Finite Group Databasebeta

← #13152042

Subgroups — #13152042

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×#6233C211yesmaximal, characteristic, complemented
2C2×#88668C211yesmaximal, characteristic, complemented
3C9×#953C211yesmaximal, characteristic, complemented
4C3×#63515C211yesmaximal, characteristic, complemented
5#11721292C212yesmaximal, complemented
6#11721275C212yesmaximal, complemented
7#11721274C211yesmaximal, characteristic, complemented
8#11721266C211yesmaximal, characteristic, complemented
9#11721262C212yesmaximal, complemented
10#11721260C211yesmaximal, characteristic, complemented
11#11721256C212yesmaximal, complemented
12#10595528—31nomaximal
13#10589691—31nomaximal
14C24×C3×C9C2211yescharacteristic, complemented
15C23×C3⋊D9C2211yescharacteristic
16C22×D3×D9C2211yescharacteristic, complemented
17C2×#6233—24no—
18C2×#6233C2212yes—
19C2×#6233C2212yescomplemented
20C22×C3⋊Dic9C2211yescharacteristic, complemented
21C2×C9×#67C2211yescharacteristic
22C2×C9×#67C2212yescomplemented
23C2×C3×#611C2211yescharacteristic
24C2×C3×#611C2212yescomplemented
25#88668—24no—
26C2×#6195C2211yescharacteristic, complemented
27C2×#6192C2211yescharacteristic, complemented
28C2×(C3⋊D9)⋊C4C2211yescharacteristic, complemented
29C2×(C3⋊D9)⋊C4C2212yescomplemented
30C9×#217C2212yescomplemented
31C9×#217C2211yescharacteristic, complemented
32C3×#3676C2212yescomplemented
33C3×#3676C2211yescharacteristic, complemented
34(C2×D3)⋊Dic9C2212yescomplemented
35#88445C2212yescomplemented
36(C2×D9)⋊Dic3C2212yescomplemented
37Dic3⋊Dic9C2212yescomplemented
38Dic9⋊Dic3C2212yescomplemented
39C22×#638—31no—
40C2×#3811—31no—
41C3×#953—31no—
42C3×#953D311yescharacteristic
43#63983—32no—
44#63975—32no—
45#63974—31no—
46#63962—31no—
47#63962—31no—
48#63958—32no—
49#63958—32no—
50C22×#611—31no—
51C22×#611—61no—
52C2×D4×D9—31no—
53C9×#120—31no—
54#63515D311yescharacteristic, complemented
55#63500—32no—
56#63499—32no—
57#63497—31no—
58#63448—31no—
59#63446—32no—
60#63445—32no—
61C22⋊C4×D9—31no—
62C23×C3×C9—22no—
63C23×C3×C9—42no—
64C23×C3×C9C2312yes—
65C23×C3×C9C2311yescharacteristic
66C22×C3⋊D9—42no—
67C22×C3⋊D9D414yescomplemented
68C22×C3⋊D9C2311yescharacteristic
69C22×C3⋊D9C2311yescharacteristic
70C22×D3×C9C2311yescharacteristic
71C22×C3×D9C2311yescharacteristic
72C2×D3×D9—22no—
73C2×D3×D9—42no—
74#6233—24no—
75#6233—24no—
76C2×C3⋊Dic9—22no—
77C2×C3⋊Dic9C2312yes—
78C9×#67—22no—
79C9×#67—42no—
80C3×#611—22no—
81C3×#611—42no—
82C2×C9×Dic3C2311yescharacteristic
83C2×C9×Dic3C2312yes—
84C2×C3×Dic9C2311yescharacteristic
85C2×C3×Dic9C2312yes—
86#6195—22no—
87#6192—22no—
88(C3⋊D9)⋊C4—22no—
89(C3⋊D9)⋊C4—42no—
90#5589—91no—
91#5589—91no—
92C24×C32C2×D311yescharacteristic
93C23×C3⋊D3—31no—
94C22×D32—31no—
95C2×#638—32no—
96C2×#638—32no—
97C2×#638—64no—
98C22×C3⋊Dic3—31no—
99C2×C3×#67—31no—
100C2×C3×#67—32no—
101C2×C3×#67C2×D311yescharacteristic
102C2×C3×#67C2×D312yes—
103#3811—64no—
104C2×#626—31no—
105C2×#626—31no—
106C2×#624—31no—
107C2×#624—32no—
108C24×C9—21no—
109C24×C9C2×D311yescharacteristic, complemented
110C23×D9—31no—
111C23×D9—31no—
112C23×D9—61no—
113C3×#217—32no—
114C3×#217—31no—
115C3×#217C2×D312yes—
116C3×#217C2×D311yescharacteristic
117Dic3⋊Dic3—32no—
118Dic3⋊Dic3—32no—
119#3722—32no—
120(C2×D3)⋊Dic3—32no—
121(C2×D3)⋊Dic3—32no—
122C2×D4×C9—31no—
123C2×D4×C9—32no—
124C2×#611—32no—
125C2×#611—32no—
126C2×#611—31no—
127C2×#611—64no—
128C2×#611—62no—
129C2×#611—62no—
130C2×#611—68no—
131C2×#611C2×D311yescharacteristic
132C2×#611C2×D312yescomplemented
133C22×Dic9—31no—
134C22×Dic9—61no—
135D4×D9—64no—
136C2×D36—31no—
137C2×C4×D9—31no—
138C2×C4×D9—32no—
139C9×C22⋊C4—31no—
140C9×C22⋊C4—32no—
141#3676—32no—
142#3676C2×D311yescharacteristic, complemented
143#3676C2×D312yescomplemented
144(C2×D9)⋊C4—32no—
145(C2×D9)⋊C4—32no—
146C4⋊Dic9—32no—
147Dic9⋊C4—32no—
148C22×C3×C9—22no—
149C22×C3×C9—22no—
150C22×C3×C9—22no—
151C22×C3×C9—22no—
152C22×C3×C9—44no—
153C22×C3×C9C2×D412yes—
154C22×C3×C9C2411yescharacteristic
155C2×C3⋊D9—22no—
156C2×C3⋊D9—22no—
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⋊Dic9—22no—
166C9×Dic3—21no—
167C9×Dic3—22no—
168C3×Dic9—21no—
169C3×Dic9—22no—
170C22×#67—91no—
171C22×#67—91no—
172C22×#67—181no—
173C2×D3×D4—91no—
174C2×D3×D4—91no—
175C3×#120—31no—
176C3×#120—91no—
177#953—31no—
178#953D911yescharacteristic, complemented
179#938—92no—
180#938—92no—
181#937—92no—
182#937—92no—
183#935—91no—
184#935—91no—
185#886—91no—
186#886—91no—
187#884—92no—
188#884—92no—
189#883—92no—
190#883—92no—
191D3×C22⋊C4—91no—
192D3×C22⋊C4—91no—
193C23×C32—22no—
194C23×C32—42no—
195C23×C32C22×D312yes—
196C23×C32C22×D311yescharacteristic
197C22×C3⋊D3—31no—
198C22×C3⋊D3—31no—
199C22×C3⋊D3—34no—
200C22×C3⋊D3—122no—
201C22×C3×D3—31no—
202C22×C3×D3C22×D311yescharacteristic
203C2×D32—62no—
204C2×D32—122no—
205#638—64no—
206#638—64no—
207C2×C3⋊Dic3—32no—
208C2×C3⋊Dic3—62no—
209C3×#67—22no—
210C3×#67—42no—
211C3×#67—62no—
212C3×#67—122no—
213C2×C3×Dic3—32no—
214C2×C3×Dic3—31no—
215C2×C3×Dic3C22×D312yes—
216C2×C3×Dic3C22×D311yescharacteristic
217#626—62no—
218#626—62no—
219#624—62no—
220#624—122no—
221C23×C9—24no—
222C23×C9—22no—
223C23×C9—21no—
224C23×C9—22no—
225C23×C9—31no—
226C23×C9—42no—
227C23×C9—44no—
228C23×C9C22×D312yes—
229C23×C9C22×D311yescharacteristic
230C22×D9—31no—
231C22×D9—31no—
232C22×D9—34no—
233C22×D9—64no—
234C22×D9—61no—
235C22×D9—61no—
236C22×D9—62no—
237C22×D9—122no—
238C22×D9—122no—
239C22×D9—124no—
240C22×D9C22×D311yescharacteristic
241D4×C9—62no—
242D4×C9—122no—
243C2×C4×C9—31no—
244C2×C4×C9—32no—
245#611—22no—
246#611—42no—
247#611—62no—
248#611—64no—
249#611—64no—
250#611—68no—
251#611—68no—
252C2×Dic9—32no—
253C2×Dic9—62no—
254C2×Dic9—64no—
255C2×Dic9—62no—
256C2×Dic9C22×D312yes—
257C2×Dic9C22×D311yescharacteristic
258D36—62no—
259C4×D9—62no—
260C4×D9—122no—
261#534—271noSylow
262C2×C3×C9—22no—
263C2×C3×C9—42no—
264C2×C3×C9C22×D411yescharacteristic
265C2×C3×C9H(5,𝔽2)12yes—
266C3⋊D9—22no—
267C3⋊D9—22no—
268D3×C9—41no—
269C3×D9—41no—
270C24×C3—21no—
271C24×C3C2×D911yescharacteristic, complemented
272C24×C3D3211yescharacteristic
273C23×D3—91no—
274C23×D3—91no—
275C23×D3—91no—
276C23×D3—91no—
277C23×D3—181no—
278C2×C3×D4—31no—
279C2×C3×D4—32no—
280C2×C3×D4—91no—
281C2×C3×D4—92no—
282C2×#67—31no—
283C2×#67—32no—
284C2×#67—91no—
285C2×#67—92no—
286C2×#67—92no—
287C2×#67—92no—
288C2×#67—92no—
289C2×#67—91no—
290C2×#67—182no—
291C2×#67—182no—
292C2×#67—184no—
293C2×#67—184no—
294C2×#67—188no—
295C2×#67C2×D911yescharacteristic
296C2×#67C2×D912yescomplemented
297C22×Dic3—91no—
298C22×Dic3—91no—
299C22×Dic3—181no—
300D3×D4—184no—
301D3×D4—184no—
302C2×D12—91no—
303C2×D12—91no—
304C2×C4×D3—91no—
305C2×C4×D3—91no—
306C2×C4×D3—92no—
307C2×C4×D3—92no—
308C3×C22⋊C4—31no—
309C3×C22⋊C4—32no—
310C3×C22⋊C4—91no—
311C3×C22⋊C4—92no—
312#217—31no—
313#217—32no—
314#217—92no—
315#217—92no—
316#217C2×D911yescharacteristic, complemented
317#217C2×D912yescomplemented
318(C2×D3)⋊C4—92no—
319(C2×D3)⋊C4—92no—
320(C2×D3)⋊C4—92no—
321(C2×D3)⋊C4—92no—
322C4⋊Dic3—92no—
323C4⋊Dic3—92no—
324Dic3⋊C4—92no—
325Dic3⋊C4—92no—
326C22×C32—22no—
327C22×C32—22no—
328C22×C32—22no—
329C22×C32—22no—
330C22×C32—44no—
331C22×C32D3×D412yes—
332C22×C32C23×D311yescharacteristic
333C2×C3⋊D3—32no—
334C2×C3⋊D3—32no—
335C2×C3⋊D3—62no—
336C2×C3⋊D3—62no—
337C2×C3⋊D3—124no—
338C2×C3×D3—21no—
339C2×C3×D3—22no—
340C2×C3×D3—61no—
341C2×C3×D3—62no—
342D32—124no—
343C3⋊Dic3—62no—
344C3×Dic3—21no—
345C3×Dic3—22no—
346C3×Dic3—61no—
347C3×Dic3—62no—
348C22×C9—24no—
349C22×C9—22no—
350C22×C9—24no—
351C22×C9—22no—
352C22×C9—22no—
353C22×C9—22no—
354C22×C9—21no—
355C22×C9—22no—
356C22×C9—42no—
357C22×C9—42no—
358C22×C9—48no—
359C22×C9—44no—
360C22×C9—61no—
361C22×C9—62no—
362C22×C9D3×D412yes—
363C22×C9C23×D311yescharacteristic
364C2×D9—21no—
365C2×D9—22no—
366C2×D9—32no—
367C2×D9—32no—
368C2×D9—62no—
369C2×D9—62no—
370C2×D9—62no—
371C2×D9—62no—
372C2×D9—122no—
373C2×D9—122no—
374C2×D9—124no—
375C2×D9—124no—
376C2×D9—128no—
377C4×C9—61no—
378C4×C9—62no—
379Dic9—21no—
380Dic9—22no—
381Dic9—64no—
382Dic9—62no—
383C22×D4—271no—
384C22×D4—271no—
385#123—272no—
386#123—271no—
387#123—271no—
388#121—272no—
389#121—272no—
390#120—31no—
391#120—91no—
392#120—272no—
393C2×C22⋊C4—271no—
394C3×C9#53411yescharacteristic, complemented, Sylow
395C23×C3—24no—
396C23×C3—22no—
397C23×C3—22no—
398C23×C3—21no—
399C23×C3—22no—
400C23×C3—31no—
401C23×C3—42no—
402C23×C3—42no—
403C23×C3—44no—
404C23×C3—91no—
405C23×C3C22×D912yes—
406C23×C3C22×D911yescharacteristic
407C23×C3C2×D3212yes—
408C23×C3C2×D3211yescharacteristic
409C22×D3—31no—
410C22×D3—91no—
411C22×D3—91no—
412C22×D3—91no—
413C22×D3—91no—
414C22×D3—94no—
415C22×D3—94no—
416C22×D3—181no—
417C22×D3—181no—
418C22×D3—184no—
419C22×D3—182no—
420C22×D3—182no—
421C22×D3—362no—
422C22×D3—362no—
423C22×D3—362no—
424C22×D3—362no—
425C22×D3—364no—
426C22×D3C22×D911yescharacteristic
427C3×D4—62no—
428C3×D4—122no—
429C3×D4—182no—
430C3×D4—362no—
431C2×C3×C4—31no—
432C2×C3×C4—32no—
433C2×C3×C4—91no—
434C2×C3×C4—92no—
435#67—22no—
436#67—42no—
437#67—62no—
438#67—122no—
439#67—182no—
440#67—182no—
441#67—184no—
442#67—184no—
443#67—184no—
444#67—184no—
445#67—188no—
446#67—188no—
447C2×Dic3—32no—
448C2×Dic3—31no—
449C2×Dic3—92no—
450C2×Dic3—92no—
451C2×Dic3—182no—
452C2×Dic3—184no—
453C2×Dic3—182no—
454C2×Dic3—182no—
455C2×Dic3C22×D912yes—
456C2×Dic3C22×D911yescharacteristic
457D12—182no—
458D12—182no—
459C4×D3—182no—
460C4×D3—182no—
461C4×D3—362no—
462C4×D3—362no—
463C2×C32—22no—
464C2×C32—42no—
465C2×C32C2×D3×D411yescharacteristic
466C2×C32#100412yes—
467C3⋊D3—62no—
468C3⋊D3—62no—
469C3×D3—41no—
470C3×D3—121no—
471C2×C9—24no—
472C2×C9—22no—
473C2×C9—21no—
474C2×C9—22no—
475C2×C9—44no—
476C2×C9—42no—
477C2×C9—121no—
478C2×C9C2×D3×D411yescharacteristic
479C2×C9#100412yes—
480D9—41no—
481D9—62no—
482D9—62no—
483D9—122no—
484D9—122no—
485C24—271no—
486C24—271no—
487C24D3×D911yescharacteristic, complemented
488C2×D4—31no—
489C2×D4—32no—
490C2×D4—91no—
491C2×D4—92no—
492C2×D4—271no—
493C2×D4—271no—
494C2×D4—272no—
495C2×D4—272no—
496C2×D4—544no—
497C2×D4—544no—
498C22×C4—271no—
499C22×C4—272no—
500C22×C4—271no—
501C4⋊C4—272no—
502C4⋊C4—272no—
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—24no—
514C22×C3—22no—
515C22×C3—22no—
516C22×C3—22no—
517C22×C3—22no—
518C22×C3—22no—
519C22×C3—22no—
520C22×C3—21no—
521C22×C3—22no—
522C22×C3—42no—
523C22×C3—42no—
524C22×C3—44no—
525C22×C3—48no—
526C22×C3—44no—
527C22×C3—61no—
528C22×C3—62no—
529C22×C3—181no—
530C22×C3—182no—
531C22×C3D4×D912yes—
532C22×C3C23×D911yescharacteristic
533C22×C3#381112yes—
534C22×C3C22×D3211yescharacteristic
535C2×D3—21no—
536C2×D3—22no—
537C2×D3—61no—
538C2×D3—62no—
539C2×D3—92no—
540C2×D3—92no—
541C2×D3—92no—
542C2×D3—92no—
543C2×D3—182no—
544C2×D3—182no—
545C2×D3—182no—
546C2×D3—182no—
547C2×D3—182no—
548C2×D3—182no—
549C2×D3—362no—
550C2×D3—362no—
551C2×D3—364no—
552C2×D3—364no—
553C2×D3—364no—
554C2×D3—364no—
555C2×D3—368no—
556C3×C4—61no—
557C3×C4—62no—
558C3×C4—181no—
559C3×C4—182no—
560Dic3—21no—
561Dic3—22no—
562Dic3—61no—
563Dic3—62no—
564Dic3—184no—
565Dic3—182no—
566Dic3—182no—
567C32#558911yescharacteristic
568C9—21no—
569C9#558911yescharacteristic, complemented
570C23—22no—
571C23—31no—
572C23—42no—
573C23—91no—
574C23—271no—
575C23—271no—
576C23—274no—
577C23—542no—
578C23—1082no—
579C23—1082no—
580C23C2×D3×D912yes—
581C23C2×D3×D911yescharacteristic
582D4—62no—
583D4—122no—
584D4—182no—
585D4—362no—
586D4—542no—
587D4—542no—
588D4—544no—
589D4—544no—
590C2×C4—32no—
591C2×C4—31no—
592C2×C4—92no—
593C2×C4—91no—
594C2×C4—272no—
595C2×C4—542no—
596C2×C4—542no—
597C2×C4—1082no—
598C2×C3—24no—
599C2×C3—22no—
600C2×C3—22no—
601C2×C3—21no—
602C2×C3—22no—
603C2×C3—44no—
604C2×C3—42no—
605C2×C3—42no—
606C2×C3—121no—
607C2×C3—361no—
608C2×C3C2×D4×D911yescharacteristic
609C2×C3#6371012yes—
610C2×C3C2×#381111yescharacteristic
611C2×C3#6433012yes—
612D3—41no—
613D3—121no—
614D3—182no—
615D3—182no—
616D3—182no—
617D3—182no—
618D3—362no—
619D3—362no—
620C22—22no—
621C22—22no—
622C22—22no—
623C22—22no—
624C22—44no—
625C22—61no—
626C22—62no—
627C22—181no—
628C22—182no—
629C22—272no—
630C22—272no—
631C22—542no—
632C22—542no—
633C22—1084no—
634C22—1084no—
635C22#8866812yes—
636C22C22×D3×D911yescharacteristic
637C4—61no—
638C4—62no—
639C4—181no—
640C4—182no—
641C4—542no—
642C3—21no—
643C3#1058969111yescharacteristic, complemented
644C3#1059552811yescharacteristic
645C2—22no—
646C2—42no—
647C2—121no—
648C2—361no—
649C2—542no—
650C2—542no—
651C2C2×#8866811yescharacteristic
652C2#1172253512yes—
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC24×C3×C9C22
socleC22×C32C23×D3
hypercenterC24D3×D9
hypocenterC3×C9#534
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C24

Complements

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

NH ≅ G/Ntype
1. C22×#6233C2semidirect
2. C2×#88668C2semidirect
3. C9×#953C2semidirect
4. C3×#63515C2semidirect
5. #11721292C2semidirect
6. #11721275C2semidirect
7. #11721274C2semidirect
8. #11721266C2semidirect
9. #11721262C2semidirect
10. #11721260C2semidirect
11. #11721256C2semidirect
14. C24×C3×C9C22semidirect
16. C22×D3×D9C22semidirect
19. C2×#6233C22semidirect
20. C22×C3⋊Dic9C22semidirect
22. C2×C9×#67C22semidirect
24. C2×C3×#611C22semidirect
26. C2×#6195C22semidirect
27. C2×#6192C22semidirect
28. C2×(C3⋊D9)⋊C4C22semidirect
29. C2×(C3⋊D9)⋊C4C22semidirect
30. C9×#217C22semidirect
31. C9×#217C22semidirect
32. C3×#3676C22semidirect
33. C3×#3676C22semidirect
34. (C2×D3)⋊Dic9C22semidirect
35. #88445C22semidirect
36. (C2×D9)⋊Dic3C22semidirect
37. Dic3⋊Dic9C22semidirect
38. Dic9⋊Dic3C22semidirect
54. #63515D3semidirect
67. C22×C3⋊D9D4semidirect
109. C24×C9C2×D3semidirect
132. C2×#611C2×D3semidirect
142. #3676C2×D3semidirect
143. #3676C2×D3semidirect
178. #953D9semidirect
271. C24×C3C2×D9semidirect
296. C2×#67C2×D9semidirect
316. #217C2×D9semidirect
317. #217C2×D9semidirect
394. C3×C9#534semidirect
487. C24D3×D9semidirect
569. C9#5589semidirect
643. C3#10589691semidirect

Maximal subgroups

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

subgroupindex
C22×#62332
C2×#886682
C9×#9532
C3×#635152
#117212922
#117212752
#117212742
#117212662
#117212622
#117212602
#117212562
#105955283
#105896913