Finite Group Databasebeta

← #13178733

Subgroups — #13178733

Order 1728.

1346 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×Q8×C3⋊D3C211yesmaximal, characteristic, complemented
2#11724618C211yesmaximal, characteristic
3C3×#64151C212yesmaximal, complemented
4#11723712C214yesmaximal, complemented
5#11723710C212yesmaximal, complemented
6#11723708C212yesmaximal, complemented
7#11723663C212yesmaximal, complemented
8#11723656C211yesmaximal, characteristic, complemented
9#10595322—31nomaximal
10D3×#942—32nomaximal
11C2×C33×Q8C2211yescharacteristic, complemented
12C3×Q8×C3⋊D3—21no—
13C3×Q8×C3⋊D3—21no—
14C3×Q8×C3⋊D3C2212yes—
15C3×Q8×C3⋊D3C2212yescomplemented
16C3×#3828C2212yes—
17C2×C3×C4×C3⋊D3—21no—
18C2×C3×C4×C3⋊D3C2211yescharacteristic
19C2×C3×#634—21no—
20C2×C3×#634C2211yescharacteristic, complemented
21#89049—21no—
22#89046—21no—
23C4⋊#1113C2211yescharacteristic
24C4×#1113—21no—
25#89042—21no—
26#89041C2212yes—
27#89040C2211yescharacteristic
28C3×#3756C2214yescomplemented
29C3×#3755C2214yescomplemented
30C3×#3748C2212yescomplemented
31#88812C2214yescomplemented
32#88811C2214yescomplemented
33#88809C2212yescomplemented
34#88808C2212yes—
35C2×Q8×C3⋊D3D311yescharacteristic, complemented
36C2×C3×D3×Q8—32no—
37C2×C3×D3×Q8—61no—
38#64306—31no—
39D3×#235—32no—
40#64151D312yescomplemented
41C3×#942—32no—
42C3×#942—32no—
43C3×#942—62no—
44#63951—34no—
45#63948—34no—
46#63946—34no—
47#63944—32no—
48D3×#216—34no—
49#63940—32no—
50D3×#215—34no—
51#63833—32no—
52#63831—32no—
53#63827—32no—
54#63818—31no—
55D3×#208—32no—
56C22×C3×C3⋊D3D411yescharacteristic
57C2×#1113—21no—
58C33×Q8—21no—
59C33×Q8C2312yes—
60C2×C33×C4—21no—
61C2×C33×C4C2311yescharacteristic
62C3×#638—22no—
63C2×C3×C3⋊Dic3—21no—
64C2×C3×C3⋊Dic3D411yescharacteristic
65C3×C4⋊(C3⋊D3)C2312yes—
66C3×C4×C3⋊D3—22no—
67C3×C4×C3⋊D3—22no—
68C3×C4×C3⋊D3—22no—
69C3×C4×C3⋊D3D411yescharacteristic
70C3×C4×C3⋊D3D411yescharacteristic
71C3×C4×C3⋊D3C2311yescharacteristic
72C3×C4×C3⋊D3C2311yescharacteristic
73C3×#634—21no—
74C3×#634—22no—
75C3×#634—22no—
76C3×#634C2312yes—
77C3×#634C2312yes—
78#6296—21no—
79#6295—22no—
80#6295—21no—
81#6294—21no—
82#6294—22no—
83C3×C3⋊(C3⋊C8)C2314yes—
84C2×#942—91no—
85D3×#137—92no—
86C2×C32×Q8—21no—
87C2×C32×Q8—22no—
88C2×C32×Q8—41no—
89C2×C32×Q8C2×D311yescharacteristic, complemented
90C2×C32×Q8C2×D312yescomplemented
91Q8×C3⋊D3—21no—
92Q8×C3⋊D3—21no—
93Q8×C3⋊D3C2×D312yes—
94Q8×C3⋊D3C2×D312yescomplemented
95#3828C2×D312yes—
96C2×C4×C3⋊D3—21no—
97C2×C4×C3⋊D3C2×D311yescharacteristic
98C2×#634—21no—
99C2×#634C2×D311yescharacteristic, complemented
100C3×D3×Q8—34no—
101C3×D3×Q8—34no—
102C3×D3×Q8—62no—
103C3×D3×Q8—62no—
104C3×D3×Q8—62no—
105C3×D3×Q8—62no—
106C3×D3×Q8—62no—
107C3×D3×Q8—62no—
108C3×#235—32no—
109C3×#235—32no—
110C3×#235—62no—
111C2×C3×C4×D3—32no—
112C2×C3×C4×D3—62no—
113C2×C3×C4×D3—61no—
114C2×C3×C4×D3—62no—
115C2×C3×Dic6—32no—
116C2×C3×Dic6—62no—
117C2×C3×Dic6—61no—
118C2×C3×Dic6—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#3719—34no—
147#3719—34no—
148#3718—34no—
149#3717—34no—
150#3716—34no—
151#3715—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⋊D3C2×D411yescharacteristic
160C2×C3×C3⋊D3C2×D411yescharacteristic
161#1113—42no—
162C33×C4—22no—
163C33×C4C2×D411yescharacteristic
164C33×C4C2411yescharacteristic
165C3×C3⋊Dic3—22no—
166C3×C3⋊Dic3—22no—
167C3×C3⋊Dic3C2×D411yescharacteristic
168C3×C3⋊Dic3C2×D411yescharacteristic
169C22×C3×Q8—91no—
170D3×#41—92no—
171C2×D3×Q8—32no—
172C2×D3×Q8—61no—
173C2×#235—91no—
174C3×#137—92no—
175#951—92no—
176C2×#216—92no—
177#942—32no—
178#942—32no—
179#942—62no—
180#942—92no—
181#942—94no—
182C2×#215—92no—
183C2×#208—91no—
184#918—94no—
185D3×Q16—94no—
186#915—94no—
187D3×SD16—94no—
188#909—92no—
189D3×M16—92no—
190C22×C3⋊D3#6711yescharacteristic
191C22×C3×D3—32no—
192C22×C3×D3—61no—
193C2×D32—62no—
194C2×D32—61no—
195C32×Q8—21no—
196C32×Q8—22no—
197C32×Q8—22no—
198C32×Q8—22no—
199C32×Q8—24no—
200C32×Q8—42no—
201C32×Q8—42no—
202C32×Q8—42no—
203C32×Q8C22×D312yes—
204C32×Q8C22×D314yes—
205C2×C32×C4—21no—
206C2×C32×C4—22no—
207C2×C32×C4—22no—
208C2×C32×C4—22no—
209C2×C32×C4—21no—
210C2×C32×C4—41no—
211C2×C32×C4—42no—
212C2×C32×C4—42no—
213C2×C32×C4C22×D312yes—
214C2×C32×C4C22×D311yescharacteristic
215#638—22no—
216C2×C3⋊Dic3—21no—
217C2×C3⋊Dic3#6711yescharacteristic
218C4⋊(C3⋊D3)C22×D312yes—
219C4×C3⋊D3—22no—
220C4×C3⋊D3—22no—
221C4×C3⋊D3—22no—
222C4×C3⋊D3#6711yescharacteristic
223C4×C3⋊D3#6711yescharacteristic
224C4×C3⋊D3C22×D311yescharacteristic
225C4×C3⋊D3C22×D311yescharacteristic
226#634—21no—
227#634—22no—
228#634—22no—
229#634C22×D312yes—
230#634C22×D312yes—
231C3×#67—62no—
232C3×#67—62no—
233C3×#67—122no—
234C2×C3×Dic3—32no—
235C2×C3×Dic3—61no—
236C2×C3×Dic3—62no—
237C2×C3×Dic3—62no—
238C3×D12—32no—
239C3×D12—32no—
240C3×D12—62no—
241C3×C4×D3—32no—
242C3×C4×D3—32no—
243C3×C4×D3—32no—
244C3×C4×D3—32no—
245C3×C4×D3—61no—
246C3×C4×D3—61no—
247C3×C4×D3—61no—
248C3×C4×D3—61no—
249C3×C4×D3—64no—
250C3×C4×D3—64no—
251C3×C4×D3—64no—
252C3×C4×D3—64no—
253C3×C4×D3—62no—
254C3×C4×D3—62no—
255C3×C4×D3—122no—
256C3×Dic6—34no—
257C3×Dic6—32no—
258C3×Dic6—32no—
259C3×Dic6—62no—
260C3×Dic6—62no—
261C3×Dic6—64no—
262C3×Dic6—64no—
263C3×Dic6—64no—
264C3×Dic6—64no—
265C3×Dic6—62no—
266C3×Dic6—62no—
267#627—62no—
268#627—61no—
269#626—62no—
270#626—62no—
271#626—62no—
272#625—62no—
273#625—61no—
274#624—62no—
275#624—61no—
276D3×Dic3—62no—
277D3×Dic3—62no—
278D3×Dic3—62no—
279C3⋊(C3⋊C8)C22×D314yes—
280C3×C3⋊C8—34no—
281C3×C3⋊C8—34no—
282C3×C3⋊C8—64no—
283C2×#137—271noSylow
284C2×C33—21no—
285C2×C33C22×D411yescharacteristic
286C3×C3⋊D3—21no—
287C3×C3⋊D3—42no—
288C3×C3⋊D3#13712yescomplemented
289C3×#41—92no—
290C2×C3×Q8—21no—
291C2×C3×Q8—22no—
292C2×C3×Q8—41no—
293C2×C3×Q8—92no—
294C2×C3×Q8—92no—
295C2×C3×Q8—91no—
296C2×C3×Q8—181no—
297C2×C3×Q8—181no—
298C2×C3×Q8—181no—
299C2×C3×Q8D3211yescharacteristic, complemented
300C2×C3×Q8D3212yescomplemented
301C22×C3×C4—91no—
302C22×C3×C4—181no—
303C2×#67—181no—
304#239—92no—
305D3×Q8—34no—
306D3×Q8—34no—
307D3×Q8—62no—
308D3×Q8—62no—
309D3×Q8—62no—
310D3×Q8—62no—
311D3×Q8—62no—
312D3×Q8—62no—
313D3×Q8—92no—
314#237—92no—
315#237—182no—
316D3×D4—92no—
317D3×D4—182no—
318#235—32no—
319#235—32no—
320#235—62no—
321#235—92no—
322#235—182no—
323#235—181no—
324#235—181no—
325C2×D12—91no—
326C2×C4×D3—32no—
327C2×C4×D3—61no—
328C2×C4×D3—62no—
329C2×C4×D3—62no—
330C2×C4×D3—182no—
331C2×C4×D3—181no—
332C2×Dic6—32no—
333C2×Dic6—61no—
334C2×Dic6—62no—
335C2×Dic6—62no—
336C2×Dic6—91no—
337C3×Q16—94no—
338C3×SD16—94no—
339C3×M16—92no—
340#216—34no—
341#216—34no—
342#216—64no—
343#216—94no—
344#216—94no—
345#215—34no—
346#215—34no—
347#215—64no—
348#215—94no—
349#214—94no—
350#208—32no—
351#208—32no—
352#208—62no—
353#208—92no—
354C2×C3⋊C8—92no—
355Dic12—94no—
356#204—94no—
357ΓL(1,𝔽25)—94no—
358D3×C8—94no—
359C22×C32—22no—
360C22×C32—21no—
361C22×C32—41no—
362C22×C32D3×D412yes—
363C22×C32C2×#6711yescharacteristic
364C2×C3⋊D3—22no—
365C2×C3⋊D3—22no—
366C2×C3⋊D3C2×#6711yescharacteristic
367C2×C3⋊D3C2×#6711yescharacteristic
368C2×C3×D3—32no—
369C2×C3×D3—32no—
370C2×C3×D3—61no—
371C2×C3×D3—61no—
372C2×C3×D3—64no—
373C2×C3×D3—64no—
374C2×C3×D3—62no—
375C2×C3×D3—62no—
376C2×C3×D3—122no—
377D32—124no—
378D32—122no—
379C32×C4—22no—
380C32×C4—24no—
381C32×C4—24no—
382C32×C4—22no—
383C32×C4—21no—
384C32×C4—22no—
385C32×C4—21no—
386C32×C4—41no—
387C32×C4—41no—
388C32×C4—44no—
389C32×C4—44no—
390C32×C4D3×D412yes—
391C32×C4C2×#6711yescharacteristic
392C32×C4C23×D312yes—
393C32×C4C23×D311yescharacteristic
394C3⋊Dic3—22no—
395C3⋊Dic3—22no—
396C3⋊Dic3C2×#6711yescharacteristic
397C3⋊Dic3C2×#6711yescharacteristic
398C3×Dic3—32no—
399C3×Dic3—32no—
400C3×Dic3—61no—
401C3×Dic3—61no—
402C3×Dic3—64no—
403C3×Dic3—62no—
404C3×Dic3—64no—
405C3×Dic3—62no—
406C3×Dic3—122no—
407C2×#41—271no—
408C22×Q8—91no—
409#137—92no—
410#137—272no—
411#137—274no—
412C2×Q16—272no—
413C2×SD16—272no—
414C2×M16—271no—
415C33C2×#13711yescharacteristic, complemented, Sylow
416C23×C3—91no—
417C22×D3—32no—
418C22×D3—61no—
419C22×D3—182no—
420C22×D3—181no—
421C3×Q8—22no—
422C3×Q8—22no—
423C3×Q8—21no—
424C3×Q8—22no—
425C3×Q8—24no—
426C3×Q8—42no—
427C3×Q8—42no—
428C3×Q8—42no—
429C3×Q8—92no—
430C3×Q8—92no—
431C3×Q8—182no—
432C3×Q8—182no—
433C3×Q8—181no—
434C3×Q8C2×D3212yes—
435C3×Q8C2×D3214yes—
436C3×D4—92no—
437C3×D4—182no—
438C2×C3×C4—22no—
439C2×C3×C4—22no—
440C2×C3×C4—21no—
441C2×C3×C4—22no—
442C2×C3×C4—21no—
443C2×C3×C4—42no—
444C2×C3×C4—42no—
445C2×C3×C4—41no—
446C2×C3×C4—91no—
447C2×C3×C4—91no—
448C2×C3×C4—91no—
449C2×C3×C4—91no—
450C2×C3×C4—91no—
451C2×C3×C4—182no—
452C2×C3×C4—182no—
453C2×C3×C4—181no—
454C2×C3×C4—182no—
455C2×C3×C4C2×D3212yes—
456C2×C3×C4C2×D3211yescharacteristic
457#67—62no—
458#67—62no—
459#67—122no—
460#67—182no—
461#67—182no—
462#67—182no—
463C2×Dic3—32no—
464C2×Dic3—61no—
465C2×Dic3—62no—
466C2×Dic3—62no—
467C2×Dic3—182no—
468C2×Dic3—181no—
469D12—32no—
470D12—32no—
471D12—62no—
472D12—181no—
473D12—182no—
474C4×D3—32no—
475C4×D3—32no—
476C4×D3—32no—
477C4×D3—32no—
478C4×D3—61no—
479C4×D3—61no—
480C4×D3—61no—
481C4×D3—61no—
482C4×D3—64no—
483C4×D3—64no—
484C4×D3—64no—
485C4×D3—64no—
486C4×D3—62no—
487C4×D3—62no—
488C4×D3—122no—
489C4×D3—182no—
490C4×D3—182no—
491C4×D3—182no—
492Dic6—34no—
493Dic6—32no—
494Dic6—32no—
495Dic6—62no—
496Dic6—62no—
497Dic6—64no—
498Dic6—64no—
499Dic6—64no—
500Dic6—64no—
501Dic6—62no—
502Dic6—62no—
503Dic6—182no—
504Dic6—181no—
505C3×C8—94no—
506C3⋊C8—34no—
507C3⋊C8—34no—
508C3⋊C8—64no—
509C2×C32—21no—
510C2×C32—22no—
511C2×C32—22no—
512C2×C32—22no—
513C2×C32—21no—
514C2×C32—41no—
515C2×C32—42no—
516C2×C32—42no—
517C2×C32C2×D3×D412yes—
518C2×C32C22×#6711yescharacteristic
519C3⋊D3—21no—
520C3⋊D3—42no—
521C3⋊D3#94212yescomplemented
522C3×D3—34no—
523C3×D3—62no—
524C3×D3—62no—
525C3×D3—62no—
526C3×D3—122no—
527C3×D3—122no—
528C3×D3—242no—
529#41—92no—
530#41—272no—
531#41—541no—
532#41—541no—
533C2×Q8—92no—
534C2×Q8—92no—
535C2×Q8—91no—
536C2×Q8—181no—
537C2×Q8—181no—
538C2×Q8—181no—
539C2×Q8—271no—
540C2×Q8#111311yescharacteristic, complemented
541C2×D4—271no—
542C2×D4—541no—
543C22×C4—91no—
544C22×C4—181no—
545C22×C4—541no—
546Q16—94no—
547Q16—274no—
548SD16—94no—
549SD16—274no—
550M16—92no—
551M16—272no—
552C2×C8—272no—
553C22×C3—21no—
554C22×C3—22no—
555C22×C3—41no—
556C22×C3—91no—
557C22×C3—91no—
558C22×C3—182no—
559C22×C3—182no—
560C22×C3D3×#6712yes—
561C22×C3#381111yescharacteristic
562C2×D3—32no—
563C2×D3—32no—
564C2×D3—61no—
565C2×D3—61no—
566C2×D3—64no—
567C2×D3—64no—
568C2×D3—62no—
569C2×D3—62no—
570C2×D3—122no—
571C2×D3—364no—
572C2×D3—362no—
573C3×C4—24no—
574C3×C4—24no—
575C3×C4—22no—
576C3×C4—22no—
577C3×C4—21no—
578C3×C4—21no—
579C3×C4—22no—
580C3×C4—44no—
581C3×C4—44no—
582C3×C4—41no—
583C3×C4—41no—
584C3×C4—91no—
585C3×C4—91no—
586C3×C4—182no—
587C3×C4—182no—
588C3×C4D3×#6712yes—
589C3×C4#381111yescharacteristic
590C3×C4C22×D3212yes—
591C3×C4C22×D3211yescharacteristic
592Dic3—32no—
593Dic3—32no—
594Dic3—61no—
595Dic3—61no—
596Dic3—64no—
597Dic3—64no—
598Dic3—62no—
599Dic3—62no—
600Dic3—122no—
601C32—22no—
602C32—21no—
603C32—41no—
604C32D3×#13712yescomplemented
605C32C2×#94211yescharacteristic, complemented
606C23—91no—
607C23—541no—
608Q8—21no—
609Q8—92no—
610Q8—92no—
611Q8—182no—
612Q8—182no—
613Q8—181no—
614Q8—541no—
615Q8C2×#111312yes—
616D4—92no—
617D4—182no—
618D4—541no—
619D4—542no—
620C2×C4—21no—
621C2×C4—91no—
622C2×C4—91no—
623C2×C4—91no—
624C2×C4—91no—
625C2×C4—91no—
626C2×C4—182no—
627C2×C4—182no—
628C2×C4—181no—
629C2×C4—182no—
630C2×C4—542no—
631C2×C4—541no—
632C2×C4C2×#111311yescharacteristic
633C8—94no—
634C2×C3—22no—
635C2×C3—22no—
636C2×C3—21no—
637C2×C3—21no—
638C2×C3—22no—
639C2×C3—42no—
640C2×C3—42no—
641C2×C3—41no—
642C2×C3—92no—
643C2×C3—181no—
644C2×C3—362no—
645C2×C3C2×D3×#6712yes—
646C2×C3C2×#381111yescharacteristic
647D3—34no—
648D3—62no—
649D3—62no—
650D3—62no—
651D3—122no—
652D3—122no—
653D3—242no—
654C22—91no—
655C22—91no—
656C22—182no—
657C22—182no—
658C22—1082no—
659C22#8904911yescharacteristic
660C4—22no—
661C4—91no—
662C4—91no—
663C4—182no—
664C4—182no—
665C4#8904911yescharacteristic
666C4C22×#111311yescharacteristic
667C3—21no—
668C3—22no—
669C3—41no—
670C3D3×#94212yescomplemented
671C3#1059532211yescharacteristic, complemented
672C2—21no—
673C2—92no—
674C2—181no—
675C2—362no—
676C2C2×#8904911yescharacteristic
subgroupG/H
centerC2C2×#89049
commutator, cosocleC33×C4C24
FrattiniC4C22×#1113
FittingC2×C33×Q8C22
socleC2×C33C22×D4
prime radical#11724618C2
hypercenterC2×Q8#1113
hypocenterC33C2×#137
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×Q8×C3⋊D3C2semidirect
3. C3×#64151C2semidirect
4. #11723712C2semidirect
5. #11723710C2semidirect
6. #11723708C2semidirect
7. #11723663C2semidirect
8. #11723656C2semidirect
11. C2×C33×Q8C22semidirect
15. C3×Q8×C3⋊D3C22semidirect
20. C2×C3×#634C22semidirect
28. C3×#3756C22semidirect
29. C3×#3755C22semidirect
30. C3×#3748C22semidirect
31. #88812C22semidirect
32. #88811C22semidirect
33. #88809C22semidirect
35. C2×Q8×C3⋊D3D3semidirect
40. #64151D3semidirect
89. C2×C32×Q8C2×D3semidirect
90. C2×C32×Q8C2×D3semidirect
94. Q8×C3⋊D3C2×D3semidirect
99. C2×#634C2×D3semidirect
134. #3756C2×D3semidirect
135. #3755C2×D3semidirect
136. #3748C2×D3semidirect
288. C3×C3⋊D3#137semidirect
299. C2×C3×Q8D32semidirect
300. C2×C3×Q8D32semidirect
415. C33C2×#137semidirect
521. C3⋊D3#942semidirect
540. C2×Q8#1113semidirect
604. C32D3×#137semidirect
605. C32C2×#942semidirect
670. C3D3×#942semidirect
671. C3#10595322semidirect

Maximal subgroups

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

subgroupindex
C2×C3×Q8×C3⋊D32
#117246182
C3×#641512
#117237122
#117237102
#117237082
#117236632
#117236562
#105953223
D3×#9423