Finite Group Databasebeta

← #13178668

Subgroups — #13178668

Order 1728.

1642 conjugacy classes of subgroups.

The subgroup lattice has 718 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3830C211yesmaximal, characteristic, complemented
2C2×#89041C211yesmaximal, characteristic, complemented
3C3×#64149C211yesmaximal, characteristic, complemented
4C3×#64145C211yesmaximal, characteristic, complemented
5#11723726C214yesmaximal, complemented
6#11723725C214yesmaximal, complemented
7C4×#6294C211yesmaximal, characteristic, complemented
8#11723678C211yesmaximal, characteristic, complemented
9#11723677C211yesmaximal, characteristic, complemented
10#10595167—31nomaximal
11#10595153—31nomaximal
12#10595152—31nomaximal
13C2×C33×D4C2211yescharacteristic
14C2×C3×#638C2214yes—
15C2×C3×#638C2212yes—
16C22×C3×C3⋊Dic3C2212yes—
17C3×#3830—24no—
18C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
19C2×C3×C4×C3⋊D3C2211yescharacteristic
20C2×C3×#634C2211yescharacteristic
21C2×C3×#634C2211yescharacteristic
22C2×#6295C2212yes—
23C2×#6294C2212yes—
24C2×#6294C2212yes—
25#89041—24no—
26C3×#3757C2214yes—
27C3×C4×C3⋊Dic3C2211yescharacteristic
28C3×C4×C3⋊Dic3C2211yescharacteristic
29#88815C2214yes—
30#88814C2214yes—
31#88813C2212yes—
32C2×#3830D311yescharacteristic, complemented
33C2×C3×#237—31no—
34C2×C3×#237—31no—
35C2×C3×#237—61no—
36C2×#3799—31no—
37C2×#3796—31no—
38C2×#3795—31no—
39#64149D311yescharacteristic, complemented
40#64145D311yescharacteristic, complemented
41C3×#940—31no—
42C3×#940—31no—
43C3×#940—61no—
44C3×#936—31no—
45C3×#936—31no—
46C3×#936—61no—
47#67⋊Dic3—34no—
48Dic3×#67—34no—
49#63971—34no—
50#63970—34no—
51#63958—34no—
52#63952—34no—
53C4×#624—31no—
54C4×D3×Dic3—31no—
55C4×D3×Dic3—31no—
56#63873—31no—
57#63866—31no—
58#63866—31no—
59#63864—31no—
60#63863—31no—
61#63860—31no—
62C23×C33C2312yes—
63C22×C3×C3⋊D3C2312yes—
64C22×C3×C3⋊D3C2311yescharacteristic
65C33×D4—22no—
66C2×C33×C4C2311yescharacteristic
67C3×#638—24no—
68C3×#638—28no—
69C2×C3×C3⋊Dic3—24no—
70C2×C3×C3⋊Dic3C2312yes—
71C2×C3×C3⋊Dic3C2312yes—
72C2×C3×C3⋊Dic3C2312yes—
73C2×C3×C3⋊Dic3C2311yescharacteristic
74C2×C3×C3⋊Dic3C2312yes—
75C3×C4⋊(C3⋊D3)—22no—
76C3×C4×C3⋊D3D414yescomplemented
77C3×#634—22no—
78C3×#634—22no—
79#6295—24no—
80#6294—24no—
81#6294—24no—
82#5832—91no—
83#5716—91no—
84#5668—91no—
85C2×C32×D4—21no—
86C2×C32×D4—21no—
87C2×C32×D4—21no—
88C2×C32×D4—41no—
89C2×C32×D4C2×D311yescharacteristic
90C2×C32×D4C2×D311yescharacteristic
91C2×C32×D4C2×D311yescharacteristic
92C2×#638C2×D312yes—
93C2×#638C2×D314yes—
94C22×C3⋊Dic3C2×D312yes—
95#3830—24no—
96C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
97C2×C4×C3⋊D3C2×D311yescharacteristic
98C2×#634C2×D311yescharacteristic
99C2×#634C2×D311yescharacteristic
100C2×C3×#67—34no—
101C2×C3×#67—34no—
102C2×C3×#67—32no—
103C2×C3×#67—32no—
104C2×C3×#67—62no—
105C2×C3×#67—64no—
106C22×C3×Dic3—32no—
107C22×C3×Dic3—32no—
108C22×C3×Dic3—62no—
109C3×#237—64no—
110C3×#237—64no—
111C3×#237—68no—
112C2×C3×D12—31no—
113C2×C3×D12—31no—
114C2×C3×D12—61no—
115C2×C3×C4×D3—31no—
116C2×C3×C4×D3—31no—
117C2×C3×C4×D3—61no—
118C2×C3×Dic6—31no—
119C2×C3×Dic6—31no—
120C2×C3×Dic6—31no—
121C2×C3×Dic6—31no—
122C2×C3×Dic6—61no—
123C2×C3×Dic6—61no—
124C2×#626—32no—
125C2×#626—32no—
126C2×#625—32no—
127C2×#624—32no—
128C2×#624—32no—
129C2×D3×Dic3—32no—
130C2×D3×Dic3—32no—
131C2×D3×Dic3—32no—
132C2×D3×Dic3—32no—
133#3799—64no—
134#3796—64no—
135#3795—64no—
136#3757C2×D314yes—
137C4×C3⋊Dic3C2×D311yescharacteristic
138C4×C3⋊Dic3C2×D311yescharacteristic
139C3×#217—34no—
140C3×#217—34no—
141C3×#217—64no—
142C3×C4×Dic3—31no—
143C3×C4×Dic3—31no—
144C3×C4×Dic3—31no—
145C3×C4×Dic3—31no—
146C3×C4×Dic3—61no—
147C3×C4×Dic3—61no—
148#3724—34no—
149Dic3⋊Dic3—34no—
150Dic3⋊Dic3—34no—
151#3722—34no—
152(C2×D3)⋊Dic3—34no—
153(C2×D3)⋊Dic3—34no—
154Dic32—32no—
155Dic32—32no—
156Dic32—32no—
157C22×C33—22no—
158C22×C33—24no—
159C22×C33C2411yescharacteristic
160C2×C3×C3⋊D3—22no—
161C2×C3×C3⋊D3—22no—
162C2×C3×C3⋊D3—24no—
163C2×C3×C3⋊D3C2×D412yes—
164C33×C4C2×D412yes—
165C3×C3⋊Dic3—22no—
166C3×C3⋊Dic3—22no—
167C3×C3⋊Dic3—22no—
168C3×C3⋊Dic3C2×D412yes—
169C3×C3⋊Dic3#4114yescomplemented
170C2×C3×#41—91no—
171C2×#239—91no—
172C2×#237—31no—
173C2×#237—31no—
174C2×#237—61no—
175C2×#237—91no—
176C2×#235—91no—
177C3×#127—91no—
178C3×#124—91no—
179#947—91no—
180#944—91no—
181#944—91no—
182#940—31no—
183#940—31no—
184#940—61no—
185#939—94no—
186#938—94no—
187#936—31no—
188#936—31no—
189#936—61no—
190#936—91no—
191D4×Dic3—94no—
192C4×#67—94no—
193C2×C4×Dic3—91no—
194#884—94no—
195#67⋊C4—94no—
196#875—91no—
197C42×D3—91no—
198C42×D3—91no—
199#869—91no—
200C23×C32—22no—
201C23×C32—22no—
202C23×C32—22no—
203C23×C32—42no—
204C23×C32C22×D312yes—
205C23×C32C22×D312yes—
206C23×C32C22×D312yes—
207C22×C3⋊D3C22×D312yes—
208C22×C3⋊D3C22×D311yescharacteristic
209C22×C3×D3—31no—
210C22×C3×D3—31no—
211C22×C3×D3—32no—
212C22×C3×D3—32no—
213C22×C3×D3—61no—
214C22×C3×D3—62no—
215C32×D4—22no—
216C32×D4—24no—
217C32×D4—22no—
218C32×D4—22no—
219C32×D4—42no—
220C32×D4—42no—
221C32×D4—44no—
222C2×C32×C4—21no—
223C2×C32×C4—21no—
224C2×C32×C4—21no—
225C2×C32×C4—41no—
226C2×C32×C4C22×D311yescharacteristic
227C2×C32×C4C22×D311yescharacteristic
228C2×C32×C4C22×D311yescharacteristic
229#638—24no—
230#638—28no—
231C2×C3⋊Dic3—24no—
232C2×C3⋊Dic3C22×D312yes—
233C2×C3⋊Dic3C22×D312yes—
234C2×C3⋊Dic3C22×D312yes—
235C2×C3⋊Dic3C22×D312yes—
236C2×C3⋊Dic3C22×D311yescharacteristic
237C4⋊(C3⋊D3)—22no—
238C4×C3⋊D3#6714yescomplemented
239#634—22no—
240#634—22no—
241C3×#67—68no—
242C3×#67—68no—
243C3×#67—64no—
244C3×#67—64no—
245C3×#67—68no—
246C3×#67—128no—
247C2×C3×Dic3—31no—
248C2×C3×Dic3—31no—
249C2×C3×Dic3—32no—
250C2×C3×Dic3—32no—
251C2×C3×Dic3—32no—
252C2×C3×Dic3—32no—
253C2×C3×Dic3—32no—
254C2×C3×Dic3—32no—
255C2×C3×Dic3—32no—
256C2×C3×Dic3—32no—
257C2×C3×Dic3—62no—
258C2×C3×Dic3—62no—
259C2×C3×Dic3—61no—
260C2×C3×Dic3—62no—
261C2×C3×Dic3—62no—
262C2×C3×Dic3—64no—
263C2×C3×Dic3—64no—
264C2×C3×Dic3—68no—
265C3×D12—62no—
266C3×D12—62no—
267C3×D12—122no—
268C3×C4×D3—34no—
269C3×C4×D3—34no—
270C3×C4×D3—64no—
271C3×Dic6—64no—
272C3×Dic6—62no—
273C3×Dic6—62no—
274C3×Dic6—62no—
275C3×Dic6—62no—
276C3×Dic6—122no—
277#626—64no—
278#626—64no—
279#625—64no—
280#624—64no—
281#624—64no—
282D3×Dic3—64no—
283D3×Dic3—64no—
284D3×Dic3—64no—
285D3×Dic3—64no—
286#532—271noSylow
287C2×C33—42no—
288C2×C33C22×D411yescharacteristic
289C2×C33C2×#4112yes—
290C3×C3⋊D3—22no—
291C3×C3⋊D3—42no—
292C3×#41—184no—
293C2×C3×Q8—91no—
294C2×C3×Q8—91no—
295C2×C3×D4—21no—
296C2×C3×D4—21no—
297C2×C3×D4—21no—
298C2×C3×D4—41no—
299C2×C3×D4—94no—
300C2×C3×D4—92no—
301C2×C3×D4—91no—
302C2×C3×D4D3211yescharacteristic
303C2×C3×D4D3211yescharacteristic
304C2×C3×D4D3211yescharacteristic
305C22×C3×C4—91no—
306C22×C3×C4—92no—
307C2×#67—34no—
308C2×#67—32no—
309C2×#67—34no—
310C2×#67—32no—
311C2×#67—62no—
312C2×#67—64no—
313C2×#67—92no—
314C2×#67—92no—
315C22×Dic3—32no—
316C22×Dic3—32no—
317C22×Dic3—62no—
318C22×Dic3—92no—
319C22×Dic3—92no—
320#239—184no—
321#237—64no—
322#237—64no—
323#237—68no—
324#237—184no—
325#235—184no—
326C2×D12—31no—
327C2×D12—31no—
328C2×D12—61no—
329C2×D12—92no—
330C2×C4×D3—31no—
331C2×C4×D3—31no—
332C2×C4×D3—61no—
333C2×C4×D3—92no—
334C2×C4×D3—92no—
335C2×C4×D3—92no—
336C2×C4×D3—92no—
337C2×Dic6—31no—
338C2×Dic6—31no—
339C2×Dic6—31no—
340C2×Dic6—31no—
341C2×Dic6—61no—
342C2×Dic6—61no—
343C3×C22⋊C4—94no—
344C3×C42—91no—
345C3×C42—91no—
346#217—34no—
347#217—34no—
348#217—64no—
349#217—94no—
350(C2×D3)⋊C4—94no—
351(C2×D3)⋊C4—94no—
352C4⋊Dic3—94no—
353Dic3⋊C4—94no—
354Dic3⋊C4—94no—
355C4×Dic3—31no—
356C4×Dic3—31no—
357C4×Dic3—31no—
358C4×Dic3—31no—
359C4×Dic3—61no—
360C4×Dic3—61no—
361C4×Dic3—92no—
362C4×Dic3—92no—
363C4×Dic3—92no—
364C22×C32—22no—
365C22×C32—24no—
366C22×C32—22no—
367C22×C32—22no—
368C22×C32—24no—
369C22×C32—24no—
370C22×C32—24no—
371C22×C32—21no—
372C22×C32—21no—
373C22×C32—21no—
374C22×C32—41no—
375C22×C32—42no—
376C22×C32—42no—
377C22×C32—44no—
378C22×C32—44no—
379C22×C32—44no—
380C22×C32—44no—
381C22×C32—84no—
382C22×C32C23×D311yescharacteristic
383C22×C32C23×D311yescharacteristic
384C22×C32C23×D311yescharacteristic
385C2×C3⋊D3—22no—
386C2×C3⋊D3—22no—
387C2×C3⋊D3—24no—
388C2×C3⋊D3C2×#6712yes—
389C2×C3×D3—32no—
390C2×C3×D3—32no—
391C2×C3×D3—62no—
392C2×C3×D3—62no—
393C2×C3×D3—62no—
394C2×C3×D3—62no—
395C2×C3×D3—64no—
396C2×C3×D3—64no—
397C2×C3×D3—62no—
398C2×C3×D3—122no—
399C2×C3×D3—122no—
400C2×C3×D3—124no—
401C32×C4—22no—
402C32×C4—22no—
403C32×C4—22no—
404C32×C4—42no—
405C32×C4D3×D412yes—
406C32×C4D3×D412yes—
407C32×C4C2×#6712yes—
408C3⋊Dic3—22no—
409C3⋊Dic3—22no—
410C3⋊Dic3—22no—
411C3⋊Dic3#23714yescomplemented
412C3⋊Dic3C2×#6712yes—
413C3×Dic3—32no—
414C3×Dic3—32no—
415C3×Dic3—34no—
416C3×Dic3—34no—
417C3×Dic3—62no—
418C3×Dic3—64no—
419C3×Dic3—64no—
420C3×Dic3—62no—
421C3×Dic3—62no—
422C3×Dic3—62no—
423C3×Dic3—62no—
424C3×Dic3—62no—
425C3×Dic3—62no—
426C3×Dic3—122no—
427C3×Dic3—122no—
428C2×#41—91no—
429C2×#41—271no—
430C4⋊Q8—271no—
431#127—91no—
432#124—91no—
433#124—271no—
434#121—274no—
435C4×D4—274no—
436C2×C42—271no—
437C33#53211yescharacteristic, complemented, Sylow
438C23×C3—22no—
439C23×C3—22no—
440C23×C3—22no—
441C23×C3—42no—
442C23×C3—91no—
443C23×C3—92no—
444C23×C3C2×D3212yes—
445C23×C3C2×D3212yes—
446C23×C3C2×D3212yes—
447C22×D3—31no—
448C22×D3—31no—
449C22×D3—32no—
450C22×D3—32no—
451C22×D3—61no—
452C22×D3—62no—
453C3×Q8—182no—
454C3×Q8—182no—
455C3×D4—24no—
456C3×D4—22no—
457C3×D4—22no—
458C3×D4—22no—
459C3×D4—42no—
460C3×D4—42no—
461C3×D4—44no—
462C3×D4—188no—
463C3×D4—184no—
464C3×D4—182no—
465C2×C3×C4—21no—
466C2×C3×C4—21no—
467C2×C3×C4—21no—
468C2×C3×C4—41no—
469C2×C3×C4—91no—
470C2×C3×C4—92no—
471C2×C3×C4—94no—
472C2×C3×C4—92no—
473C2×C3×C4—92no—
474C2×C3×C4—92no—
475C2×C3×C4—184no—
476C2×C3×C4C2×D3211yescharacteristic
477C2×C3×C4C2×D3211yescharacteristic
478C2×C3×C4C2×D3211yescharacteristic
479#67—68no—
480#67—68no—
481#67—64no—
482#67—64no—
483#67—68no—
484#67—128no—
485#67—184no—
486#67—184no—
487C2×Dic3—32no—
488C2×Dic3—32no—
489C2×Dic3—31no—
490C2×Dic3—31no—
491C2×Dic3—32no—
492C2×Dic3—32no—
493C2×Dic3—32no—
494C2×Dic3—32no—
495C2×Dic3—32no—
496C2×Dic3—32no—
497C2×Dic3—62no—
498C2×Dic3—62no—
499C2×Dic3—61no—
500C2×Dic3—62no—
501C2×Dic3—62no—
502C2×Dic3—64no—
503C2×Dic3—64no—
504C2×Dic3—68no—
505C2×Dic3—184no—
506C2×Dic3—184no—
507D12—62no—
508D12—62no—
509D12—122no—
510D12—184no—
511C4×D3—34no—
512C4×D3—34no—
513C4×D3—64no—
514C4×D3—184no—
515C4×D3—184no—
516C4×D3—184no—
517C4×D3—184no—
518Dic6—64no—
519Dic6—62no—
520Dic6—62no—
521Dic6—62no—
522Dic6—62no—
523Dic6—122no—
524C2×C32—22no—
525C2×C32—22no—
526C2×C32—22no—
527C2×C32—21no—
528C2×C32—21no—
529C2×C32—21no—
530C2×C32—41no—
531C2×C32—42no—
532C2×C32—42no—
533C2×C32—44no—
534C2×C32—42no—
535C2×C32—42no—
536C2×C32—82no—
537C2×C32—82no—
538C2×C32—84no—
539C2×C32C2×D3×D411yescharacteristic
540C2×C32C2×D3×D411yescharacteristic
541C2×C32C2×#23712yes—
542C2×C32D3×#4112yes—
543C2×C32D3×#4112yes—
544C2×C32C22×#6711yescharacteristic
545C3⋊D3—22no—
546C3⋊D3—42no—
547C3×D3—62no—
548C3×D3—62no—
549C3×D3—122no—
550C3×D3—122no—
551C3×D3—122no—
552C3×D3—242no—
553#41—184no—
554#41—544no—
555C2×Q8—91no—
556C2×Q8—91no—
557C2×D4—94no—
558C2×D4—92no—
559C2×D4—91no—
560C2×D4—272no—
561C2×D4#111311yescharacteristic
562C22×C4—91no—
563C22×C4—92no—
564C22×C4—272no—
565C22×C4—272no—
566C4⋊C4—274no—
567C22⋊C4—94no—
568C22⋊C4—274no—
569C42—91no—
570C42—91no—
571C42—272no—
572C22×C3—24no—
573C22×C3—24no—
574C22×C3—24no—
575C22×C3—24no—
576C22×C3—22no—
577C22×C3—22no—
578C22×C3—22no—
579C22×C3—21no—
580C22×C3—21no—
581C22×C3—21no—
582C22×C3—44no—
583C22×C3—44no—
584C22×C3—42no—
585C22×C3—42no—
586C22×C3—44no—
587C22×C3—44no—
588C22×C3—41no—
589C22×C3—84no—
590C22×C3—92no—
591C22×C3—182no—
592C22×C3—184no—
593C22×C3—182no—
594C22×C3C22×D3211yescharacteristic
595C22×C3C22×D3211yescharacteristic
596C22×C3C22×D3211yescharacteristic
597C2×D3—32no—
598C2×D3—32no—
599C2×D3—62no—
600C2×D3—62no—
601C2×D3—62no—
602C2×D3—64no—
603C2×D3—64no—
604C2×D3—62no—
605C2×D3—62no—
606C2×D3—122no—
607C2×D3—122no—
608C2×D3—124no—
609C3×C4—22no—
610C3×C4—22no—
611C3×C4—22no—
612C3×C4—42no—
613C3×C4—94no—
614C3×C4—92no—
615C3×C4—182no—
616C3×C4—182no—
617C3×C4—182no—
618C3×C4D3×#6712yes—
619C3×C4D3×#6712yes—
620C3×C4#381112yes—
621Dic3—34no—
622Dic3—34no—
623Dic3—32no—
624Dic3—32no—
625Dic3—62no—
626Dic3—64no—
627Dic3—64no—
628Dic3—62no—
629Dic3—62no—
630Dic3—62no—
631Dic3—62no—
632Dic3—62no—
633Dic3—62no—
634Dic3—122no—
635Dic3—122no—
636C32—21no—
637C32—21no—
638C32—21no—
639C32—41no—
640C32#566811yescharacteristic, complemented
641C32#571611yescharacteristic, complemented
642C32#583211yescharacteristic, complemented
643C23—91no—
644C23—92no—
645C23C2×#111312yes—
646Q8—182no—
647Q8—182no—
648D4—22no—
649D4—188no—
650D4—184no—
651D4—182no—
652D4—544no—
653C2×C4—92no—
654C2×C4—91no—
655C2×C4—94no—
656C2×C4—92no—
657C2×C4—92no—
658C2×C4—92no—
659C2×C4—184no—
660C2×C4—544no—
661C2×C4—544no—
662C2×C4C2×#111311yescharacteristic
663C2×C3—22no—
664C2×C3—22no—
665C2×C3—22no—
666C2×C3—21no—
667C2×C3—21no—
668C2×C3—21no—
669C2×C3—44no—
670C2×C3—42no—
671C2×C3—42no—
672C2×C3—42no—
673C2×C3—42no—
674C2×C3—41no—
675C2×C3—82no—
676C2×C3—82no—
677C2×C3—84no—
678C2×C3—182no—
679C2×C3—362no—
680C2×C3D3×#23712yes—
681C2×C3D3×#23712yes—
682C2×C3#6431212yes—
683C2×C3C2×D3×#6711yescharacteristic
684C2×C3C2×D3×#6711yescharacteristic
685C2×C3C2×#381111yescharacteristic
686D3—62no—
687D3—62no—
688D3—122no—
689D3—122no—
690D3—122no—
691D3—242no—
692C22—24no—
693C22—22no—
694C22—92no—
695C22—182no—
696C22—184no—
697C22—182no—
698C22C22×#111311yescharacteristic
699C4—94no—
700C4—92no—
701C4—182no—
702C4—182no—
703C4—182no—
704C4#8904912yes—
705C3—21no—
706C3—21no—
707C3—21no—
708C3—41no—
709C3#1059515211yescharacteristic, complemented
710C3#1059515311yescharacteristic, complemented
711C3#1059516711yescharacteristic, complemented
712C2—42no—
713C2—182no—
714C2—362no—
715C2#1172462212yes—
716C2C2×#8904911yescharacteristic
subgroupG/H
center, FrattiniC22C22×#1113
commutator, socle, cosocleC22×C33C24
FittingC2×C33×D4C22
hypercenterC2×D4#1113
hypocenterC33#532
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×C3×#3830C2semidirect
2. C2×#89041C2semidirect
3. C3×#64149C2semidirect
4. C3×#64145C2semidirect
5. #11723726C2semidirect
6. #11723725C2semidirect
7. C4×#6294C2semidirect
8. #11723678C2semidirect
9. #11723677C2semidirect
32. C2×#3830D3semidirect
39. #64149D3semidirect
40. #64145D3semidirect
76. C3×C4×C3⋊D3D4semidirect
169. C3×C3⋊Dic3#41semidirect
238. C4×C3⋊D3#67semidirect
411. C3⋊Dic3#237semidirect
437. C33#532semidirect
640. C32#5668semidirect
641. C32#5716semidirect
642. C32#5832semidirect
709. C3#10595152semidirect
710. C3#10595153semidirect
711. C3#10595167semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#38302
C2×#890412
C3×#641492
C3×#641452
#117237262
#117237252
C4×#62942
#117236782
#117236772
#105951673
#105951533
#105951523