Finite Group Databasebeta

← #13155339

Subgroups — #13155339

Order 1728.

772 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3796C211yesmaximal, characteristic
2#11723656C211yesmaximal, characteristic, complemented
3C3⋊D3×#208C211yesmaximal, characteristic, complemented
4#11722028C211yesmaximal, characteristic, complemented
5#11722025C211yesmaximal, characteristic, complemented
6C3×#63560C211yesmaximal, characteristic, complemented
7C3×#63558C211yesmaximal, characteristic, complemented
8#10591637—31nomaximal
9#10591501—31nomaximal
10#10591010—31nomaximal
11C2×C3×C4×C3⋊D3C2211yescharacteristic
12C2×C32×D12C2211yescharacteristic
13C2×C32×Dic6C2211yescharacteristic
14C2×C3×#626C412yes—
15C2×C3×D3×Dic3C412yes—
16C3×#3796—21no—
17C3×#3796—21no—
18C3×#3796—21no—
19C3×#3796—21no—
20#88855C2211yescharacteristic
21C3×#3748C2211yescharacteristic
22C3×#3748C2211yescharacteristic
23C32×#208C2211yescharacteristic
24#88809—21no—
25#88808—21no—
26#88787—21no—
27C3⋊D3×C3⋊C8—21no—
28C2×C3×#239—31no—
29C2×C3×#237—31no—
30C2×#3796D311yescharacteristic
31M16×C3⋊D3—31no—
32#63818—31no—
33D3×#208—31no—
34D3×#208—31no—
35D3×#208—31no—
36D3×#208—31no—
37D3×#208—61no—
38#63663—31no—
39#63660—31no—
40C3×#836—31no—
41C3×#833—31no—
42C3×#824—31no—
43C3×#823—31no—
44#63560—31no—
45#63560—31no—
46#63560—61no—
47#63560D311yescharacteristic, complemented
48#63558—31no—
49#63558—31no—
50#63558—61no—
51#63558D311yescharacteristic, complemented
52C22×C3×C3⋊D3C2×C411yescharacteristic
53C22×C32×D3C2×C412yes—
54C2×C33×C4C2311yescharacteristic
55C2×C3×C3⋊Dic3C2×C411yescharacteristic
56C3×C4×C3⋊D3D411yescharacteristic
57C3×C4×C3⋊D3D411yescharacteristic
58C3×C4×C3⋊D3D411yescharacteristic
59C3×C4×C3⋊D3D411yescharacteristic
60C2×C32×Dic3C2×C412yes—
61C32×D12—21no—
62C32×D12—21no—
63C32×Dic6—21no—
64C32×Dic6—21no—
65C3×#626—22no—
66C3×#626—22no—
67C3×D3×Dic3—22no—
68C3×D3×Dic3—22no—
69C3⋊(C3⋊(C3⋊C8))—21no—
70C3×C3⋊(C3⋊C8)—21no—
71C3×C3⋊(C3⋊C8)—21no—
72C32×C3⋊C8—21no—
73#5237—91no—
74#4751—91no—
75#4745—91no—
76C2×C32×Q8—31no—
77C2×C32×D4—31no—
78C2×C4×C3⋊D3C2×D311yescharacteristic
79C2×C3×#67—32no—
80C22×C3×Dic3—32no—
81C3×#239—61no—
82C3×#239—61no—
83C3×#239—61no—
84C3×#239—61no—
85C3×#237—61no—
86C3×#237—61no—
87C3×#237—61no—
88C3×#237—61no—
89C2×C3×D12—21no—
90C2×C3×D12—32no—
91C2×C3×D12C2×D311yescharacteristic
92C2×C3×D12C2×D311yescharacteristic
93C2×C3×C4×D3—31no—
94C2×C3×C4×D3—31no—
95C2×C3×C4×D3—32no—
96C2×C3×C4×D3—61no—
97C2×C3×Dic6—21no—
98C2×C3×Dic6C2×D311yescharacteristic
99C2×C3×Dic6C2×D311yescharacteristic
100C2×#626Dic312yes—
101C2×D3×Dic3Dic312yes—
102#3796—21no—
103#3796—21no—
104#3796—21no—
105#3796—21no—
106C32×M16—31no—
107#3748—31no—
108#3748—31no—
109#3748—31no—
110#3748—61no—
111#3748—61no—
112#3748—61no—
113#3748—121no—
114#3748C2×D311yescharacteristic
115#3748C2×D311yescharacteristic
116#3743—61no—
117C8×C3⋊D3—61no—
118C3×#208—21no—
119C3×#208—31no—
120C3×#208—31no—
121C3×#208—31no—
122C3×#208—31no—
123C3×#208—61no—
124C3×#208—61no—
125C3×#208C2×D311yescharacteristic
126C3×#208C2×D311yescharacteristic
127#3712—61no—
128#3711—61no—
129#3711—61no—
130#3711—62no—
131#3711—61no—
132#3711—61no—
133#3710—61no—
134D3×C3⋊C8—61no—
135D3×C3⋊C8—61no—
136D3×C3⋊C8—62no—
137D3×C3⋊C8—61no—
138D3×C3⋊C8—61no—
139C22×C33C22×C411yescharacteristic
140C2×C3×C3⋊D3—41no—
141C2×C3×C3⋊D3C22⋊C411yescharacteristic
142C2×C3×C3⋊D3C22⋊C411yescharacteristic
143C2×C32×D3—22no—
144C2×C32×D3—42no—
145C33×C4C2×D411yescharacteristic
146C33×C4C2×D411yescharacteristic
147C3×C3⋊Dic3C22⋊C411yescharacteristic
148C3×C3⋊Dic3C22⋊C411yescharacteristic
149C32×Dic3—22no—
150C2×C3×#41—91no—
151C2×#239—31no—
152C2×#237—31no—
153C2×#208—91no—
154C2×#208—91no—
155D3×M16—91no—
156D3×M16—91no—
157D3×M16—91no—
158D3×M16—91no—
159D3×M16—181no—
160C3×#101—91no—
161C3×#100—91no—
162#836—31no—
163#836—91no—
164#836—91no—
165#836—181no—
166#833—31no—
167#833—91no—
168#833—91no—
169#833—181no—
170#824—31no—
171#824—91no—
172#823—31no—
173#823—91no—
174C23×C32—32no—
175C22×C3⋊D3C2×Dic311yescharacteristic
176C22×C3×D3—22no—
177C22×C3×D3—31no—
178C22×C3×D3—31no—
179C22×C3×D3—61no—
180C22×C3×D3C4×D312yes—
181C22×C3×D3C2×Dic312yes—
182C32×Q8—61no—
183C32×Q8—61no—
184C32×D4—61no—
185C32×D4—61no—
186C2×C32×C4—21no—
187C2×C32×C4—21no—
188C2×C32×C4—21no—
189C2×C32×C4—32no—
190C2×C32×C4—41no—
191C2×C32×C4C22×D311yescharacteristic
192C2×C32×C4C22×D311yescharacteristic
193C2×C32×C4C22×D311yescharacteristic
194C2×C3⋊Dic3C2×Dic311yescharacteristic
195C4×C3⋊D3#6711yescharacteristic
196C4×C3⋊D3#6711yescharacteristic
197C4×C3⋊D3#6711yescharacteristic
198C4×C3⋊D3#6711yescharacteristic
199C3×#67—62no—
200C3×#67—62no—
201C2×C3×Dic3—22no—
202C2×C3×Dic3—31no—
203C2×C3×Dic3—31no—
204C2×C3×Dic3—61no—
205C2×C3×Dic3—62no—
206C2×C3×Dic3—62no—
207C2×C3×Dic3C4×D312yes—
208C2×C3×Dic3C2×Dic312yes—
209C3×D12—21no—
210C3×D12—21no—
211C3×D12—21no—
212C3×D12—21no—
213C3×D12—41no—
214C3×D12—41no—
215C3×D12—62no—
216C3×D12—62no—
217C3×C4×D3—31no—
218C3×C4×D3—31no—
219C3×C4×D3—31no—
220C3×C4×D3—31no—
221C3×C4×D3—31no—
222C3×C4×D3—31no—
223C3×C4×D3—31no—
224C3×C4×D3—31no—
225C3×C4×D3—61no—
226C3×C4×D3—61no—
227C3×C4×D3—61no—
228C3×C4×D3—61no—
229C3×C4×D3—62no—
230C3×C4×D3—62no—
231C3×Dic6—21no—
232C3×Dic6—21no—
233C3×Dic6—21no—
234C3×Dic6—21no—
235C3×Dic6—41no—
236C3×Dic6—41no—
237#626—22no—
238#626—22no—
239D3×Dic3—22no—
240D3×Dic3—22no—
241C32×C8—61no—
242C3⋊(C3⋊C8)—21no—
243C3⋊(C3⋊C8)—21no—
244C3⋊(C3⋊C8)—61no—
245C3⋊(C3⋊C8)—62no—
246C3⋊(C3⋊C8)—61no—
247C3⋊(C3⋊C8)—61no—
248C3⋊(C3⋊C8)—121no—
249C3⋊(C3⋊C8)—121no—
250C3⋊(C3⋊C8)—122no—
251C3×C3⋊C8—22no—
252C3×C3⋊C8—21no—
253C3×C3⋊C8—21no—
254C3×C3⋊C8—61no—
255C3×C3⋊C8—61no—
256C3×C3⋊C8—61no—
257C3×C3⋊C8—61no—
258C3×C3⋊C8—121no—
259C3×C3⋊C8—121no—
260#413—271noSylow
261C2×C33—21no—
262C2×C33C2×C22⋊C411yescharacteristic
263C3×C3⋊D3—21no—
264C3×C3⋊D3—21no—
265C32×D3—42no—
266C3×#41—181no—
267C3×#41—181no—
268C3×#41—181no—
269C3×#41—181no—
270C2×C3×Q8—31no—
271C2×C3×Q8—31no—
272C2×C3×Q8—61no—
273C2×C3×D4—31no—
274C2×C3×D4—31no—
275C2×C3×D4—61no—
276C2×C3×D4—92no—
277C22×C3×C4—92no—
278C22×C3×C4—91no—
279C2×#67—32no—
280C22×Dic3—32no—
281#239—61no—
282#239—61no—
283#239—61no—
284#239—61no—
285#237—61no—
286#237—61no—
287#237—61no—
288#237—61no—
289C2×D12—32no—
290C2×D12D3211yescharacteristic
291C2×C4×D3—31no—
292C2×C4×D3—31no—
293C2×C4×D3—32no—
294C2×C4×D3—61no—
295C2×Dic6D3211yescharacteristic
296C3×M16—31no—
297C3×M16—31no—
298C3×M16—61no—
299C3×M16—91no—
300C3×M16—91no—
301#208—31no—
302#208—31no—
303#208—31no—
304#208—31no—
305#208—61no—
306#208—61no—
307#208—91no—
308#208—91no—
309#208—91no—
310#208—181no—
311#208—181no—
312#208—181no—
313#208—181no—
314#208—181no—
315#208—361no—
316#208D3211yescharacteristic
317C2×C3⋊C8—181no—
318C2×C3⋊C8—181no—
319ΓL(1,𝔽25)—181no—
320ΓL(1,𝔽25)—181no—
321ΓL(1,𝔽25)—181no—
322ΓL(1,𝔽25)—181no—
323ΓL(1,𝔽25)—182no—
324D3×C8—181no—
325D3×C8—181no—
326D3×C8—181no—
327D3×C8—181no—
328D3×C8—182no—
329C22×C32—21no—
330C22×C32—21no—
331C22×C32—21no—
332C22×C32—41no—
333C22×C32—62no—
334C22×C32—122no—
335C22×C32C2×C4×D311yescharacteristic
336C22×C32C2×C4×D311yescharacteristic
337C22×C32C22×Dic311yescharacteristic
338C2×C3⋊D3—41no—
339C2×C3⋊D3#21711yescharacteristic
340C2×C3⋊D3#21711yescharacteristic
341C2×C3×D3—22no—
342C2×C3×D3—22no—
343C2×C3×D3—31no—
344C2×C3×D3—31no—
345C2×C3×D3—31no—
346C2×C3×D3—31no—
347C2×C3×D3—42no—
348C2×C3×D3—42no—
349C2×C3×D3—42no—
350C2×C3×D3—61no—
351C2×C3×D3—61no—
352C2×C3×D3—82no—
353C2×C3×D3—121no—
354C2×C3×D3—121no—
355C2×C3×D3—122no—
356C32×C4—21no—
357C32×C4—21no—
358C32×C4—21no—
359C32×C4—21no—
360C32×C4—21no—
361C32×C4—21no—
362C32×C4—41no—
363C32×C4—41no—
364C32×C4—62no—
365C32×C4D3×D411yescharacteristic
366C32×C4D3×D411yescharacteristic
367C32×C4D3×D411yescharacteristic
368C32×C4D3×D411yescharacteristic
369C32×C4C2×#6711yescharacteristic
370C32×C4C2×#6711yescharacteristic
371C3⋊Dic3#21711yescharacteristic
372C3⋊Dic3#21711yescharacteristic
373C3×Dic3—22no—
374C3×Dic3—22no—
375C3×Dic3—31no—
376C3×Dic3—31no—
377C3×Dic3—31no—
378C3×Dic3—31no—
379C3×Dic3—42no—
380C3×Dic3—61no—
381C3×Dic3—61no—
382C2×#41—91no—
383C2×M16—271no—
384C2×M16—271no—
385#101—91no—
386#101—271no—
387#100—91no—
388#100—271no—
389C33#41311yescharacteristic, complemented, Sylow
390C23×C3—32no—
391C23×C3—32no—
392C23×C3—62no—
393C23×C3—91no—
394C22×D3—31no—
395C22×D3—31no—
396C22×D3—61no—
397C22×D3D3×Dic312yes—
398C3×Q8—61no—
399C3×Q8—61no—
400C3×Q8—61no—
401C3×Q8—61no—
402C3×Q8—121no—
403C3×Q8—121no—
404C3×D4—61no—
405C3×D4—61no—
406C3×D4—61no—
407C3×D4—61no—
408C3×D4—121no—
409C3×D4—121no—
410C3×D4—182no—
411C3×D4—182no—
412C2×C3×C4—21no—
413C2×C3×C4—21no—
414C2×C3×C4—21no—
415C2×C3×C4—32no—
416C2×C3×C4—32no—
417C2×C3×C4—41no—
418C2×C3×C4—62no—
419C2×C3×C4—91no—
420C2×C3×C4—91no—
421C2×C3×C4—91no—
422C2×C3×C4—91no—
423C2×C3×C4—91no—
424C2×C3×C4—182no—
425C2×C3×C4—182no—
426C2×C3×C4C2×D3211yescharacteristic
427C2×C3×C4C2×D3211yescharacteristic
428C2×C3×C4C2×D3211yescharacteristic
429#67—62no—
430#67—62no—
431C2×Dic3—31no—
432C2×Dic3—31no—
433C2×Dic3—61no—
434C2×Dic3—62no—
435C2×Dic3—62no—
436C2×Dic3D3×Dic312yes—
437D12—21no—
438D12—21no—
439D12—62no—
440D12—62no—
441C4×D3—31no—
442C4×D3—31no—
443C4×D3—31no—
444C4×D3—31no—
445C4×D3—31no—
446C4×D3—31no—
447C4×D3—31no—
448C4×D3—31no—
449C4×D3—61no—
450C4×D3—61no—
451C4×D3—61no—
452C4×D3—61no—
453C4×D3—62no—
454C4×D3—62no—
455Dic6—21no—
456Dic6—21no—
457C3×C8—62no—
458C3×C8—61no—
459C3×C8—61no—
460C3×C8—181no—
461C3×C8—181no—
462C3⋊C8—21no—
463C3⋊C8—61no—
464C3⋊C8—61no—
465C3⋊C8—61no—
466C3⋊C8—61no—
467C3⋊C8—121no—
468C3⋊C8—121no—
469C3⋊C8—182no—
470C3⋊C8—181no—
471C3⋊C8—181no—
472C3⋊C8—181no—
473C3⋊C8—361no—
474C3⋊C8—361no—
475C3⋊C8—362no—
476C2×C32—21no—
477C2×C32—21no—
478C2×C32—21no—
479C2×C32—21no—
480C2×C32—21no—
481C2×C32—21no—
482C2×C32—41no—
483C2×C32—41no—
484C2×C32—41no—
485C2×C32—41no—
486C2×C32—81no—
487C2×C32—122no—
488C2×C32D3×C22⋊C411yescharacteristic
489C2×C32D3×C22⋊C411yescharacteristic
490C2×C32C2×#21711yescharacteristic
491C3⋊D3—21no—
492C3⋊D3—21no—
493C3×D3—42no—
494C3×D3—42no—
495C3×D3—62no—
496C3×D3—61no—
497C3×D3—61no—
498C3×D3—61no—
499C3×D3—61no—
500C3×D3—82no—
501C3×D3—121no—
502#41—181no—
503#41—181no—
504#41—181no—
505#41—181no—
506C2×Q8—31no—
507C2×D4—31no—
508C2×D4—92no—
509C22×C4—92no—
510C22×C4—91no—
511M16—31no—
512M16—91no—
513M16—91no—
514M16—271no—
515M16—541no—
516M16—541no—
517C2×C8—541no—
518C2×C8—541no—
519C22×C3—21no—
520C22×C3—21no—
521C22×C3—21no—
522C22×C3—41no—
523C22×C3—62no—
524C22×C3—62no—
525C22×C3—91no—
526C22×C3—91no—
527C22×C3—122no—
528C22×C3—122no—
529C22×C3—122no—
530C22×C3—242no—
531C22×C3—361no—
532C22×C3C4×D3211yescharacteristic
533C22×C3C2×D3×Dic311yescharacteristic
534C22×C3C2×D3×Dic311yescharacteristic
535C2×D3—22no—
536C2×D3—31no—
537C2×D3—31no—
538C2×D3—31no—
539C2×D3—31no—
540C2×D3—42no—
541C2×D3—61no—
542C2×D3—61no—
543C2×D3—121no—
544C2×D3—121no—
545C2×D3—122no—
546C3×C4—21no—
547C3×C4—21no—
548C3×C4—21no—
549C3×C4—21no—
550C3×C4—21no—
551C3×C4—21no—
552C3×C4—41no—
553C3×C4—41no—
554C3×C4—62no—
555C3×C4—62no—
556C3×C4—91no—
557C3×C4—91no—
558C3×C4—122no—
559C3×C4#380211yescharacteristic
560C3×C4D3×#6711yescharacteristic
561C3×C4D3×#6711yescharacteristic
562C3×C4D3×#6711yescharacteristic
563C3×C4D3×#6711yescharacteristic
564C3×C4#381111yescharacteristic
565Dic3—22no—
566Dic3—31no—
567Dic3—31no—
568Dic3—31no—
569Dic3—31no—
570Dic3—61no—
571Dic3—61no—
572C32—21no—
573C32—21no—
574C32—21no—
575C32—41no—
576C32#474511yescharacteristic, complemented
577C32#475111yescharacteristic, complemented
578C32#523711yescharacteristic, complemented
579C23—32no—
580C23—91no—
581Q8—61no—
582Q8—61no—
583D4—61no—
584D4—61no—
585D4—182no—
586D4—182no—
587C2×C4—32no—
588C2×C4—91no—
589C2×C4—91no—
590C2×C4—91no—
591C2×C4—91no—
592C2×C4—91no—
593C2×C4—182no—
594C2×C4—182no—
595C2×C4D3311yescharacteristic
596C8—61no—
597C8—181no—
598C8—181no—
599C8—541no—
600C2×C3—21no—
601C2×C3—21no—
602C2×C3—21no—
603C2×C3—21no—
604C2×C3—21no—
605C2×C3—21no—
606C2×C3—41no—
607C2×C3—41no—
608C2×C3—41no—
609C2×C3—41no—
610C2×C3—81no—
611C2×C3—122no—
612C2×C3—122no—
613C2×C3—181no—
614C2×C3—181no—
615C2×C3—242no—
616C2×C3#6392111yescharacteristic
617C2×C3D3×#21711yescharacteristic
618C2×C3D3×#21711yescharacteristic
619D3—42no—
620D3—62no—
621D3—61no—
622D3—61no—
623D3—61no—
624D3—61no—
625D3—121no—
626C22—62no—
627C22—91no—
628C22—91no—
629C22—122no—
630C22—361no—
631C22D32×Dic311yescharacteristic
632C4—62no—
633C4—91no—
634C4—91no—
635C4#8895211yescharacteristic
636C4#8895311yescharacteristic
637C3—21no—
638C3—21no—
639C3—21no—
640C3—41no—
641C3#1059101011yescharacteristic, complemented
642C3#1059150111yescharacteristic, complemented
643C3#1059163711yescharacteristic, complemented
644C2—21no—
645C2—122no—
646C2—181no—
647C2—181no—
648C2#1172290211yescharacteristic
subgroupG/H
centerC2#11722902
commutatorC22×C33C22×C4
Frattini, hypercenterC2×C4D33
Fitting, cosocleC2×C33×C4C23
socleC2×C33C2×C22⋊C4
prime radicalC2×C3×#3796C2
hypocenterC33#413
LCSG ▷ C22×C33 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. #11723656C2semidirect
3. C3⋊D3×#208C2semidirect
4. #11722028C2semidirect
5. #11722025C2semidirect
6. C3×#63560C2semidirect
7. C3×#63558C2semidirect
47. #63560D3semidirect
51. #63558D3semidirect
389. C33#413semidirect
576. C32#4745semidirect
577. C32#4751semidirect
578. C32#5237semidirect
641. C3#10591010semidirect
642. C3#10591501semidirect
643. C3#10591637semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37962
#117236562
C3⋊D3×#2082
#117220282
#117220252
C3×#635602
C3×#635582
#105916373
#105915013
#105910103