Finite Group Databasebeta

← #13155861

Subgroups — #13155861

Order 1728.

776 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×D3×Dic6C211yesmaximal, characteristic
2C2×#88812C211yesmaximal, characteristic, complemented
3#11723425C211yesmaximal, characteristic
4C2×C3×#3719C211yesmaximal, characteristic, complemented
5#11722040C211yesmaximal, characteristic, complemented
6#11722038C211yesmaximal, characteristic, complemented
7C3×(C2×D3)⋊(C3⋊C8)C211yesmaximal, characteristic
8#10591594—31nomaximal
9#10591186—31nomaximal
10#10590949—31nomaximal
11C2×C3×#634C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C2×C32×C4×D3C2211yescharacteristic
14C2×C32×Dic6C2211yescharacteristic
15C2×C3×#627—21no—
16C2×C3×D3×Dic3—21no—
17C3×D3×Dic6—22no—
18C3×D3×Dic6—22no—
19C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
20C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
21C2×C32×C3⋊C8C2211yescharacteristic
22#88812—22no—
23#88806—21no—
24(C2×D3)⋊(C3⋊Dic3)—21no—
25C3×#3719—22no—
26C2×C3×D3×Q8—31no—
27C22×C3×Dic6—31no—
28C2×D3×Dic6D311yescharacteristic
29#64133—31no—
30C2×C3×#216—31no—
31C2×C3×Dic12—31no—
32#63856—31no—
33#63856—31no—
34#63856—61no—
35C2×#3719—31no—
36C2×#3719—31no—
37C2×#3719D311yescharacteristic, complemented
38C2×#3718—31no—
39#63662—31no—
40#63643—31no—
41C3×#830—31no—
42C3×(C2×D3)⋊C8—31no—
43#63566—31no—
44#63566—31no—
45#63566—31no—
46#63566—31no—
47#63566—61no—
48#63566—61no—
49(C2×D3)⋊(C3⋊C8)D311yescharacteristic
50C22×C32×D3D411yescharacteristic
51C2×C33×C4C2311yescharacteristic
52C2×C3⋊(C3⋊Dic3)—21no—
53C2×C3×C3⋊Dic3—21no—
54C2×C3×C3⋊Dic3—21no—
55C3×#634—21no—
56C3×#634—21no—
57C3×#634—21no—
58C3×#634D412yes—
59C2×C32×Dic3—21no—
60C2×C32×Dic3D411yescharacteristic
61C32×C4×D3—21no—
62C32×C4×D3—21no—
63C32×Dic6—21no—
64C32×Dic6D412yes—
65C3×#627—24no—
66C3×D3×Dic3—24no—
67C3×C3⋊(C3⋊C8)—21no—
68C32×C3⋊C8—21no—
69#5228—91no—
70#5186—91no—
71#4739—91no—
72C2×C32×Q8—31no—
73C22×C32×C4—31no—
74C2×#634C2×D311yescharacteristic
75C2×#634C2×D311yescharacteristic
76C22×C3×Dic3—61no—
77C3×D3×Q8—62no—
78C3×D3×Q8—62no—
79C2×C3×C4×D3—21no—
80C2×C3×C4×D3—61no—
81C2×C3×C4×D3C2×D311yescharacteristic
82C2×C3×C4×D3C2×D311yescharacteristic
83C2×C3×Dic6—21no—
84C2×C3×Dic6—31no—
85C2×C3×Dic6—31no—
86C2×C3×Dic6—31no—
87C2×C3×Dic6—31no—
88C2×C3×Dic6—62no—
89C2×C3×Dic6—61no—
90C2×C3×Dic6—61no—
91C2×C3×Dic6—62no—
92C2×C3×Dic6—61no—
93C2×C3×Dic6—61no—
94C2×C3×Dic6C2×D311yescharacteristic
95C2×C3×Dic6C2×D311yescharacteristic
96C2×#627—21no—
97C2×D3×Dic3—21no—
98D3×Dic6—22no—
99D3×Dic6—22no—
100C2×C32×C8—31no—
101#3757—61no—
102C4⋊(C3⋊Dic3)—31no—
103C4⋊(C3⋊Dic3)—31no—
104C4⋊(C3⋊Dic3)—31no—
105C4⋊(C3⋊Dic3)—61no—
106C4⋊(C3⋊Dic3)—61no—
107C4⋊(C3⋊Dic3)—61no—
108C4⋊(C3⋊Dic3)—121no—
109#3750—61no—
110C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
111C3×#216—62no—
112C2×C3×C3⋊C8—21no—
113C2×C3×C3⋊C8—31no—
114C2×C3×C3⋊C8—31no—
115C2×C3×C3⋊C8—61no—
116C2×C3×C3⋊C8C2×D311yescharacteristic
117C2×C3×C3⋊C8C2×D311yescharacteristic
118C3×Dic12—62no—
119#3724—62no—
120#3724—61no—
121#3724—61no—
122(C2×D3)⋊Dic3—62no—
123(C2×D3)⋊Dic3—61no—
124(C2×D3)⋊Dic3—61no—
125#3719—22no—
126#3719—62no—
127#3719—62no—
128#3718—62no—
129C22×C33C2×D411yescharacteristic
130C2×C32×D3—21no—
131C2×C32×D3—21no—
132C2×C32×D3Q1612yescomplemented
133C33×C4C2×D411yescharacteristic
134C33×C4C2×D411yescharacteristic
135C3⋊(C3⋊Dic3)—41no—
136C3×C3⋊Dic3—22no—
137C3×C3⋊Dic3—41no—
138C32×Dic3—22no—
139C32×Dic3—21no—
140C22×C3×Q8—91no—
141C2×D3×Q8—31no—
142C22×Dic6—31no—
143C2×C3×Q16—91no—
144#946—91no—
145C2×#216—31no—
146C2×#216—91no—
147C2×#216—91no—
148#924—91no—
149#924—91no—
150#924—181no—
151C2×Dic12—31no—
152C2×Dic12—91no—
153C3×C22⋊C8—91no—
154#835—91no—
155#835—91no—
156#835—91no—
157#835—181no—
158#830—31no—
159(C2×D3)⋊C8—31no—
160#816—91no—
161#816—91no—
162#816—91no—
163#816—181no—
164C23×C32—31no—
165C22×C3×D3—21no—
166C22×C3×D3D1211yescharacteristic
167C22×C3×D3#6711yescharacteristic
168C32×Q8—32no—
169C32×Q8—61no—
170C2×C32×C4—21no—
171C2×C32×C4—21no—
172C2×C32×C4—21no—
173C2×C32×C4—31no—
174C2×C32×C4—41no—
175C2×C32×C4—61no—
176C2×C32×C4—61no—
177C2×C32×C4—61no—
178C2×C32×C4C22×D311yescharacteristic
179C2×C32×C4C22×D311yescharacteristic
180C2×C32×C4C22×D311yescharacteristic
181C2×C3⋊Dic3—21no—
182C2×C3⋊Dic3—21no—
183C2×C3⋊Dic3—61no—
184C2×C3⋊Dic3—62no—
185C2×C3⋊Dic3—61no—
186C2×C3⋊Dic3—61no—
187C2×C3⋊Dic3—121no—
188C2×C3⋊Dic3—121no—
189C2×C3⋊Dic3—122no—
190#634—21no—
191#634—21no—
192#634—21no—
193#634#6712yes—
194C2×C3×Dic3—21no—
195C2×C3×Dic3—21no—
196C2×C3×Dic3—21no—
197C2×C3×Dic3—41no—
198C2×C3×Dic3—64no—
199C2×C3×Dic3—61no—
200C2×C3×Dic3—61no—
201C2×C3×Dic3—62no—
202C2×C3×Dic3—61no—
203C2×C3×Dic3—61no—
204C2×C3×Dic3—121no—
205C2×C3×Dic3D1211yescharacteristic
206C2×C3×Dic3#6711yescharacteristic
207C3×C4×D3—21no—
208C3×C4×D3—21no—
209C3×C4×D3—21no—
210C3×C4×D3—21no—
211C3×C4×D3—41no—
212C3×C4×D3—41no—
213C3×C4×D3—64no—
214C3×Dic6—21no—
215C3×Dic6—21no—
216C3×Dic6—22no—
217C3×Dic6—32no—
218C3×Dic6—32no—
219C3×Dic6—41no—
220C3×Dic6—62no—
221C3×Dic6—64no—
222C3×Dic6—64no—
223C3×Dic6—61no—
224C3×Dic6—61no—
225C3×Dic6—61no—
226C3×Dic6—61no—
227C3×Dic6—61no—
228C3×Dic6—61no—
229C3×Dic6—121no—
230C3×Dic6—121no—
231C3×Dic6—121no—
232C3×Dic6#6712yes—
233C3×Dic6#6712yes—
234#627—24no—
235D3×Dic3—24no—
236C32×C8—61no—
237C3⋊(C3⋊C8)—21no—
238C3×C3⋊C8—21no—
239C3×C3⋊C8—21no—
240C3×C3⋊C8—41no—
241C3×C3⋊C8—61no—
242C3×C3⋊C8—61no—
243C3×C3⋊C8—121no—
244#451—271noSylow
245C2×C33#12011yescharacteristic
246C2×C33C2×Q1611yescharacteristic
247C2×C33#13711yescharacteristic
248C32×D3—22no—
249C2×C3×Q8—31no—
250C2×C3×Q8—31no—
251C2×C3×Q8—61no—
252C2×C3×Q8—91no—
253C2×C3×Q8—91no—
254C2×C3×Q8—181no—
255C2×C3×Q8—182no—
256C2×C3×Q8—182no—
257C22×C3×C4—31no—
258C22×C3×C4—31no—
259C22×C3×C4—61no—
260C22×C3×C4—181no—
261C22×Dic3—61no—
262D3×Q8—62no—
263D3×Q8—62no—
264C2×C4×D3—61no—
265C2×C4×D3D3211yescharacteristic
266C2×Dic6—31no—
267C2×Dic6—31no—
268C2×Dic6—31no—
269C2×Dic6—31no—
270C2×Dic6—61no—
271C2×Dic6—61no—
272C2×Dic6—62no—
273C2×Dic6—61no—
274C2×Dic6—61no—
275C2×Dic6—62no—
276C2×Dic6D3211yescharacteristic
277C3×Q16—182no—
278C2×C3×C8—31no—
279C2×C3×C8—31no—
280C2×C3×C8—61no—
281C2×C3×C8—91no—
282#217—181no—
283#217—181no—
284#217—182no—
285#216—62no—
286#216—182no—
287#216—182no—
288(C2×D3)⋊C4—181no—
289C4⋊Dic3—91no—
290C4⋊Dic3—91no—
291C4⋊Dic3—91no—
292C4⋊Dic3—181no—
293C4⋊Dic3—181no—
294C4⋊Dic3—181no—
295C4⋊Dic3—361no—
296Dic3⋊C4—181no—
297Dic3⋊C4—181no—
298Dic3⋊C4—182no—
299Dic3⋊C4—181no—
300C2×C3⋊C8—31no—
301C2×C3⋊C8—31no—
302C2×C3⋊C8—61no—
303C2×C3⋊C8D3211yescharacteristic
304Dic12—62no—
305Dic12—182no—
306C22×C32—21no—
307C22×C32—21no—
308C22×C32—21no—
309C22×C32—32no—
310C22×C32—41no—
311C22×C32—61no—
312C22×C32—61no—
313C22×C32C2×D1211yescharacteristic
314C22×C32D3×D411yescharacteristic
315C22×C32C2×#6711yescharacteristic
316C2×C3×D3—21no—
317C2×C3×D3—21no—
318C2×C3×D3—21no—
319C2×C3×D3—21no—
320C2×C3×D3—22no—
321C2×C3×D3—41no—
322C2×C3×D3—41no—
323C2×C3×D3Dic1212yescomplemented
324C2×C3×D3#21612yescomplemented
325C32×C4—21no—
326C32×C4—21no—
327C32×C4—21no—
328C32×C4—21no—
329C32×C4—21no—
330C32×C4—21no—
331C32×C4—41no—
332C32×C4—41no—
333C32×C4—62no—
334C32×C4—61no—
335C32×C4D3×D411yescharacteristic
336C32×C4D3×D411yescharacteristic
337C32×C4D3×D411yescharacteristic
338C32×C4C2×#6711yescharacteristic
339C32×C4C2×#6711yescharacteristic
340C32×C4C2×#6711yescharacteristic
341C3⋊Dic3—22no—
342C3⋊Dic3—41no—
343C3⋊Dic3—121no—
344C3⋊Dic3—122no—
345C3⋊Dic3—121no—
346C3⋊Dic3—121no—
347C3⋊Dic3—241no—
348C3⋊Dic3—241no—
349C3⋊Dic3—242no—
350C3×Dic3—21no—
351C3×Dic3—22no—
352C3×Dic3—22no—
353C3×Dic3—21no—
354C3×Dic3—42no—
355C3×Dic3—41no—
356C3×Dic3—62no—
357C3×Dic3—62no—
358C3×Dic3—122no—
359C3×Dic3—122no—
360C3×Dic3—121no—
361C3×Dic3—121no—
362C22×Q8—91no—
363C2×Q16—91no—
364C2×Q16—271no—
365C22⋊Q8—271no—
366#103—271no—
367#103—271no—
368C22⋊C8—91no—
369C33#45111yescharacteristic, complemented, Sylow
370C23×C3—31no—
371C23×C3—31no—
372C23×C3—61no—
373C22×D3#62611yescharacteristic
374C3×Q8—32no—
375C3×Q8—32no—
376C3×Q8—62no—
377C3×Q8—61no—
378C3×Q8—61no—
379C3×Q8—92no—
380C3×Q8—121no—
381C3×Q8—184no—
382C3×Q8—181no—
383C3×Q8—181no—
384C3×Q8—181no—
385C2×C3×C4—21no—
386C2×C3×C4—21no—
387C2×C3×C4—21no—
388C2×C3×C4—31no—
389C2×C3×C4—31no—
390C2×C3×C4—41no—
391C2×C3×C4—61no—
392C2×C3×C4—61no—
393C2×C3×C4—61no—
394C2×C3×C4—61no—
395C2×C3×C4—61no—
396C2×C3×C4—61no—
397C2×C3×C4—61no—
398C2×C3×C4—121no—
399C2×C3×C4—121no—
400C2×C3×C4—121no—
401C2×C3×C4—184no—
402C2×C3×C4—181no—
403C2×C3×C4—181no—
404C2×C3×C4C2×D3211yescharacteristic
405C2×C3×C4C2×D3211yescharacteristic
406C2×C3×C4C2×D3211yescharacteristic
407C2×Dic3—21no—
408C2×Dic3—64no—
409C2×Dic3—61no—
410C2×Dic3—61no—
411C2×Dic3—62no—
412C2×Dic3—61no—
413C2×Dic3—61no—
414C2×Dic3—121no—
415C2×Dic3—182no—
416C2×Dic3—181no—
417C2×Dic3—181no—
418C2×Dic3—181no—
419C2×Dic3—361no—
420C2×Dic3—361no—
421C2×Dic3—362no—
422C2×Dic3#62611yescharacteristic
423C4×D3—21no—
424C4×D3—21no—
425C4×D3—64no—
426Dic6—21no—
427Dic6—32no—
428Dic6—32no—
429Dic6—62no—
430Dic6—64no—
431Dic6—64no—
432Dic6—61no—
433Dic6—61no—
434Dic6—61no—
435Dic6—61no—
436Dic6—61no—
437Dic6—61no—
438Dic6—121no—
439Dic6—121no—
440Dic6—121no—
441Dic6#62512yes—
442C3×C8—61no—
443C3×C8—61no—
444C3×C8—121no—
445C3×C8—181no—
446C3⋊C8—21no—
447C3⋊C8—61no—
448C3⋊C8—61no—
449C3⋊C8—121no—
450C2×C32—21no—
451C2×C32—21no—
452C2×C32—21no—
453C2×C32—21no—
454C2×C32—21no—
455C2×C32—21no—
456C2×C32—21no—
457C2×C32—21no—
458C2×C32—21no—
459C2×C32—41no—
460C2×C32—41no—
461C2×C32—41no—
462C2×C32—62no—
463C2×C32#88211yescharacteristic
464C2×C32C2×Dic1211yescharacteristic
465C2×C32#90911yescharacteristic
466C2×C32D3×Q1611yescharacteristic
467C2×C32#91811yescharacteristic
468C2×C32#93711yescharacteristic
469C2×C32#94211yescharacteristic
470C2×C32C2×#21611yescharacteristic
471C2×C32#95311yescharacteristic
472C3×D3—22no—
473C3×D3—22no—
474C3×D3—42no—
475C2×Q8—31no—
476C2×Q8—91no—
477C2×Q8—91no—
478C2×Q8—181no—
479C2×Q8—182no—
480C2×Q8—182no—
481C22×C4—31no—
482C22×C4—181no—
483Q16—182no—
484Q16—542no—
485C2×C8—31no—
486C2×C8—91no—
487C4⋊C4—271no—
488C4⋊C4—541no—
489C22⋊C4—541no—
490C22×C3—21no—
491C22×C3—21no—
492C22×C3—21no—
493C22×C3—32no—
494C22×C3—32no—
495C22×C3—41no—
496C22×C3—62no—
497C22×C3—61no—
498C22×C3—61no—
499C22×C3—61no—
500C22×C3—61no—
501C22×C3—121no—
502C22×C3—121no—
503C22×C3D3×D1211yescharacteristic
504C22×C3C2×#62611yescharacteristic
505C22×C3D3×#6711yescharacteristic
506C2×D3—21no—
507C2×D3—21no—
508C2×D3#371912yescomplemented
509C3×C4—21no—
510C3×C4—21no—
511C3×C4—21no—
512C3×C4—21no—
513C3×C4—21no—
514C3×C4—21no—
515C3×C4—41no—
516C3×C4—41no—
517C3×C4—61no—
518C3×C4—62no—
519C3×C4—61no—
520C3×C4—62no—
521C3×C4—121no—
522C3×C4—122no—
523C3×C4—182no—
524C3×C4—361no—
525C3×C4#380211yescharacteristic
526C3×C4C2×#62511yescharacteristic
527C3×C4D3×#6711yescharacteristic
528C3×C4D3×#6711yescharacteristic
529C3×C4D3×#6711yescharacteristic
530C3×C4D3×#6711yescharacteristic
531Dic3—22no—
532Dic3—21no—
533Dic3—62no—
534Dic3—62no—
535Dic3—122no—
536Dic3—122no—
537Dic3—121no—
538Dic3—121no—
539Dic3—362no—
540Dic3—361no—
541Dic3—361no—
542Dic3—361no—
543Dic3—721no—
544Dic3—721no—
545Dic3—722no—
546C32—21no—
547C32—21no—
548C32—21no—
549C32—41no—
550C32#473911yescharacteristic, complemented
551C32#518611yescharacteristic, complemented
552C32#522811yescharacteristic, complemented
553C23—31no—
554Q8—32no—
555Q8—61no—
556Q8—92no—
557Q8—184no—
558Q8—181no—
559Q8—181no—
560Q8—181no—
561C2×C4—31no—
562C2×C4—61no—
563C2×C4—61no—
564C2×C4—61no—
565C2×C4—184no—
566C2×C4—181no—
567C2×C4—181no—
568C2×C4—541no—
569C2×C4D3311yescharacteristic
570C8—61no—
571C8—181no—
572C2×C3—21no—
573C2×C3—21no—
574C2×C3—21no—
575C2×C3—21no—
576C2×C3—21no—
577C2×C3—21no—
578C2×C3—21no—
579C2×C3—21no—
580C2×C3—21no—
581C2×C3—41no—
582C2×C3—41no—
583C2×C3—41no—
584C2×C3—62no—
585C2×C3—62no—
586C2×C3—122no—
587C2×C3D3×Dic1211yescharacteristic
588C2×C3#6380111yescharacteristic
589C2×C3#6383311yescharacteristic
590C2×C3C2×#371911yescharacteristic
591C2×C3#6392211yescharacteristic
592C2×C3D3×#21611yescharacteristic
593C2×C3#6394811yescharacteristic
594C2×C3#6397611yescharacteristic
595C2×C3#6397711yescharacteristic
596D3—22no—
597C22—32no—
598C22—61no—
599C22—61no—
600C22D3×#62611yescharacteristic
601C4—61no—
602C4—62no—
603C4—182no—
604C4—361no—
605C4—1081no—
606C4D3×#62511yescharacteristic
607C4#8895311yescharacteristic
608C3—21no—
609C3—21no—
610C3—21no—
611C3—41no—
612C3#1059094911yescharacteristic, complemented
613C3#1059118611yescharacteristic, complemented
614C3#1059159411yescharacteristic, complemented
615C2—62no—
616C2D3×#371911yescharacteristic
617C2#1172288511yescharacteristic
618C2#1172294711yescharacteristic
subgroupG/H
centerC22D3×#626
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radicalC22×C32×D3D4
hypocenterC33#451
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C2×#88812C2semidirect
4. C2×C3×#3719C2semidirect
5. #11722040C2semidirect
6. #11722038C2semidirect
37. C2×#3719D3semidirect
132. C2×C32×D3Q16semidirect
323. C2×C3×D3Dic12semidirect
324. C2×C3×D3#216semidirect
369. C33#451semidirect
508. C2×D3#3719semidirect
550. C32#4739semidirect
551. C32#5186semidirect
552. C32#5228semidirect
612. C3#10590949semidirect
613. C3#10591186semidirect
614. C3#10591594semidirect

Maximal subgroups

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

subgroupindex
C2×C3×D3×Dic62
C2×#888122
#117234252
C2×C3×#37192
#117220402
#117220382
C3×(C2×D3)⋊(C3⋊C8)2
#105915943
#105911863
#105909493