Finite Group Databasebeta

← #13178734

Subgroups — #13178734

Order 1728.

1370 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3832C211yesmaximal, characteristic, complemented
2#11724618C211yesmaximal, characteristic, complemented
3C3×#64151C212yesmaximal, complemented
4#11723713C214yesmaximal, complemented
5#11723711C212yesmaximal, complemented
6#11723709C212yesmaximal, complemented
7#11723661C212yesmaximal, complemented
8#11723656C211yesmaximal, characteristic, complemented
9#10595323—31nomaximal
10#10595317—32nomaximal
11C2×C33×Q8C2211yescharacteristic, complemented
12C3×#3832—21no—
13C3×#3832—21no—
14C3×#3832C2212yes—
15C3×#3832C2212yescomplemented
16C3×#3828C2212yes—
17C2×C3×C4⋊(C3⋊D3)—21no—
18C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic, complemented
19C2×C3×C4×C3⋊D3—21no—
20C2×C3×C4×C3⋊D3C2211yescharacteristic
21#89049—21no—
22#89046—21no—
23C4⋊#1113C2211yescharacteristic, complemented
24C4×#1113—21no—
25#89042—21no—
26#89041C2212yescomplemented
27#89040C2211yescharacteristic, complemented
28C3×#3756C2214yescomplemented
29C3×#3755C2214yescomplemented
30C3×#3748C2212yescomplemented
31#88811C2214yescomplemented
32#88810C2214yescomplemented
33#88809C2212yescomplemented
34#88808C2212yescomplemented
35C2×#3832D311yescharacteristic, complemented
36C2×C3×#239—32no—
37C2×C3×#239—61no—
38#64306—31no—
39D3×#235—32no—
40#64151D312yescomplemented
41C3×#942—32no—
42C3×#942—32no—
43C3×#942—62no—
44#63950—34no—
45#63949—34no—
46#63947—34no—
47#63945—32no—
48#63943—34no—
49#63941—32no—
50#63939—34no—
51#63830—32no—
52#63825—32no—
53#63823—32no—
54#63818—31no—
55D3×#208—32no—
56C22×C3×C3⋊D3—21no—
57C22×C3×C3⋊D3D411yescharacteristic
58C2×#1113—21no—
59C33×Q8—21no—
60C33×Q8C2312yes—
61C2×C33×C4—21no—
62C2×C33×C4C2311yescharacteristic
63C3×#638—22no—
64C2×C3×C3⋊Dic3D411yescharacteristic
65C3×C4⋊(C3⋊D3)—22no—
66C3×C4⋊(C3⋊D3)—22no—
67C3×C4⋊(C3⋊D3)—21no—
68C3×C4⋊(C3⋊D3)C2312yes—
69C3×C4⋊(C3⋊D3)C2312yes—
70C3×C4×C3⋊D3—22no—
71C3×C4×C3⋊D3—22no—
72C3×C4×C3⋊D3—22no—
73C3×C4×C3⋊D3D411yescharacteristic
74C3×C4×C3⋊D3D411yescharacteristic
75C3×C4×C3⋊D3C2311yescharacteristic
76C3×C4×C3⋊D3C2311yescharacteristic
77C3×#634C2312yes—
78#6296—21no—
79#6295—22no—
80#6295—21no—
81#6294—21no—
82#6294—22no—
83C3×C3⋊(C3⋊C8)C2314yes—
84#5822—91no—
85#5777—92no—
86C2×C32×Q8—21no—
87C2×C32×Q8—22no—
88C2×C32×Q8—41no—
89C2×C32×Q8C2×D311yescharacteristic, complemented
90C2×C32×Q8C2×D312yescomplemented
91#3832—21no—
92#3832—21no—
93#3832C2×D312yes—
94#3832C2×D312yescomplemented
95#3828C2×D312yes—
96C2×C4⋊(C3⋊D3)—21no—
97C2×C4⋊(C3⋊D3)C2×D311yescharacteristic, complemented
98C2×C4×C3⋊D3—21no—
99C2×C4×C3⋊D3C2×D311yescharacteristic
100C3×#239—34no—
101C3×#239—34no—
102C3×#239—62no—
103C3×#239—62no—
104C3×#239—62no—
105C3×#239—62no—
106C3×#239—62no—
107C3×#239—62no—
108C3×#235—32no—
109C3×#235—32no—
110C3×#235—62no—
111C2×C3×D12—32no—
112C2×C3×D12—62no—
113C2×C3×D12—61no—
114C2×C3×D12—62no—
115C2×C3×C4×D3—32no—
116C2×C3×C4×D3—62no—
117C2×C3×C4×D3—61no—
118C2×C3×C4×D3—62no—
119#3811—61no—
120D3×#67—62no—
121#3805—61no—
122#3804—62no—
123#3802—31no—
124D3×D12—32no—
125C4×D32—62no—
126C4×D32—61no—
127#3799—32no—
128#3798—62no—
129#3798—61no—
130#3797—31no—
131#3796—32no—
132#3795—32no—
133D3×Dic6—32no—
134#3756C2×D314yescomplemented
135#3755C2×D314yescomplemented
136#3748C2×D312yescomplemented
137C3×#216—34no—
138C3×#216—34no—
139C3×#216—64no—
140C3×#215—34no—
141C3×#215—34no—
142C3×#215—64no—
143C3×#208—32no—
144C3×#208—32no—
145C3×#208—62no—
146#3717—34no—
147#3716—34no—
148#3715—34no—
149#3714—34no—
150#3714—34no—
151#3713—34no—
152#3712—32no—
153#3711—34no—
154#3710—32no—
155D3×C3⋊C8—34no—
156C22×C33C2×D411yescharacteristic
157C2×C3×C3⋊D3—22no—
158C2×C3×C3⋊D3—22no—
159C2×C3×C3⋊D3—41no—
160C2×C3×C3⋊D3—42no—
161C2×C3×C3⋊D3C2×D411yescharacteristic
162C2×C3×C3⋊D3C2×D411yescharacteristic
163#1113—42no—
164C33×C4—22no—
165C33×C4C2×D411yescharacteristic
166C33×C4C2411yescharacteristic
167C3×C3⋊Dic3—22no—
168C3×C3⋊Dic3C2×D411yescharacteristic
169C3×C3⋊Dic3C2×D411yescharacteristic
170C2×C3×#41—91no—
171D3×#41—92no—
172C2×#239—32no—
173C2×#239—61no—
174C2×#235—91no—
175C3×#137—92no—
176#951—92no—
177#950—94no—
178#949—92no—
179#949—92no—
180#942—32no—
181#942—32no—
182#942—62no—
183#932—92no—
184C2×#208—91no—
185#919—94no—
186#918—94no—
187#916—94no—
188#914—94no—
189#908—92no—
190D3×M16—92no—
191C22×C3⋊D3—21no—
192C22×C3⋊D3#6711yescharacteristic
193C22×C3×D3—32no—
194C22×C3×D3—61no—
195C22×C3×D3—62no—
196C22×C3×D3—62no—
197C2×D32—62no—
198C2×D32—61no—
199C32×Q8—21no—
200C32×Q8—22no—
201C32×Q8—22no—
202C32×Q8—22no—
203C32×Q8—24no—
204C32×Q8—42no—
205C32×Q8—42no—
206C32×Q8—42no—
207C32×Q8C22×D312yes—
208C32×Q8C22×D314yes—
209C2×C32×C4—21no—
210C2×C32×C4—22no—
211C2×C32×C4—22no—
212C2×C32×C4—22no—
213C2×C32×C4—21no—
214C2×C32×C4—41no—
215C2×C32×C4—42no—
216C2×C32×C4—42no—
217C2×C32×C4C22×D312yes—
218C2×C32×C4C22×D311yescharacteristic
219#638—22no—
220C2×C3⋊Dic3#6711yescharacteristic
221C4⋊(C3⋊D3)—22no—
222C4⋊(C3⋊D3)—22no—
223C4⋊(C3⋊D3)—21no—
224C4⋊(C3⋊D3)C22×D312yes—
225C4⋊(C3⋊D3)C22×D312yes—
226C4×C3⋊D3—22no—
227C4×C3⋊D3—22no—
228C4×C3⋊D3—22no—
229C4×C3⋊D3#6711yescharacteristic
230C4×C3⋊D3#6711yescharacteristic
231C4×C3⋊D3C22×D311yescharacteristic
232C4×C3⋊D3C22×D311yescharacteristic
233#634C22×D312yes—
234C3×#67—62no—
235C3×#67—62no—
236C3×#67—122no—
237C2×C3×Dic3—32no—
238C2×C3×Dic3—61no—
239C3×D12—32no—
240C3×D12—34no—
241C3×D12—32no—
242C3×D12—62no—
243C3×D12—62no—
244C3×D12—64no—
245C3×D12—64no—
246C3×D12—64no—
247C3×D12—64no—
248C3×D12—62no—
249C3×D12—62no—
250C3×C4×D3—32no—
251C3×C4×D3—32no—
252C3×C4×D3—32no—
253C3×C4×D3—32no—
254C3×C4×D3—61no—
255C3×C4×D3—61no—
256C3×C4×D3—61no—
257C3×C4×D3—61no—
258C3×C4×D3—64no—
259C3×C4×D3—64no—
260C3×C4×D3—64no—
261C3×C4×D3—64no—
262C3×C4×D3—62no—
263C3×C4×D3—62no—
264C3×C4×D3—122no—
265C3×Dic6—32no—
266C3×Dic6—32no—
267C3×Dic6—62no—
268#627—62no—
269#627—61no—
270#626—62no—
271#626—62no—
272#626—62no—
273#625—62no—
274#625—61no—
275#624—62no—
276#624—61no—
277D3×Dic3—62no—
278D3×Dic3—62no—
279D3×Dic3—62no—
280C3⋊(C3⋊C8)C22×D314yes—
281C3×C3⋊C8—34no—
282C3×C3⋊C8—34no—
283C3×C3⋊C8—64no—
284#575—271noSylow
285C2×C33—21no—
286C2×C33C22×D411yescharacteristic
287C3×C3⋊D3—21no—
288C3×C3⋊D3—21no—
289C3×C3⋊D3—42no—
290C3×C3⋊D3—42no—
291C3×#41—92no—
292C3×#41—92no—
293C3×#41—92no—
294C3×#41—181no—
295C3×#41—181no—
296C2×C3×Q8—21no—
297C2×C3×Q8—22no—
298C2×C3×Q8—41no—
299C2×C3×Q8D3211yescharacteristic, complemented
300C2×C3×Q8D3212yescomplemented
301C2×C3×D4—91no—
302C2×C3×D4—181no—
303C22×C3×C4—91no—
304C22×C3×C4—181no—
305C2×#67—181no—
306#239—34no—
307#239—34no—
308#239—62no—
309#239—62no—
310#239—62no—
311#239—62no—
312#239—62no—
313#239—62no—
314#239—92no—
315D3×Q8—92no—
316#237—92no—
317#237—182no—
318D3×D4—92no—
319D3×D4—182no—
320#235—32no—
321#235—32no—
322#235—62no—
323#235—92no—
324#235—182no—
325#235—181no—
326#235—181no—
327C2×D12—32no—
328C2×D12—61no—
329C2×D12—62no—
330C2×D12—62no—
331C2×D12—91no—
332C2×C4×D3—32no—
333C2×C4×D3—61no—
334C2×C4×D3—62no—
335C2×C4×D3—62no—
336C2×C4×D3—182no—
337C2×C4×D3—181no—
338C2×Dic6—91no—
339C3×Q16—94no—
340C3×SD16—94no—
341C3×M16—92no—
342#216—34no—
343#216—34no—
344#216—64no—
345#215—34no—
346#215—34no—
347#215—64no—
348#215—94no—
349#214—94no—
350#213—94no—
351#213—94no—
352#208—32no—
353#208—32no—
354#208—62no—
355#208—92no—
356C2×C3⋊C8—92no—
357D24—94no—
358#204—94no—
359ΓL(1,𝔽25)—94no—
360D3×C8—94no—
361C22×C32—22no—
362C22×C32—21no—
363C22×C32—41no—
364C22×C32D3×D412yes—
365C22×C32C2×#6711yescharacteristic
366C2×C3⋊D3—22no—
367C2×C3⋊D3—22no—
368C2×C3⋊D3—41no—
369C2×C3⋊D3—42no—
370C2×C3⋊D3C2×#6711yescharacteristic
371C2×C3⋊D3C2×#6711yescharacteristic
372C2×C3×D3—32no—
373C2×C3×D3—32no—
374C2×C3×D3—61no—
375C2×C3×D3—61no—
376C2×C3×D3—64no—
377C2×C3×D3—62no—
378C2×C3×D3—64no—
379C2×C3×D3—62no—
380C2×C3×D3—122no—
381C2×C3×D3—122no—
382C2×C3×D3—122no—
383C2×C3×D3—124no—
384C2×C3×D3—124no—
385D32—124no—
386D32—122no—
387C32×C4—22no—
388C32×C4—24no—
389C32×C4—24no—
390C32×C4—22no—
391C32×C4—21no—
392C32×C4—22no—
393C32×C4—21no—
394C32×C4—41no—
395C32×C4—41no—
396C32×C4—44no—
397C32×C4—44no—
398C32×C4D3×D412yes—
399C32×C4C2×#6711yescharacteristic
400C32×C4C23×D312yes—
401C32×C4C23×D311yescharacteristic
402C3⋊Dic3—22no—
403C3⋊Dic3C2×#6711yescharacteristic
404C3⋊Dic3C2×#6711yescharacteristic
405C3×Dic3—32no—
406C3×Dic3—32no—
407C3×Dic3—61no—
408C3×Dic3—61no—
409C3×Dic3—62no—
410C3×Dic3—62no—
411C3×Dic3—122no—
412C2×#41—91no—
413C2×#41—271no—
414#137—92no—
415#137—272no—
416AGL(1,ℤ/8)—272no—
417AGL(1,ℤ/8)—272no—
418#135—274no—
419C2×M16—271no—
420C33#57511yescharacteristic, complemented, Sylow
421C23×C3—91no—
422C23×C3—181no—
423C22×D3—32no—
424C22×D3—61no—
425C22×D3—62no—
426C22×D3—62no—
427C22×D3—182no—
428C22×D3—181no—
429C3×Q8—22no—
430C3×Q8—22no—
431C3×Q8—21no—
432C3×Q8—22no—
433C3×Q8—24no—
434C3×Q8—42no—
435C3×Q8—42no—
436C3×Q8—42no—
437C3×Q8—92no—
438C3×Q8C2×D3212yes—
439C3×Q8C2×D3214yes—
440C3×D4—92no—
441C3×D4—92no—
442C3×D4—182no—
443C3×D4—182no—
444C3×D4—182no—
445C3×D4—181no—
446C2×C3×C4—22no—
447C2×C3×C4—22no—
448C2×C3×C4—21no—
449C2×C3×C4—22no—
450C2×C3×C4—21no—
451C2×C3×C4—42no—
452C2×C3×C4—42no—
453C2×C3×C4—41no—
454C2×C3×C4—91no—
455C2×C3×C4—91no—
456C2×C3×C4—91no—
457C2×C3×C4—91no—
458C2×C3×C4—91no—
459C2×C3×C4—182no—
460C2×C3×C4—182no—
461C2×C3×C4—182no—
462C2×C3×C4C2×D3212yes—
463C2×C3×C4C2×D3211yescharacteristic
464#67—62no—
465#67—62no—
466#67—122no—
467#67—182no—
468#67—182no—
469#67—182no—
470C2×Dic3—32no—
471C2×Dic3—61no—
472C2×Dic3—182no—
473C2×Dic3—181no—
474D12—34no—
475D12—32no—
476D12—32no—
477D12—62no—
478D12—62no—
479D12—64no—
480D12—64no—
481D12—64no—
482D12—64no—
483D12—62no—
484D12—62no—
485D12—181no—
486D12—182no—
487C4×D3—32no—
488C4×D3—32no—
489C4×D3—32no—
490C4×D3—32no—
491C4×D3—61no—
492C4×D3—61no—
493C4×D3—61no—
494C4×D3—61no—
495C4×D3—64no—
496C4×D3—64no—
497C4×D3—64no—
498C4×D3—64no—
499C4×D3—62no—
500C4×D3—62no—
501C4×D3—122no—
502C4×D3—182no—
503C4×D3—182no—
504C4×D3—182no—
505Dic6—32no—
506Dic6—32no—
507Dic6—62no—
508Dic6—182no—
509Dic6—181no—
510C3×C8—94no—
511C3⋊C8—34no—
512C3⋊C8—34no—
513C3⋊C8—64no—
514C2×C32—21no—
515C2×C32—22no—
516C2×C32—22no—
517C2×C32—22no—
518C2×C32—21no—
519C2×C32—41no—
520C2×C32—42no—
521C2×C32—42no—
522C2×C32C2×D3×D412yes—
523C2×C32C22×#6711yescharacteristic
524C3⋊D3—21no—
525C3⋊D3—21no—
526C3⋊D3—42no—
527C3⋊D3—42no—
528C3×D3—62no—
529C3×D3—62no—
530C3×D3—121no—
531C3×D3—121no—
532C3×D3—124no—
533C3×D3—122no—
534C3×D3—124no—
535C3×D3—122no—
536C3×D3—242no—
537#41—92no—
538#41—92no—
539#41—92no—
540#41—181no—
541#41—181no—
542#41—272no—
543#41—541no—
544#41—541no—
545C2×Q8—271no—
546C2×Q8#111311yescharacteristic, complemented
547C2×D4—91no—
548C2×D4—181no—
549C2×D4—271no—
550C2×D4—541no—
551C22×C4—91no—
552C22×C4—181no—
553C22×C4—541no—
554Q16—94no—
555SD16—94no—
556SD16—274no—
557D8—274no—
558M16—92no—
559M16—272no—
560C2×C8—272no—
561C22×C3—21no—
562C22×C3—22no—
563C22×C3—41no—
564C22×C3—91no—
565C22×C3—91no—
566C22×C3—182no—
567C22×C3—182no—
568C22×C3—361no—
569C22×C3—362no—
570C22×C3D3×#6712yes—
571C22×C3#381111yescharacteristic
572C2×D3—32no—
573C2×D3—32no—
574C2×D3—61no—
575C2×D3—61no—
576C2×D3—64no—
577C2×D3—64no—
578C2×D3—62no—
579C2×D3—62no—
580C2×D3—122no—
581C2×D3—122no—
582C2×D3—122no—
583C2×D3—124no—
584C2×D3—124no—
585C2×D3—364no—
586C2×D3—362no—
587C3×C4—24no—
588C3×C4—24no—
589C3×C4—22no—
590C3×C4—22no—
591C3×C4—21no—
592C3×C4—21no—
593C3×C4—22no—
594C3×C4—44no—
595C3×C4—44no—
596C3×C4—41no—
597C3×C4—41no—
598C3×C4—91no—
599C3×C4—91no—
600C3×C4—182no—
601C3×C4D3×#6712yes—
602C3×C4#381111yescharacteristic
603C3×C4C22×D3212yes—
604C3×C4C22×D3211yescharacteristic
605Dic3—32no—
606Dic3—32no—
607Dic3—61no—
608Dic3—61no—
609Dic3—62no—
610Dic3—62no—
611Dic3—122no—
612C32—22no—
613C32—21no—
614C32—41no—
615C32#577712yescomplemented
616C32#582211yescharacteristic, complemented
617C23—91no—
618C23—181no—
619C23—541no—
620Q8—21no—
621Q8—92no—
622Q8—541no—
623Q8C2×#111312yes—
624D4—92no—
625D4—92no—
626D4—182no—
627D4—182no—
628D4—182no—
629D4—181no—
630D4—541no—
631D4—542no—
632C2×C4—21no—
633C2×C4—91no—
634C2×C4—91no—
635C2×C4—91no—
636C2×C4—91no—
637C2×C4—91no—
638C2×C4—182no—
639C2×C4—182no—
640C2×C4—182no—
641C2×C4—542no—
642C2×C4—541no—
643C2×C4C2×#111311yescharacteristic
644C8—94no—
645C2×C3—22no—
646C2×C3—22no—
647C2×C3—21no—
648C2×C3—21no—
649C2×C3—22no—
650C2×C3—42no—
651C2×C3—42no—
652C2×C3—41no—
653C2×C3—181no—
654C2×C3—181no—
655C2×C3—362no—
656C2×C3—362no—
657C2×C3C2×D3×#6712yes—
658C2×C3C2×#381111yescharacteristic
659D3—62no—
660D3—62no—
661D3—121no—
662D3—121no—
663D3—124no—
664D3—124no—
665D3—122no—
666D3—122no—
667D3—242no—
668C22—91no—
669C22—91no—
670C22—182no—
671C22—182no—
672C22—361no—
673C22—362no—
674C22—1082no—
675C22#8904911yescharacteristic
676C4—22no—
677C4—91no—
678C4—91no—
679C4—182no—
680C4#8904911yescharacteristic
681C4C22×#111311yescharacteristic
682C3—21no—
683C3—22no—
684C3—41no—
685C3#1059531712yescomplemented
686C3#1059532311yescharacteristic, complemented
687C2—21no—
688C2—181no—
689C2—181no—
690C2—362no—
691C2—362no—
692C2C2×#8904911yescharacteristic
subgroupG/H
centerC2C2×#89049
commutator, cosocleC33×C4C24
FrattiniC4C22×#1113
FittingC2×C33×Q8C22
socleC2×C33C22×D4
hypercenterC2×Q8#1113
hypocenterC33#575
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. C2×C3×#3832C2semidirect
2. #11724618C2semidirect
3. C3×#64151C2semidirect
4. #11723713C2semidirect
5. #11723711C2semidirect
6. #11723709C2semidirect
7. #11723661C2semidirect
8. #11723656C2semidirect
11. C2×C33×Q8C22semidirect
15. C3×#3832C22semidirect
18. C2×C3×C4⋊(C3⋊D3)C22semidirect
23. C4⋊#1113C22semidirect
26. #89041C22semidirect
27. #89040C22semidirect
28. C3×#3756C22semidirect
29. C3×#3755C22semidirect
30. C3×#3748C22semidirect
31. #88811C22semidirect
32. #88810C22semidirect
33. #88809C22semidirect
34. #88808C22semidirect
35. C2×#3832D3semidirect
40. #64151D3semidirect
89. C2×C32×Q8C2×D3semidirect
90. C2×C32×Q8C2×D3semidirect
94. #3832C2×D3semidirect
97. C2×C4⋊(C3⋊D3)C2×D3semidirect
134. #3756C2×D3semidirect
135. #3755C2×D3semidirect
136. #3748C2×D3semidirect
299. C2×C3×Q8D32semidirect
300. C2×C3×Q8D32semidirect
420. C33#575semidirect
546. C2×Q8#1113semidirect
615. C32#5777semidirect
616. C32#5822semidirect
685. C3#10595317semidirect
686. C3#10595323semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#38322
#117246182
C3×#641512
#117237132
#117237112
#117237092
#117236612
#117236562
#105953233
#105953173