Finite Group Databasebeta

← #13150567

Subgroups — #13150567

Order 1728.

1274 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4⋊(C3⋊D9)C211yesmaximal, characteristic, complemented
2C2×#88668C211yesmaximal, characteristic, complemented
3C9×#929C211yesmaximal, characteristic, complemented
4C3×#63491C211yesmaximal, characteristic, complemented
5#11721291C211yesmaximal, characteristic, complemented
6#11721278C211yesmaximal, characteristic, complemented
7#11721274C211yesmaximal, characteristic, complemented
8#11721051C212yesmaximal, complemented
9#11721039C212yesmaximal, complemented
10#11721005C212yesmaximal, complemented
11#11720997C212yesmaximal, complemented
12#10595080—31nomaximal
13#10589755—31nomaximal
14C23×C3⋊D9C2211yescharacteristic
15C23×C3⋊D9C2211yescharacteristic, complemented
16C22×D3×D9C2211yescharacteristic, complemented
17C22×C3×C4×C9C2211yescharacteristic, complemented
18C2×C4⋊(C3⋊D9)—24no—
19C2×C4⋊(C3⋊D9)C2212yes—
20C2×C4⋊(C3⋊D9)C2212yescomplemented
21C2×C9×#67C2211yescharacteristic
22C2×C3×#611C2211yescharacteristic
23#88668—24no—
24C2×#6195C2211yescharacteristic, complemented
25C2×#6195C2212yescomplemented
26C2×#6192C2211yescharacteristic, complemented
27C2×#6192C2212yescomplemented
28C2×(C3⋊D9)⋊C4C2211yescharacteristic, complemented
29C2×(C3⋊D9)⋊C4C2212yescomplemented
30C9×#217C2211yescharacteristic, complemented
31C3×#3676C2211yescharacteristic, complemented
32C9×(C2×D3)⋊C4C2212yescomplemented
33C3×(C2×D9)⋊C4C2212yescomplemented
34C9×Dic3⋊C4C2212yescomplemented
35C3×Dic9⋊C4C2212yescomplemented
36#88445C2212yescomplemented
37#88445C2212yescomplemented
38#88445C2212yescomplemented
39C22×C4⋊(C3⋊D3)—31no—
40C2×#3811—31no—
41C3×#929—31no—
42C3×#929D311yescharacteristic
43#63981—31no—
44#63981—31no—
45#63974—31no—
46#63923—32no—
47#63910—32no—
48#63910—32no—
49#63897—32no—
50C2×D4×D9—31no—
51C22×D36—31no—
52C22×D36—61no—
53C9×#123—31no—
54#63499—31no—
55#63491D311yescharacteristic, complemented
56#63457—32no—
57#63456—32no—
58#63446—32no—
59#63444—31no—
60#63444—32no—
61C22⋊C4×D9—31no—
62C23×C3×C9C2311yescharacteristic
63C22×C3⋊D9—22no—
64C22×C3⋊D9—42no—
65C22×C3⋊D9—42no—
66C22×C3⋊D9D414yescomplemented
67C22×C3⋊D9C2312yes—
68C22×C3⋊D9C2311yescharacteristic
69C22×C3⋊D9C2311yescharacteristic
70C22×D3×C9C2311yescharacteristic
71C22×C3×D9C2311yescharacteristic
72C2×D3×D9—22no—
73C2×D3×D9—42no—
74C2×C3×C4×C9—22no—
75C2×C3×C4×C9C2312yes—
76C4⋊(C3⋊D9)—24no—
77C4⋊(C3⋊D9)—24no—
78C9×#67—22no—
79C3×#611—22no—
80C2×C9×Dic3C2311yescharacteristic
81C2×C9×Dic3C2312yes—
82C2×C3×Dic9C2311yescharacteristic
83C2×C3×Dic9C2312yes—
84#6195—22no—
85#6195—42no—
86#6192—22no—
87#6192—42no—
88(C3⋊D9)⋊C4—22no—
89(C3⋊D9)⋊C4—42no—
90#5653—91no—
91#5653—91no—
92C23×C3⋊D3—31no—
93C23×C3⋊D3—31no—
94C22×D32—31no—
95C22×C32×C4C2×D311yescharacteristic
96C2×C4⋊(C3⋊D3)—32no—
97C2×C4⋊(C3⋊D3)—32no—
98C2×C4⋊(C3⋊D3)—64no—
99C2×C3×#67—31no—
100C2×C3×#67C2×D311yescharacteristic
101#3811—64no—
102C2×#626—31no—
103C2×#626—31no—
104C2×#626—32no—
105C2×#626—32no—
106C2×#624—31no—
107C2×#624—32no—
108C23×D9—31no—
109C23×D9—31no—
110C23×D9—31no—
111C23×D9—61no—
112C23×D9—61no—
113C3×#217—31no—
114C3×#217C2×D311yescharacteristic
115C3×(C2×D3)⋊C4—32no—
116C3×(C2×D3)⋊C4C2×D312yes—
117C3×Dic3⋊C4—32no—
118C3×Dic3⋊C4C2×D312yes—
119#3722—32no—
120#3722—32no—
121#3722—32no—
122C2×D4×C9—31no—
123C22×C4×C9—21no—
124C22×C4×C9C2×D311yescharacteristic, complemented
125C2×#611—31no—
126C2×#611—32no—
127C2×#611C2×D311yescharacteristic
128D4×D9—64no—
129C2×D36—31no—
130C2×D36—32no—
131C2×D36—32no—
132C2×D36—32no—
133C2×D36—64no—
134C2×D36—62no—
135C2×D36—62no—
136C2×D36—68no—
137C2×C4×D9—31no—
138C2×C4×D9—32no—
139C9×C4⋊C4—32no—
140C9×C22⋊C4—31no—
141C9×C22⋊C4—32no—
142#3676C2×D311yescharacteristic, complemented
143(C2×D9)⋊C4—32no—
144(C2×D9)⋊C4—32no—
145(C2×D9)⋊C4—32no—
146(C2×D9)⋊C4C2×D312yescomplemented
147Dic9⋊C4C2×D312yescomplemented
148C22×C3×C9—22no—
149C22×C3×C9C2×D412yes—
150C22×C3×C9C2411yescharacteristic
151C2×C3⋊D9—22no—
152C2×C3⋊D9—22no—
153C2×C3⋊D9—22no—
154C2×C3⋊D9—22no—
155C2×C3⋊D9—22no—
156C2×C3⋊D9—44no—
157C2×C3⋊D9—44no—
158C2×C3⋊D9C2×D412yes—
159C2×C3⋊D9C2×D412yes—
160C2×D3×C9—21no—
161C2×D3×C9—22no—
162C2×C3×D9—21no—
163C2×C3×D9—22no—
164D3×D9—44no—
165C3×C4×C9—22no—
166C9×Dic3—21no—
167C9×Dic3—22no—
168C3×Dic9—21no—
169C3×Dic9—22no—
170C2×D3×D4—91no—
171C2×D3×D4—91no—
172C22×D12—91no—
173C22×D12—91no—
174C22×D12—181no—
175C3×#123—31no—
176C3×#123—91no—
177#937—91no—
178#937—91no—
179#929—31no—
180#929D911yescharacteristic, complemented
181#895—92no—
182#895—92no—
183#894—92no—
184#894—92no—
185#884—92no—
186#884—92no—
187#882—92no—
188#882—91no—
189#882—91no—
190#882—92no—
191D3×C22⋊C4—91no—
192D3×C22⋊C4—91no—
193C23×C32C22×D311yescharacteristic
194C22×C3⋊D3—31no—
195C22×C3⋊D3—31no—
196C22×C3⋊D3—34no—
197C22×C3⋊D3—32no—
198C22×C3⋊D3—62no—
199C22×C3⋊D3—122no—
200C22×C3⋊D3—122no—
201C22×C3×D3—31no—
202C22×C3×D3C22×D311yescharacteristic
203C2×D32—62no—
204C2×D32—122no—
205C2×C32×C4—22no—
206C2×C32×C4C22×D312yes—
207C4⋊(C3⋊D3)—64no—
208C4⋊(C3⋊D3)—64no—
209C3×#67—22no—
210C3×#67—62no—
211C2×C3×Dic3—31no—
212C2×C3×Dic3—32no—
213C2×C3×Dic3C22×D311yescharacteristic
214C2×C3×Dic3C22×D312yes—
215#626—62no—
216#626—62no—
217#626—122no—
218#626—122no—
219#624—62no—
220#624—122no—
221C23×C9—21no—
222C23×C9—31no—
223C23×C9C22×D311yescharacteristic
224C22×D9—31no—
225C22×D9—31no—
226C22×D9—34no—
227C22×D9—32no—
228C22×D9—64no—
229C22×D9—61no—
230C22×D9—61no—
231C22×D9—62no—
232C22×D9—62no—
233C22×D9—64no—
234C22×D9—62no—
235C22×D9—122no—
236C22×D9—122no—
237C22×D9—122no—
238C22×D9—124no—
239C22×D9—124no—
240C22×D9C22×D311yescharacteristic
241D4×C9—62no—
242C2×C4×C9—24no—
243C2×C4×C9—22no—
244C2×C4×C9—22no—
245C2×C4×C9—32no—
246C2×C4×C9—31no—
247C2×C4×C9C22×D312yes—
248#611—22no—
249#611—62no—
250#611—122no—
251C2×Dic9C22×D311yescharacteristic
252C2×Dic9C22×D312yes—
253D36—64no—
254D36—64no—
255D36—68no—
256D36—68no—
257D36—62no—
258D36—122no—
259C4×D9—62no—
260C4×D9—122no—
261#534—271noSylow
262C2×C3×C9—22no—
263C2×C3×C9C22×D411yescharacteristic
264C2×C3×C9H(5,𝔽2)12yes—
265C3⋊D9—22no—
266C3⋊D9—22no—
267C3⋊D9—42no—
268D3×C9—41no—
269C3×D9—41no—
270C23×D3—91no—
271C23×D3—91no—
272C23×D3—91no—
273C23×D3—91no—
274C23×D3—91no—
275C23×D3—91no—
276C23×D3—181no—
277C23×D3—181no—
278C2×C3×D4—31no—
279C2×C3×D4—91no—
280C22×C3×C4—21no—
281C22×C3×C4C2×D911yescharacteristic, complemented
282C22×C3×C4D3211yescharacteristic
283C2×#67—31no—
284C2×#67—91no—
285C2×#67—92no—
286C2×#67—91no—
287C2×#67—92no—
288C2×#67C2×D911yescharacteristic
289D3×D4—184no—
290D3×D4—184no—
291C2×D12—91no—
292C2×D12—91no—
293C2×D12—92no—
294C2×D12—92no—
295C2×D12—92no—
296C2×D12—92no—
297C2×D12—92no—
298C2×D12—92no—
299C2×D12—182no—
300C2×D12—182no—
301C2×D12—184no—
302C2×D12—184no—
303C2×D12—188no—
304C2×C4×D3—91no—
305C2×C4×D3—91no—
306C2×C4×D3—92no—
307C2×C4×D3—92no—
308C3×C4⋊C4—32no—
309C3×C4⋊C4—92no—
310C3×C22⋊C4—31no—
311C3×C22⋊C4—32no—
312C3×C22⋊C4—91no—
313C3×C22⋊C4—92no—
314#217—31no—
315#217C2×D911yescharacteristic, complemented
316(C2×D3)⋊C4—32no—
317(C2×D3)⋊C4—92no—
318(C2×D3)⋊C4—92no—
319(C2×D3)⋊C4—92no—
320(C2×D3)⋊C4—92no—
321(C2×D3)⋊C4—92no—
322(C2×D3)⋊C4—92no—
323(C2×D3)⋊C4C2×D912yescomplemented
324Dic3⋊C4—32no—
325Dic3⋊C4C2×D912yescomplemented
326C22×C32—22no—
327C22×C32D3×D412yes—
328C22×C32C23×D311yescharacteristic
329C2×C3⋊D3—32no—
330C2×C3⋊D3—32no—
331C2×C3⋊D3—62no—
332C2×C3⋊D3—62no—
333C2×C3⋊D3—62no—
334C2×C3⋊D3—62no—
335C2×C3⋊D3—62no—
336C2×C3⋊D3—124no—
337C2×C3⋊D3—124no—
338C2×C3×D3—21no—
339C2×C3×D3—22no—
340C2×C3×D3—61no—
341C2×C3×D3—62no—
342D32—124no—
343C32×C4—22no—
344C3×Dic3—21no—
345C3×Dic3—22no—
346C3×Dic3—61no—
347C3×Dic3—62no—
348C22×C9—24no—
349C22×C9—22no—
350C22×C9—21no—
351C22×C9—22no—
352C22×C9—61no—
353C22×C9—62no—
354C22×C9D3×D412yes—
355C22×C9C23×D311yescharacteristic
356C2×D9—21no—
357C2×D9—22no—
358C2×D9—32no—
359C2×D9—32no—
360C2×D9—62no—
361C2×D9—62no—
362C2×D9—62no—
363C2×D9—62no—
364C2×D9—64no—
365C2×D9—62no—
366C2×D9—62no—
367C2×D9—62no—
368C2×D9—122no—
369C2×D9—122no—
370C2×D9—122no—
371C2×D9—122no—
372C2×D9—124no—
373C2×D9—124no—
374C2×D9—128no—
375C2×D9—128no—
376C2×D9—124no—
377C4×C9—24no—
378C4×C9—22no—
379C4×C9—61no—
380C4×C9—62no—
381Dic9—21no—
382Dic9—22no—
383C22×D4—271no—
384C22×D4—271no—
385#123—31no—
386#123—91no—
387#123—272no—
388#121—272no—
389#121—272no—
390#120—272no—
391#120—271no—
392#120—271no—
393C2×C22⋊C4—271no—
394C3×C9#53411yescharacteristic, complemented, Sylow
395C23×C3—21no—
396C23×C3—31no—
397C23×C3—91no—
398C23×C3C22×D911yescharacteristic
399C23×C3C2×D3211yescharacteristic
400C22×D3—31no—
401C22×D3—91no—
402C22×D3—91no—
403C22×D3—91no—
404C22×D3—91no—
405C22×D3—94no—
406C22×D3—94no—
407C22×D3—92no—
408C22×D3—92no—
409C22×D3—181no—
410C22×D3—181no—
411C22×D3—184no—
412C22×D3—182no—
413C22×D3—182no—
414C22×D3—182no—
415C22×D3—184no—
416C22×D3—182no—
417C22×D3—182no—
418C22×D3—362no—
419C22×D3—362no—
420C22×D3—362no—
421C22×D3—362no—
422C22×D3—362no—
423C22×D3—362no—
424C22×D3—364no—
425C22×D3—364no—
426C22×D3C22×D911yescharacteristic
427C3×D4—62no—
428C3×D4—182no—
429C2×C3×C4—24no—
430C2×C3×C4—22no—
431C2×C3×C4—22no—
432C2×C3×C4—22no—
433C2×C3×C4—32no—
434C2×C3×C4—31no—
435C2×C3×C4—92no—
436C2×C3×C4—91no—
437C2×C3×C4C22×D912yes—
438C2×C3×C4C2×D3212yes—
439#67—22no—
440#67—62no—
441#67—182no—
442#67—182no—
443#67—362no—
444#67—362no—
445C2×Dic3—31no—
446C2×Dic3—32no—
447C2×Dic3C22×D911yescharacteristic
448C2×Dic3C22×D912yes—
449D12—184no—
450D12—184no—
451D12—184no—
452D12—184no—
453D12—188no—
454D12—188no—
455D12—182no—
456D12—182no—
457D12—362no—
458D12—362no—
459C4×D3—182no—
460C4×D3—182no—
461C4×D3—362no—
462C4×D3—362no—
463C2×C32—22no—
464C2×C32C2×D3×D411yescharacteristic
465C2×C32#100912yes—
466C3⋊D3—62no—
467C3⋊D3—62no—
468C3⋊D3—122no—
469C3×D3—41no—
470C3×D3—121no—
471C2×C9—24no—
472C2×C9—22no—
473C2×C9—22no—
474C2×C9—21no—
475C2×C9—121no—
476C2×C9C2×D3×D411yescharacteristic
477C2×C9#100912yes—
478D9—41no—
479D9—62no—
480D9—62no—
481D9—122no—
482D9—122no—
483D9—124no—
484D9—122no—
485C24—271no—
486C24—271no—
487C24—271no—
488C2×D4—31no—
489C2×D4—91no—
490C2×D4—271no—
491C2×D4—272no—
492C2×D4—271no—
493C2×D4—272no—
494C2×D4—272no—
495C2×D4—272no—
496C2×D4—544no—
497C2×D4—544no—
498C22×C4—271no—
499C22×C4—272no—
500C22×C4D3×D911yescharacteristic, complemented
501C4⋊C4—32no—
502C4⋊C4—92no—
503C22⋊C4—31no—
504C22⋊C4—32no—
505C22⋊C4—91no—
506C22⋊C4—92no—
507C22⋊C4—272no—
508C22⋊C4—272no—
509C22⋊C4—272no—
510C22×C3—24no—
511C22×C3—22no—
512C22×C3—22no—
513C22×C3—21no—
514C22×C3—22no—
515C22×C3—62no—
516C22×C3—61no—
517C22×C3—181no—
518C22×C3—182no—
519C22×C3D4×D912yes—
520C22×C3C23×D911yescharacteristic
521C22×C3#381112yes—
522C22×C3C22×D3211yescharacteristic
523C2×D3—21no—
524C2×D3—22no—
525C2×D3—62no—
526C2×D3—61no—
527C2×D3—92no—
528C2×D3—92no—
529C2×D3—92no—
530C2×D3—92no—
531C2×D3—182no—
532C2×D3—182no—
533C2×D3—182no—
534C2×D3—182no—
535C2×D3—182no—
536C2×D3—182no—
537C2×D3—184no—
538C2×D3—182no—
539C2×D3—182no—
540C2×D3—182no—
541C2×D3—182no—
542C2×D3—182no—
543C2×D3—182no—
544C2×D3—362no—
545C2×D3—362no—
546C2×D3—362no—
547C2×D3—362no—
548C2×D3—364no—
549C2×D3—364no—
550C2×D3—364no—
551C2×D3—364no—
552C2×D3—364no—
553C2×D3—368no—
554C2×D3—368no—
555C2×D3—364no—
556C3×C4—24no—
557C3×C4—22no—
558C3×C4—22no—
559C3×C4—62no—
560C3×C4—61no—
561C3×C4—181no—
562C3×C4—182no—
563Dic3—21no—
564Dic3—22no—
565Dic3—62no—
566Dic3—61no—
567C32#565311yescharacteristic
568C9—21no—
569C9#565311yescharacteristic, complemented
570C23—31no—
571C23—91no—
572C23—271no—
573C23—271no—
574C23—274no—
575C23—272no—
576C23—542no—
577C23—542no—
578C23—1082no—
579C23—1082no—
580C23—1082no—
581C23C2×D3×D911yescharacteristic
582D4—62no—
583D4—182no—
584D4—544no—
585D4—544no—
586D4—542no—
587D4—542no—
588D4—1082no—
589D4—1082no—
590C2×C4—22no—
591C2×C4—32no—
592C2×C4—31no—
593C2×C4—92no—
594C2×C4—91no—
595C2×C4—542no—
596C2×C4—1082no—
597C2×C4C2×D3×D912yes—
598C2×C3—24no—
599C2×C3—22no—
600C2×C3—22no—
601C2×C3—21no—
602C2×C3—22no—
603C2×C3—121no—
604C2×C3—361no—
605C2×C3C2×D4×D911yescharacteristic
606C2×C3#6371512yes—
607C2×C3#6430812yes—
608C2×C3C2×#381111yescharacteristic
609D3—41no—
610D3—121no—
611D3—182no—
612D3—182no—
613D3—182no—
614D3—182no—
615D3—362no—
616D3—362no—
617D3—364no—
618D3—362no—
619D3—362no—
620C22—22no—
621C22—61no—
622C22—62no—
623C22—181no—
624C22—182no—
625C22—272no—
626C22—272no—
627C22—542no—
628C22—542no—
629C22—542no—
630C22—542no—
631C22—542no—
632C22—1084no—
633C22—1084no—
634C22—1084no—
635C22#8866812yes—
636C22C22×D3×D911yescharacteristic
637C4—22no—
638C4—61no—
639C4—62no—
640C4—181no—
641C4—182no—
642C3—21no—
643C3#1058975511yescharacteristic, complemented
644C3#1059508011yescharacteristic
645C2—22no—
646C2—121no—
647C2—361no—
648C2—542no—
649C2—542no—
650C2—1082no—
651C2#1172247812yes—
652C2C2×#8866811yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC22×C3×C4×C9C22
socleC22×C32C23×D3
hypercenterC22×C4D3×D9
hypocenterC3×C9#534
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C22×C4⋊(C3⋊D9)C2semidirect
2. C2×#88668C2semidirect
3. C9×#929C2semidirect
4. C3×#63491C2semidirect
5. #11721291C2semidirect
6. #11721278C2semidirect
7. #11721274C2semidirect
8. #11721051C2semidirect
9. #11721039C2semidirect
10. #11721005C2semidirect
11. #11720997C2semidirect
15. C23×C3⋊D9C22semidirect
16. C22×D3×D9C22semidirect
17. C22×C3×C4×C9C22semidirect
20. C2×C4⋊(C3⋊D9)C22semidirect
24. C2×#6195C22semidirect
25. C2×#6195C22semidirect
26. C2×#6192C22semidirect
27. C2×#6192C22semidirect
28. C2×(C3⋊D9)⋊C4C22semidirect
29. C2×(C3⋊D9)⋊C4C22semidirect
30. C9×#217C22semidirect
31. C3×#3676C22semidirect
32. C9×(C2×D3)⋊C4C22semidirect
33. C3×(C2×D9)⋊C4C22semidirect
34. C9×Dic3⋊C4C22semidirect
35. C3×Dic9⋊C4C22semidirect
36. #88445C22semidirect
37. #88445C22semidirect
38. #88445C22semidirect
55. #63491D3semidirect
66. C22×C3⋊D9D4semidirect
124. C22×C4×C9C2×D3semidirect
142. #3676C2×D3semidirect
146. (C2×D9)⋊C4C2×D3semidirect
147. Dic9⋊C4C2×D3semidirect
180. #929D9semidirect
281. C22×C3×C4C2×D9semidirect
315. #217C2×D9semidirect
323. (C2×D3)⋊C4C2×D9semidirect
325. Dic3⋊C4C2×D9semidirect
394. C3×C9#534semidirect
500. C22×C4D3×D9semidirect
569. C9#5653semidirect
643. C3#10589755semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊(C3⋊D9)2
C2×#886682
C9×#9292
C3×#634912
#117212912
#117212782
#117212742
#117210512
#117210392
#117210052
#117209972
#105950803
#105897553