Finite Group Databasebeta

← #13172316

Subgroups — #13172316

Order 1728.

2014 conjugacy classes of subgroups; only the 196 normal subgroups are shown.

normal subgroups only

This group has 2014 conjugacy classes of subgroups — too many to draw. Only its 196 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6292C211yesmaximal, characteristic, complemented
2C2×C3×#3811C211yesmaximal, characteristic, complemented
3#11723723C211yesmaximal, characteristic, complemented
4#11723638C212yesmaximal, complemented
5C3×#63961C212yesmaximal, complemented
6#11722993C212yesmaximal, complemented
7#11722966C214yesmaximal, complemented
8#11722948C212yesmaximal, complemented
9#10595234—32nomaximal
10#10595080—31nomaximal
11C23×C3⋊(C3⋊D3)C2211yescharacteristic, complemented
12C23×C3×C3⋊D3C2211yescharacteristic
13C22×C3×D32C2211yescharacteristic, complemented
14C2×C32×#67C2212yes—
15C22×C32×Dic3C2211yescharacteristic, complemented
16C2×#6294C2212yescomplemented
17C2×#6292—24no—
18C2×#6292C2212yes—
19C2×#6292C2212yescomplemented
20C2×#6291C2214yescomplemented
21C3×#3811—24no—
22C2×C3×#626C2212yescomplemented
23C2×C3×#624C2211yescharacteristic, complemented
24C3×#3757C2212yescomplemented
25#88814C2212yescomplemented
26#88804C2214yescomplemented
27C3×#3724C2214yescomplemented
28C3×(C2×D3)⋊Dic3C2214yescomplemented
29C22×C4⋊(C3⋊D3)—31no—
30C2×C3×D3×D4—32no—
31C2×#3811D311yescharacteristic, complemented
32C22×#626—32no—
33C22×#626—61no—
34#64146—32no—
35C3×#935—32no—
36C3×#929—32no—
37#63981—32no—
38#63981—32no—
39#63981—62no—
40#63977—34no—
41#63974—31no—
42#63971—34no—
43D3×#217—32no—
44#63961D312yescomplemented
45#63923—32no—
46#63910—34no—
47#63897—32no—
48#63886—34no—
49#63850—34no—
50C23×C33C2311yescharacteristic
51C22×C3⋊(C3⋊D3)—22no—
52C22×C3⋊(C3⋊D3)—42no—
53C22×C3⋊(C3⋊D3)C2312yes—
54C22×C3×C3⋊D3—42no—
55C22×C3×C3⋊D3D414yescomplemented
56C22×C3×C3⋊D3C2311yescharacteristic
57C22×C3×C3⋊D3C2311yescharacteristic
58C22×C32×D3C2312yes—
59C2×C3×D32—22no—
60C2×C3×D32—42no—
61C2×C3×C3⋊Dic3C2314yes—
62C32×#67—24no—
63C2×C32×Dic3—22no—
64C2×C32×Dic3C2312yes—
65C2×C32×Dic3C2312yes—
66#6294—42no—
67#6292—24no—
68#6292—24no—
69#6291—44no—
70C3×#626—24no—
71C3×#624—22no—
72#5804—91no—
73#5653—92no—
74C23×C3⋊D3—32no—
75C23×C3⋊D3—31no—
76C23×C3⋊D3—61no—
77C23×C3⋊D3—62no—
78C23×C3⋊D3—121no—
79C23×C3⋊D3C2×D311yescharacteristic
80C23×C3×D3—32no—
81C23×C3×D3—32no—
82C23×C3×D3—61no—
83C22×D32C2×D311yescharacteristic, complemented
84C2×C32×D4—32no—
85C22×C32×C4—31no—
86C2×#638—34no—
87C2×C4⋊(C3⋊D3)—32no—
88C2×C4⋊(C3⋊D3)—32no—
89C2×C4⋊(C3⋊D3)—64no—
90C2×C3×#67—22no—
91C2×C3×#67—32no—
92C2×C3×#67C2×D312yes—
93C2×C3×#67C2×D312yes—
94C22×C3×Dic3—21no—
95C22×C3×Dic3C2×D312yescomplemented
96C3×D3×D4—68no—
97C2×C3×D12—32no—
98C2×C3×C4×D3—32no—
99#3811—24no—
100C2×#626—34no—
101C2×#626—34no—
102C2×#626—34no—
103C2×#626—34no—
104C2×#626—68no—
105C2×#626—62no—
106C2×#626—62no—
107C2×#626—68no—
108C2×#626—64no—
109C2×#626C2×D312yescomplemented
110C2×#624—32no—
111C2×#624C2×D311yescharacteristic, complemented
112C2×D3×Dic3—34no—
113#3757C2×D312yescomplemented
114#3752—34no—
115C3×#217—32no—
116C3×#217—32no—
117C3×#217—34no—
118C3×#217—62no—
119C3×(C2×D3)⋊C4—34no—
120C3×Dic3⋊C4—34no—
121C3×Dic3⋊C4—34no—
122#3724C2×D314yescomplemented
123#3722—32no—
124#3722—34no—
125#3722—34no—
126#3722—64no—
127(C2×D3)⋊Dic3—34no—
128(C2×D3)⋊Dic3C2×D314yescomplemented
129C22×C33—22no—
130C22×C33C2×D412yes—
131C22×C33C2411yescharacteristic
132C2×C3⋊(C3⋊D3)—22no—
133C2×C3⋊(C3⋊D3)—24no—
134C2×C3⋊(C3⋊D3)—44no—
135C2×C3×C3⋊D3—22no—
136C2×C3×C3⋊D3—22no—
137C2×C3×C3⋊D3—44no—
138C2×C3×C3⋊D3C2×D412yes—
139C2×C3×C3⋊D3C2×D412yes—
140C2×C32×D3—22no—
141C2×C32×D3—24no—
142C3×D32—44no—
143C3×C3⋊Dic3—24no—
144C32×Dic3—22no—
145C32×Dic3—22no—
146C22×C3×D4—91no—
147C22×#67—91no—
148C2×D3×D4—32no—
149C22×D12—92no—
150C22×D12—181no—
151C3×#123—92no—
152#953—92no—
153C2×#217—91no—
154#939—94no—
155#937—92no—
156#937—92no—
157#937—182no—
158#935—32no—
159#929—32no—
160#929—92no—
161#895—94no—
162#894—94no—
163#884—94no—
164#882—94no—
165#882—92no—
166D3×C22⋊C4—92no—
167C23×C32—21no—
168C23×C32—22no—
169C23×C32—32no—
170C23×C32—41no—
171C23×C32C22×D312yes—
172C23×C32C22×D311yescharacteristic
173C22×C3⋊D3—32no—
174C22×C3⋊D3—34no—
175C22×C3⋊D3—42no—
176C22×C3⋊D3—64no—
177C22×C3⋊D3—62no—
178C22×C3⋊D3—62no—
179C22×C3⋊D3—64no—
180C22×C3⋊D3—64no—
181C22×C3⋊D3—122no—
182C22×C3⋊D3—124no—
183C22×C3⋊D3—124no—
184C22×C3⋊D3—124no—
185C22×C3⋊D3—122no—
186C22×C3⋊D3—128no—
187C22×C3⋊D3—124no—
188C22×C3⋊D3—128no—
189C22×C3⋊D3#6714yescomplemented
190C22×C3⋊D3C22×D311yescharacteristic
191C22×C3⋊D3C22×D311yescharacteristic
192C22×C3×D3—22no—
193C22×C3×D3—32no—
194C22×C3×D3—32no—
195C22×C3×D3—38no—
196C22×C3×D3—64no—
197C22×C3×D3—61no—
198C22×C3×D3—61no—
199C22×C3×D3—64no—
200C22×C3×D3—124no—
201C22×C3×D3—124no—
202C22×C3×D3—124no—
203C22×C3×D3C22×D312yes—
204C22×C3×D3C22×D312yes—
205C2×D32—22no—
206C2×D32—42no—
207C32×D4—64no—
208C2×C32×C4—32no—
209C2×C32×C4—32no—
210C2×C32×C4—62no—
211#638—124no—
212C2×C3⋊Dic3C22×D314yes—
213C4⋊(C3⋊D3)—64no—
214C4⋊(C3⋊D3)—64no—
215C3×#67—24no—
216C3×#67—24no—
217C3×#67—44no—
218C3×#67—64no—
219C2×C3×Dic3—24no—
220C2×C3×Dic3—24no—
221C2×C3×Dic3—22no—
222C2×C3×Dic3—22no—
223C2×C3×Dic3—34no—
224C2×C3×Dic3—34no—
225C2×C3×Dic3—64no—
226C2×C3×Dic3C22×D314yes—
227C2×C3×Dic3C22×D312yes—
228C2×C3×Dic3C22×D312yes—
229C3×D12—64no—
230C3×C4×D3—64no—
231#626—24no—
232#626—68no—
233#626—68no—
234#626—68no—
235#626—68no—
236#626—124no—
237#626—124no—
238#626—128no—
239#624—22no—
240#624—122no—
241D3×Dic3—124no—
242#534—271noSylow
243C2×C33—22no—
244C2×C33C22×D411yescharacteristic
245C2×C33H(5,𝔽2)12yes—
246C3⋊(C3⋊D3)—42no—
247C3×C3⋊D3—22no—
248C3×C3⋊D3—22no—
249C32×D3—42no—
250C24×C3—91no—
251C24×C3—91no—
252C23×D3—32no—
253C23×D3—32no—
254C23×D3—61no—
255C23×D3—92no—
256C23×D3—91no—
257C23×D3—181no—
258C23×D3—182no—
259C23×D3—361no—
260C2×C3×D4—32no—
261C2×C3×D4—32no—
262C2×C3×D4—62no—
263C2×C3×D4—92no—
264C2×C3×D4—184no—
265C22×C3×C4—32no—
266C22×C3×C4—61no—
267C22×C3×C4—91no—
268C2×#67—32no—
269C2×#67—94no—
270C2×#67—94no—
271C2×#67—92no—
272C2×#67—92no—
273C2×#67—184no—
274C2×#67—184no—
275C2×#67D3212yes—
276C22×Dic3—92no—
277C22×Dic3D3211yescharacteristic, complemented
278D3×D4—68no—
279C2×D12—32no—
280C2×D12—94no—
281C2×D12—94no—
282C2×D12—94no—
283C2×D12—182no—
284C2×D12—182no—
285C2×D12—188no—
286C2×D12—188no—
287C2×C4×D3—32no—
288C2×C4×D3—94no—
289C3×C4⋊C4—94no—
290C3×C22⋊C4—94no—
291C3×C22⋊C4—92no—
292#217—32no—
293#217—32no—
294#217—34no—
295#217—62no—
296#217—92no—
297(C2×D3)⋊C4—34no—
298(C2×D3)⋊C4—94no—
299(C2×D3)⋊C4—94no—
300(C2×D3)⋊C4—94no—
301(C2×D3)⋊C4—94no—
302(C2×D3)⋊C4—184no—
303Dic3⋊C4—34no—
304Dic3⋊C4—34no—
305C22×C32—22no—
306C22×C32—24no—
307C22×C32—24no—
308C22×C32—21no—
309C22×C32—22no—
310C22×C32—24no—
311C22×C32—22no—
312C22×C32—42no—
313C22×C32—41no—
314C22×C32—44no—
315C22×C32—44no—
316C22×C32—62no—
317C22×C32—64no—
318C22×C32D3×D414yes—
319C22×C32C2×#6712yes—
320C22×C32C23×D312yes—
321C22×C32C23×D311yescharacteristic
322C2×C3⋊D3—22no—
323C2×C3⋊D3—22no—
324C2×C3⋊D3—44no—
325C2×C3⋊D3—64no—
326C2×C3⋊D3—62no—
327C2×C3⋊D3—64no—
328C2×C3⋊D3—68no—
329C2×C3⋊D3—64no—
330C2×C3⋊D3—64no—
331C2×C3⋊D3—64no—
332C2×C3⋊D3—124no—
333C2×C3⋊D3—124no—
334C2×C3⋊D3—124no—
335C2×C3⋊D3—124no—
336C2×C3⋊D3—128no—
337C2×C3⋊D3—124no—
338C2×C3⋊D3—128no—
339C2×C3⋊D3—128no—
340C2×C3⋊D3—1216no—
341C2×C3⋊D3—1216no—
342C2×C3⋊D3C2×#6712yes—
343C2×C3⋊D3C2×#6712yes—
344C2×C3×D3—22no—
345C2×C3×D3—24no—
346C2×C3×D3—24no—
347C2×C3×D3—22no—
348C2×C3×D3—34no—
349C2×C3×D3—34no—
350C2×C3×D3—44no—
351C2×C3×D3—42no—
352C2×C3×D3—62no—
353C2×C3×D3—62no—
354C2×C3×D3—64no—
355C2×C3×D3—64no—
356C2×C3×D3—122no—
357C2×C3×D3—122no—
358C2×C3×D3—128no—
359C2×C3×D3—128no—
360C2×C3×D3—128no—
361D32—44no—
362C32×C4—62no—
363C32×C4—62no—
364C3⋊Dic3—24no—
365C3×Dic3—24no—
366C3×Dic3—24no—
367C3×Dic3—22no—
368C3×Dic3—22no—
369C3×Dic3—42no—
370C3×Dic3—64no—
371C3×Dic3—64no—
372C3×Dic3—124no—
373C22×D4—91no—
374C22×D4—271no—
375#123—92no—
376#123—272no—
377#121—274no—
378#120—272no—
379#120—272no—
380C2×C22⋊C4—271no—
381C33#53411yescharacteristic, complemented, Sylow
382C23×C3—21no—
383C23×C3—22no—
384C23×C3—32no—
385C23×C3—32no—
386C23×C3—41no—
387C23×C3—62no—
388C23×C3—91no—
389C23×C3—91no—
390C23×C3—94no—
391C23×C3—182no—
392C23×C3—362no—
393C23×C3—362no—
394C23×C3C2×D3212yes—
395C23×C3C2×D3211yescharacteristic
396C22×D3—32no—
397C22×D3—32no—
398C22×D3—38no—
399C22×D3—64no—
400C22×D3—61no—
401C22×D3—61no—
402C22×D3—64no—
403C22×D3—94no—
404C22×D3—92no—
405C22×D3—124no—
406C22×D3—124no—
407C22×D3—124no—
408C22×D3—184no—
409C22×D3—182no—
410C22×D3—184no—
411C22×D3—184no—
412C22×D3—182no—
413C22×D3—362no—
414C22×D3—364no—
415C22×D3—364no—
416C22×D3—362no—
417C22×D3—364no—
418C22×D3—368no—
419C22×D3—364no—
420C22×D3—368no—
421C22×D3C2×D3212yes—
422C3×D4—64no—
423C3×D4—64no—
424C3×D4—124no—
425C3×D4—184no—
426C2×C3×C4—32no—
427C2×C3×C4—34no—
428C2×C3×C4—32no—
429C2×C3×C4—62no—
430C2×C3×C4—62no—
431C2×C3×C4—64no—
432C2×C3×C4—64no—
433C2×C3×C4—94no—
434C2×C3×C4—182no—
435#67—24no—
436#67—64no—
437#67—184no—
438#67—184no—
439#67—364no—
440#67—364no—
441#67—368no—
442C2×Dic3—22no—
443C2×Dic3—34no—
444C2×Dic3—34no—
445C2×Dic3—64no—
446C2×Dic3—362no—
447C2×Dic3C2×D3212yes—
448C2×Dic3C2×D3212yes—
449D12—64no—
450D12—188no—
451D12—188no—
452D12—188no—
453D12—188no—
454D12—364no—
455C4×D3—64no—
456C4×D3—364no—
457C2×C32—22no—
458C2×C32—24no—
459C2×C32—24no—
460C2×C32—22no—
461C2×C32—24no—
462C2×C32—22no—
463C2×C32—21no—
464C2×C32—41no—
465C2×C32—42no—
466C2×C32—44no—
467C2×C32—44no—
468C2×C32—122no—
469C2×C32C2×D3×D412yes—
470C2×C32#100412yes—
471C2×C32#100914yes—
472C2×C32C22×#6711yescharacteristic
473C3⋊D3—22no—
474C3⋊D3—22no—
475C3⋊D3—124no—
476C3⋊D3—128no—
477C3⋊D3—128no—
478C3⋊D3—124no—
479C3⋊D3—122no—
480C3×D3—42no—
481C3×D3—42no—
482C3×D3—64no—
483C3×D3—64no—
484C3×D3—82no—
485C3×D3—122no—
486C3×D3—122no—
487C24—91no—
488C24—91no—
489C24—271no—
490C2×D4—32no—
491C2×D4—92no—
492C2×D4—184no—
493C2×D4—272no—
494C2×D4—272no—
495C2×D4—274no—
496C2×D4—544no—
497C22×C4—31no—
498C22×C4—91no—
499C22×C4—272no—
500C4⋊C4—94no—
501C22⋊C4—94no—
502C22⋊C4—92no—
503C22⋊C4—274no—
504C22⋊C4—272no—
505C22×C3—24no—
506C22×C3—22no—
507C22×C3—24no—
508C22×C3—22no—
509C22×C3—21no—
510C22×C3—22no—
511C22×C3—24no—
512C22×C3—44no—
513C22×C3—44no—
514C22×C3—41no—
515C22×C3—42no—
516C22×C3—64no—
517C22×C3—64no—
518C22×C3—62no—
519C22×C3—62no—
520C22×C3—92no—
521C22×C3—92no—
522C22×C3—124no—
523C22×C3—122no—
524C22×C3—182no—
525C22×C3—182no—
526C22×C3—364no—
527C22×C3—364no—
528C22×C3D3×#6714yes—
529C22×C3#381112yes—
530C22×C3C22×D3211yescharacteristic
531C22×C3C22×D3212yes—
532C2×D3—24no—
533C2×D3—22no—
534C2×D3—34no—
535C2×D3—34no—
536C2×D3—62no—
537C2×D3—62no—
538C2×D3—64no—
539C2×D3—64no—
540C2×D3—122no—
541C2×D3—122no—
542C2×D3—128no—
543C2×D3—128no—
544C2×D3—128no—
545C2×D3—184no—
546C2×D3—188no—
547C2×D3—184no—
548C2×D3—184no—
549C2×D3—184no—
550C2×D3—184no—
551C2×D3—182no—
552C2×D3—364no—
553C2×D3—364no—
554C2×D3—364no—
555C2×D3—368no—
556C2×D3—364no—
557C2×D3—368no—
558C2×D3—364no—
559C2×D3—368no—
560C2×D3—3616no—
561C2×D3—3616no—
562C3×C4—64no—
563C3×C4—62no—
564C3×C4—64no—
565C3×C4—62no—
566C3×C4—122no—
567C3×C4—184no—
568Dic3—22no—
569Dic3—22no—
570Dic3—64no—
571Dic3—64no—
572Dic3—124no—
573C32—21no—
574C32—22no—
575C32—41no—
576C32#565312yescomplemented
577C32#580411yescharacteristic, complemented
578C23—32no—
579C23—91no—
580C23—91no—
581C23—94no—
582C23—182no—
583C23—272no—
584C23—362no—
585C23—362no—
586C23—542no—
587C23—1082no—
588C23D3311yescharacteristic
589D4—64no—
590D4—184no—
591D4—544no—
592D4—544no—
593D4—1084no—
594C2×C4—32no—
595C2×C4—32no—
596C2×C4—62no—
597C2×C4—94no—
598C2×C4—182no—
599C2×C4—1082no—
600C2×C3—24no—
601C2×C3—22no—
602C2×C3—24no—
603C2×C3—24no—
604C2×C3—22no—
605C2×C3—21no—
606C2×C3—22no—
607C2×C3—44no—
608C2×C3—44no—
609C2×C3—42no—
610C2×C3—41no—
611C2×C3—122no—
612C2×C3—122no—
613C2×C3—182no—
614C2×C3—182no—
615C2×C3—242no—
616C2×C3#6430812yes—
617C2×C3#6431414yes—
618C2×C3C2×D3×#6712yes—
619C2×C3C2×#381111yescharacteristic
620D3—42no—
621D3—64no—
622D3—64no—
623D3—122no—
624D3—122no—
625D3—364no—
626D3—368no—
627D3—368no—
628D3—362no—
629D3—364no—
630C22—22no—
631C22—64no—
632C22—62no—
633C22—92no—
634C22—92no—
635C22—182no—
636C22—182no—
637C22—364no—
638C22—364no—
639C22—544no—
640C22—542no—
641C22—1084no—
642C22#8895212yes—
643C22C2×D3311yescharacteristic
644C4—62no—
645C4—62no—
646C4—184no—
647C3—21no—
648C3—22no—
649C3—41no—
650C3#1059508011yescharacteristic, complemented
651C3#1059523412yescomplemented
652C2—22no—
653C2—122no—
654C2—182no—
655C2—182no—
656C2—1082no—
657C2#1172442012yes—
658C2C2×#8895211yescharacteristic
subgroupG/H
center, FrattiniC22C2×D33
commutator, socle, cosocleC22×C33C24
FittingC23×C33C23
hypercenterC23D33
hypocenterC33#534
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
1. C22×#6292C2semidirect
2. C2×C3×#3811C2semidirect
3. #11723723C2semidirect
4. #11723638C2semidirect
5. C3×#63961C2semidirect
6. #11722993C2semidirect
7. #11722966C2semidirect
8. #11722948C2semidirect
11. C23×C3⋊(C3⋊D3)C22semidirect
13. C22×C3×D32C22semidirect
15. C22×C32×Dic3C22semidirect
16. C2×#6294C22semidirect
19. C2×#6292C22semidirect
20. C2×#6291C22semidirect
22. C2×C3×#626C22semidirect
23. C2×C3×#624C22semidirect
24. C3×#3757C22semidirect
25. #88814C22semidirect
26. #88804C22semidirect
27. C3×#3724C22semidirect
28. C3×(C2×D3)⋊Dic3C22semidirect
31. C2×#3811D3semidirect
44. #63961D3semidirect
55. C22×C3×C3⋊D3D4semidirect
83. C22×D32C2×D3semidirect
95. C22×C3×Dic3C2×D3semidirect
109. C2×#626C2×D3semidirect
111. C2×#624C2×D3semidirect
113. #3757C2×D3semidirect
122. #3724C2×D3semidirect
128. (C2×D3)⋊Dic3C2×D3semidirect
189. C22×C3⋊D3#67semidirect
277. C22×Dic3D32semidirect
381. C33#534semidirect
576. C32#5653semidirect
577. C32#5804semidirect
650. C3#10595080semidirect
651. C3#10595234semidirect

Maximal subgroups

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

subgroupindex
C22×#62922
C2×C3×#38112
#117237232
#117236382
C3×#639612
#117229932
#117229662
#117229482
#105952343
#105950803