Finite Group Databasebeta

← #13150644

Subgroups — #13150644

Order 1728.

1081 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88740C211yesmaximal, characteristic, complemented
2C2×#3828⋊C3C211yesmaximal, characteristic, complemented
3C2×#3748⋊C3C211yesmaximal, characteristic, complemented
4#11721074C212yesmaximal, complemented
5#11721073C214yesmaximal, complemented
6#11721071C212yesmaximal, complemented
7#11721069C212yesmaximal, complemented
8#11721068C212yesmaximal, complemented
9#10594795—31nomaximal
10#10594786—31nomaximal
11C22×C4×H(3,𝔽3)C2211yescharacteristic
12C2×#6235—21no—
13#88740—21no—
14#88740—21no—
15#88740C2212yes—
16#88740C2212yescomplemented
17C2×C4⋊#272C2211yescharacteristic, complemented
18C2×C4×#272—21no—
19C2×#6229C2211yescharacteristic, complemented
20C2×#638⋊C3—21no—
21#3828⋊C3—21no—
22#3828⋊C3—21no—
23#3828⋊C3C2212yes—
24#3828⋊C3C2212yescomplemented
25C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic, complemented
26C2×C4×(C3⋊D3)⋊C3—21no—
27C2×#634⋊C3C2211yescharacteristic, complemented
28#3748⋊C3C2212yescomplemented
29#3748⋊C3C2212yescomplemented
30C2×(C3⋊(C3⋊C8))⋊C3C2212yescomplemented
31#88439C2214yescomplemented
32#88438C2214yescomplemented
33#88437C2214yescomplemented
34#88436C2214yescomplemented
35C2×#3828D311yescharacteristic, complemented
36C2×C3×#235—31no—
37C2×C3×#235—31no—
38C2×C3×#235—31no—
39C2×C3×#235—61no—
40C2×#3748D311yescharacteristic, complemented
41C2×C3×#208—31no—
42#63833—32no—
43#63831—32no—
44#63830—32no—
45#63829—34no—
46#63827—32no—
47#63827—32no—
48#63825—32no—
49#63823—32no—
50#63823—32no—
51#63822—34no—
52C23×H(3,𝔽3)D411yescharacteristic
53C22×#272—21no—
54C22×(C3⋊D3)⋊C3—21no—
55C2×C4×H(3,𝔽3)D411yescharacteristic
56C2×C4×H(3,𝔽3)D411yescharacteristic
57C2×C4×H(3,𝔽3)D411yescharacteristic
58C2×C4×H(3,𝔽3)C2311yescharacteristic
59C2×C4×H(3,𝔽3)C2311yescharacteristic
60C2×C4×H(3,𝔽3)C2311yescharacteristic
61#6235—22no—
62#6235—22no—
63C2×#1084—21no—
64C4⋊#272—21no—
65C4⋊#272C2312yes—
66C4×#272—22no—
67C4×#272—22no—
68#6229—21no—
69#6229C2312yes—
70#638⋊C3—22no—
71#638⋊C3—22no—
72C2×(C3⋊Dic3)⋊C3—21no—
73C4⋊((C3⋊D3)⋊C3)—21no—
74C4⋊((C3⋊D3)⋊C3)C2312yes—
75C4×(C3⋊D3)⋊C3—22no—
76C4×(C3⋊D3)⋊C3—22no—
77#634⋊C3—21no—
78#634⋊C3C2312yes—
79(C3⋊(C3⋊C8))⋊C3C2314yes—
80#5822—91no—
81#5822—91no—
82#5748—91no—
83C22×C32×C4—21no—
84C22×C32×C4C2×D311yescharacteristic
85C22×C32×C4C2×D311yescharacteristic
86C2×#638—21no—
87#3828—21no—
88#3828—21no—
89#3828C2×D312yes—
90#3828C2×D312yescomplemented
91C2×C4⋊(C3⋊D3)C2×D311yescharacteristic, complemented
92C2×C4×C3⋊D3—21no—
93C2×#634C2×D311yescharacteristic, complemented
94C2×C3×#67—62no—
95C2×C3×#67—61no—
96C2×C3×#67—61no—
97C2×C3×#67—61no—
98C3×#235—32no—
99C3×#235—32no—
100C3×#235—32no—
101C3×#235—32no—
102C3×#235—32no—
103C3×#235—32no—
104C3×#235—62no—
105C3×#235—62no—
106C3×#235—62no—
107C3×#235—62no—
108C3×#235—61no—
109C3×#235—61no—
110C3×#235—61no—
111C3×#235—61no—
112C3×#235—61no—
113C3×#235—61no—
114C2×C3×D12—31no—
115C2×C3×D12—31no—
116C2×C3×D12—31no—
117C2×C3×D12—61no—
118C2×C3×C4×D3—62no—
119C2×C3×C4×D3—61no—
120C2×C3×C4×D3—61no—
121C2×C3×C4×D3—61no—
122C2×C3×Dic6—31no—
123C2×C3×Dic6—31no—
124C2×C3×Dic6—31no—
125C2×C3×Dic6—61no—
126#3748C2×D312yescomplemented
127#3748C2×D312yescomplemented
128C2×C3⋊(C3⋊C8)C2×D312yescomplemented
129C3×#208—32no—
130C3×#208—32no—
131C2×C3×C3⋊C8—32no—
132#3719—34no—
133#3718—34no—
134#3717—34no—
135#3716—34no—
136#3715—34no—
137#3715—34no—
138#3714—34no—
139#3713—34no—
140C22×H(3,𝔽3)—41no—
141C22×H(3,𝔽3)C2×D411yescharacteristic
142C22×H(3,𝔽3)C2×D411yescharacteristic
143C22×H(3,𝔽3)C2×D411yescharacteristic
144C2×#272—22no—
145C2×#272—42no—
146C2×(C3⋊D3)⋊C3—22no—
147C2×(C3⋊D3)⋊C3—42no—
148C4×H(3,𝔽3)C2×D411yescharacteristic
149C4×H(3,𝔽3)C2×D411yescharacteristic
150C4×H(3,𝔽3)C2×D411yescharacteristic
151C4×H(3,𝔽3)C2411yescharacteristic
152#1084—22no—
153(C3⋊Dic3)⋊C3—22no—
154C2×C3×#41—91no—
155C2×C3×#41—91no—
156C2×#235—31no—
157C2×#235—31no—
158C2×#235—31no—
159C2×#235—61no—
160C2×#235—91no—
161C2×C3×M16—91no—
162#951—92no—
163#951—92no—
164#950—94no—
165#950—94no—
166#949—92no—
167#949—92no—
168#942—92no—
169#942—92no—
170#932—92no—
171#932—92no—
172C2×#208—31no—
173C2×#208—91no—
174#909—92no—
175#909—92no—
176#908—92no—
177#908—92no—
178#904—94no—
179C23×C32—21no—
180C23×C32D1211yescharacteristic
181C23×C32#6711yescharacteristic
182C22×C3⋊D3—21no—
183C22×C3×D3—62no—
184C22×C3×D3—61no—
185C22×C3×D3—61no—
186C22×C3×D3—61no—
187C2×C32×C4—21no—
188C2×C32×C4—21no—
189C2×C32×C4—21no—
190C2×C32×C4—21no—
191C2×C32×C4—21no—
192C2×C32×C4—21no—
193C2×C32×C4D1211yescharacteristic
194C2×C32×C4D1211yescharacteristic
195C2×C32×C4D1211yescharacteristic
196C2×C32×C4#6711yescharacteristic
197C2×C32×C4#6711yescharacteristic
198C2×C32×C4#6711yescharacteristic
199C2×C32×C4C22×D311yescharacteristic
200C2×C32×C4C22×D311yescharacteristic
201C2×C32×C4C22×D311yescharacteristic
202C2×C32×C4C22×D311yescharacteristic
203C2×C32×C4C22×D311yescharacteristic
204C2×C32×C4C22×D311yescharacteristic
205#638—22no—
206#638—22no—
207C2×C3⋊Dic3—21no—
208C4⋊(C3⋊D3)—21no—
209C4⋊(C3⋊D3)C22×D312yes—
210C4×C3⋊D3—22no—
211C4×C3⋊D3—22no—
212#634—21no—
213#634C22×D312yes—
214C3×#67—64no—
215C3×#67—64no—
216C3×#67—62no—
217C3×#67—62no—
218C3×#67—62no—
219C3×#67—62no—
220C3×#67—62no—
221C3×#67—62no—
222C2×C3×Dic3—62no—
223C2×C3×Dic3—61no—
224C2×C3×Dic3—61no—
225C2×C3×Dic3—61no—
226C3×D12—32no—
227C3×D12—32no—
228C3×D12—32no—
229C3×D12—62no—
230C3×D12—62no—
231C3×D12—61no—
232C3×D12—61no—
233C3×D12—61no—
234C3×C4×D3—64no—
235C3×C4×D3—64no—
236C3×C4×D3—62no—
237C3×C4×D3—62no—
238C3×C4×D3—62no—
239C3×C4×D3—62no—
240C3×C4×D3—62no—
241C3×C4×D3—62no—
242C3×Dic6—32no—
243C3×Dic6—32no—
244C3×Dic6—32no—
245C3×Dic6—62no—
246C3×Dic6—62no—
247C3×Dic6—61no—
248C3×Dic6—61no—
249C3×Dic6—61no—
250C3⋊(C3⋊C8)C22×D314yes—
251C3×C3⋊C8—34no—
252#575—271noSylow
253C2×H(3,𝔽3)—21no—
254C2×H(3,𝔽3)—21no—
255C2×H(3,𝔽3)—21no—
256C2×H(3,𝔽3)C22×D411yescharacteristic
257#272—42no—
258(C3⋊D3)⋊C3—42no—
259C3×#41—92no—
260C3×#41—92no—
261C3×#41—92no—
262C3×#41—92no—
263C3×#41—181no—
264C3×#41—181no—
265C3×#41—181no—
266C3×#41—181no—
267C2×C3×Q8—91no—
268C2×C3×Q8—91no—
269C2×C3×D4—91no—
270C2×C3×D4—91no—
271C2×C3×D4—181no—
272C2×C3×D4—181no—
273C22×C3×C4—31no—
274C22×C3×C4—31no—
275C22×C3×C4—61no—
276C22×C3×C4—181no—
277C22×C3×C4—181no—
278C22×C3×C4D3211yescharacteristic
279C2×#67—61no—
280C2×#67—61no—
281C2×#67—62no—
282C2×#67—61no—
283C2×#67—181no—
284#235—32no—
285#235—32no—
286#235—32no—
287#235—32no—
288#235—32no—
289#235—32no—
290#235—62no—
291#235—62no—
292#235—61no—
293#235—61no—
294#235—61no—
295#235—61no—
296#235—62no—
297#235—62no—
298#235—61no—
299#235—61no—
300#235—92no—
301#235—92no—
302#235—181no—
303#235—181no—
304C2×D12—31no—
305C2×D12—31no—
306C2×D12—31no—
307C2×D12—61no—
308C2×D12—91no—
309C2×C4×D3—61no—
310C2×C4×D3—61no—
311C2×C4×D3—62no—
312C2×C4×D3—61no—
313C2×C4×D3—181no—
314C2×Dic6—31no—
315C2×Dic6—31no—
316C2×Dic6—31no—
317C2×Dic6—61no—
318C2×Dic6—91no—
319C3×M16—92no—
320C3×M16—92no—
321C2×C3×C8—92no—
322#216—94no—
323#216—94no—
324#215—94no—
325#215—94no—
326#214—94no—
327#214—94no—
328#213—94no—
329#213—94no—
330#208—32no—
331#208—32no—
332#208—92no—
333#208—92no—
334C2×C3⋊C8—32no—
335C2×C3⋊C8—92no—
336Dic12—94no—
337D24—94no—
338#204—94no—
339#204—94no—
340C22×C32—21no—
341C22×C32—21no—
342C22×C32—21no—
343C22×C32—41no—
344C22×C32—41no—
345C22×C32—42no—
346C22×C32C2×D1211yescharacteristic
347C22×C32C2×D1211yescharacteristic
348C22×C32C2×D1211yescharacteristic
349C22×C32C2×#6711yescharacteristic
350C22×C32C2×#6711yescharacteristic
351C22×C32C2×#6711yescharacteristic
352C2×C3⋊D3—22no—
353C2×C3⋊D3—42no—
354C2×C3×D3—64no—
355C2×C3×D3—62no—
356C2×C3×D3—62no—
357C2×C3×D3—62no—
358C2×C3×D3—122no—
359C2×C3×D3—124no—
360C2×C3×D3—122no—
361C2×C3×D3—122no—
362C32×C4—21no—
363C32×C4—21no—
364C32×C4—21no—
365C32×C4—21no—
366C32×C4C2×D1211yescharacteristic
367C32×C4C2×D1211yescharacteristic
368C32×C4C2×D1211yescharacteristic
369C32×C4C2×#6711yescharacteristic
370C32×C4C2×#6711yescharacteristic
371C32×C4C2×#6711yescharacteristic
372C32×C4C23×D311yescharacteristic
373C32×C4C23×D311yescharacteristic
374C3⋊Dic3—22no—
375C3×Dic3—64no—
376C3×Dic3—62no—
377C3×Dic3—62no—
378C3×Dic3—62no—
379C2×#41—91no—
380C2×#41—91no—
381#137—272no—
382#137—272no—
383AGL(1,ℤ/8)—272no—
384AGL(1,ℤ/8)—272no—
385#135—274no—
386C2×M16—91no—
387H(3,𝔽3)#57511yescharacteristic, complemented, Sylow
388C23×C3—31no—
389C23×C3—31no—
390C23×C3—61no—
391C23×C3—181no—
392C23×C3—181no—
393C23×C3#62611yescharacteristic
394C22×D3—61no—
395C22×D3—61no—
396C22×D3—62no—
397C22×D3—61no—
398C22×D3—181no—
399C3×Q8—92no—
400C3×Q8—92no—
401C3×Q8—181no—
402C3×Q8—181no—
403C3×D4—92no—
404C3×D4—92no—
405C3×D4—181no—
406C3×D4—182no—
407C3×D4—182no—
408C3×D4—182no—
409C3×D4—182no—
410C3×D4—181no—
411C2×C3×C4—31no—
412C2×C3×C4—31no—
413C2×C3×C4—31no—
414C2×C3×C4—31no—
415C2×C3×C4—31no—
416C2×C3×C4—31no—
417C2×C3×C4—31no—
418C2×C3×C4—31no—
419C2×C3×C4—31no—
420C2×C3×C4—31no—
421C2×C3×C4—31no—
422C2×C3×C4—31no—
423C2×C3×C4—61no—
424C2×C3×C4—61no—
425C2×C3×C4—61no—
426C2×C3×C4—61no—
427C2×C3×C4—61no—
428C2×C3×C4—61no—
429C2×C3×C4—182no—
430C2×C3×C4—182no—
431C2×C3×C4—181no—
432C2×C3×C4—181no—
433C2×C3×C4—182no—
434C2×C3×C4—182no—
435C2×C3×C4#62611yescharacteristic
436C2×C3×C4#62611yescharacteristic
437C2×C3×C4#62611yescharacteristic
438C2×C3×C4C2×D3211yescharacteristic
439C2×C3×C4C2×D3211yescharacteristic
440C2×C3×C4C2×D3211yescharacteristic
441#67—62no—
442#67—62no—
443#67—62no—
444#67—62no—
445#67—64no—
446#67—64no—
447#67—62no—
448#67—62no—
449#67—182no—
450#67—182no—
451C2×Dic3—61no—
452C2×Dic3—61no—
453C2×Dic3—62no—
454C2×Dic3—61no—
455C2×Dic3—181no—
456D12—32no—
457D12—32no—
458D12—32no—
459D12—62no—
460D12—61no—
461D12—61no—
462D12—62no—
463D12—61no—
464D12—92no—
465D12—181no—
466C4×D3—62no—
467C4×D3—62no—
468C4×D3—62no—
469C4×D3—62no—
470C4×D3—64no—
471C4×D3—64no—
472C4×D3—62no—
473C4×D3—62no—
474C4×D3—182no—
475C4×D3—182no—
476Dic6—32no—
477Dic6—32no—
478Dic6—32no—
479Dic6—62no—
480Dic6—61no—
481Dic6—61no—
482Dic6—62no—
483Dic6—61no—
484Dic6—92no—
485Dic6—181no—
486C3×C8—94no—
487C3⋊C8—34no—
488C3⋊C8—94no—
489C2×C32—22no—
490C2×C32—21no—
491C2×C32—21no—
492C2×C32—21no—
493C2×C32—21no—
494C2×C32—21no—
495C2×C32—21no—
496C2×C32—21no—
497C2×C32—41no—
498C2×C32—41no—
499C2×C32C22×D1211yescharacteristic
500C2×C32C22×#6711yescharacteristic
501C3⋊D3—42no—
502C3×D3—124no—
503C3×D3—122no—
504C3×D3—122no—
505C3×D3—122no—
506#41—92no—
507#41—92no—
508#41—92no—
509#41—92no—
510#41—181no—
511#41—181no—
512#41—181no—
513#41—181no—
514C2×Q8—91no—
515C2×Q8—91no—
516C2×D4—91no—
517C2×D4—91no—
518C2×D4—181no—
519C2×D4—181no—
520C22×C4—181no—
521C22×C4—181no—
522C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
523Q16—274no—
524SD16—274no—
525SD16—274no—
526D8—274no—
527M16—92no—
528M16—92no—
529C2×C8—92no—
530C22×C3—31no—
531C22×C3—31no—
532C22×C3—31no—
533C22×C3—31no—
534C22×C3—31no—
535C22×C3—31no—
536C22×C3—41no—
537C22×C3—61no—
538C22×C3—61no—
539C22×C3—61no—
540C22×C3—121no—
541C22×C3—121no—
542C22×C3—122no—
543C22×C3—182no—
544C22×C3—182no—
545C22×C3—362no—
546C22×C3—362no—
547C22×C3C2×#62611yescharacteristic
548C22×C3C2×#62611yescharacteristic
549C22×C3C2×#62611yescharacteristic
550C2×D3—62no—
551C2×D3—62no—
552C2×D3—64no—
553C2×D3—62no—
554C2×D3—122no—
555C2×D3—122no—
556C2×D3—122no—
557C2×D3—124no—
558C2×D3—182no—
559C2×D3—362no—
560C3×C4—31no—
561C3×C4—31no—
562C3×C4—31no—
563C3×C4—31no—
564C3×C4—31no—
565C3×C4—31no—
566C3×C4—31no—
567C3×C4—31no—
568C3×C4—61no—
569C3×C4—61no—
570C3×C4—61no—
571C3×C4—61no—
572C3×C4—182no—
573C3×C4—182no—
574C3×C4C2×#62611yescharacteristic
575C3×C4C2×#62611yescharacteristic
576C3×C4C2×#62611yescharacteristic
577C3×C4C22×D3211yescharacteristic
578Dic3—62no—
579Dic3—62no—
580Dic3—64no—
581Dic3—62no—
582Dic3—182no—
583C32—21no—
584C32#574811yescharacteristic, complemented
585C32#582211yescharacteristic, complemented
586C23—181no—
587C23—181no—
588C23#620011yescharacteristic
589Q8—92no—
590Q8—92no—
591Q8—181no—
592Q8—181no—
593D4—92no—
594D4—92no—
595D4—182no—
596D4—182no—
597D4—182no—
598D4—182no—
599D4—181no—
600D4—181no—
601C2×C4—181no—
602C2×C4—181no—
603C2×C4—182no—
604C2×C4—182no—
605C2×C4—182no—
606C2×C4—182no—
607C2×C4#620011yescharacteristic
608C2×C4#620011yescharacteristic
609C2×C4#620011yescharacteristic
610C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
611C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
612C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
613C8—94no—
614C2×C3—21no—
615C2×C3—21no—
616C2×C3—21no—
617C2×C3—31no—
618C2×C3—31no—
619C2×C3—61no—
620C2×C3—61no—
621C2×C3—61no—
622C2×C3—61no—
623C2×C3—61no—
624C2×C3—61no—
625C2×C3—62no—
626C2×C3—61no—
627C2×C3—121no—
628C2×C3—121no—
629C2×C3—362no—
630C2×C3—362no—
631C2×C3C22×#62611yescharacteristic
632D3—122no—
633D3—122no—
634D3—124no—
635D3—122no—
636D3—362no—
637C22—41no—
638C22—182no—
639C22—182no—
640C22—362no—
641C22—362no—
642C22C2×#620011yescharacteristic
643C22C2×#620011yescharacteristic
644C22C2×#620011yescharacteristic
645C4—182no—
646C4—182no—
647C4C2×#620011yescharacteristic
648C4C2×#620011yescharacteristic
649C4C2×#620011yescharacteristic
650C4C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
651C3—31no—
652C3—31no—
653C3—61no—
654C3#1059479511yescharacteristic
655C2—21no—
656C2—21no—
657C2—21no—
658C2—362no—
659C2—362no—
660C2C22×#620011yescharacteristic
subgroupG/H
centerC4C2×#6200
commutator, cosocleC4×H(3,𝔽3)C24
FrattiniC3×C4C22×D32
FittingC22×C4×H(3,𝔽3)C22
socleC2×C3C22×#626
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#575
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C2×#88740C2semidirect
2. C2×#3828⋊C3C2semidirect
3. C2×#3748⋊C3C2semidirect
4. #11721074C2semidirect
5. #11721073C2semidirect
6. #11721071C2semidirect
7. #11721069C2semidirect
8. #11721068C2semidirect
16. #88740C22semidirect
17. C2×C4⋊#272C22semidirect
19. C2×#6229C22semidirect
24. #3828⋊C3C22semidirect
25. C2×C4⋊((C3⋊D3)⋊C3)C22semidirect
27. C2×#634⋊C3C22semidirect
28. #3748⋊C3C22semidirect
29. #3748⋊C3C22semidirect
30. C2×(C3⋊(C3⋊C8))⋊C3C22semidirect
31. #88439C22semidirect
32. #88438C22semidirect
33. #88437C22semidirect
34. #88436C22semidirect
35. C2×#3828D3semidirect
40. C2×#3748D3semidirect
90. #3828C2×D3semidirect
91. C2×C4⋊(C3⋊D3)C2×D3semidirect
93. C2×#634C2×D3semidirect
126. #3748C2×D3semidirect
127. #3748C2×D3semidirect
128. C2×C3⋊(C3⋊C8)C2×D3semidirect
387. H(3,𝔽3)#575semidirect
584. C32#5748semidirect
585. C32#5822semidirect

Maximal subgroups

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

subgroupindex
C2×#887402
C2×#3828⋊C32
C2×#3748⋊C32
#117210742
#117210732
#117210712
#117210692
#117210682
#105947953
#105947863