Finite Group Databasebeta

← #72222963

Subgroups — #72222963

Order 2592.

672 conjugacy classes of subgroups; only the 48 normal subgroups are shown.

normal subgroups only

This group has 672 conjugacy classes of subgroups — too many to draw. Only its 48 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×D3×C22⋊H(3,𝔽3)C211yesmaximal, characteristic, complemented
2C2×C3×(C3⋊S4)⋊C3C211yesmaximal, characteristic, complemented
3C3⋊((C3⋊SL(2,ℤ/4))⋊C3)C211yesmaximal, characteristic, complemented
4#11724204C311yesmaximal, characteristic, complemented
5C3×#63976—31nomaximal
6#64269⋊C3—31nomaximal
7C2×C3×C22⋊H(3,𝔽3)C2211yescharacteristic
8D3×C22⋊H(3,𝔽3)—21no—
9C3×(C3⋊S4)⋊C3—21no—
10#6290⋊C3—41nomaximal
11C23×C32×D3D311yescharacteristic, complemented
12C2×C3×D3×A4—21no—
13C2×C3×D3×A4C2×C311yescharacteristic, complemented
14C2×C3×C3⋊S4C2×C311yescharacteristic, complemented
15C2×C32×#67—31no—
16C2×C3×#625—31no—
17C2×C3×#625—31no—
18C3⋊(C3⋊SL(2,ℤ/4))C2×C311yescharacteristic, complemented
19C22×C22⋊H(3,𝔽3)—31no—
20C2×(C3⋊S4)⋊C3—21no—
21C2×(C3⋊S4)⋊C3D311yescharacteristic, complemented
22C3×#3757—31no—
23C3×(C2×D3)⋊Dic3—31no—
24C3×(C2×D3)⋊Dic3—31no—
25(C3⋊SL(2,ℤ/4))⋊C3—31no—
26(C3⋊SL(2,ℤ/4))⋊C3—61no—
27C2×D3×H(3,𝔽3)—41no—
28C2×C3×(C3⋊D3)⋊C3—41no—
29C3×C22⋊H(3,𝔽3)D411yescharacteristic, complemented
30C3⋊((C3⋊Dic3)⋊C3)—41no—
31#64269—31no—
32#64209—91no—
33C3×#953—91no—
34C3×#937—91no—
35#63976—31no—
36C23×C33C2×D311yescharacteristic
37C22×C32×D3—21no—
38C22×C32×D3—31no—
39C22×C32×D3—31no—
40C22×C32×D3—31no—
41C22×C32×D3—61no—
42C2×C32×A4—21no—
43C2×C32×A4C22×C311yescharacteristic
44C3×D3×A4—22no—
45C3×D3×A4—21no—
46C3×C3⋊S4—21no—
47C2×C3×C3⋊Dic3—31no—
48C2×C3×C3⋊Dic3—31no—
49C32×#67—61no—
50C32×#67—61no—
51C2×C32×Dic3—31no—
52#6290—41no—
53C3×#625—62no—
54C3×#625—61no—
55C3×#625—61no—
56C2×C22⋊H(3,𝔽3)—21no—
57C2×C22⋊H(3,𝔽3)—21no—
58C2×C22⋊H(3,𝔽3)—41no—
59C2×C22⋊H(3,𝔽3)—61no—
60C2×C22⋊H(3,𝔽3)C2×D311yescharacteristic
61(C3⋊S4)⋊C3—22no—
62(C3⋊S4)⋊C3—21no—
63#638⋊C3—121no—
64C2×C3×H(3,𝔽3)—41no—
65D3×H(3,𝔽3)—81no—
66C3×(C3⋊D3)⋊C3—81no—
67C24×C32—31no—
68C23×C3×D3—31no—
69C23×C3×D3C3×D311yescharacteristic
70C22×C3×A4—31no—
71C22×C3×A4—61no—
72C2×D3×A4—31no—
73C2×D3×A4—61no—
74C2×C3⋊S4C3×D311yescharacteristic, complemented
75C2×C3×S4—91no—
76C2×C32×D4—91no—
77C2×C3×#67—31no—
78C2×C3×#67—31no—
79C2×C3×#67—61no—
80C2×C3×#67—91no—
81C2×C3×#67—91no—
82C2×C3×#67—91no—
83C2×C3×#67—91no—
84C2×#625—31no—
85C2×#625—31no—
86C3⋊SL(2,ℤ/4)—31no—
87C3⋊SL(2,ℤ/4)—61no—
88C3⋊SL(2,ℤ/4)—91no—
89C3⋊SL(2,ℤ/4)—181no—
90#3757—31no—
91C3×#217—91no—
92C3×#217—91no—
93C3×#217—91no—
94C3×#217—91no—
95C3×#217—181no—
96C3×(C2×D3)⋊C4—91no—
97C3×(C2×D3)⋊C4—91no—
98(C2×D3)⋊Dic3—31no—
99(C2×D3)⋊Dic3—31no—
100C22×C33—31no—
101C22×C33—31no—
102C22×C33#6711yescharacteristic, complemented
103C2×C3×C3⋊D3—41no—
104C2×C32×D3—32no—
105C2×C32×D3—31no—
106C2×C32×D3—41no—
107C2×C32×D3—61no—
108C2×C32×D3—61no—
109C2×C32×D3—61no—
110C2×C32×D3—61no—
111C2×C32×D3—61no—
112C2×C32×D3—61no—
113C2×C32×D3—81no—
114C2×C32×D3S411yescharacteristic, complemented
115C32×A4—21no—
116C32×A4C3×D411yescharacteristic, complemented
117C3⋊(C3⋊Dic3)—41no—
118C3×C3⋊Dic3—61no—
119C3×C3⋊Dic3—61no—
120C32×Dic3—61no—
121C22×H(3,𝔽3)—121no—
122C2×(C3⋊D3)⋊C3—41no—
123C2×(C3⋊D3)⋊C3—81no—
124C22⋊H(3,𝔽3)—21no—
125C22⋊H(3,𝔽3)—21no—
126C22⋊H(3,𝔽3)—41no—
127C22⋊H(3,𝔽3)#6711yescharacteristic, complemented
128(C3⋊Dic3)⋊C3—121no—
129(C3⋊Dic3)⋊C3—241no—
130GL(2,ℤ/4)—271no—
131C3×#120—271no—
132#953—91no—
133#937—91no—
134C3×H(3,𝔽3)—41noSylow
135C23×C32—21no—
136C23×C32—31no—
137C23×C32—61no—
138C23×C32—61no—
139C23×C32—91no—
140C23×C32—91no—
141C23×C32—91no—
142C23×C32—181no—
143C23×C32D3211yescharacteristic
144C23×C32C2×C3×D311yescharacteristic
145C22×C3×D3—21no—
146C22×C3×D3—31no—
147C22×C3×D3—31no—
148C22×C3×D3—31no—
149C22×C3×D3—31no—
150C22×C3×D3—31no—
151C22×C3×D3—31no—
152C22×C3×D3—61no—
153C22×C3×D3—61no—
154C22×C3×D3—61no—
155C22×C3×D3—61no—
156C22×C3×D3—61no—
157C22×C3×D3—62no—
158C22×C3×D3—61no—
159C2×C3×A4—21no—
160C2×C3×A4—21no—
161C2×C3×A4—31no—
162C2×C3×A4—41no—
163C2×C3×A4—61no—
164C2×C3×A4—61no—
165C2×C3×A4—62no—
166C2×C3×A4—61no—
167C2×C3×A4—121no—
168C2×C3×A4C2×C3×D311yescharacteristic
169D3×A4—61no—
170D3×A4—62no—
171C3⋊S4—21no—
172C3×S4—181no—
173C32×D4—181no—
174C32×D4—181no—
175C2×C32×C4—91no—
176#638—121no—
177C2×C3⋊Dic3—31no—
178C2×C3⋊Dic3—31no—
179C3×#67—62no—
180C3×#67—61no—
181C3×#67—61no—
182C3×#67—62no—
183C3×#67—61no—
184C3×#67—61no—
185C3×#67—182no—
186C3×#67—181no—
187C3×#67—181no—
188C3×#67—181no—
189C3×#67—182no—
190C3×#67—181no—
191C2×C3×Dic3—31no—
192C2×C3×Dic3—31no—
193C2×C3×Dic3—61no—
194C2×C3×Dic3—91no—
195C2×C3×Dic3—91no—
196C2×C3×Dic3—91no—
197C2×C3×Dic3—91no—
198C2×C3×Dic3—181no—
199C2×C3×Dic3—181no—
200#625—62no—
201#625—61no—
202#625—61no—
203#625—361no—
204C2×C33—31no—
205C2×C33—31no—
206C2×C33—41no—
207C2×C33—81no—
208C2×C33C2×S411yescharacteristic
209C3×C3⋊D3—81no—
210C32×D3—21no—
211C32×D3—61no—
212C32×D3—81no—
213C32×D3—82no—
214C32×D3—121no—
215C2×H(3,𝔽3)—41no—
216C2×H(3,𝔽3)—81no—
217C2×H(3,𝔽3)—81no—
218C2×H(3,𝔽3)—161no—
219C2×H(3,𝔽3)—241no—
220(C3⋊D3)⋊C3—81no—
221(C3⋊D3)⋊C3—82no—
222C24×C3—31no—
223C24×C3—91no—
224C23×D3(C3⋊D3)⋊C311yescharacteristic, complemented
225C22×A4—91no—
226C22×A4—181no—
227C2×S4—91no—
228C2×C3×D4—91no—
229C2×C3×D4—91no—
230C2×C3×D4—181no—
231C2×C3×D4—271no—
232C2×C3×D4—271no—
233C2×#67—31no—
234C2×#67—91no—
235C2×#67—91no—
236C2×#67—91no—
237C2×#67—91no—
238SL(2,ℤ/4)—271no—
239SL(2,ℤ/4)—541no—
240C3×C22⋊C4—271no—
241C3×C22⋊C4—271no—
242C3×C22⋊C4—271no—
243#217—91no—
244#217—91no—
245#217—91no—
246#217—91no—
247#217—181no—
248(C2×D3)⋊C4—91no—
249(C2×D3)⋊C4—91no—
250C22×C32—21no—
251C22×C32—31no—
252C22×C32—31no—
253C22×C32—31no—
254C22×C32—31no—
255C22×C32—31no—
256C22×C32—31no—
257C22×C32—31no—
258C22×C32—31no—
259C22×C32—61no—
260C22×C32—61no—
261C22×C32—61no—
262C22×C32—61no—
263C22×C32—61no—
264C22×C32—61no—
265C22×C32—61no—
266C22×C32—92no—
267C22×C32—91no—
268C22×C32—121no—
269C22×C32—121no—
270C22×C32—121no—
271C22×C32—181no—
272C22×C32—181no—
273C22×C32—181no—
274C22×C32—181no—
275C22×C32—181no—
276C22×C32—181no—
277C22×C32—241no—
278C22×C32#62511yescharacteristic, complemented
279C22×C32C3×#6711yescharacteristic
280C2×C3⋊D3—41no—
281C2×C3×D3—32no—
282C2×C3×D3—32no—
283C2×C3×D3—31no—
284C2×C3×D3—31no—
285C2×C3×D3—31no—
286C2×C3×D3—61no—
287C2×C3×D3—62no—
288C2×C3×D3—61no—
289C2×C3×D3—61no—
290C2×C3×D3—61no—
291C2×C3×D3—61no—
292C2×C3×D3—61no—
293C2×C3×D3—62no—
294C2×C3×D3—62no—
295C2×C3×D3—62no—
296C2×C3×D3—61no—
297C2×C3×D3—61no—
298C2×C3×D3—61no—
299C2×C3×D3—61no—
300C2×C3×D3—61no—
301C2×C3×D3—62no—
302C2×C3×D3—61no—
303C2×C3×D3—61no—
304C2×C3×D3—121no—
305C2×C3×D3—121no—
306C2×C3×D3—121no—
307C2×C3×D3—241no—
308C2×C3×D3—361no—
309C2×C3×D3C3×S411yescharacteristic
310C3×A4—21no—
311C3×A4—21no—
312C3×A4—31no—
313C3×A4—41no—
314C3×A4—61no—
315C3×A4—61no—
316C3×A4—121no—
317C3×A4C3×#6711yescharacteristic, complemented
318C32×C4—181no—
319C3⋊Dic3—61no—
320C3⋊Dic3—61no—
321C3⋊Dic3—121no—
322C3⋊Dic3—241no—
323C3⋊Dic3—361no—
324C3⋊Dic3—721no—
325C3×Dic3—61no—
326C3×Dic3—61no—
327C3×Dic3—62no—
328C3×Dic3—181no—
329C3×Dic3—181no—
330C3×Dic3—181no—
331C3×Dic3—181no—
332C3×Dic3—361no—
333C3×Dic3—361no—
334#120—271noSylow
335C33—41no—
336C33—81no—
337C33GL(2,ℤ/4)11yescharacteristic, complemented
338H(3,𝔽3)—41no—
339H(3,𝔽3)—81no—
340H(3,𝔽3)—81no—
341H(3,𝔽3)—161no—
342C23×C3—21no—
343C23×C3—31no—
344C23×C3—61no—
345C23×C3—61no—
346C23×C3—91no—
347C23×C3—91no—
348C23×C3—91no—
349C23×C3—91no—
350C23×C3—91no—
351C23×C3—91no—
352C23×C3—181no—
353C23×C3—181no—
354C23×C3—181no—
355C23×C3—181no—
356C23×C3—181no—
357C23×C3—181no—
358C23×C3—182no—
359C23×C3C2×(C3⋊D3)⋊C311yescharacteristic
360C23×C3C3×D3211yescharacteristic
361C22×D3—21no—
362C22×D3—31no—
363C22×D3—31no—
364C22×D3—31no—
365C22×D3—61no—
366C2×A4—31no—
367C2×A4—61no—
368C2×A4—61no—
369C2×A4—121no—
370C2×A4—181no—
371C2×A4—182no—
372S4—181no—
373C3×D4—181no—
374C3×D4—181no—
375C3×D4—181no—
376C3×D4—182no—
377C3×D4—182no—
378C3×D4—181no—
379C3×D4—541no—
380C3×D4—542no—
381C3×D4—541no—
382C2×C3×C4—91no—
383C2×C3×C4—91no—
384C2×C3×C4—181no—
385C2×C3×C4—271no—
386C2×C3×C4—271no—
387#67—61no—
388#67—61no—
389#67—182no—
390#67—181no—
391#67—181no—
392#67—182no—
393#67—181no—
394#67—181no—
395#67—1081no—
396C2×Dic3—31no—
397C2×Dic3—91no—
398C2×Dic3—91no—
399C2×Dic3—91no—
400C2×Dic3—91no—
401C2×Dic3—181no—
402C2×Dic3—181no—
403C2×C32—21no—
404C2×C32—31no—
405C2×C32—31no—
406C2×C32—31no—
407C2×C32—31no—
408C2×C32—31no—
409C2×C32—31no—
410C2×C32—31no—
411C2×C32—41no—
412C2×C32—61no—
413C2×C32—61no—
414C2×C32—61no—
415C2×C32—61no—
416C2×C32—61no—
417C2×C32—61no—
418C2×C32—61no—
419C2×C32—61no—
420C2×C32—81no—
421C2×C32—81no—
422C2×C32—121no—
423C2×C32—121no—
424C2×C32—121no—
425C2×C32—161no—
426C2×C32—181no—
427C2×C32—241no—
428C2×C32—241no—
429C2×C32—242no—
430C2×C32—241no—
431C2×C32—361no—
432C2×C32—481no—
433C2×C32D3×S411yescharacteristic
434C2×C32C2×C3×S411yescharacteristic
435C3⋊D3—81no—
436C3×D3—21no—
437C3×D3—61no—
438C3×D3—62no—
439C3×D3—61no—
440C3×D3—61no—
441C3×D3—122no—
442C3×D3—121no—
443C3×D3—121no—
444C3×D3—241no—
445C3×D3—242no—
446C3×D3—721no—
447C24—31no—
448C2×D4—91no—
449C2×D4—271no—
450C2×D4—271no—
451C22⋊C4—271no—
452C22⋊C4—271no—
453C22⋊C4—271no—
454C22×C3—21no—
455C22×C3—31no—
456C22×C3—31no—
457C22×C3—31no—
458C22×C3—31no—
459C22×C3—31no—
460C22×C3—31no—
461C22×C3—31no—
462C22×C3—31no—
463C22×C3—61no—
464C22×C3—61no—
465C22×C3—61no—
466C22×C3—61no—
467C22×C3—61no—
468C22×C3—61no—
469C22×C3—61no—
470C22×C3—91no—
471C22×C3—92no—
472C22×C3—92no—
473C22×C3—91no—
474C22×C3—91no—
475C22×C3—121no—
476C22×C3—121no—
477C22×C3—181no—
478C22×C3—182no—
479C22×C3—181no—
480C22×C3—181no—
481C22×C3—181no—
482C22×C3—181no—
483C22×C3—181no—
484C22×C3—181no—
485C22×C3—181no—
486C22×C3—181no—
487C22×C3—181no—
488C22×C3—181no—
489C22×C3—181no—
490C22×C3—182no—
491C22×C3—182no—
492C22×C3—182no—
493C22×C3—182no—
494C22×C3—181no—
495C22×C3—361no—
496C22×C3—361no—
497C22×C3—361no—
498C22×C3—721no—
499C22×C3#638⋊C311yescharacteristic, complemented
500C22×C3C3×#62511yescharacteristic
501C2×D3—32no—
502C2×D3—31no—
503C2×D3—61no—
504C2×D3—61no—
505C2×D3—61no—
506C2×D3—61no—
507C2×D3—61no—
508C2×D3—61no—
509C2×D3—361no—
510C2×D3(C3⋊S4)⋊C311yescharacteristic, complemented
511A4—31no—
512A4—61no—
513A4—61no—
514A4—121no—
515C3×C4—181no—
516C3×C4—181no—
517C3×C4—182no—
518C3×C4—541no—
519C3×C4—541no—
520Dic3—61no—
521Dic3—181no—
522Dic3—181no—
523Dic3—181no—
524Dic3—181no—
525Dic3—361no—
526Dic3—361no—
527Dic3—1081no—
528Dic3—2161no—
529C32—21no—
530C32—31no—
531C32—41no—
532C32—61no—
533C32—81no—
534C32—81no—
535C32—121no—
536C32—161no—
537C32—241no—
538C32—241no—
539C32—481no—
540C32#6420911yescharacteristic, complemented
541C32C3×GL(2,ℤ/4)11yescharacteristic
542C23—61no—
543C23—91no—
544C23—91no—
545C23—91no—
546C23—181no—
547C23D3×(C3⋊D3)⋊C311yescharacteristic
548D4—181no—
549D4—181no—
550D4—541no—
551D4—542no—
552D4—541no—
553C2×C4—91no—
554C2×C4—271no—
555C2×C4—271no—
556C2×C3—21no—
557C2×C3—31no—
558C2×C3—31no—
559C2×C3—31no—
560C2×C3—31no—
561C2×C3—31no—
562C2×C3—31no—
563C2×C3—31no—
564C2×C3—61no—
565C2×C3—61no—
566C2×C3—61no—
567C2×C3—61no—
568C2×C3—61no—
569C2×C3—61no—
570C2×C3—61no—
571C2×C3—61no—
572C2×C3—121no—
573C2×C3—121no—
574C2×C3—121no—
575C2×C3—181no—
576C2×C3—182no—
577C2×C3—181no—
578C2×C3—181no—
579C2×C3—241no—
580C2×C3—241no—
581C2×C3—361no—
582C2×C3—362no—
583C2×C3—361no—
584C2×C3—481no—
585C2×C3—721no—
586C2×C3—722no—
587C2×C3C2×(C3⋊S4)⋊C311yescharacteristic
588C2×C3C3×D3×S411yescharacteristic
589D3—21no—
590D3—61no—
591D3—121no—
592D3—721no—
593C22—31no—
594C22—31no—
595C22—31no—
596C22—92no—
597C22—91no—
598C22—181no—
599C22—181no—
600C22—181no—
601C22—181no—
602C22—181no—
603C22—181no—
604C22#6290⋊C311yescharacteristic, complemented
605C4—181no—
606C4—541no—
607C4—541no—
608C3—21no—
609C3—31no—
610C3—61no—
611C3—121no—
612C3—241no—
613C3—241no—
614C3—481no—
615C3#64269⋊C311yescharacteristic, complemented
616C3C3×#6420911yescharacteristic
617C2—31no—
618C2—31no—
619C2—61no—
620C2—181no—
621C2—361no—
622C2D3×(C3⋊S4)⋊C311yescharacteristic
subgroupG/H
center, hypercenterC2D3×(C3⋊S4)⋊C3
commutator, cosocleC2×C32×A4C22×C3
FrattiniC2×C3C3×D3×S4
FittingC23×C33C2×D3
socleC23×C32C2×C3×D3
hypocenterC32×A4C3×D4
LCSG ▷ C2×C32×A4 ▷ C32×A4
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
1. C2×D3×C22⋊H(3,𝔽3)C2semidirect
2. C2×C3×(C3⋊S4)⋊C3C2semidirect
3. C3⋊((C3⋊SL(2,ℤ/4))⋊C3)C2semidirect
4. #11724204C3semidirect
11. C23×C32×D3D3semidirect
13. C2×C3×D3×A4C2×C3semidirect
14. C2×C3×C3⋊S4C2×C3semidirect
18. C3⋊(C3⋊SL(2,ℤ/4))C2×C3semidirect
21. C2×(C3⋊S4)⋊C3D3semidirect
29. C3×C22⋊H(3,𝔽3)D4semidirect
74. C2×C3⋊S4C3×D3semidirect
102. C22×C33#67semidirect
114. C2×C32×D3S4semidirect
116. C32×A4C3×D4semidirect
127. C22⋊H(3,𝔽3)#67semidirect
224. C23×D3(C3⋊D3)⋊C3semidirect
278. C22×C32#625semidirect
317. C3×A4C3×#67semidirect
337. C33GL(2,ℤ/4)semidirect
499. C22×C3#638⋊C3semidirect
510. C2×D3(C3⋊S4)⋊C3semidirect
540. C32#64209semidirect
604. C22#6290⋊C3semidirect
615. C3#64269⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×D3×C22⋊H(3,𝔽3)2
C2×C3×(C3⋊S4)⋊C32
C3⋊((C3⋊SL(2,ℤ/4))⋊C3)2
#117242043
C3×#639763
#64269⋊C33
#6290⋊C34