Finite Group Databasebeta

← #13178246

Subgroups — #13178246

Order 1728.

1258 conjugacy classes of subgroups.

The subgroup lattice has 724 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64093C212yesmaximal, complemented
2C3×#64086C211yesmaximal, characteristic, complemented
3#11723725C212yesmaximal, complemented
4#11723717C211yesmaximal, characteristic, complemented
5#11723717C211yesmaximal, characteristic, complemented
6#11723697C211yesmaximal, characteristic, complemented
7#11723694C212yesmaximal, complemented
8C4×#6295C211yesmaximal, characteristic, complemented
9#11723686C212yesmaximal, complemented
10#11723685C211yesmaximal, characteristic, complemented
11#11723679C211yesmaximal, characteristic, complemented
12#10594223—31nomaximal
13#10594206—32nomaximal
14C2×C3×#638C2212yes—
15C2×C3×#638C2212yes—
16C22×C3×C3⋊Dic3C2211yescharacteristic
17C2×C3×C4⋊(C3⋊D3)C2212yes—
18C2×C3×C4×C3⋊D3C2212yes—
19C2×#6295C2211yescharacteristic
20C2×#6295C2211yescharacteristic
21C2×#6295C2212yes—
22C2×#6294C2212yes—
23C33×C22⋊C4C2211yescharacteristic
24C3×#3757C2211yescharacteristic
25C3×#3752C2212yes—
26C3×#3750C2211yescharacteristic
27C3×#3750C2211yescharacteristic
28C3×#3750C2212yes—
29C3×C4×C3⋊Dic3C2211yescharacteristic
30C3×C4×C3⋊Dic3C2211yescharacteristic
31#88815C2211yescharacteristic
32#88815C2211yescharacteristic
33#88815C2212yes—
34#88814C2212yes—
35#88814C2212yes—
36#88814C2212yes—
37#64093D312yescomplemented
38#64086D311yescharacteristic, complemented
39C3×#884—32no—
40C3×#884—32no—
41C3×#884—62no—
42C3×#877—32no—
43C3×#877—61no—
44#63980—31no—
45#63980—31no—
46#63978—32no—
47#63978—32no—
48#63971—32no—
49#63970—32no—
50#63958—32no—
51#63914—31no—
52#63911—32no—
53#63905—32no—
54C4×#626—32no—
55#63902—31no—
56C4×#625—31no—
57#63897—32no—
58#63884—32no—
59#63884—32no—
60#63880—32no—
61#63870—31no—
62#63867—32no—
63#63853—32no—
64#63850—32no—
65C23×C33C2311yescharacteristic
66C22×C3×C3⋊D3C2312yes—
67C22×C3×C3⋊D3C2312yes—
68C2×C33×C4C2311yescharacteristic
69C2×C33×C4C2311yescharacteristic
70C3×#638—42no—
71C3×#638—42no—
72C2×C3×C3⋊Dic3—22no—
73C2×C3×C3⋊Dic3C2311yescharacteristic
74C2×C3×C3⋊Dic3C2311yescharacteristic
75C2×C3×C3⋊Dic3C2311yescharacteristic
76C2×C3×C3⋊Dic3C2312yes—
77C2×C3×C3⋊Dic3C2312yes—
78C2×C3×C3⋊Dic3C2311yescharacteristic
79C3×C4⋊(C3⋊D3)—42no—
80C3×C4×C3⋊D3—42no—
81#6295—22no—
82#6295—22no—
83#6295—42no—
84#6294—42no—
85#5615—92no—
86#5535—91no—
87C2×#638C2×D312yes—
88C2×#638C2×D312yes—
89C22×C3⋊Dic3C2×D311yescharacteristic
90C2×C4⋊(C3⋊D3)C2×D312yes—
91C2×C4×C3⋊D3C2×D312yes—
92C2×C3×#67—32no—
93C2×C3×#67—32no—
94C2×C3×#67—32no—
95C2×C3×#67—32no—
96C2×C3×#67—62no—
97C2×C3×#67—62no—
98C22×C3×Dic3—32no—
99C22×C3×Dic3—61no—
100C2×C3×D12—32no—
101C2×C3×D12—32no—
102C2×C3×D12—62no—
103C2×C3×C4×D3—32no—
104C2×C3×C4×D3—32no—
105C2×C3×C4×D3—62no—
106C2×#626—32no—
107C2×#626—32no—
108C2×#626—32no—
109C2×#626—32no—
110C2×#625—31no—
111C2×#625—31no—
112C2×#625—32no—
113C2×#624—32no—
114C2×D3×Dic3—32no—
115C2×D3×Dic3—32no—
116C32×C22⋊C4—21no—
117C32×C22⋊C4—22no—
118C32×C22⋊C4—41no—
119C32×C22⋊C4C2×D312yes—
120C32×C22⋊C4C2×D311yescharacteristic
121#3757C2×D311yescharacteristic
122#3752C2×D312yes—
123#3750C2×D311yescharacteristic
124#3750C2×D311yescharacteristic
125#3750C2×D312yes—
126C4×C3⋊Dic3C2×D311yescharacteristic
127C4×C3⋊Dic3C2×D311yescharacteristic
128C3×#217—32no—
129C3×#217—61no—
130C3×(C2×D3)⋊C4—32no—
131C3×(C2×D3)⋊C4—32no—
132C3×(C2×D3)⋊C4—62no—
133C3×Dic3⋊C4—32no—
134C3×Dic3⋊C4—32no—
135C3×Dic3⋊C4—32no—
136C3×Dic3⋊C4—32no—
137C3×Dic3⋊C4—61no—
138C3×Dic3⋊C4—61no—
139C3×Dic3⋊C4—62no—
140C3×C4×Dic3—32no—
141C3×C4×Dic3—32no—
142C3×C4×Dic3—61no—
143C3×C4×Dic3—61no—
144#3724—32no—
145#3724—31no—
146#3724—31no—
147Dic3⋊Dic3—32no—
148Dic3⋊Dic3—32no—
149Dic3⋊Dic3—32no—
150Dic3⋊Dic3—32no—
151#3722—32no—
152#3722—32no—
153#3722—32no—
154(C2×D3)⋊Dic3—32no—
155(C2×D3)⋊Dic3—32no—
156(C2×D3)⋊Dic3—32no—
157(C2×D3)⋊Dic3—32no—
158(C2×D3)⋊Dic3—32no—
159(C2×D3)⋊Dic3—32no—
160C22×C33—21no—
161C22×C33—21no—
162C22×C33—21no—
163C22×C33C2411yescharacteristic
164C2×C3×C3⋊D3—22no—
165C2×C3×C3⋊D3—22no—
166C2×C3×C3⋊D3—22no—
167C2×C3×C3⋊D3—22no—
168C2×C3×C3⋊D3—22no—
169C2×C3×C3⋊D3—22no—
170C33×C4—21no—
171C33×C4—21no—
172C3×C3⋊Dic3—21no—
173C3×C3⋊Dic3—22no—
174C3×C3⋊Dic3—21no—
175C3×C3⋊Dic3—22no—
176C3×C3⋊Dic3—21no—
177C3×C3⋊Dic3#4112yes—
178C3×#121—92no—
179C3×#117—91no—
180#940—92no—
181#939—92no—
182#939—92no—
183#939—92no—
184#938—92no—
185#935—92no—
186#930—91no—
187#930—91no—
188#929—92no—
189C4×#67—92no—
190#894—92no—
191#894—92no—
192#894—92no—
193#889—92no—
194#884—32no—
195#884—32no—
196#884—62no—
197#884—92no—
198#883—92no—
199#67⋊C4—92no—
200#877—32no—
201#877—61no—
202#874—91no—
203C4×D12—91no—
204C4×D12—91no—
205#870—91no—
206C23×C32—21no—
207C23×C32—22no—
208C23×C32—41no—
209C23×C32C22×D312yes—
210C23×C32C22×D311yescharacteristic
211C22×C3⋊D3C22×D312yes—
212C22×C3⋊D3C22×D312yes—
213C22×C3×D3—32no—
214C22×C3×D3—32no—
215C22×C3×D3—32no—
216C22×C3×D3—32no—
217C22×C3×D3—62no—
218C22×C3×D3—62no—
219C2×C32×C4—21no—
220C2×C32×C4—21no—
221C2×C32×C4—22no—
222C2×C32×C4—22no—
223C2×C32×C4—41no—
224C2×C32×C4—41no—
225C2×C32×C4C22×D312yes—
226C2×C32×C4C22×D312yes—
227C2×C32×C4C22×D311yescharacteristic
228C2×C32×C4C22×D311yescharacteristic
229#638—42no—
230#638—42no—
231C2×C3⋊Dic3—22no—
232C2×C3⋊Dic3C22×D311yescharacteristic
233C2×C3⋊Dic3C22×D312yes—
234C2×C3⋊Dic3C22×D311yescharacteristic
235C2×C3⋊Dic3C22×D312yes—
236C2×C3⋊Dic3C22×D311yescharacteristic
237C2×C3⋊Dic3C22×D311yescharacteristic
238C4⋊(C3⋊D3)—42no—
239C4×C3⋊D3—42no—
240C3×#67—122no—
241C3×#67—122no—
242C3×#67—122no—
243C3×#67—122no—
244C3×#67—124no—
245C3×#67—124no—
246C2×C3×Dic3—32no—
247C2×C3×Dic3—32no—
248C2×C3×Dic3—32no—
249C2×C3×Dic3—32no—
250C2×C3×Dic3—32no—
251C2×C3×Dic3—32no—
252C2×C3×Dic3—32no—
253C2×C3×Dic3—32no—
254C2×C3×Dic3—61no—
255C2×C3×Dic3—61no—
256C2×C3×Dic3—62no—
257C2×C3×Dic3—61no—
258C2×C3×Dic3—61no—
259C2×C3×Dic3—62no—
260C2×C3×Dic3—64no—
261C2×C3×Dic3—64no—
262C3×D12—122no—
263C3×D12—122no—
264C3×D12—124no—
265C3×C4×D3—122no—
266C3×C4×D3—122no—
267C3×C4×D3—124no—
268#626—64no—
269#626—64no—
270#626—122no—
271#626—122no—
272#625—62no—
273#625—62no—
274#625—122no—
275#624—122no—
276D3×Dic3—122no—
277D3×Dic3—122no—
278#540—271noSylow
279C2×C33—41no—
280C2×C33C2×#4111yescharacteristic
281C2×C33H(5,𝔽2)11yescharacteristic
282C2×C33H(5,𝔽2)11yescharacteristic
283C3×C3⋊D3—42no—
284C3×C3⋊D3—42no—
285C2×C3×D4—92no—
286C2×C3×D4—92no—
287C2×C3×D4—92no—
288C22×C3×C4—91no—
289C22×C3×C4—92no—
290C2×#67—32no—
291C2×#67—32no—
292C2×#67—32no—
293C2×#67—32no—
294C2×#67—62no—
295C2×#67—62no—
296C2×#67—92no—
297C2×#67—92no—
298C2×#67—92no—
299C2×#67—92no—
300C22×Dic3—32no—
301C22×Dic3—61no—
302C22×Dic3—92no—
303C2×D12—32no—
304C2×D12—32no—
305C2×D12—62no—
306C2×D12—91no—
307C2×D12—91no—
308C2×D12—92no—
309C2×C4×D3—32no—
310C2×C4×D3—32no—
311C2×C4×D3—62no—
312C2×C4×D3—92no—
313C2×C4×D3—92no—
314C3×C4⋊C4—92no—
315C3×C4⋊C4—91no—
316C3×C4⋊C4—91no—
317C3×C22⋊C4—21no—
318C3×C22⋊C4—22no—
319C3×C22⋊C4—41no—
320C3×C22⋊C4—91no—
321C3×C22⋊C4—92no—
322C3×C22⋊C4D3212yes—
323C3×C22⋊C4D3211yescharacteristic
324C3×C42—91no—
325C3×C42—91no—
326#217—32no—
327#217—61no—
328#217—92no—
329#217—92no—
330#217—92no—
331(C2×D3)⋊C4—32no—
332(C2×D3)⋊C4—32no—
333(C2×D3)⋊C4—62no—
334(C2×D3)⋊C4—92no—
335(C2×D3)⋊C4—92no—
336(C2×D3)⋊C4—92no—
337(C2×D3)⋊C4—92no—
338(C2×D3)⋊C4—92no—
339(C2×D3)⋊C4—92no—
340C4⋊Dic3—91no—
341C4⋊Dic3—91no—
342C4⋊Dic3—92no—
343Dic3⋊C4—32no—
344Dic3⋊C4—32no—
345Dic3⋊C4—32no—
346Dic3⋊C4—32no—
347Dic3⋊C4—61no—
348Dic3⋊C4—61no—
349Dic3⋊C4—62no—
350Dic3⋊C4—92no—
351Dic3⋊C4—92no—
352Dic3⋊C4—92no—
353Dic3⋊C4—92no—
354C4×Dic3—32no—
355C4×Dic3—32no—
356C4×Dic3—61no—
357C4×Dic3—61no—
358C22×C32—21no—
359C22×C32—22no—
360C22×C32—22no—
361C22×C32—22no—
362C22×C32—22no—
363C22×C32—21no—
364C22×C32—21no—
365C22×C32—21no—
366C22×C32—22no—
367C22×C32—41no—
368C22×C32—42no—
369C22×C32—42no—
370C22×C32—41no—
371C22×C32—41no—
372C22×C32—42no—
373C22×C32—42no—
374C22×C32—81no—
375C22×C32—81no—
376C22×C32C23×D311yescharacteristic
377C22×C32C23×D312yes—
378C2×C3⋊D3—22no—
379C2×C3⋊D3—22no—
380C2×C3⋊D3—22no—
381C2×C3⋊D3—22no—
382C2×C3⋊D3—22no—
383C2×C3⋊D3—22no—
384C2×C3×D3—64no—
385C2×C3×D3—62no—
386C2×C3×D3—62no—
387C2×C3×D3—62no—
388C2×C3×D3—62no—
389C2×C3×D3—62no—
390C2×C3×D3—62no—
391C2×C3×D3—62no—
392C2×C3×D3—62no—
393C2×C3×D3—62no—
394C2×C3×D3—62no—
395C2×C3×D3—62no—
396C2×C3×D3—62no—
397C2×C3×D3—122no—
398C2×C3×D3—122no—
399C2×C3×D3—122no—
400C2×C3×D3—122no—
401C2×C3×D3—122no—
402C32×C4—22no—
403C32×C4—22no—
404C32×C4—21no—
405C32×C4—21no—
406C32×C4—42no—
407C32×C4—42no—
408C32×C4—41no—
409C32×C4—41no—
410C32×C4—81no—
411C32×C4—81no—
412C3⋊Dic3—21no—
413C3⋊Dic3—22no—
414C3⋊Dic3—22no—
415C3⋊Dic3—21no—
416C3⋊Dic3—21no—
417C3⋊Dic3#23512yes—
418C3×Dic3—34no—
419C3×Dic3—62no—
420C3×Dic3—64no—
421C3×Dic3—62no—
422C3×Dic3—62no—
423C3×Dic3—62no—
424C3×Dic3—62no—
425C3×Dic3—62no—
426C3×Dic3—62no—
427C3×Dic3—62no—
428C3×Dic3—62no—
429C3×Dic3—121no—
430C3×Dic3—121no—
431C3×Dic3—122no—
432#127—271no—
433#125—271no—
434#123—272no—
435#123—272no—
436#121—92no—
437#121—272no—
438#121—271no—
439#121—271no—
440C4×D4—271no—
441C4×D4—271no—
442#117—91no—
443C33#54011yescharacteristic, complemented, Sylow
444C23×C3—21no—
445C23×C3—22no—
446C23×C3—41no—
447C23×C3—92no—
448C23×C3—92no—
449C23×C3C2×D3212yes—
450C23×C3C2×D3211yescharacteristic
451C22×D3—32no—
452C22×D3—32no—
453C22×D3—32no—
454C22×D3—32no—
455C22×D3—62no—
456C22×D3—62no—
457C3×D4—362no—
458C3×D4—362no—
459C3×D4—362no—
460C2×C3×C4—21no—
461C2×C3×C4—21no—
462C2×C3×C4—22no—
463C2×C3×C4—22no—
464C2×C3×C4—41no—
465C2×C3×C4—41no—
466C2×C3×C4—91no—
467C2×C3×C4—92no—
468C2×C3×C4—91no—
469C2×C3×C4—92no—
470C2×C3×C4—91no—
471C2×C3×C4—91no—
472C2×C3×C4—182no—
473C2×C3×C4—362no—
474C2×C3×C4C2×D3212yes—
475C2×C3×C4C2×D3212yes—
476C2×C3×C4C2×D3211yescharacteristic
477C2×C3×C4C2×D3211yescharacteristic
478#67—122no—
479#67—122no—
480#67—122no—
481#67—122no—
482#67—124no—
483#67—124no—
484#67—184no—
485#67—184no—
486#67—362no—
487#67—362no—
488C2×Dic3—32no—
489C2×Dic3—32no—
490C2×Dic3—32no—
491C2×Dic3—32no—
492C2×Dic3—32no—
493C2×Dic3—32no—
494C2×Dic3—32no—
495C2×Dic3—32no—
496C2×Dic3—61no—
497C2×Dic3—61no—
498C2×Dic3—62no—
499C2×Dic3—61no—
500C2×Dic3—61no—
501C2×Dic3—62no—
502C2×Dic3—64no—
503C2×Dic3—64no—
504C2×Dic3—362no—
505D12—122no—
506D12—122no—
507D12—124no—
508D12—182no—
509D12—182no—
510D12—362no—
511C4×D3—122no—
512C4×D3—122no—
513C4×D3—124no—
514C4×D3—362no—
515C4×D3—362no—
516C2×C32—22no—
517C2×C32—21no—
518C2×C32—21no—
519C2×C32—21no—
520C2×C32—22no—
521C2×C32—22no—
522C2×C32—41no—
523C2×C32—41no—
524C2×C32—41no—
525C2×C32—41no—
526C2×C32—42no—
527C2×C32—42no—
528C2×C32—82no—
529C2×C32—82no—
530C2×C32C2×#23511yescharacteristic
531C2×C32#100412yes—
532C2×C32#100412yes—
533C2×C32D3×#4112yes—
534C2×C32#100911yescharacteristic
535C2×C32#100911yescharacteristic
536C3⋊D3—42no—
537C3⋊D3—42no—
538C3×D3—124no—
539C3×D3—122no—
540C3×D3—122no—
541C3×D3—122no—
542C3×D3—122no—
543C3×D3—242no—
544C2×D4—92no—
545C2×D4—92no—
546C2×D4—92no—
547C2×D4—271no—
548C2×D4—271no—
549C2×D4—272no—
550C22×C4—91no—
551C22×C4—92no—
552C22×C4—272no—
553C4⋊C4—91no—
554C4⋊C4—91no—
555C4⋊C4—92no—
556C4⋊C4—271no—
557C4⋊C4—271no—
558C4⋊C4—272no—
559C22⋊C4—91no—
560C22⋊C4—92no—
561C22⋊C4—272no—
562C22⋊C4—272no—
563C22⋊C4—272no—
564C22⋊C4#111311yescharacteristic
565C42—91no—
566C42—91no—
567C22×C3—22no—
568C22×C3—22no—
569C22×C3—21no—
570C22×C3—21no—
571C22×C3—21no—
572C22×C3—22no—
573C22×C3—22no—
574C22×C3—21no—
575C22×C3—22no—
576C22×C3—41no—
577C22×C3—41no—
578C22×C3—42no—
579C22×C3—42no—
580C22×C3—42no—
581C22×C3—42no—
582C22×C3—41no—
583C22×C3—81no—
584C22×C3—81no—
585C22×C3—182no—
586C22×C3—182no—
587C22×C3—182no—
588C22×C3—182no—
589C22×C3—182no—
590C22×C3—182no—
591C22×C3C22×D3212yes—
592C22×C3C22×D3211yescharacteristic
593C2×D3—64no—
594C2×D3—62no—
595C2×D3—62no—
596C2×D3—62no—
597C2×D3—62no—
598C2×D3—62no—
599C2×D3—62no—
600C2×D3—62no—
601C2×D3—62no—
602C2×D3—62no—
603C2×D3—62no—
604C2×D3—62no—
605C2×D3—62no—
606C2×D3—122no—
607C2×D3—122no—
608C2×D3—122no—
609C2×D3—122no—
610C2×D3—122no—
611C3×C4—21no—
612C3×C4—21no—
613C3×C4—22no—
614C3×C4—22no—
615C3×C4—42no—
616C3×C4—42no—
617C3×C4—41no—
618C3×C4—41no—
619C3×C4—81no—
620C3×C4—81no—
621C3×C4—92no—
622C3×C4—181no—
623C3×C4—181no—
624C3×C4—181no—
625C3×C4—182no—
626C3×C4—182no—
627Dic3—34no—
628Dic3—62no—
629Dic3—64no—
630Dic3—62no—
631Dic3—62no—
632Dic3—62no—
633Dic3—62no—
634Dic3—62no—
635Dic3—62no—
636Dic3—62no—
637Dic3—62no—
638Dic3—122no—
639Dic3—121no—
640Dic3—121no—
641C32—22no—
642C32—21no—
643C32—41no—
644C32#553511yescharacteristic, complemented
645C32#561512yescomplemented
646C23—92no—
647C23—92no—
648C23C2×#111311yescharacteristic
649D4—362no—
650D4—362no—
651D4—362no—
652D4—542no—
653D4—542no—
654D4—1082no—
655C2×C4—91no—
656C2×C4—91no—
657C2×C4—91no—
658C2×C4—92no—
659C2×C4—91no—
660C2×C4—92no—
661C2×C4—182no—
662C2×C4—362no—
663C2×C4—1082no—
664C2×C4C2×#111311yescharacteristic
665C2×C4C2×#111311yescharacteristic
666C2×C3—21no—
667C2×C3—22no—
668C2×C3—22no—
669C2×C3—22no—
670C2×C3—21no—
671C2×C3—21no—
672C2×C3—42no—
673C2×C3—42no—
674C2×C3—41no—
675C2×C3—41no—
676C2×C3—41no—
677C2×C3—41no—
678C2×C3—82no—
679C2×C3—82no—
680C2×C3—362no—
681C2×C3—362no—
682C2×C3D3×#23512yes—
683C2×C3#6430611yescharacteristic
684C2×C3#6431311yescharacteristic
685C2×C3#6431311yescharacteristic
686C2×C3#6431412yes—
687C2×C3#6431412yes—
688D3—124no—
689D3—122no—
690D3—122no—
691D3—122no—
692D3—122no—
693D3—242no—
694C22—21no—
695C22—21no—
696C22—21no—
697C22—182no—
698C22—182no—
699C22—182no—
700C22—182no—
701C22—182no—
702C22—182no—
703C22C22×#111311yescharacteristic
704C4—21no—
705C4—21no—
706C4—92no—
707C4—181no—
708C4—182no—
709C4—181no—
710C4—182no—
711C4—181no—
712C3—22no—
713C3—21no—
714C3—41no—
715C3#1059420612yescomplemented
716C3#1059422311yescharacteristic, complemented
717C2—41no—
718C2—362no—
719C2—362no—
720C2#1172461811yescharacteristic
721C2#1172462311yescharacteristic
722C2#1172462311yescharacteristic
subgroupG/H
center, FrattiniC22C22×#1113
commutator, socle, cosocleC22×C33C24
FittingC33×C22⋊C4C22
hypercenterC22⋊C4#1113
hypocenterC33#540
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C3×#64093C2semidirect
2. C3×#64086C2semidirect
3. #11723725C2semidirect
4. #11723717C2semidirect
5. #11723717C2semidirect
6. #11723697C2semidirect
7. #11723694C2semidirect
8. C4×#6295C2semidirect
9. #11723686C2semidirect
10. #11723685C2semidirect
11. #11723679C2semidirect
37. #64093D3semidirect
38. #64086D3semidirect
443. C33#540semidirect
644. C32#5535semidirect
645. C32#5615semidirect
715. C3#10594206semidirect
716. C3#10594223semidirect

Maximal subgroups

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

subgroupindex
C3×#640932
C3×#640862
#117237252
#117237172
#117236972
#117236942
C4×#62952
#117236862
#117236852
#117236792
#105942233
#105942063