Finite Group Databasebeta

← #13155936

Subgroups — #13155936

Order 1728.

778 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 778 conjugacy classes of subgroups — too many to draw. Only its 98 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
1C3×#64308C211yesmaximal, characteristic
2#11723677C211yesmaximal, characteristic, complemented
3#11723373C211yesmaximal, characteristic, complemented
4C3×#63831C211yesmaximal, characteristic, complemented
5#11722044C211yesmaximal, characteristic, complemented
6#11722026C211yesmaximal, characteristic, complemented
7C3×#63568C211yesmaximal, characteristic, complemented
8#10591688—31nomaximal
9#10591664—31nomaximal
10#10591011—31nomaximal
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic, complemented
13C32×#235C2211yescharacteristic
14C32×#235C2211yescharacteristic
15C3×#3811—22no—
16C3×D3×D12—21no—
17C3×D3×D12—21no—
18C3×#3799—21no—
19C3×#3799—21no—
20#88855C2211yescharacteristic, complemented
21C3×C4×C3⋊Dic3C2211yescharacteristic, complemented
22C32×#208C2211yescharacteristic, complemented
23#88814—22no—
24#88798—21no—
25#88792—21no—
26C3×#3719—21no—
27C3×#3716—21no—
28C3×#1009—31no—
29C3×#1009—31no—
30#64308D311yescharacteristic
31#64160—31no—
32C3×#951—31no—
33C3×#909—31no—
34#63864—31no—
35#63863—31no—
36#63860—31no—
37#63831D311yescharacteristic, complemented
38#63827—31no—
39#63827—31no—
40#63827—61no—
41#63664—31no—
42#63650—31no—
43C3×#837—31no—
44C3×#825—31no—
45#63569—31no—
46#63569—31no—
47#63569—61no—
48#63568D311yescharacteristic, complemented
49#63558—31no—
50#63558—31no—
51#63558—61no—
52C22×C3×C3⋊D3D412yes—
53C2×C3×D32—22no—
54C2×C33×C4C2311yescharacteristic
55C2×C3×C3⋊Dic3—21no—
56C2×C3×C3⋊Dic3—21no—
57C3×C4⋊(C3⋊D3)—21no—
58C3×C4⋊(C3⋊D3)—21no—
59C3×#634—21no—
60C3×#634—21no—
61C32×#67—21no—
62C32×#67—21no—
63C32×D12D411yescharacteristic
64C32×D12D411yescharacteristic
65C32×C4×D3—21no—
66C32×C4×D3—21no—
67C32×Dic6D411yescharacteristic
68C32×Dic6D411yescharacteristic
69C3×#626—22no—
70C3×#626—22no—
71C3×#624—22no—
72C3⋊(C3⋊(C3⋊C8))—21no—
73C32×C3⋊C8—21no—
74#5242—91no—
75#5202—91no—
76#4825—91no—
77C32×#41—31no—
78C32×#41—31no—
79C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
80C2×#634C2×D311yescharacteristic, complemented
81C3×#239—61no—
82C3×#239—61no—
83C3×D3×D4—61no—
84C3×D3×D4—61no—
85C3×D3×D4—62no—
86C3×D3×D4—62no—
87C3×#235—21no—
88C3×#235—21no—
89C3×#235—61no—
90C3×#235—61no—
91C3×#235C2×D311yescharacteristic
92C3×#235C2×D311yescharacteristic
93C3×#235C2×D311yescharacteristic
94C3×#235C2×D311yescharacteristic
95C2×C3×D12—31no—
96C2×C3×D12—31no—
97C2×C3×D12—61no—
98C2×C3×D12—61no—
99C2×C3×D12—61no—
100C2×C3×Dic6—31no—
101C2×C3×Dic6—31no—
102C2×C3×Dic6—61no—
103#3811—22no—
104D3×D12—21no—
105D3×D12—21no—
106#3799—21no—
107#3799—21no—
108C32×M16—31no—
109#3756—61no—
110#3754—61no—
111C4×C3⋊Dic3C2×D311yescharacteristic, complemented
112#3748—31no—
113#3748—31no—
114#3748—31no—
115#3748—61no—
116#3748—61no—
117#3748—61no—
118#3748—121no—
119C3×#216—61no—
120C3×#214—61no—
121C3×C4×Dic3—31no—
122C3×C4×Dic3—31no—
123C3×C4×Dic3—61no—
124C3×#208—21no—
125C3×#208C2×D311yescharacteristic, complemented
126C3×#208C2×D311yescharacteristic, complemented
127C3×Dic12—61no—
128C3×#204—61no—
129#3722—62no—
130(C2×D3)⋊Dic3—62no—
131(C2×D3)⋊Dic3—62no—
132#3719—21no—
133#3718—62no—
134#3718—61no—
135#3718—61no—
136#3716—21no—
137#3715—62no—
138#3715—61no—
139#3715—61no—
140C22×C33C2×D411yescharacteristic
141C2×C3×C3⋊D3—22no—
142C2×C3×C3⋊D3—42no—
143C2×C32×D3—21no—
144C2×C32×D3—21no—
145C3×D32—82no—
146C33×C4C2×D411yescharacteristic
147C33×C4C2×D411yescharacteristic
148C3×C3⋊Dic3—41no—
149C3×C3⋊Dic3—41no—
150C32×Dic3—21no—
151C32×Dic3—21no—
152C3×H(5,𝔽2)—91no—
153#1009—31no—
154#1009—31no—
155C3×#137—91no—
156#951—31no—
157#951—91no—
158#951—91no—
159#951—181no—
160#947—91no—
161#942—91no—
162#936—91no—
163#909—31no—
164#875—91no—
165C3×#104—91no—
166#837—31no—
167#837—91no—
168#837—91no—
169#837—181no—
170#833—91no—
171#825—31no—
172#823—91no—
173#823—91no—
174#823—181no—
175#805—91no—
176C22×C3⋊D3#6712yes—
177C22×C3×D3—32no—
178C22×C3×D3—32no—
179C22×C3×D3—62no—
180C22×C3×D3—62no—
181C22×C3×D3—62no—
182C2×D32—22no—
183C32×Q8—31no—
184C32×Q8—31no—
185C32×D4—31no—
186C32×D4—31no—
187C32×D4—61no—
188C32×D4—61no—
189C2×C32×C4—21no—
190C2×C32×C4—21no—
191C2×C32×C4—21no—
192C2×C32×C4—41no—
193C2×C32×C4—61no—
194C2×C32×C4—61no—
195C2×C32×C4C22×D311yescharacteristic
196C2×C32×C4C22×D311yescharacteristic
197C2×C32×C4C22×D311yescharacteristic
198C2×C3⋊Dic3—21no—
199C2×C3⋊Dic3—21no—
200C4⋊(C3⋊D3)—21no—
201C4⋊(C3⋊D3)—21no—
202#634—21no—
203#634—21no—
204C3×#67—21no—
205C3×#67—21no—
206C3×#67—21no—
207C3×#67—21no—
208C3×#67—41no—
209C3×#67—41no—
210C3×#67—62no—
211C3×#67—62no—
212C2×C3×Dic3—61no—
213C2×C3×Dic3—61no—
214C2×C3×Dic3—61no—
215C2×C3×Dic3—61no—
216C2×C3×Dic3—121no—
217C2×C3×Dic3—121no—
218C3×D12—21no—
219C3×D12—21no—
220C3×D12—62no—
221C3×D12—62no—
222C3×D12—61no—
223C3×D12—61no—
224C3×D12—61no—
225C3×D12—61no—
226C3×D12—121no—
227C3×D12—121no—
228C3×D12D1211yescharacteristic
229C3×D12#6711yescharacteristic
230C3×D12#6711yescharacteristic
231C3×D12#6711yescharacteristic
232C3×C4×D3—21no—
233C3×C4×D3—21no—
234C3×C4×D3—21no—
235C3×C4×D3—21no—
236C3×C4×D3—41no—
237C3×C4×D3—41no—
238C3×C4×D3—62no—
239C3×C4×D3—62no—
240C3×Dic6—21no—
241C3×Dic6—21no—
242C3×Dic6—62no—
243C3×Dic6—61no—
244C3×Dic6—61no—
245C3×Dic6—61no—
246C3×Dic6—61no—
247C3×Dic6—121no—
248C3×Dic6D1211yescharacteristic
249C3×Dic6#6711yescharacteristic
250C3×Dic6#6711yescharacteristic
251C3×Dic6#6711yescharacteristic
252#626—22no—
253#626—22no—
254#624—22no—
255C32×C8—61no—
256C3⋊(C3⋊C8)—61no—
257C3⋊(C3⋊C8)—62no—
258C3⋊(C3⋊C8)—61no—
259C3⋊(C3⋊C8)—61no—
260C3⋊(C3⋊C8)—121no—
261C3⋊(C3⋊C8)—121no—
262C3⋊(C3⋊C8)—122no—
263C3×C3⋊C8—21no—
264C3×C3⋊C8—21no—
265C3×C3⋊C8—41no—
266#455—271noSylow
267C2×C33—21no—
268C2×C33#12011yescharacteristic
269C3×C3⋊D3—42no—
270C32×D3—41no—
271C32×D3—41no—
272C3×#41—31no—
273C3×#41—31no—
274C3×#41—31no—
275C3×#41—31no—
276C3×#41—61no—
277C3×#41—61no—
278C3×#41—181no—
279C3×#41—181no—
280C2×C3×Q8—91no—
281C2×C3×D4—91no—
282C2×C3×D4—181no—
283C2×C3×D4—181no—
284C2×C3×D4—182no—
285#239—61no—
286#239—61no—
287D3×D4—61no—
288D3×D4—61no—
289D3×D4—62no—
290D3×D4—62no—
291#235—61no—
292#235—61no—
293#235D3211yescharacteristic
294#235D3211yescharacteristic
295C2×D12—31no—
296C2×D12—31no—
297C2×D12—61no—
298C2×D12—61no—
299C2×D12—61no—
300C2×Dic6—31no—
301C2×Dic6—31no—
302C2×Dic6—61no—
303C3×Q16—181no—
304C3×SD16—181no—
305C3×M16—31no—
306C3×M16—31no—
307C3×M16—61no—
308C3×C42—91no—
309#217—182no—
310#216—61no—
311#216—181no—
312#216—181no—
313#216—182no—
314#216—181no—
315#215—181no—
316#214—61no—
317#214—181no—
318#214—181no—
319#214—182no—
320(C2×D3)⋊C4—182no—
321(C2×D3)⋊C4—182no—
322C4×Dic3—31no—
323C4×Dic3—31no—
324C4×Dic3—61no—
325#208—91no—
326#208—91no—
327#208—91no—
328#208—181no—
329#208—181no—
330#208—181no—
331#208—361no—
332#208D3211yescharacteristic, complemented
333Dic12—61no—
334#204—61no—
335C22×C32—21no—
336C22×C32—21no—
337C22×C32—21no—
338C22×C32—41no—
339C22×C32—61no—
340C22×C32—61no—
341C22×C32D3×D411yescharacteristic
342C22×C32D3×D411yescharacteristic
343C22×C32C2×#6711yescharacteristic
344C2×C3⋊D3—22no—
345C2×C3⋊D3—42no—
346C2×C3×D3—21no—
347C2×C3×D3—21no—
348C2×C3×D3—21no—
349C2×C3×D3—21no—
350C2×C3×D3—41no—
351C2×C3×D3—41no—
352C2×C3×D3—62no—
353C2×C3×D3—62no—
354C2×C3×D3—122no—
355C2×C3×D3—122no—
356C2×C3×D3—122no—
357C2×C3×D3—242no—
358C2×C3×D3—242no—
359C2×C3×D3—242no—
360D32—82no—
361C32×C4—21no—
362C32×C4—21no—
363C32×C4—21no—
364C32×C4—21no—
365C32×C4—21no—
366C32×C4—21no—
367C32×C4—41no—
368C32×C4—41no—
369C32×C4—61no—
370C32×C4—61no—
371C32×C4C2×D1211yescharacteristic
372C32×C4D3×D411yescharacteristic
373C32×C4D3×D411yescharacteristic
374C32×C4C2×#6711yescharacteristic
375C32×C4C2×#6711yescharacteristic
376C32×C4C2×#6711yescharacteristic
377C3⋊Dic3—41no—
378C3⋊Dic3—41no—
379C3×Dic3—21no—
380C3×Dic3—21no—
381C3×Dic3—21no—
382C3×Dic3—21no—
383C3×Dic3—41no—
384C3×Dic3—41no—
385C3×Dic3—122no—
386C3×Dic3—121no—
387C3×Dic3—121no—
388C3×Dic3—122no—
389C3×Dic3—121no—
390C3×Dic3—121no—
391H(5,𝔽2)—91no—
392#137—91no—
393#137—271no—
394#124—271no—
395#104—91no—
396#104—271no—
397#100—271no—
398C33#45511yescharacteristic, complemented, Sylow
399C23×C3—92no—
400C23×C3—182no—
401C22×D3—32no—
402C22×D3—32no—
403C22×D3—62no—
404C22×D3—62no—
405C22×D3—62no—
406C3×Q8—31no—
407C3×Q8—31no—
408C3×Q8—31no—
409C3×Q8—31no—
410C3×Q8—61no—
411C3×Q8—61no—
412C3×Q8—181no—
413C3×Q8—181no—
414C3×D4—31no—
415C3×D4—31no—
416C3×D4—31no—
417C3×D4—31no—
418C3×D4—61no—
419C3×D4—61no—
420C3×D4—61no—
421C3×D4—61no—
422C3×D4—61no—
423C3×D4—61no—
424C3×D4—121no—
425C3×D4—121no—
426C3×D4—182no—
427C3×D4—182no—
428C3×D4—181no—
429C3×D4—181no—
430C2×C3×C4—21no—
431C2×C3×C4—21no—
432C2×C3×C4—21no—
433C2×C3×C4—41no—
434C2×C3×C4—61no—
435C2×C3×C4—61no—
436C2×C3×C4—61no—
437C2×C3×C4—61no—
438C2×C3×C4—121no—
439C2×C3×C4—121no—
440C2×C3×C4—182no—
441C2×C3×C4—181no—
442C2×C3×C4—181no—
443C2×C3×C4C2×D3211yescharacteristic
444C2×C3×C4C2×D3211yescharacteristic
445C2×C3×C4C2×D3211yescharacteristic
446#67—21no—
447#67—21no—
448#67—62no—
449#67—62no—
450C2×Dic3—61no—
451C2×Dic3—61no—
452C2×Dic3—61no—
453C2×Dic3—61no—
454C2×Dic3—121no—
455C2×Dic3—121no—
456D12—62no—
457D12—62no—
458D12—61no—
459D12—61no—
460D12—61no—
461D12—61no—
462D12—121no—
463D12—121no—
464D12#62511yescharacteristic
465D12#62611yescharacteristic
466C4×D3—21no—
467C4×D3—21no—
468C4×D3—62no—
469C4×D3—62no—
470Dic6—62no—
471Dic6—61no—
472Dic6—61no—
473Dic6—61no—
474Dic6—61no—
475Dic6—121no—
476Dic6#62511yescharacteristic
477Dic6#62611yescharacteristic
478C3×C8—61no—
479C3×C8—61no—
480C3×C8—121no—
481C3⋊C8—21no—
482C3⋊C8—182no—
483C3⋊C8—181no—
484C3⋊C8—181no—
485C3⋊C8—181no—
486C3⋊C8—361no—
487C3⋊C8—361no—
488C3⋊C8—362no—
489C2×C32—21no—
490C2×C32—21no—
491C2×C32—21no—
492C2×C32—21no—
493C2×C32—21no—
494C2×C32—21no—
495C2×C32—41no—
496C2×C32—41no—
497C2×C32—41no—
498C2×C32—41no—
499C2×C32—81no—
500C2×C32—121no—
501C2×C32—121no—
502C2×C32#88211yescharacteristic
503C2×C32#93711yescharacteristic
504C2×C32#95311yescharacteristic
505C3⋊D3—42no—
506C3×D3—41no—
507C3×D3—41no—
508C3×D3—41no—
509C3×D3—41no—
510C3×D3—81no—
511C3×D3—81no—
512C3×D3—122no—
513C3×D3—122no—
514C3×D3—242no—
515#41—31no—
516#41—31no—
517#41—181no—
518#41—181no—
519C2×Q8—91no—
520C2×D4—91no—
521C2×D4—181no—
522C2×D4—181no—
523C2×D4—182no—
524Q16—181no—
525Q16—541no—
526SD16—181no—
527SD16—541no—
528M16—31no—
529M16—271no—
530C22⋊C4—542no—
531C42—91no—
532C22×C3—21no—
533C22×C3—21no—
534C22×C3—21no—
535C22×C3—41no—
536C22×C3—61no—
537C22×C3—61no—
538C22×C3—61no—
539C22×C3—61no—
540C22×C3—121no—
541C22×C3—121no—
542C22×C3—182no—
543C22×C3—362no—
544C22×C3—722no—
545C22×C3#380211yescharacteristic
546C22×C3D3×#6711yescharacteristic
547C22×C3D3×#6711yescharacteristic
548C2×D3—21no—
549C2×D3—21no—
550C2×D3—62no—
551C2×D3—62no—
552C2×D3—122no—
553C2×D3—122no—
554C2×D3—122no—
555C2×D3—242no—
556C2×D3—242no—
557C2×D3—242no—
558C3×C4—21no—
559C3×C4—21no—
560C3×C4—21no—
561C3×C4—21no—
562C3×C4—21no—
563C3×C4—21no—
564C3×C4—41no—
565C3×C4—41no—
566C3×C4—61no—
567C3×C4—61no—
568C3×C4—61no—
569C3×C4—61no—
570C3×C4—121no—
571C3×C4—121no—
572C3×C4—361no—
573C3×C4—361no—
574C3×C4D3×D1211yescharacteristic
575C3×C4C2×#62511yescharacteristic
576C3×C4C2×#62611yescharacteristic
577C3×C4D3×#6711yescharacteristic
578C3×C4D3×#6711yescharacteristic
579C3×C4D3×#6711yescharacteristic
580Dic3—21no—
581Dic3—21no—
582Dic3—122no—
583Dic3—121no—
584Dic3—121no—
585Dic3—122no—
586Dic3—121no—
587Dic3—121no—
588C32—21no—
589C32—21no—
590C32—21no—
591C32—41no—
592C32#482511yescharacteristic, complemented
593C32#520211yescharacteristic, complemented
594C32#524211yescharacteristic, complemented
595C23—92no—
596C23—182no—
597Q8—31no—
598Q8—31no—
599Q8—181no—
600Q8—181no—
601D4—31no—
602D4—31no—
603D4—61no—
604D4—61no—
605D4—182no—
606D4—182no—
607D4—181no—
608D4—181no—
609C2×C4—61no—
610C2×C4—61no—
611C2×C4—182no—
612C2×C4—181no—
613C2×C4—181no—
614C2×C4D3311yescharacteristic
615C8—61no—
616C8—541no—
617C2×C3—21no—
618C2×C3—21no—
619C2×C3—21no—
620C2×C3—21no—
621C2×C3—21no—
622C2×C3—21no—
623C2×C3—41no—
624C2×C3—41no—
625C2×C3—41no—
626C2×C3—41no—
627C2×C3—81no—
628C2×C3—121no—
629C2×C3—121no—
630C2×C3—121no—
631C2×C3—121no—
632C2×C3—241no—
633C2×C3—241no—
634C2×C3—362no—
635C2×C3#6392211yescharacteristic
636C2×C3#6397611yescharacteristic
637C2×C3#6397711yescharacteristic
638D3—41no—
639D3—41no—
640D3—122no—
641D3—122no—
642D3—242no—
643C22—61no—
644C22—61no—
645C22—182no—
646C22—362no—
647C22—722no—
648C22#8895311yescharacteristic
649C4—61no—
650C4—61no—
651C4—361no—
652C4—361no—
653C4D3×#62511yescharacteristic
654C4D3×#62611yescharacteristic
655C3—21no—
656C3—21no—
657C3—21no—
658C3—41no—
659C3#1059101111yescharacteristic, complemented
660C3#1059166411yescharacteristic, complemented
661C3#1059168811yescharacteristic, complemented
662C2—21no—
663C2—121no—
664C2—121no—
665C2—362no—
666C2#1172294711yescharacteristic
subgroupG/H
centerC2#11722947
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC2×C33#120
prime radicalC3×#64308C2
hypocenterC33#455
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. #11723677C2semidirect
3. #11723373C2semidirect
4. C3×#63831C2semidirect
5. #11722044C2semidirect
6. #11722026C2semidirect
7. C3×#63568C2semidirect
12. C2×C3×#634C22semidirect
20. #88855C22semidirect
21. C3×C4×C3⋊Dic3C22semidirect
22. C32×#208C22semidirect
37. #63831D3semidirect
48. #63568D3semidirect
80. C2×#634C2×D3semidirect
111. C4×C3⋊Dic3C2×D3semidirect
125. C3×#208C2×D3semidirect
126. C3×#208C2×D3semidirect
332. #208D32semidirect
398. C33#455semidirect
592. C32#4825semidirect
593. C32#5202semidirect
594. C32#5242semidirect
659. C3#10591011semidirect
660. C3#10591664semidirect
661. C3#10591688semidirect

Maximal subgroups

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

subgroupindex
C3×#643082
#117236772
#117233732
C3×#638312
#117220442
#117220262
C3×#635682
#105916883
#105916643
#105910113