Finite Group Databasebeta

← #13170541

Subgroups — #13170541

Order 1728.

1416 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4⋊#1113C211yesmaximal, characteristic, complemented
2C3×#64306C211yesmaximal, characteristic, complemented
3#11723365C212yesmaximal, complemented
4C3⋊D3×#208C211yesmaximal, characteristic, complemented
5C3×#63825C212yesmaximal, complemented
6#11722876C214yesmaximal, complemented
7#11722840C212yesmaximal, complemented
8#11722831C212yesmaximal, complemented
9D3×#949—32nomaximal
10#10594716—31nomaximal
11C22×#1113—21no—
12C2×C3×C4⋊(C3⋊D3)C2212yescomplemented
13C2×C3×C4×C3⋊D3C2211yescharacteristic
14C32×#235C2212yescomplemented
15C2×#6295—21no—
16C4⋊#1113—21no—
17C4⋊#1113—21no—
18C4⋊#1113C2212yes—
19C4⋊#1113C2212yescomplemented
20C3×#3811—21no—
21C3×#3805—21no—
22C3×#3802C2211yescharacteristic, complemented
23C3×C4×D32—21no—
24C3×#3798—21no—
25C3×#3797C2211yescharacteristic, complemented
26C3×#3796C2212yescomplemented
27#88855C2211yescharacteristic, complemented
28C32×#208C2211yescharacteristic, complemented
29#88793C2214yescomplemented
30#88789C2214yescomplemented
31#88787C2212yescomplemented
32C3⋊D3×C3⋊C8C2212yescomplemented
33C3×#3717C2214yescomplemented
34C3×#3714C2214yescomplemented
35C3×D3×#41—32no—
36#64306D311yescharacteristic, complemented
37C2×#3802—31no—
38C2×D3×D12—32no—
39#64158—32no—
40M16×C3⋊D3—31no—
41C3×#949—32no—
42C3×#908—32no—
43#63939—34no—
44D3×#215—34no—
45#63925—34no—
46D3×#213—34no—
47#63825D312yescomplemented
48#63823—32no—
49#63823—32no—
50#63823—62no—
51D3×#208—32no—
52D3×#208—61no—
53#63798—34no—
54#63797—32no—
55#63796—32no—
56C22×C3×C3⋊D3—22no—
57C22×C3×C3⋊D3D411yescharacteristic
58C2×#1113—22no—
59C2×#1113—22no—
60C2×#1113—44no—
61C2×C3×D32—21no—
62C2×C33×C4C2311yescharacteristic
63C2×C3×C3⋊Dic3D411yescharacteristic
64C3×C4⋊(C3⋊D3)—22no—
65C3×C4⋊(C3⋊D3)C2314yes—
66C3×C4×C3⋊D3D411yescharacteristic
67C3×C4×C3⋊D3D411yescharacteristic
68C3×C4×C3⋊D3C2311yescharacteristic
69C3×C4×C3⋊D3C2311yescharacteristic
70C32×#67—22no—
71C32×D12C2312yescomplemented
72C32×C4×D3—22no—
73C32×Dic6C2312yescomplemented
74#6295—22no—
75#6295—22no—
76C3×#627—21no—
77C3×#626—22no—
78C3×#625—21no—
79C3×#624—21no—
80C3×D3×Dic3—22no—
81C3⋊(C3⋊(C3⋊C8))C2312yescomplemented
82C32×C3⋊C8C2312yescomplemented
83C2×#949—91no—
84D3×AGL(1,ℤ/8)—92no—
85C22×D32—62no—
86C22×D32—61no—
87C32×#41—32no—
88C2×C4⋊(C3⋊D3)C2×D312yescomplemented
89C2×C4×C3⋊D3C2×D311yescharacteristic
90C3×#239—32no—
91C3×D3×Q8—32no—
92C3×#237—32no—
93C3×#237—62no—
94C3×D3×D4—32no—
95C3×D3×D4—62no—
96C3×#235—22no—
97C3×#235—62no—
98C3×#235C2×D312yescomplemented
99C3×#235C2×D312yescomplemented
100C2×C3×D12—32no—
101C2×C3×D12—32no—
102C2×C3×D12—62no—
103C2×C3×C4×D3—32no—
104C2×C3×C4×D3—62no—
105C2×C3×C4×D3—61no—
106#3811—21no—
107C2×#626—62no—
108C2×#625—61no—
109#3805—21no—
110#3802—32no—
111#3802—32no—
112#3802—61no—
113#3802—61no—
114#3802C2×D311yescharacteristic, complemented
115D3×D12—34no—
116D3×D12—34no—
117D3×D12—62no—
118D3×D12—62no—
119C4×D32—21no—
120#3798—21no—
121#3797C2×D311yescharacteristic, complemented
122#3796C2×D312yescomplemented
123C32×M16—31no—
124#3755—34no—
125#3753—34no—
126#3748—32no—
127#3748—31no—
128#3748—61no—
129#3748—62no—
130#3748—121no—
131#3743—32no—
132C8×C3⋊D3—32no—
133C3×#215—34no—
134C3×#213—34no—
135C3×#208—21no—
136C3×#208C2×D312yescomplemented
137C3×D24—34no—
138C3×#204—34no—
139#3717C2×D314yescomplemented
140#3715—34no—
141#3715—34no—
142#3715—64no—
143#3714C2×D314yescomplemented
144#3713—34no—
145#3713—34no—
146#3713—64no—
147#3711—34no—
148#3711—62no—
149D3×C3⋊C8—34no—
150D3×C3⋊C8—62no—
151C22×C33C2×D411yescharacteristic
152C2×C3×C3⋊D3—22no—
153C2×C3×C3⋊D3—24no—
154C2×C3×C3⋊D3—44no—
155C2×C3×C3⋊D3C2×D411yescharacteristic
156C2×C3×C3⋊D3C2×D411yescharacteristic
157C2×C32×D3—22no—
158#1113—44no—
159#1113—44no—
160C3×D32—42no—
161C33×C4C2×D411yescharacteristic
162C33×C4C2411yescharacteristic
163C3×C3⋊Dic3C2×D411yescharacteristic
164C3×C3⋊Dic3C2×D411yescharacteristic
165C32×Dic3—22no—
166C2×C3×#41—91no—
167D3×#41—32no—
168C2×D3×D4—92no—
169C22×D12—91no—
170C3×AGL(1,ℤ/8)—92no—
171#949—32no—
172#949—94no—
173#949—92no—
174#949—92no—
175#949—182no—
176C2×#215—92no—
177#932—92no—
178C2×#213—92no—
179C2×#208—91no—
180#914—94no—
181D3×SD16—94no—
182#911—94no—
183D3×D8—94no—
184#908—32no—
185D3×M16—92no—
186D3×M16—181no—
187C22×C3⋊D3—22no—
188C22×C3⋊D3#6711yescharacteristic
189C22×C3×D3—32no—
190C22×C3×D3—61no—
191C22×C3×D3—62no—
192C22×C3×D3—62no—
193C22×C3×D3—62no—
194C22×C3×D3—122no—
195C2×D32—21no—
196C2×D32—64no—
197C2×D32—64no—
198C2×D32—62no—
199C2×D32—62no—
200C2×D32—128no—
201C2×D32—124no—
202C32×Q8—32no—
203C32×D4—32no—
204C32×D4—62no—
205C2×C32×C4—21no—
206C2×C32×C4—22no—
207C2×C32×C4—41no—
208C2×C32×C4—62no—
209C2×C32×C4C22×D312yes—
210C2×C32×C4C22×D311yescharacteristic
211C2×C3⋊Dic3#6711yescharacteristic
212C4⋊(C3⋊D3)—22no—
213C4⋊(C3⋊D3)C22×D314yes—
214C4×C3⋊D3#6711yescharacteristic
215C4×C3⋊D3#6711yescharacteristic
216C4×C3⋊D3C22×D311yescharacteristic
217C4×C3⋊D3C22×D311yescharacteristic
218C3×#67—22no—
219C3×#67—22no—
220C3×#67—42no—
221C3×#67—62no—
222C3×#67—62no—
223C2×C3×Dic3—32no—
224C2×C3×Dic3—61no—
225C2×C3×Dic3—62no—
226C3×D12—22no—
227C3×D12—34no—
228C3×D12—34no—
229C3×D12—64no—
230C3×D12—62no—
231C3×D12—62no—
232C3×D12—62no—
233C3×D12—122no—
234C3×D12C22×D312yescomplemented
235C3×D12C22×D312yescomplemented
236C3×C4×D3—22no—
237C3×C4×D3—22no—
238C3×C4×D3—32no—
239C3×C4×D3—32no—
240C3×C4×D3—32no—
241C3×C4×D3—32no—
242C3×C4×D3—42no—
243C3×C4×D3—61no—
244C3×C4×D3—61no—
245C3×C4×D3—61no—
246C3×C4×D3—61no—
247C3×C4×D3—62no—
248C3×C4×D3—62no—
249C3×Dic6—22no—
250C3×Dic6—62no—
251C3×Dic6C22×D312yescomplemented
252C3×Dic6C22×D312yescomplemented
253#627—21no—
254#626—22no—
255#626—64no—
256#626—64no—
257#625—21no—
258#625—62no—
259#625—62no—
260#624—21no—
261D3×Dic3—22no—
262C32×C8—32no—
263C3⋊(C3⋊C8)—34no—
264C3⋊(C3⋊C8)—32no—
265C3⋊(C3⋊C8)—62no—
266C3⋊(C3⋊C8)—64no—
267C3⋊(C3⋊C8)—122no—
268C3×C3⋊C8—22no—
269C3×C3⋊C8C22×D314yescomplemented
270C2×AGL(1,ℤ/8)—271noSylow
271C2×C33—21no—
272C2×C33C22×D411yescharacteristic
273C3×C3⋊D3—21no—
274C3×C3⋊D3—44no—
275C3×C3⋊D3AGL(1,ℤ/8)12yescomplemented
276C32×D3—42no—
277C23×D3—182no—
278C23×D3—181no—
279C3×#41—32no—
280C3×#41—32no—
281C3×#41—62no—
282C3×#41—92no—
283C3×#41—181no—
284C3×#41—181no—
285C2×C3×Q8—91no—
286C2×C3×D4—91no—
287C2×C3×D4—92no—
288C2×C3×D4—181no—
289C22×C3×C4—91no—
290C22×C3×C4—181no—
291C2×#67—182no—
292#239—32no—
293D3×Q8—32no—
294#237—32no—
295#237—62no—
296D3×D4—32no—
297D3×D4—62no—
298D3×D4—94no—
299D3×D4—94no—
300D3×D4—182no—
301D3×D4—182no—
302#235—62no—
303#235D3212yescomplemented
304C2×D12—32no—
305C2×D12—32no—
306C2×D12—62no—
307C2×D12—92no—
308C2×D12—92no—
309C2×D12—181no—
310C2×D12—181no—
311C2×D12—181no—
312C2×C4×D3—32no—
313C2×C4×D3—61no—
314C2×C4×D3—62no—
315C3×SD16—94no—
316C3×D8—94no—
317C3×M16—32no—
318C3×M16—61no—
319#215—34no—
320#215—94no—
321#215—94no—
322#215—184no—
323#214—94no—
324#213—34no—
325#213—94no—
326#213—94no—
327#213—94no—
328#213—184no—
329#208—92no—
330#208—91no—
331#208—92no—
332#208—181no—
333#208—182no—
334#208—361no—
335#208D3211yescharacteristic, complemented
336C2×C3⋊C8—92no—
337D24—34no—
338#204—34no—
339ΓL(1,𝔽25)—94no—
340ΓL(1,𝔽25)—182no—
341D3×C8—94no—
342D3×C8—182no—
343C22×C32—21no—
344C22×C32—22no—
345C22×C32—41no—
346C22×C32—62no—
347C22×C32D3×D412yes—
348C22×C32C2×#6711yescharacteristic
349C2×C3⋊D3—22no—
350C2×C3⋊D3—24no—
351C2×C3⋊D3—44no—
352C2×C3⋊D3C2×#6711yescharacteristic
353C2×C3⋊D3C2×#6711yescharacteristic
354C2×C3×D3—22no—
355C2×C3×D3—22no—
356C2×C3×D3—32no—
357C2×C3×D3—32no—
358C2×C3×D3—42no—
359C2×C3×D3—61no—
360C2×C3×D3—61no—
361C2×C3×D3—64no—
362C2×C3×D3—64no—
363C2×C3×D3—64no—
364C2×C3×D3—122no—
365C2×C3×D3—124no—
366C2×C3×D3—124no—
367C2×C3×D3—124no—
368C2×C3×D3—124no—
369C2×C3×D3—244no—
370D32—42no—
371D32—128no—
372D32—128no—
373D32—124no—
374D32—124no—
375C32×C4—21no—
376C32×C4—22no—
377C32×C4—21no—
378C32×C4—22no—
379C32×C4—41no—
380C32×C4—41no—
381C32×C4—62no—
382C32×C4D3×D412yes—
383C32×C4C2×#6711yescharacteristic
384C32×C4C23×D312yes—
385C32×C4C23×D311yescharacteristic
386C3⋊Dic3C2×#6711yescharacteristic
387C3⋊Dic3C2×#6711yescharacteristic
388C3×Dic3—22no—
389C3×Dic3—22no—
390C3×Dic3—32no—
391C3×Dic3—32no—
392C3×Dic3—42no—
393C3×Dic3—61no—
394C3×Dic3—61no—
395C2×#41—91no—
396C22×D4—271no—
397AGL(1,ℤ/8)—92no—
398AGL(1,ℤ/8)—274no—
399AGL(1,ℤ/8)—272no—
400C2×SD16—272no—
401C2×D8—272no—
402C2×M16—271no—
403C33C2×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
404C23×C3—91no—
405C23×C3—181no—
406C23×C3—182no—
407C22×D3—32no—
408C22×D3—61no—
409C22×D3—62no—
410C22×D3—62no—
411C22×D3—62no—
412C22×D3—122no—
413C22×D3—184no—
414C22×D3—184no—
415C22×D3—182no—
416C22×D3—182no—
417C22×D3—368no—
418C22×D3—364no—
419C3×Q8—32no—
420C3×Q8—32no—
421C3×Q8—62no—
422C3×Q8—181no—
423C3×D4—32no—
424C3×D4—32no—
425C3×D4—62no—
426C3×D4—62no—
427C3×D4—62no—
428C3×D4—94no—
429C3×D4—122no—
430C3×D4—182no—
431C3×D4—181no—
432C3×D4—182no—
433C2×C3×C4—21no—
434C2×C3×C4—22no—
435C2×C3×C4—41no—
436C2×C3×C4—62no—
437C2×C3×C4—62no—
438C2×C3×C4—91no—
439C2×C3×C4—91no—
440C2×C3×C4—91no—
441C2×C3×C4—91no—
442C2×C3×C4—91no—
443C2×C3×C4—122no—
444C2×C3×C4—181no—
445C2×C3×C4—182no—
446C2×C3×C4C2×D3211yescharacteristic
447C2×C3×C4C2×D3212yes—
448#67—22no—
449#67—62no—
450#67—62no—
451#67—184no—
452#67—184no—
453C2×Dic3—32no—
454C2×Dic3—61no—
455C2×Dic3—62no—
456D12—34no—
457D12—34no—
458D12—64no—
459D12—62no—
460D12—62no—
461D12—62no—
462D12—122no—
463D12—182no—
464D12—182no—
465D12C2×D3212yescomplemented
466C4×D3—22no—
467C4×D3—32no—
468C4×D3—32no—
469C4×D3—32no—
470C4×D3—32no—
471C4×D3—61no—
472C4×D3—61no—
473C4×D3—61no—
474C4×D3—61no—
475C4×D3—62no—
476C4×D3—62no—
477Dic6—62no—
478Dic6C2×D3212yescomplemented
479C3×C8—34no—
480C3×C8—62no—
481C3⋊C8—94no—
482C3⋊C8—92no—
483C3⋊C8—182no—
484C3⋊C8—184no—
485C3⋊C8—362no—
486C3⋊C8C2×D3212yescomplemented
487C2×C32—22no—
488C2×C32—21no—
489C2×C32—21no—
490C2×C32—22no—
491C2×C32—41no—
492C2×C32—41no—
493C2×C32—42no—
494C2×C32—81no—
495C2×C32—122no—
496C2×C32C2×D3×D412yes—
497C2×C32C22×#6711yescharacteristic
498C3⋊D3—21no—
499C3⋊D3—44no—
500C3⋊D3#94912yescomplemented
501C3×D3—34no—
502C3×D3—42no—
503C3×D3—42no—
504C3×D3—62no—
505C3×D3—62no—
506C3×D3—82no—
507C3×D3—121no—
508C3×D3—124no—
509C3×D3—124no—
510C3×D3—244no—
511C24—541no—
512#41—32no—
513#41—92no—
514#41—181no—
515#41—181no—
516C2×Q8—91no—
517C2×D4—91no—
518C2×D4—92no—
519C2×D4—181no—
520C2×D4—272no—
521C2×D4—272no—
522C2×D4—541no—
523C2×D4—541no—
524C2×D4—541no—
525C22×C4—91no—
526C22×C4—181no—
527SD16—94no—
528SD16—274no—
529D8—94no—
530D8—274no—
531M16—31no—
532M16—272no—
533M16—271no—
534C2×C8—272no—
535C22×C3—21no—
536C22×C3—22no—
537C22×C3—41no—
538C22×C3—62no—
539C22×C3—62no—
540C22×C3—91no—
541C22×C3—91no—
542C22×C3—122no—
543C22×C3—182no—
544C22×C3—184no—
545C22×C3—364no—
546C22×C3—362no—
547C22×C3#380211yescharacteristic
548C22×C3D3×#6712yes—
549C2×D3—22no—
550C2×D3—32no—
551C2×D3—32no—
552C2×D3—61no—
553C2×D3—61no—
554C2×D3—64no—
555C2×D3—64no—
556C2×D3—64no—
557C2×D3—122no—
558C2×D3—124no—
559C2×D3—124no—
560C2×D3—124no—
561C2×D3—124no—
562C2×D3—244no—
563C2×D3—368no—
564C2×D3—368no—
565C2×D3—364no—
566C2×D3—364no—
567C3×C4—21no—
568C3×C4—22no—
569C3×C4—21no—
570C3×C4—22no—
571C3×C4—41no—
572C3×C4—41no—
573C3×C4—62no—
574C3×C4—62no—
575C3×C4—91no—
576C3×C4—91no—
577C3×C4—122no—
578C3×C4#380211yescharacteristic
579C3×C4D3×#6712yes—
580C3×C4C22×D3211yescharacteristic
581C3×C4C22×D3212yes—
582Dic3—22no—
583Dic3—32no—
584Dic3—32no—
585Dic3—61no—
586Dic3—61no—
587C32—22no—
588C32—21no—
589C32—41no—
590C32D3×AGL(1,ℤ/8)12yescomplemented
591C32C2×#94911yescharacteristic, complemented
592C23—91no—
593C23—181no—
594C23—182no—
595C23—542no—
596C23—542no—
597C23—1084no—
598Q8—32no—
599Q8—181no—
600D4—32no—
601D4—62no—
602D4—94no—
603D4—182no—
604D4—181no—
605D4—182no—
606D4—542no—
607D4—542no—
608C2×C4—62no—
609C2×C4—91no—
610C2×C4—91no—
611C2×C4—91no—
612C2×C4—91no—
613C2×C4—91no—
614C2×C4—181no—
615C2×C4—182no—
616C2×C4D3311yescharacteristic
617C8—32no—
618C8—272no—
619C2×C3—22no—
620C2×C3—21no—
621C2×C3—21no—
622C2×C3—22no—
623C2×C3—41no—
624C2×C3—42no—
625C2×C3—41no—
626C2×C3—81no—
627C2×C3—92no—
628C2×C3—122no—
629C2×C3—122no—
630C2×C3—181no—
631C2×C3—242no—
632C2×C3—364no—
633C2×C3C2×#380211yescharacteristic
634C2×C3C2×D3×#6712yes—
635D3—34no—
636D3—42no—
637D3—62no—
638D3—62no—
639D3—121no—
640D3—124no—
641D3—124no—
642D3—244no—
643C22—62no—
644C22—91no—
645C22—91no—
646C22—182no—
647C22—184no—
648C22—364no—
649C22—362no—
650C22—1084no—
651C22—1084no—
652C22#8895311yescharacteristic
653C4—62no—
654C4—91no—
655C4—91no—
656C4#8895311yescharacteristic
657C4C2×D3311yescharacteristic
658C3—21no—
659C3—22no—
660C3—41no—
661C3#1059471611yescharacteristic, complemented
662C3D3×#94912yescomplemented
663C2—21no—
664C2—92no—
665C2—122no—
666C2—181no—
667C2—364no—
668C2C2×#8895311yescharacteristic
subgroupG/H
centerC2C2×#88953
commutator, cosocleC33×C4C24
FrattiniC4C2×D33
FittingC2×C33×C4C23
socleC2×C33C22×D4
hypercenterC2×C4D33
hypocenterC33C2×AGL(1,ℤ/8)
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×C4⋊#1113C2semidirect
2. C3×#64306C2semidirect
3. #11723365C2semidirect
4. C3⋊D3×#208C2semidirect
5. C3×#63825C2semidirect
6. #11722876C2semidirect
7. #11722840C2semidirect
8. #11722831C2semidirect
12. C2×C3×C4⋊(C3⋊D3)C22semidirect
14. C32×#235C22semidirect
19. C4⋊#1113C22semidirect
22. C3×#3802C22semidirect
25. C3×#3797C22semidirect
26. C3×#3796C22semidirect
27. #88855C22semidirect
28. C32×#208C22semidirect
29. #88793C22semidirect
30. #88789C22semidirect
31. #88787C22semidirect
32. C3⋊D3×C3⋊C8C22semidirect
33. C3×#3717C22semidirect
34. C3×#3714C22semidirect
36. #64306D3semidirect
47. #63825D3semidirect
71. C32×D12C23semidirect
73. C32×Dic6C23semidirect
81. C3⋊(C3⋊(C3⋊C8))C23semidirect
82. C32×C3⋊C8C23semidirect
88. C2×C4⋊(C3⋊D3)C2×D3semidirect
98. C3×#235C2×D3semidirect
99. C3×#235C2×D3semidirect
114. #3802C2×D3semidirect
121. #3797C2×D3semidirect
122. #3796C2×D3semidirect
136. C3×#208C2×D3semidirect
139. #3717C2×D3semidirect
143. #3714C2×D3semidirect
234. C3×D12C22×D3semidirect
235. C3×D12C22×D3semidirect
251. C3×Dic6C22×D3semidirect
252. C3×Dic6C22×D3semidirect
269. C3×C3⋊C8C22×D3semidirect
275. C3×C3⋊D3AGL(1,ℤ/8)semidirect
303. #235D32semidirect
335. #208D32semidirect
403. C33C2×AGL(1,ℤ/8)semidirect
465. D12C2×D32semidirect
478. Dic6C2×D32semidirect
486. C3⋊C8C2×D32semidirect
500. C3⋊D3#949semidirect
590. C32D3×AGL(1,ℤ/8)semidirect
591. C32C2×#949semidirect
661. C3#10594716semidirect
662. C3D3×#949semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊#11132
C3×#643062
#117233652
C3⋊D3×#2082
C3×#638252
#117228762
#117228402
#117228312
D3×#9493
#105947163