Finite Group Databasebeta

← #13162045

Subgroups — #13162045

Order 1728.

918 conjugacy classes of subgroups.

The subgroup lattice has 724 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64366C211yesmaximal, characteristic
2C3×#64160C211yesmaximal, characteristic, complemented
3#11723677C211yesmaximal, characteristic, complemented
4#11723663C211yesmaximal, characteristic, complemented
5C3×#63664C211yesmaximal, characteristic, complemented
6#11722077C211yesmaximal, characteristic, complemented
7#11722073C211yesmaximal, characteristic, complemented
8#10591690—31nomaximal
9#10591688—31nomaximal
10#10591664—31nomaximal
11C33×#41C2211yescharacteristic
12C3×#3832—21no—
13C3×D4×C3⋊D3—22no—
14C3×D4×C3⋊D3—21no—
15C3×#3828—21no—
16C3×#3828C2211yescharacteristic
17C2×C3×C4⋊(C3⋊D3)—21no—
18C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
19C2×C3×#634C2211yescharacteristic, complemented
20C3×#3756—21no—
21C3×#3754—21no—
22C3×C4×C3⋊Dic3C2211yescharacteristic, complemented
23C3×#3748C2211yescharacteristic, complemented
24C3×#3748C2211yescharacteristic, complemented
25#88814—22no—
26#88812—21no—
27#88811—21no—
28#64366D311yescharacteristic
29C3×#1009—31no—
30C3×#1009—31no—
31C3×#1009—61no—
32#64160D311yescharacteristic, complemented
33C3×#951—31no—
34C3×#951—31no—
35C3×#951—61no—
36#63864—31no—
37#63863—31no—
38#63860—31no—
39#63833—31no—
40#63831—31no—
41#63827—31no—
42#63664D311yescharacteristic, complemented
43C3×#837—31no—
44C3×#837—31no—
45C3×#837—61no—
46#63570—31no—
47#63569—31no—
48#63568—31no—
49#63559—31no—
50#63558—31no—
51#63558—31no—
52C22×C3×C3⋊D3—22no—
53C22×C3×C3⋊D3D412yes—
54C33×Q8D411yescharacteristic
55C33×D4—21no—
56C33×D4D411yescharacteristic
57C2×C33×C4—21no—
58C2×C33×C4C2311yescharacteristic
59C3×#638—22no—
60C3×#638—21no—
61C2×C3×C3⋊Dic3—21no—
62C2×C3×C3⋊Dic3—21no—
63C3×C4⋊(C3⋊D3)—22no—
64C3×C4⋊(C3⋊D3)—21no—
65C3×C4⋊(C3⋊D3)—21no—
66C3×C4⋊(C3⋊D3)D411yescharacteristic
67C3×C4×C3⋊D3—22no—
68C3×C4×C3⋊D3—21no—
69C3×#634—21no—
70C3×#634—21no—
71C3×#634D411yescharacteristic
72C3×C3⋊(C3⋊C8)—21no—
73C3×C3⋊(C3⋊C8)—21no—
74#5242—91no—
75#5202—91no—
76#4825—91no—
77C32×#41—21no—
78C32×#41—21no—
79C32×#41—21no—
80C32×#41—41no—
81C32×#41C2×D311yescharacteristic
82C32×#41C2×D311yescharacteristic
83C32×#41C2×D311yescharacteristic
84#3832—21no—
85D4×C3⋊D3—22no—
86D4×C3⋊D3—21no—
87#3828—21no—
88#3828C2×D311yescharacteristic
89C2×C4⋊(C3⋊D3)—21no—
90C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
91C2×#634C2×D311yescharacteristic, complemented
92C3×#239—62no—
93C3×#239—61no—
94C3×#239—61no—
95C3×D3×D4—62no—
96C3×D3×D4—62no—
97C3×D3×D4—64no—
98C3×D3×D4—62no—
99C3×D3×D4—61no—
100C3×D3×D4—61no—
101C3×#235—31no—
102C3×#235—31no—
103C3×#235—61no—
104C3×#235—61no—
105C3×#235—61no—
106C3×#235—62no—
107C2×C3×D12—31no—
108C2×C3×D12—31no—
109C2×C3×D12—61no—
110C2×C3×D12—61no—
111C2×C3×D12—61no—
112C2×C3×D12—62no—
113C2×C3×Dic6—31no—
114C2×C3×Dic6—31no—
115C2×C3×Dic6—61no—
116#3756—21no—
117#3754—21no—
118C4×C3⋊Dic3C2×D311yescharacteristic, complemented
119#3748C2×D311yescharacteristic, complemented
120#3748C2×D311yescharacteristic, complemented
121C3×#216—62no—
122C3×#216—61no—
123C3×#216—61no—
124C3×#214—62no—
125C3×#214—61no—
126C3×#214—61no—
127C3×C4×Dic3—31no—
128C3×C4×Dic3—31no—
129C3×C4×Dic3—61no—
130C3×#208—31no—
131C3×#208—31no—
132C3×#208—31no—
133C3×#208—31no—
134C3×#208—61no—
135C3×#208—61no—
136#3722—62no—
137(C2×D3)⋊Dic3—62no—
138(C2×D3)⋊Dic3—62no—
139#3719—61no—
140#3719—61no—
141#3718—61no—
142#3717—61no—
143#3716—61no—
144#3715—61no—
145C22×C33—21no—
146C22×C33C2×D411yescharacteristic
147C2×C3×C3⋊D3—22no—
148C2×C3×C3⋊D3—21no—
149C2×C3×C3⋊D3—42no—
150C2×C3×C3⋊D3—82no—
151C33×C4—21no—
152C33×C4C2×D411yescharacteristic
153C33×C4C2×D411yescharacteristic
154C3×C3⋊Dic3—21no—
155C3×C3⋊Dic3—41no—
156C3×C3⋊Dic3—41no—
157C3×H(5,𝔽2)—91no—
158#1009—31no—
159#1009—31no—
160#1009—61no—
161C3×#137—91no—
162#951—31no—
163#951—31no—
164#951—61no—
165#951—91no—
166#947—91no—
167#942—91no—
168#936—91no—
169#909—91no—
170#875—91no—
171C3×#104—91no—
172#837—31no—
173#837—31no—
174#837—61no—
175#837—91no—
176#833—91no—
177#825—91no—
178#823—91no—
179#823—91no—
180#805—91no—
181C22×C3⋊D3—22no—
182C22×C3⋊D3#6712yes—
183C22×C3×D3—32no—
184C22×C3×D3—32no—
185C22×C3×D3—62no—
186C22×C3×D3—64no—
187C22×C3×D3—62no—
188C22×C3×D3—62no—
189C32×Q8—21no—
190C32×Q8—21no—
191C32×Q8—21no—
192C32×Q8—41no—
193C32×Q8D1211yescharacteristic
194C32×Q8#6711yescharacteristic
195C32×Q8#6711yescharacteristic
196C32×D4—21no—
197C32×D4—22no—
198C32×D4—21no—
199C32×D4—21no—
200C32×D4—21no—
201C32×D4—21no—
202C32×D4—21no—
203C32×D4—41no—
204C32×D4—41no—
205C32×D4—41no—
206C32×D4—42no—
207C32×D4D1211yescharacteristic
208C32×D4#6711yescharacteristic
209C32×D4#6711yescharacteristic
210C2×C32×C4—21no—
211C2×C32×C4—22no—
212C2×C32×C4—21no—
213C2×C32×C4—21no—
214C2×C32×C4—21no—
215C2×C32×C4—21no—
216C2×C32×C4—21no—
217C2×C32×C4—41no—
218C2×C32×C4—41no—
219C2×C32×C4—41no—
220C2×C32×C4—42no—
221C2×C32×C4C22×D311yescharacteristic
222C2×C32×C4C22×D311yescharacteristic
223C2×C32×C4C22×D311yescharacteristic
224#638—22no—
225#638—21no—
226C2×C3⋊Dic3—21no—
227C2×C3⋊Dic3—21no—
228C4⋊(C3⋊D3)—22no—
229C4⋊(C3⋊D3)—21no—
230C4⋊(C3⋊D3)—21no—
231C4⋊(C3⋊D3)#6711yescharacteristic
232C4×C3⋊D3—22no—
233C4×C3⋊D3—21no—
234#634—21no—
235#634—21no—
236#634#6711yescharacteristic
237C3×#67—64no—
238C3×#67—62no—
239C3×#67—62no—
240C3×#67—62no—
241C3×#67—61no—
242C3×#67—61no—
243C2×C3×Dic3—61no—
244C2×C3×Dic3—61no—
245C2×C3×Dic3—61no—
246C2×C3×Dic3—61no—
247C2×C3×Dic3—121no—
248C2×C3×Dic3—121no—
249C3×D12—31no—
250C3×D12—31no—
251C3×D12—61no—
252C3×D12—62no—
253C3×D12—64no—
254C3×D12—62no—
255C3×D12—62no—
256C3×D12—62no—
257C3×D12—61no—
258C3×D12—61no—
259C3×D12—61no—
260C3×D12—61no—
261C3×C4×D3—64no—
262C3×C4×D3—62no—
263C3×C4×D3—62no—
264C3×C4×D3—62no—
265C3×C4×D3—61no—
266C3×C4×D3—61no—
267C3×Dic6—31no—
268C3×Dic6—31no—
269C3×Dic6—61no—
270C3×Dic6—62no—
271C3×Dic6—61no—
272C3×Dic6—61no—
273C3×Dic6—61no—
274C3×Dic6—61no—
275C3×Dic6—121no—
276C3⋊(C3⋊C8)—21no—
277C3⋊(C3⋊C8)—21no—
278C3×C3⋊C8—62no—
279C3×C3⋊C8—61no—
280C3×C3⋊C8—61no—
281C3×C3⋊C8—61no—
282C3×C3⋊C8—61no—
283C3×C3⋊C8—121no—
284#455—271noSylow
285C2×C33—21no—
286C2×C33—41no—
287C2×C33#12011yescharacteristic
288C3×C3⋊D3—42no—
289C3×C3⋊D3—41no—
290C3×#41—21no—
291C3×#41—21no—
292C3×#41—21no—
293C3×#41—41no—
294C3×#41—91no—
295C3×#41—181no—
296C3×#41—181no—
297C3×#41D3211yescharacteristic
298C3×#41D3211yescharacteristic
299C3×#41D3211yescharacteristic
300C2×C3×Q8—91no—
301C2×C3×D4—91no—
302C2×C3×D4—181no—
303C2×C3×D4—182no—
304C2×C3×D4—181no—
305#239—62no—
306#239—61no—
307#239—61no—
308D3×D4—62no—
309D3×D4—62no—
310D3×D4—64no—
311D3×D4—62no—
312D3×D4—61no—
313D3×D4—61no—
314#235—31no—
315#235—31no—
316#235—61no—
317#235—61no—
318#235—61no—
319#235—62no—
320C2×D12—31no—
321C2×D12—31no—
322C2×D12—61no—
323C2×D12—61no—
324C2×D12—61no—
325C2×D12—62no—
326C2×Dic6—31no—
327C2×Dic6—31no—
328C2×Dic6—61no—
329C3×Q16—181no—
330C3×SD16—181no—
331C3×M16—91no—
332C3×M16—91no—
333C3×C42—91no—
334#217—182no—
335#216—62no—
336#216—61no—
337#216—61no—
338#216—181no—
339#216—181no—
340#215—181no—
341#214—62no—
342#214—61no—
343#214—61no—
344#214—181no—
345(C2×D3)⋊C4—182no—
346(C2×D3)⋊C4—182no—
347C4×Dic3—31no—
348C4×Dic3—31no—
349C4×Dic3—61no—
350#208—31no—
351#208—31no—
352#208—31no—
353#208—31no—
354#208—61no—
355#208—61no—
356Dic12—181no—
357#204—181no—
358C22×C32—21no—
359C22×C32—22no—
360C22×C32—21no—
361C22×C32—21no—
362C22×C32—21no—
363C22×C32—21no—
364C22×C32—21no—
365C22×C32—41no—
366C22×C32—41no—
367C22×C32—41no—
368C22×C32—42no—
369C22×C32D3×D411yescharacteristic
370C22×C32D3×D411yescharacteristic
371C22×C32C2×#6711yescharacteristic
372C2×C3⋊D3—22no—
373C2×C3⋊D3—21no—
374C2×C3⋊D3—42no—
375C2×C3⋊D3—82no—
376C2×C3×D3—62no—
377C2×C3×D3—64no—
378C2×C3×D3—61no—
379C2×C3×D3—62no—
380C2×C3×D3—61no—
381C2×C3×D3—62no—
382C2×C3×D3—122no—
383C2×C3×D3—122no—
384C2×C3×D3—242no—
385C2×C3×D3—244no—
386C2×C3×D3—242no—
387C2×C3×D3—242no—
388C32×C4—21no—
389C32×C4—22no—
390C32×C4—21no—
391C32×C4—21no—
392C32×C4—21no—
393C32×C4—21no—
394C32×C4—21no—
395C32×C4—21no—
396C32×C4—21no—
397C32×C4—21no—
398C32×C4—41no—
399C32×C4—41no—
400C32×C4—41no—
401C32×C4—41no—
402C32×C4—42no—
403C32×C4C2×D1211yescharacteristic
404C32×C4D3×D411yescharacteristic
405C32×C4D3×D411yescharacteristic
406C32×C4C2×#6711yescharacteristic
407C32×C4C2×#6711yescharacteristic
408C32×C4C2×#6711yescharacteristic
409C3⋊Dic3—21no—
410C3⋊Dic3—41no—
411C3⋊Dic3—41no—
412C3×Dic3—62no—
413C3×Dic3—61no—
414C3×Dic3—61no—
415C3×Dic3—122no—
416C3×Dic3—121no—
417C3×Dic3—121no—
418C3×Dic3—122no—
419C3×Dic3—121no—
420C3×Dic3—121no—
421H(5,𝔽2)—91no—
422#137—91no—
423#137—271no—
424#124—271no—
425#104—91no—
426#104—271no—
427#100—271no—
428C33#45511yescharacteristic, complemented, Sylow
429C23×C3—92no—
430C23×C3—182no—
431C22×D3—32no—
432C22×D3—32no—
433C22×D3—62no—
434C22×D3—64no—
435C22×D3—62no—
436C22×D3—62no—
437C3×Q8—21no—
438C3×Q8—21no—
439C3×Q8—21no—
440C3×Q8—41no—
441C3×Q8—91no—
442C3×Q8—181no—
443C3×Q8—181no—
444C3×Q8#62511yescharacteristic
445C3×Q8#62611yescharacteristic
446C3×Q8#62611yescharacteristic
447C3×D4—22no—
448C3×D4—21no—
449C3×D4—21no—
450C3×D4—21no—
451C3×D4—21no—
452C3×D4—21no—
453C3×D4—21no—
454C3×D4—41no—
455C3×D4—41no—
456C3×D4—42no—
457C3×D4—41no—
458C3×D4—91no—
459C3×D4—182no—
460C3×D4—182no—
461C3×D4—181no—
462C3×D4—181no—
463C3×D4—181no—
464C3×D4#62511yescharacteristic
465C3×D4#62611yescharacteristic
466C3×D4#62611yescharacteristic
467C2×C3×C4—22no—
468C2×C3×C4—21no—
469C2×C3×C4—21no—
470C2×C3×C4—21no—
471C2×C3×C4—21no—
472C2×C3×C4—21no—
473C2×C3×C4—21no—
474C2×C3×C4—41no—
475C2×C3×C4—41no—
476C2×C3×C4—42no—
477C2×C3×C4—41no—
478C2×C3×C4—182no—
479C2×C3×C4—181no—
480C2×C3×C4—181no—
481C2×C3×C4—181no—
482C2×C3×C4C2×D3211yescharacteristic
483C2×C3×C4C2×D3211yescharacteristic
484C2×C3×C4C2×D3211yescharacteristic
485#67—64no—
486#67—62no—
487#67—62no—
488#67—62no—
489#67—61no—
490#67—61no—
491C2×Dic3—61no—
492C2×Dic3—61no—
493C2×Dic3—61no—
494C2×Dic3—61no—
495C2×Dic3—121no—
496C2×Dic3—121no—
497D12—31no—
498D12—31no—
499D12—61no—
500D12—62no—
501D12—64no—
502D12—62no—
503D12—62no—
504D12—62no—
505D12—61no—
506D12—61no—
507D12—61no—
508D12—61no—
509C4×D3—64no—
510C4×D3—62no—
511C4×D3—62no—
512C4×D3—62no—
513C4×D3—61no—
514C4×D3—61no—
515Dic6—31no—
516Dic6—31no—
517Dic6—61no—
518Dic6—62no—
519Dic6—61no—
520Dic6—61no—
521Dic6—61no—
522Dic6—61no—
523Dic6—121no—
524C3×C8—181no—
525C3×C8—181no—
526C3⋊C8—62no—
527C3⋊C8—61no—
528C3⋊C8—61no—
529C3⋊C8—61no—
530C3⋊C8—61no—
531C3⋊C8—121no—
532C2×C32—21no—
533C2×C32—21no—
534C2×C32—21no—
535C2×C32—21no—
536C2×C32—21no—
537C2×C32—21no—
538C2×C32—41no—
539C2×C32—41no—
540C2×C32—41no—
541C2×C32—41no—
542C2×C32—41no—
543C2×C32—42no—
544C2×C32—41no—
545C2×C32—41no—
546C2×C32—81no—
547C2×C32—81no—
548C2×C32—81no—
549C2×C32—82no—
550C2×C32#88211yescharacteristic
551C2×C32#93711yescharacteristic
552C2×C32#95311yescharacteristic
553C3⋊D3—42no—
554C3⋊D3—41no—
555C3×D3—124no—
556C3×D3—122no—
557C3×D3—122no—
558C3×D3—121no—
559C3×D3—122no—
560C3×D3—121no—
561#41—91no—
562#41—181no—
563#41—181no—
564#41#111311yescharacteristic
565C2×Q8—91no—
566C2×D4—91no—
567C2×D4—181no—
568C2×D4—182no—
569C2×D4—181no—
570Q16—181no—
571Q16—541no—
572SD16—181no—
573SD16—541no—
574M16—91no—
575M16—91no—
576C22⋊C4—542no—
577C42—91no—
578C22×C3—22no—
579C22×C3—21no—
580C22×C3—21no—
581C22×C3—21no—
582C22×C3—21no—
583C22×C3—21no—
584C22×C3—21no—
585C22×C3—41no—
586C22×C3—41no—
587C22×C3—42no—
588C22×C3—41no—
589C22×C3—181no—
590C22×C3—182no—
591C22×C3—362no—
592C22×C3—722no—
593C22×C3D3×#6711yescharacteristic
594C22×C3D3×#6711yescharacteristic
595C22×C3#381111yescharacteristic
596C2×D3—62no—
597C2×D3—64no—
598C2×D3—61no—
599C2×D3—62no—
600C2×D3—61no—
601C2×D3—62no—
602C2×D3—122no—
603C2×D3—122no—
604C2×D3—242no—
605C2×D3—244no—
606C2×D3—242no—
607C2×D3—242no—
608C3×C4—22no—
609C3×C4—21no—
610C3×C4—21no—
611C3×C4—21no—
612C3×C4—21no—
613C3×C4—21no—
614C3×C4—21no—
615C3×C4—21no—
616C3×C4—21no—
617C3×C4—21no—
618C3×C4—41no—
619C3×C4—41no—
620C3×C4—42no—
621C3×C4—41no—
622C3×C4—41no—
623C3×C4—181no—
624C3×C4—361no—
625C3×C4—361no—
626C3×C4C2×#62511yescharacteristic
627C3×C4C2×#62611yescharacteristic
628C3×C4C2×#62611yescharacteristic
629C3×C4D3×#6711yescharacteristic
630C3×C4D3×#6711yescharacteristic
631C3×C4#381111yescharacteristic
632Dic3—62no—
633Dic3—61no—
634Dic3—61no—
635Dic3—122no—
636Dic3—121no—
637Dic3—121no—
638Dic3—122no—
639Dic3—121no—
640Dic3—121no—
641C32—21no—
642C32—21no—
643C32—21no—
644C32—41no—
645C32#482511yescharacteristic, complemented
646C32#520211yescharacteristic, complemented
647C32#524211yescharacteristic, complemented
648C23—92no—
649C23—182no—
650Q8—91no—
651Q8—181no—
652Q8—181no—
653Q8#629511yescharacteristic
654D4—21no—
655D4—91no—
656D4—182no—
657D4—182no—
658D4—181no—
659D4—181no—
660D4—181no—
661D4#629511yescharacteristic
662C2×C4—21no—
663C2×C4—182no—
664C2×C4—181no—
665C2×C4—181no—
666C2×C4—181no—
667C2×C4C2×#111311yescharacteristic
668C8—181no—
669C8—181no—
670C2×C3—21no—
671C2×C3—21no—
672C2×C3—21no—
673C2×C3—21no—
674C2×C3—21no—
675C2×C3—21no—
676C2×C3—41no—
677C2×C3—41no—
678C2×C3—41no—
679C2×C3—42no—
680C2×C3—41no—
681C2×C3—41no—
682C2×C3—41no—
683C2×C3—41no—
684C2×C3—81no—
685C2×C3—81no—
686C2×C3—81no—
687C2×C3—82no—
688C2×C3—362no—
689C2×C3—361no—
690C2×C3#6397611yescharacteristic
691C2×C3#6397711yescharacteristic
692C2×C3#6398111yescharacteristic
693D3—124no—
694D3—122no—
695D3—122no—
696D3—121no—
697D3—122no—
698D3—121no—
699C22—21no—
700C22—181no—
701C22—182no—
702C22—362no—
703C22—722no—
704C22#8904911yescharacteristic
705C4—21no—
706C4—181no—
707C4—361no—
708C4—361no—
709C4C2×#629511yescharacteristic
710C4#8904911yescharacteristic
711C3—21no—
712C3—21no—
713C3—21no—
714C3—41no—
715C3#1059166411yescharacteristic, complemented
716C3#1059168811yescharacteristic, complemented
717C3#1059169011yescharacteristic, complemented
718C2—21no—
719C2—41no—
720C2—362no—
721C2—361no—
722C2#1172372711yescharacteristic
subgroupG/H
centerC2#11723727
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×#1113
FittingC33×#41C22
socleC2×C33#120
prime radicalC3×#64366C2
hypercenter#41#1113
hypocenterC33#455
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4 ◁ #41

Complements

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

NH ≅ G/Ntype
2. C3×#64160C2semidirect
3. #11723677C2semidirect
4. #11723663C2semidirect
5. C3×#63664C2semidirect
6. #11722077C2semidirect
7. #11722073C2semidirect
19. C2×C3×#634C22semidirect
22. C3×C4×C3⋊Dic3C22semidirect
23. C3×#3748C22semidirect
24. C3×#3748C22semidirect
32. #64160D3semidirect
42. #63664D3semidirect
91. C2×#634C2×D3semidirect
118. C4×C3⋊Dic3C2×D3semidirect
119. #3748C2×D3semidirect
120. #3748C2×D3semidirect
428. C33#455semidirect
645. C32#4825semidirect
646. C32#5202semidirect
647. C32#5242semidirect
715. C3#10591664semidirect
716. C3#10591688semidirect
717. C3#10591690semidirect

Maximal subgroups

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

subgroupindex
C3×#643662
C3×#641602
#117236772
#117236632
C3×#636642
#117220772
#117220732
#105916903
#105916883
#105916643