Finite Group Databasebeta

← #13181644

Subgroups — #13181644

Order 1728.

5444 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11724685C211yesmaximal, characteristic, complemented
2C3×#41×C3⋊D3C211yesmaximal, characteristic, complemented
3C32×#1009C211yesmaximal, characteristic, complemented
4#11724596C212yesmaximal, complemented
5#11724590C212yesmaximal, complemented
6#11724587C216yesmaximal, complemented
7D3×#3830C216yesmaximal, complemented
8#11724579C213yesmaximal, complemented
9C3⋊D3×#235C213yesmaximal, complemented
10#11724563C213yesmaximal, complemented
11C2×#89014C213yesmaximal, complemented
12#10596758—31nomaximal
13#10596696—34nomaximal
14C33×#41C2211yescharacteristic, complemented
15Q8×C3⋊(C3⋊D3)C2212yescomplemented
16#89078C2216yescomplemented
17#89076C2213yescomplemented
18C2×#6308C2213yescomplemented
19C3×#3832C2211yescharacteristic, complemented
20C3×Q8×C3⋊D3C2211yescharacteristic, complemented
21C3×#3830C2213yescomplemented
22C3×D4×C3⋊D3C2213yescomplemented
23C3×#3828C2213yescomplemented
24C2×C3×C4×C3⋊D3C2213yescomplemented
25C32×#239C2212yescomplemented
26C32×D3×D4C2216yescomplemented
27C32×#235C2213yescomplemented
28C2×C32×D12C2213yescomplemented
29C3⋊D3×#67C2216yescomplemented
30D3×#638C22112yescomplemented
31C2×#6290C2216yescomplemented
32#89029C22112yescomplemented
33#89028C2216yescomplemented
34C2×D3×C3⋊Dic3C2216yescomplemented
35#89026C2216yescomplemented
36C3⋊D3×D12C2213yescomplemented
37C4×D3×C3⋊D3C2216yescomplemented
38#89021C2216yescomplemented
39#89020C2216yescomplemented
40#89018C2211yescharacteristic, complemented
41C3⋊D3×Dic6C2213yescomplemented
42#89015C2216yescomplemented
43#89014C2213yescomplemented
44#89014C2216yescomplemented
45#89014C2216yescomplemented
46#89014C2216yescomplemented
47D3×#634C2216yescomplemented
48C32×H(5,𝔽2)—31no—
49#64367—31no—
50#64367—32no—
51#64367—33no—
52#64367—34no—
53#64367—64no—
54#41×C3⋊D3—32no—
55#41×C3⋊D3—33no—
56#41×C3⋊D3D311yescharacteristic, complemented
57#64361—36no—
58#64361—33no—
59C2×#3830—36no—
60C2×#3830—33no—
61C3×#1009D314yescomplemented
62C3×D3×#41—34no—
63#64319—38no—
64#64315—38no—
65#64313—324no—
66D3×#237—324no—
67#64307—312no—
68D3×#235—312no—
69#64298—312no—
70C2×#3795—312no—
71C22×C3×C3⋊D3C2313yes—
72C22×C32×D3C2316yes—
73C2×D3×C3⋊D3C2316yes—
74C33×Q8C2311yescharacteristic
75C33×D4C2313yes—
76C2×C33×C4C2313yes—
77#6312C2316yes—
78C2×C3⋊(C3⋊Dic3)C2316yes—
79C4⋊(C3⋊(C3⋊D3))C2311yescharacteristic
80C4×C3⋊(C3⋊D3)C2316yes—
81#6308C2316yes—
82#6308C2313yes—
83C3×#638C2316yes—
84C2×C3×C3⋊Dic3C2313yes—
85C3×C4⋊(C3⋊D3)C2313yes—
86C3×C4×C3⋊D3C2311yescharacteristic
87C3×C4×C3⋊D3C2313yes—
88C3×C4×C3⋊D3C2313yes—
89C3×C4×C3⋊D3C2313yes—
90C3×#634C2313yes—
91C32×#67C2316yes—
92C32×D12C2313yes—
93C32×D12C2316yes—
94C32×C4×D3C2316yes—
95C32×Dic6C2311yescharacteristic
96#6293C2316yes—
97#6292C2312yes—
98#6291C2316yes—
99#6290C2316yes—
100#6290C23112yes—
101#6289C2312yes—
102Dic3×C3⋊D3C2316yes—
103D3×C3⋊Dic3C23112yes—
104D3×C3⋊Dic3C2316yes—
105#5966—94no—
106#5964—91no—
107C32×#41—24no—
108C32×#41—32no—
109C32×#41—33no—
110C32×#41C2×D314yescomplemented
111C32×#41C2×D311yescharacteristic, complemented
112C2×C32×D4—36no—
113C2×C32×D4—33no—
114C2×#638—36no—
115C2×#638—312no—
116C22×C3⋊Dic3—36no—
117#3832—31no—
118#3832C2×D311yescharacteristic, complemented
119Q8×C3⋊D3—32no—
120Q8×C3⋊D3—38no—
121Q8×C3⋊D3—33no—
122Q8×C3⋊D3—68no—
123Q8×C3⋊D3C2×D311yescharacteristic, complemented
124#3830—36no—
125#3830—324no—
126#3830—312no—
127#3830—36no—
128#3830—36no—
129#3830—33no—
130#3830—36no—
131#3830—36no—
132#3830—36no—
133#3830—624no—
134#3830C2×D313yescomplemented
135D4×C3⋊D3—36no—
136D4×C3⋊D3—36no—
137D4×C3⋊D3—33no—
138D4×C3⋊D3C2×D313yescomplemented
139#3828—33no—
140#3828—312no—
141#3828—36no—
142#3828—36no—
143#3828—612no—
144#3828C2×D313yescomplemented
145C2×C4×C3⋊D3—36no—
146C2×C4×C3⋊D3C2×D313yescomplemented
147C2×#634—33no—
148C2×#634—312no—
149C2×#634—36no—
150C2×#634—612no—
151C3×#239—34no—
152C3×#239C2×D318yescomplemented
153C3×D3×Q8—34no—
154C3×#237—312no—
155C3×D3×D4—312no—
156C3×D3×D4C2×D3124yescomplemented
157C3×#235—312no—
158C3×#235C2×D3112yescomplemented
159C2×C3×D12C2×D3112yescomplemented
160C2×C3×C4×D3—312no—
161D3×#67—348no—
162D3×#67—324no—
163C2×#625—324no—
164#3805—348no—
165#3804—324no—
166C2×D3×Dic3—324no—
167#3802—324no—
168D3×D12—312no—
169C4×D32—324no—
170#3799—34no—
171#3798—324no—
172#3796—324no—
173#3795—312no—
174#3795—324no—
175#3795—324no—
176#3795—324no—
177#3795—324no—
178D3×Dic6—324no—
179D3×Dic6—312no—
180C22×C33C2413yes—
181C2×C3⋊(C3⋊D3)C2412yes—
182C2×C3×C3⋊D3—43no—
183C2×C3×C3⋊D3C2413yes—
184C2×C3×C3⋊D3C2411yescharacteristic
185C2×C32×D3—46no—
186C2×C32×D3C2416yes—
187D3×C3⋊D3—46no—
188C33×C4C2411yescharacteristic
189C33×C4C2413yes—
190C3⋊(C3⋊Dic3)C2416yes—
191C3×C3⋊Dic3C2411yescharacteristic
192C3×C3⋊Dic3C2413yes—
193C32×Dic3C2412yes—
194C3×H(5,𝔽2)—34no—
195C2×C3×#41—91no—
196#1010—912no—
197#1010—98no—
198#1010—94no—
199#1010—91no—
200#1010—93no—
201#1010—184no—
202#1009—93no—
203#1009C3⋊D311yescharacteristic, complemented
204D3×#41—34no—
205D3×#41—912no—
206D3×#41—98no—
207D3×#41—92no—
208D3×#41—96no—
209#1007—92no—
210#1004—912no—
211#1004—924no—
212#1004—96no—
213C2×#237—912no—
214C2×#237—924no—
215C2×#235—93no—
216C2×#235—93no—
217C23×C32—36no—
218C22×C3⋊D3—36no—
219C22×C3⋊D3C22×D313yes—
220C22×C3×D3—312no—
221C22×C3×D3C22×D3124yes—
222C2×D32—324no—
223C32×Q8—24no—
224C32×Q8—31no—
225C32×Q8C22×D314yes—
226C32×Q8C22×D311yescharacteristic
227C32×D4—212no—
228C32×D4—36no—
229C32×D4—33no—
230C32×D4—36no—
231C32×D4C22×D3112yes—
232C32×D4C22×D313yes—
233C2×C32×C4—212no—
234C2×C32×C4—36no—
235C2×C32×C4C22×D3112yes—
236C2×C32×C4C22×D313yes—
237#638—36no—
238#638—312no—
239#638—36no—
240#638—36no—
241#638—324no—
242#638—624no—
243#638C22×D316yes—
244C2×C3⋊Dic3—36no—
245C2×C3⋊Dic3—312no—
246C2×C3⋊Dic3—36no—
247C2×C3⋊Dic3—324no—
248C2×C3⋊Dic3—624no—
249C2×C3⋊Dic3C22×D313yes—
250C4⋊(C3⋊D3)—32no—
251C4⋊(C3⋊D3)—31no—
252C4⋊(C3⋊D3)—34no—
253C4⋊(C3⋊D3)—64no—
254C4⋊(C3⋊D3)C22×D313yes—
255C4×C3⋊D3—36no—
256C4×C3⋊D3—36no—
257C4×C3⋊D3—32no—
258C4×C3⋊D3—324no—
259C4×C3⋊D3—624no—
260C4×C3⋊D3C22×D313yes—
261C4×C3⋊D3C22×D311yescharacteristic
262C4×C3⋊D3C22×D313yes—
263C4×C3⋊D3C22×D313yes—
264#634—33no—
265#634—36no—
266#634—36no—
267#634—312no—
268#634—324no—
269#634—612no—
270#634—624no—
271#634C22×D313yes—
272C3×#67—324no—
273C3×#67C22×D3124yes—
274C2×C3×Dic3—312no—
275C3×D12—312no—
276C3×D12C22×D3112yes—
277C3×D12C22×D3124yes—
278C3×C4×D3—312no—
279C3×C4×D3—34no—
280C3×C4×D3—312no—
281C3×C4×D3—312no—
282C3×C4×D3C22×D3124yes—
283C3×Dic6—312no—
284C3×Dic6C22×D314yes—
285#627—324no—
286#626—324no—
287#626—38no—
288#625—324no—
289#625—348no—
290#624—38no—
291D3×Dic3—348no—
292D3×Dic3—324no—
293D3×Dic3—324no—
294#585—271noSylow
295C2×C33—23no—
296C2×C33C2511yescharacteristic
297C3⋊(C3⋊D3)—22no—
298C3×C3⋊D3—23no—
299C3×C3⋊D3—21no—
300C32×D3—26no—
301C3×#41—24no—
302C3×#41—38no—
303C3×#41—312no—
304C3×#41—91no—
305C3×#41—93no—
306C3×#41—93no—
307C3×#41D3214yescomplemented
308C3×#41C2×C3⋊D311yescharacteristic, complemented
309C2×C3×Q8—91no—
310C2×C3×D4—324no—
311C2×C3×D4—312no—
312C2×C3×D4—93no—
313C22×C3×C4—93no—
314C2×#67—948no—
315C2×#67—924no—
316C2×#67—96no—
317C2×#67—96no—
318C22×Dic3—924no—
319#239—34no—
320#239—94no—
321#239—96no—
322#239C2×C3⋊D312yescomplemented
323D3×Q8—34no—
324D3×Q8—912no—
325D3×Q8—98no—
326D3×Q8—92no—
327D3×Q8—96no—
328D3×Q8—188no—
329#237—312no—
330#237—924no—
331#237—912no—
332#237—924no—
333#237—924no—
334#237—924no—
335#237—924no—
336#237—948no—
337#237—924no—
338#237—96no—
339#237—912no—
340#237—96no—
341#237—1824no—
342D3×D4—312no—
343D3×D4—912no—
344D3×D4—924no—
345D3×D4—924no—
346D3×D4—912no—
347D3×D4—96no—
348D3×D4C2×C3⋊D316yescomplemented
349#235—312no—
350#235—924no—
351#235—924no—
352#235—912no—
353#235—93no—
354#235—96no—
355#235—93no—
356#235—96no—
357#235—96no—
358#235—96no—
359#235—96no—
360#235—91no—
361#235—1812no—
362#235C2×C3⋊D313yescomplemented
363C2×D12—93no—
364C2×D12C2×C3⋊D313yescomplemented
365C2×C4×D3—312no—
366C2×C4×D3—924no—
367C2×C4×D3—96no—
368C2×C4×D3—96no—
369C2×Dic6—924no—
370C2×Dic6—912no—
371C2×Dic6—93no—
372C2×Dic6—93no—
373C2×Dic6—1812no—
374C22×C32—212no—
375C22×C32—36no—
376C22×C32—126no—
377C22×C32C23×D3112yes—
378C22×C32C23×D313yes—
379C2×C3⋊D3—32no—
380C2×C3⋊D3—38no—
381C2×C3⋊D3—43no—
382C2×C3⋊D3—68no—
383C2×C3⋊D3—126no—
384C2×C3⋊D3C23×D313yes—
385C2×C3⋊D3C23×D311yescharacteristic
386C2×C3×D3—312no—
387C2×C3×D3—34no—
388C2×C3×D3—424no—
389C2×C3×D3—1212no—
390C2×C3×D3C23×D3124yes—
391D32—1224no—
392C32×C4—24no—
393C32×C4—212no—
394C32×C4—32no—
395C32×C4C23×D3112yes—
396C32×C4C23×D314yes—
397C32×C4C23×D311yescharacteristic
398C32×C4C23×D313yes—
399C3⋊Dic3—36no—
400C3⋊Dic3—324no—
401C3⋊Dic3—624no—
402C3⋊Dic3C23×D313yes—
403C3⋊Dic3C23×D311yescharacteristic
404C3×Dic3—34no—
405C3×Dic3—312no—
406C3×Dic3C23×D318yes—
4072−1+4—272no—
4082−1+4—273no—
4092−1+4—271no—
410H(5,𝔽2)—31no—
411H(5,𝔽2)—276no—
412H(5,𝔽2)—273no—
413C2×#41—91no—
414C2×#41—273no—
415C2×#41—273no—
416C2×#41—276no—
417C2×#41—272no—
418C33#58511yescharacteristic, complemented, Sylow
419C23×C3—324no—
420C23×C3—93no—
421C22×D3—312no—
422C22×D3—924no—
423C22×D3—96no—
424C22×D3C22×C3⋊D316yes—
425C3×Q8—24no—
426C3×Q8—34no—
427C3×Q8—93no—
428C3×Q8C2×D3214yes—
429C3×Q8C22×C3⋊D311yescharacteristic
430C3×D4—212no—
431C3×D4—324no—
432C3×D4—324no—
433C3×D4—312no—
434C3×D4—96no—
435C3×D4—93no—
436C3×D4C2×D32112yes—
437C3×D4C22×C3⋊D313yes—
438C2×C3×C4—212no—
439C2×C3×C4—324no—
440C2×C3×C4—93no—
441C2×C3×C4—93no—
442C2×C3×C4—93no—
443C2×C3×C4—91no—
444C2×C3×C4—93no—
445C2×C3×C4C2×D32112yes—
446C2×C3×C4C22×C3⋊D313yes—
447#67—324no—
448#67—924no—
449#67—924no—
450#67—948no—
451#67—924no—
452#67—96no—
453#67—96no—
454#67—912no—
455#67—92no—
456#67—1824no—
457#67C22×C3⋊D316yes—
458C2×Dic3—312no—
459C2×Dic3—948no—
460C2×Dic3—924no—
461C2×Dic3—924no—
462C2×Dic3—96no—
463C2×Dic3—96no—
464C2×Dic3—1824no—
465D12—312no—
466D12—98no—
467D12—96no—
468D12—94no—
469D12—91no—
470D12—184no—
471D12C22×C3⋊D313yes—
472D12C22×C3⋊D316yes—
473C4×D3—34no—
474C4×D3—312no—
475C4×D3—312no—
476C4×D3—312no—
477C4×D3—98no—
478C4×D3—924no—
479C4×D3—96no—
480C4×D3—912no—
481C4×D3—924no—
482C4×D3—96no—
483C4×D3—92no—
484C4×D3—1824no—
485C4×D3C22×C3⋊D316yes—
486Dic6—312no—
487Dic6—924no—
488Dic6—924no—
489Dic6—912no—
490Dic6—93no—
491Dic6—96no—
492Dic6—96no—
493Dic6—1812no—
494Dic6—1824no—
495Dic6C22×C3⋊D311yescharacteristic
496C2×C32—23no—
497C2×C32—24no—
498C2×C32—212no—
499C2×C32—412no—
500C2×C32—66no—
501C2×C32C24×D314yes—
502C2×C32C24×D311yescharacteristic
503C3⋊D3—21no—
504C3⋊D3—23no—
505C3⋊D3—62no—
506C3⋊D3—68no—
507C3⋊D3—128no—
508C3×D3—224no—
509C3×D3—612no—
510C3×D3—64no—
511#41—32no—
512#41—33no—
513#41—91no—
514#41—93no—
515#41—93no—
516#41—276no—
517#41—276no—
518#41—276no—
519#41—273no—
520#41—276no—
521#41—276no—
522#41—276no—
523#41—271no—
524#41—2712no—
525#41—276no—
526#41—276no—
527#41—273no—
528#41D3×C3⋊D311yescharacteristic, complemented
529C2×Q8—91no—
530C2×Q8—276no—
531C2×Q8—273no—
532C2×Q8—272no—
533C2×Q8—273no—
534C2×D4—36no—
535C2×D4—33no—
536C2×D4—93no—
537C2×D4—273no—
538C2×D4—276no—
539C2×D4—276no—
540C2×D4—2712no—
541C2×D4—276no—
542C22×C4—93no—
543C22×C4—276no—
544C22×C4—276no—
545C22×C3—212no—
546C22×C3—324no—
547C22×C3—93no—
548C22×C3—91no—
549C22×C3—1224no—
550C22×C3—363no—
551C22×C3C22×D32112yes—
552C22×C3C23×C3⋊D313yes—
553C2×D3—34no—
554C2×D3—312no—
555C2×D3—46no—
556C2×D3—98no—
557C2×D3—92no—
558C2×D3—1212no—
559C2×D3—188no—
560C2×D3—366no—
561C2×D3—3624no—
562C2×D3C23×C3⋊D316yes—
563C3×C4—24no—
564C3×C4—212no—
565C3×C4—38no—
566C3×C4—93no—
567C3×C4—91no—
568C3×C4C22×D32112yes—
569C3×C4C22×D3214yes—
570C3×C4C23×C3⋊D311yescharacteristic
571C3×C4C23×C3⋊D313yes—
572Dic3—312no—
573Dic3—34no—
574Dic3—924no—
575Dic3—96no—
576Dic3—1824no—
577Dic3C23×C3⋊D312yes—
578C32—24no—
579C32#596411yescharacteristic, complemented
580C32#596614yescomplemented
581C23—36no—
582C23—93no—
583C23—276no—
584Q8—31no—
585Q8—93no—
586Q8—276no—
587Q8—276no—
588Q8—273no—
589Q8C2×D3×C3⋊D311yescharacteristic
590D4—36no—
591D4—36no—
592D4—33no—
593D4—96no—
594D4—93no—
595D4—272no—
596D4—276no—
597D4—276no—
598D4—2712no—
599D4—276no—
600D4—271no—
601D4C2×D3×C3⋊D313yes—
602C2×C4—36no—
603C2×C4—93no—
604C2×C4—91no—
605C2×C4—93no—
606C2×C4—93no—
607C2×C4—93no—
608C2×C4—272no—
609C2×C4—276no—
610C2×C4—276no—
611C2×C4—2712no—
612C2×C4—276no—
613C2×C4—276no—
614C2×C4C2×D3×C3⋊D313yes—
615C2×C3—212no—
616C2×C3—23no—
617C2×C3—24no—
618C2×C3—412no—
619C2×C3—624no—
620C2×C3—183no—
621C2×C3—181no—
622C2×C3C23×D3214yes—
623C2×C3C24×C3⋊D311yescharacteristic
624D3—26no—
625D3—64no—
626D3—612no—
627D3—188no—
628D3—182no—
629D3—368no—
630C22—36no—
631C22—91no—
632C22—93no—
633C22—126no—
634C22—272no—
635C22—363no—
636C22—1086no—
637C22C22×D3×C3⋊D313yes—
638C4—32no—
639C4—91no—
640C4—93no—
641C4—276no—
642C4C22×D3×C3⋊D313yes—
643C4C22×D3×C3⋊D311yescharacteristic
644C3—24no—
645C3#1059669614yescomplemented
646C3#1059675811yescharacteristic, complemented
647C2—23no—
648C2—66no—
649C2—181no—
650C2—183no—
651C2—542no—
652C2C23×D3×C3⋊D311yescharacteristic
subgroupG/H
center, FrattiniC2C23×D3×C3⋊D3
commutator, socle, cosocleC2×C33C25
FittingC33×#41C22
hypercenter#41D3×C3⋊D3
hypocenterC33#585
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ #41

Complements

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

NH ≅ G/Ntype
1. #11724685C2semidirect
2. C3×#41×C3⋊D3C2semidirect
3. C32×#1009C2semidirect
4. #11724596C2semidirect
5. #11724590C2semidirect
6. #11724587C2semidirect
7. D3×#3830C2semidirect
8. #11724579C2semidirect
9. C3⋊D3×#235C2semidirect
10. #11724563C2semidirect
11. C2×#89014C2semidirect
14. C33×#41C22semidirect
15. Q8×C3⋊(C3⋊D3)C22semidirect
16. #89078C22semidirect
17. #89076C22semidirect
18. C2×#6308C22semidirect
19. C3×#3832C22semidirect
20. C3×Q8×C3⋊D3C22semidirect
21. C3×#3830C22semidirect
22. C3×D4×C3⋊D3C22semidirect
23. C3×#3828C22semidirect
24. C2×C3×C4×C3⋊D3C22semidirect
25. C32×#239C22semidirect
26. C32×D3×D4C22semidirect
27. C32×#235C22semidirect
28. C2×C32×D12C22semidirect
29. C3⋊D3×#67C22semidirect
30. D3×#638C22semidirect
31. C2×#6290C22semidirect
32. #89029C22semidirect
33. #89028C22semidirect
34. C2×D3×C3⋊Dic3C22semidirect
35. #89026C22semidirect
36. C3⋊D3×D12C22semidirect
37. C4×D3×C3⋊D3C22semidirect
38. #89021C22semidirect
39. #89020C22semidirect
40. #89018C22semidirect
41. C3⋊D3×Dic6C22semidirect
42. #89015C22semidirect
43. #89014C22semidirect
44. #89014C22semidirect
45. #89014C22semidirect
46. #89014C22semidirect
47. D3×#634C22semidirect
56. #41×C3⋊D3D3semidirect
61. C3×#1009D3semidirect
110. C32×#41C2×D3semidirect
111. C32×#41C2×D3semidirect
118. #3832C2×D3semidirect
123. Q8×C3⋊D3C2×D3semidirect
134. #3830C2×D3semidirect
138. D4×C3⋊D3C2×D3semidirect
144. #3828C2×D3semidirect
146. C2×C4×C3⋊D3C2×D3semidirect
152. C3×#239C2×D3semidirect
156. C3×D3×D4C2×D3semidirect
158. C3×#235C2×D3semidirect
159. C2×C3×D12C2×D3semidirect
203. #1009C3⋊D3semidirect
307. C3×#41D32semidirect
308. C3×#41C2×C3⋊D3semidirect
322. #239C2×C3⋊D3semidirect
348. D3×D4C2×C3⋊D3semidirect
362. #235C2×C3⋊D3semidirect
364. C2×D12C2×C3⋊D3semidirect
418. C33#585semidirect
528. #41D3×C3⋊D3semidirect
579. C32#5964semidirect
580. C32#5966semidirect
645. C3#10596696semidirect
646. C3#10596758semidirect

Maximal subgroups

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

subgroupindex
#117246852
C3×#41×C3⋊D32
C32×#10092
#117245962
#117245902
#117245872
D3×#38302
#117245792
C3⋊D3×#2352
#117245632
C2×#890142
#105967583
#105966963