4—91no—
492C3×C4—181no—
493C3×C4—181no—
494C3×C4—181no—
495C3×C4—181no—
496C3×C4—182no—
497C3×C4—182no—
498C3×C4—271no—
499C3×C4—361no—
500C3×C4—721no—
501Dic3—31no—
502Dic3—61no—
503Dic3—61no—
504Dic3—91no—
505Dic3—91no—
506Dic3—91no—
507Dic3—181no—
508C32—21no—
509C32—31no—
510C32—41no—
511C32—61no—
512C32—81no—
513C32—81no—
514C32—121no—
515C32—161no—
516C32—241no—
517C32—242no—
518C32—241no—
519C32#6420711yescharacteristic
520C32C3×C4⋊S411yescharacteristic, complemented
521C23—91no—
522C23—91no—
523C23C3×((C3⋊D3)⋊C3)⋊C211yescharacteristic
524D4—181no—
525D4—181no—
526D4—181no—
527D4—181no—
528D4—541no—
529D4—541no—
530C2×C4—91no—
531C2×C4—91no—
532C2×C4—271no—
533C2×C4—271no—
534C2×C3—21no—
535C2×C3—31no—
536C2×C3—31no—
537C2×C3—31no—
538C2×C3—31no—
539C2×C3—31no—
540C2×C3—31no—
541C2×C3—31no—
542C2×C3—61no—
543C2×C3—61no—
544C2×C3—61no—
545C2×C3—61no—
546C2×C3—61no—
547C2×C3—61no—
548C2×C3—61no—
549C2×C3—121no—
550C2×C3—121no—
551C2×C3—121no—
552C2×C3—241no—
553C2×C3—241no—
554C2×C3—362no—
555C2×C3—361no—
556C2×C3—362no—
557C2×C3—361no—
558C2×C3—361no—
559C2×C3—361no—
560C2×C3—481no—
561C2×C3((C3⋊S4)⋊C3)⋊C211yescharacteristic
562C2×C3C3×D3×S411yescharacteristic
563D3—121no—
564D3—121no—
565D3—241no—
566D3—242no—
567D3—721no—
568C22—31no—
569C22—31no—
570C22—181no—
571C22—181no—
572C22—181no—
573C22—181no—
574C22—181no—
575C22—181no—
576C22C3×#620011yescharacteristic, complemented
577C4—91no—
578C4—181no—
579C4—181no—
580C4—271no—
581C3—21no—
582C3—31no—
583C3—61no—
584C3—121no—
585C3—241no—
586C3—241no—
587C3—481no—
588C3#1172227311yescharacteristic, direct product factor
589C3C3×#6420711yescharacteristic
590C2—31no—
591C2—31no—
592C2—361no—
593C2—361no—
594C2C3×((C3⋊S4)⋊C3)⋊C211yescharacteristic
subgroupG/H
center, hypercenterC2×C3((C3⋊S4)⋊C3)⋊C2
commutator, cosocleC2×C22⋊H(3,𝔽3)C22×C3
FrattiniC2×C3C3×D3×S4
FittingC23×C33C2×D3
socleC23×C32D32
hypocenterC22⋊H(3,𝔽3)C3×D4
LCSG ▷ C2×C22⋊H(3,𝔽3) ▷ C22⋊H(3,𝔽3)
UCSC1 ◁ C2×C3

Complements

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

NH ≅ G/Ntype
1. C2×C3×#6258C2semidirect
2. C2×C3×(C3⋊S4)⋊C3C2semidirect
3. C3×(C3⋊Dic3)⋊A4C2semidirect
6. #11722273C3direct (N × H)
11. C2×C3×C3⋊S4D3semidirect
14. C22×C3×C3⋊Dic3D3semidirect
19. C2×#6258C2×C3semidirect
21. C2×(C3⋊S4)⋊C3C2×C3semidirect
26. (C3⋊Dic3)⋊A4C2×C3semidirect
29. C3×C22⋊H(3,𝔽3)D4semidirect
66. C2×C3⋊S4C3×D3semidirect
76. C22×C3⋊Dic3C3×D3semidirect
102. C22×C33D12semidirect
113. C32×A4#67semidirect
115. C3×C3⋊Dic3S4semidirect
125. C22⋊H(3,𝔽3)C3×D4semidirect
271. C22×C32C3×D12semidirect
305. C3×A4C3×#67semidirect
310. C3⋊Dic3C3×S4semidirect
327. C33C4⋊S4semidirect
476. C22×C3#6200semidirect
520. C32C3×C4⋊S4semidirect
576. C22C3×#6200semidirect
588. C3#11722273direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#62582
C2×C3×(C3⋊S4)⋊C32
C3×(C3⋊Dic3)⋊A42
C3×#642073
C3×#639803
#117222733
C3×#62004