Finite Group Databasebeta

← #13151006

Subgroups — #13151006

Order 1728.

1262 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D4×C3⋊D9C211yesmaximal, characteristic
2C22×(C3⋊D9)⋊C4C211yesmaximal, characteristic, complemented
3C9×#935C211yesmaximal, characteristic, complemented
4C3×#63497C211yesmaximal, characteristic, complemented
5#11721275C212yesmaximal, complemented
6#11721274C212yesmaximal, complemented
7#11721274C211yesmaximal, characteristic, complemented
8#11721255C211yesmaximal, characteristic, complemented
9#11721251C211yesmaximal, characteristic, complemented
10#11720997C212yesmaximal, complemented
11#11720980C212yesmaximal, complemented
12#10595169—31nomaximal
13D9×#123—31nomaximal
14C23×C3⋊D9C2211yescharacteristic
15C23×C3⋊D9C2211yescharacteristic
16C2×C3×D4×C9C2211yescharacteristic
17C2×#6233C2211yescharacteristic
18C2×#6233C2211yescharacteristic
19D4×C3⋊D9—24no—
20C2×C4⋊(C3⋊D9)C2211yescharacteristic
21C2×C4×C3⋊D9C2211yescharacteristic
22C22×C9×Dic3C2211yescharacteristic, complemented
23C22×C3×Dic9C2211yescharacteristic, complemented
24C2×(C3⋊D9)⋊C4—24no—
25C2×(C3⋊D9)⋊C4C2212yes—
26C2×(C3⋊D9)⋊C4C2212yes—
27C2×(C3⋊D9)⋊C4C2212yescomplemented
28C9×#217C2211yescharacteristic, complemented
29C9×#217C2212yescomplemented
30C3×#3676C2211yescharacteristic, complemented
31C3×#3676C2212yescomplemented
32C9×Dic3⋊C4C2212yescomplemented
33C3×Dic9⋊C4C2212yescomplemented
34#88445C2212yescomplemented
35#88445C2212yescomplemented
36#88445C2212yescomplemented
37Dic3⋊Dic9C2212yescomplemented
38Dic9⋊Dic3C2212yescomplemented
39C2×D4×C3⋊D3—31no—
40C22×#624—31no—
41C3×#935—31no—
42C3×#935D311yescharacteristic
43#63975—32no—
44#63974—31no—
45#63974—32no—
46#63966—31no—
47#63966—31no—
48#63897—32no—
49#63883—32no—
50C2×D4×D9—31no—
51C2×D4×D9—61no—
52C22×C4×D9—31no—
53C9×#123—31no—
54#63497D311yescharacteristic, complemented
55#63491—31no—
56#63456—32no—
57C4⋊C4×D9—32no—
58#63448—31no—
59#63445—32no—
60C22⋊C4×D9—32no—
61C22⋊C4×D9—31no—
62C23×C3×C9C2311yescharacteristic
63C23×C3×C9C2311yescharacteristic
64C22×C3⋊D9—24no—
65C22×C3⋊D9—22no—
66C22×C3⋊D9—24no—
67C22×C3⋊D9D414yes—
68C22×C3⋊D9C2311yescharacteristic
69C22×C3⋊D9C2311yescharacteristic
70C22×C3⋊D9C2311yescharacteristic
71C3×D4×C9—22no—
72C2×C3×C4×C9C2311yescharacteristic
73#6233—22no—
74#6233—22no—
75C2×C3⋊Dic9C2311yescharacteristic
76C4⋊(C3⋊D9)—22no—
77C4×C3⋊D9—22no—
78C2×C9×Dic3—22no—
79C2×C9×Dic3C2312yes—
80C2×C9×Dic3C2312yes—
81C2×C3×Dic9—22no—
82C2×C3×Dic9C2312yes—
83C2×C3×Dic9C2312yes—
84(C3⋊D9)⋊C4—24no—
85(C3⋊D9)⋊C4—24no—
86(C3⋊D9)⋊C4—24no—
87D3×#123—91no—
88D3×#123—91no—
89C23×C3⋊D3—31no—
90C23×C3⋊D3—31no—
91C2×C32×D4C2×D311yescharacteristic
92C2×#638—31no—
93C2×#638—31no—
94D4×C3⋊D3—64no—
95C2×C4⋊(C3⋊D3)—31no—
96C2×C4×C3⋊D3—31no—
97C22×C3×Dic3—31no—
98C22×C3×Dic3C2×D311yescharacteristic
99C2×#624—32no—
100C2×#624—32no—
101C2×#624—32no—
102C2×#624—64no—
103C23×D9—31no—
104C23×D9—31no—
105C23×D9—61no—
106C23×D9—61no—
107C3×#217—31no—
108C3×#217—32no—
109C3×#217C2×D311yescharacteristic
110C3×#217C2×D312yes—
111C3×Dic3⋊C4—32no—
112C3×Dic3⋊C4C2×D312yes—
113Dic3⋊Dic3—32no—
114Dic3⋊Dic3—32no—
115#3722—32no—
116#3722—32no—
117#3722—32no—
118C2×D4×C9—21no—
119C2×D4×C9C2×D311yescharacteristic
120C22×C4×C9—31no—
121C2×#611—31no—
122C2×#611—31no—
123C2×#611—61no—
124C2×#611—61no—
125C22×Dic9C2×D311yescharacteristic, complemented
126D4×D9—64no—
127D4×D9—68no—
128C2×D36—31no—
129C2×D36—61no—
130C2×C4×D9—31no—
131C2×C4×D9—32no—
132C2×C4×D9—32no—
133C2×C4×D9—32no—
134C2×C4×D9—64no—
135C2×C4×D9—61no—
136C9×C4⋊C4—32no—
137C9×C22⋊C4—32no—
138C9×C22⋊C4—31no—
139#3676C2×D311yescharacteristic, complemented
140#3676C2×D312yescomplemented
141(C2×D9)⋊C4—32no—
142(C2×D9)⋊C4—32no—
143(C2×D9)⋊C4—32no—
144C4⋊Dic9—32no—
145Dic9⋊C4—32no—
146Dic9⋊C4C2×D312yescomplemented
147C22×C3×C9—22no—
148C22×C3×C9—21no—
149C22×C3×C9—22no—
150C22×C3×C9C2×D412yes—
151C22×C3×C9C2411yescharacteristic
152C2×C3⋊D9—22no—
153C2×C3⋊D9—28no—
154C2×C3⋊D9—21no—
155C2×C3⋊D9—22no—
156C2×C3⋊D9—44no—
157C2×C3⋊D9C2×D412yes—
158C2×C3⋊D9C2×D412yes—
159C2×C3⋊D9#4114yes—
160C3×C4×C9—21no—
161C3⋊Dic9—21no—
162C9×Dic3—22no—
163C9×Dic3—22no—
164C3×Dic9—22no—
165C3×Dic9—22no—
166C2×D3×D4—91no—
167C2×D3×D4—91no—
168C2×D3×D4—181no—
169C22×C4×D3—91no—
170C22×C4×D3—91no—
171C3×#123—31no—
172C3×#123—91no—
173#935—31no—
174#935D911yescharacteristic, complemented
175#929—91no—
176#929—91no—
177#894—92no—
178#894—92no—
179D3×C4⋊C4—92no—
180D3×C4⋊C4—92no—
181#886—91no—
182#886—91no—
183#883—92no—
184#883—92no—
185D3×C22⋊C4—91no—
186D3×C22⋊C4—92no—
187D3×C22⋊C4—91no—
188D3×C22⋊C4—92no—
189C23×C32C22×D311yescharacteristic
190C23×C32C22×D311yescharacteristic
191C22×C3⋊D3—31no—
192C22×C3⋊D3—34no—
193C22×C3⋊D3—31no—
194C22×C3⋊D3—31no—
195C22×C3⋊D3—62no—
196C22×C3⋊D3—64no—
197C22×C3⋊D3—64no—
198C32×D4—22no—
199C2×C32×C4C22×D311yescharacteristic
200#638—62no—
201#638—62no—
202C2×C3⋊Dic3—31no—
203C4⋊(C3⋊D3)—62no—
204C4×C3⋊D3—62no—
205C2×C3×Dic3—22no—
206C2×C3×Dic3—32no—
207C2×C3×Dic3—32no—
208C2×C3×Dic3—62no—
209C2×C3×Dic3C22×D312yes—
210C2×C3×Dic3C22×D312yes—
211#624—64no—
212#624—64no—
213#624—64no—
214C23×C9—21no—
215C23×C9—21no—
216C23×C9C22×D311yescharacteristic
217C23×C9C22×D311yescharacteristic
218C22×D9—31no—
219C22×D9—34no—
220C22×D9—31no—
221C22×D9—31no—
222C22×D9—61no—
223C22×D9—61no—
224C22×D9—61no—
225C22×D9—64no—
226C22×D9—64no—
227C22×D9—62no—
228C22×D9—64no—
229C22×D9—64no—
230C22×D9—124no—
231C22×D9—124no—
232D4×C9—24no—
233D4×C9—22no—
234C2×C4×C9—21no—
235C2×C4×C9—32no—
236C2×C4×C9—32no—
237C2×C4×C9—62no—
238C2×C4×C9C22×D311yescharacteristic
239#611—62no—
240#611—64no—
241#611—64no—
242#611—62no—
243C2×Dic9—22no—
244C2×Dic9—31no—
245C2×Dic9—61no—
246C2×Dic9C22×D312yes—
247C2×Dic9C22×D312yes—
248D36—62no—
249D36—64no—
250C4×D9—64no—
251C4×D9—64no—
252C4×D9—64no—
253C4×D9—62no—
254C4×D9—64no—
255C2×#123—271noSylow
256C2×C3×C9—22no—
257C2×C3×C9—41no—
258C2×C3×C9C22×D411yescharacteristic
259C2×C3×C9C2×#4112yes—
260C3⋊D9—22no—
261C3⋊D9—41no—
262C3⋊D9#12314yescomplemented
263C23×D3—91no—
264C23×D3—91no—
265C23×D3—91no—
266C23×D3—91no—
267C23×D3—181no—
268C23×D3—181no—
269C2×C3×D4—21no—
270C2×C3×D4C2×D911yescharacteristic
271C2×C3×D4D3211yescharacteristic
272C22×C3×C4—31no—
273C22×C3×C4—91no—
274C2×#67—91no—
275C2×#67—91no—
276C2×#67—91no—
277C2×#67—91no—
278C2×#67—181no—
279C2×#67—181no—
280C22×Dic3—31no—
281C22×Dic3C2×D911yescharacteristic, complemented
282D3×D4—184no—
283D3×D4—184no—
284D3×D4—188no—
285C2×D12—91no—
286C2×D12—91no—
287C2×D12—181no—
288C2×C4×D3—92no—
289C2×C4×D3—92no—
290C2×C4×D3—92no—
291C2×C4×D3—91no—
292C2×C4×D3—91no—
293C2×C4×D3—92no—
294C2×C4×D3—92no—
295C2×C4×D3—92no—
296C2×C4×D3—181no—
297C2×C4×D3—184no—
298C2×C4×D3—184no—
299C3×C4⋊C4—32no—
300C3×C4⋊C4—92no—
301C3×C22⋊C4—32no—
302C3×C22⋊C4—31no—
303C3×C22⋊C4—92no—
304C3×C22⋊C4—91no—
305#217—32no—
306#217—31no—
307#217C2×D912yescomplemented
308#217C2×D911yescharacteristic, complemented
309(C2×D3)⋊C4—92no—
310(C2×D3)⋊C4—92no—
311(C2×D3)⋊C4—92no—
312(C2×D3)⋊C4—92no—
313(C2×D3)⋊C4—92no—
314(C2×D3)⋊C4—92no—
315C4⋊Dic3—92no—
316C4⋊Dic3—92no—
317Dic3⋊C4—32no—
318Dic3⋊C4—92no—
319Dic3⋊C4—92no—
320Dic3⋊C4C2×D912yescomplemented
321C22×C32—21no—
322C22×C32—22no—
323C22×C32—22no—
324C22×C32D3×D412yes—
325C22×C32C23×D311yescharacteristic
326C2×C3⋊D3—32no—
327C2×C3⋊D3—34no—
328C2×C3⋊D3—32no—
329C2×C3⋊D3—61no—
330C2×C3⋊D3—68no—
331C2×C3⋊D3—62no—
332C2×C3⋊D3—62no—
333C2×C3⋊D3—124no—
334C32×C4—21no—
335C3⋊Dic3—61no—
336C3×Dic3—22no—
337C3×Dic3—22no—
338C3×Dic3—62no—
339C3×Dic3—62no—
340C22×C9—22no—
341C22×C9—22no—
342C22×C9—21no—
343C22×C9—22no—
344C22×C9—21no—
345C22×C9—22no—
346C22×C9—42no—
347C22×C9—42no—
348C22×C9D3×D412yes—
349C22×C9C23×D311yescharacteristic
350C2×D9—32no—
351C2×D9—34no—
352C2×D9—32no—
353C2×D9—62no—
354C2×D9—64no—
355C2×D9—62no—
356C2×D9—68no—
357C2×D9—62no—
358C2×D9—62no—
359C2×D9—61no—
360C2×D9—62no—
361C2×D9—128no—
362C2×D9—122no—
363C2×D9—122no—
364C2×D9—124no—
365C2×D9—128no—
366C4×C9—22no—
367C4×C9—21no—
368C4×C9—62no—
369C4×C9—62no—
370Dic9—22no—
371Dic9—22no—
372Dic9—62no—
373Dic9—61no—
374C22×D4—271no—
375C23×C4—271no—
376#123—31no—
377#123—91no—
378#123—272no—
379#123—271no—
380#123—271no—
381#123—272no—
382C2×C4⋊C4—272no—
383C2×C22⋊C4—271no—
384C2×C22⋊C4—272no—
385C3×C9C2×#12311yescharacteristic, complemented, Sylow
386C23×C3—21no—
387C23×C3—21no—
388C23×C3C22×D911yescharacteristic
389C23×C3C22×D911yescharacteristic
390C23×C3C2×D3211yescharacteristic
391C23×C3C2×D3211yescharacteristic
392C22×D3—91no—
393C22×D3—91no—
394C22×D3—91no—
395C22×D3—94no—
396C22×D3—91no—
397C22×D3—94no—
398C22×D3—91no—
399C22×D3—91no—
400C22×D3—181no—
401C22×D3—181no—
402C22×D3—181no—
403C22×D3—184no—
404C22×D3—184no—
405C22×D3—184no—
406C22×D3—182no—
407C22×D3—184no—
408C22×D3—182no—
409C22×D3—184no—
410C22×D3—184no—
411C22×D3—364no—
412C22×D3—364no—
413C3×D4—24no—
414C3×D4—22no—
415C3×D4—22no—
416C2×C3×C4—21no—
417C2×C3×C4—32no—
418C2×C3×C4—32no—
419C2×C3×C4—62no—
420C2×C3×C4—92no—
421C2×C3×C4—92no—
422C2×C3×C4—182no—
423C2×C3×C4C22×D911yescharacteristic
424C2×C3×C4C2×D3211yescharacteristic
425#67—182no—
426#67—182no—
427#67—184no—
428#67—184no—
429#67—182no—
430#67—182no—
431C2×Dic3—22no—
432C2×Dic3—32no—
433C2×Dic3—32no—
434C2×Dic3—62no—
435C2×Dic3—91no—
436C2×Dic3—91no—
437C2×Dic3—181no—
438C2×Dic3C22×D912yes—
439C2×Dic3C22×D912yes—
440D12—182no—
441D12—182no—
442D12—184no—
443C4×D3—184no—
444C4×D3—184no—
445C4×D3—184no—
446C4×D3—184no—
447C4×D3—184no—
448C4×D3—184no—
449C4×D3—182no—
450C4×D3—182no—
451C4×D3—184no—
452C2×C32—22no—
453C2×C32—41no—
454C2×C32C2×D3×D411yescharacteristic
455C2×C32D3×#4112yes—
456C3⋊D3—34no—
457C3⋊D3—62no—
458C3⋊D3—121no—
459C2×C9—22no—
460C2×C9—22no—
461C2×C9—21no—
462C2×C9—42no—
463C2×C9—42no—
464C2×C9—41no—
465C2×C9C2×D3×D411yescharacteristic
466C2×C9D3×#4112yes—
467D9—34no—
468D9—64no—
469D9—62no—
470D9—122no—
471D9—122no—
472D9—121no—
473C24—271no—
474C24—271no—
475C2×D4—271no—
476C2×D4—271no—
477C2×D4—271no—
478C2×D4—544no—
479C2×D4D3×D911yescharacteristic
480C22×C4—31no—
481C22×C4—91no—
482C22×C4—272no—
483C22×C4—272no—
484C22×C4—272no—
485C22×C4—271no—
486C22×C4—544no—
487C4⋊C4—32no—
488C4⋊C4—92no—
489C4⋊C4—272no—
490C4⋊C4—272no—
491C22⋊C4—31no—
492C22⋊C4—32no—
493C22⋊C4—91no—
494C22⋊C4—92no—
495C22⋊C4—272no—
496C22⋊C4—272no—
497C22⋊C4—272no—
498C22×C3—22no—
499C22×C3—22no—
500C22×C3—22no—
501C22×C3—22no—
502C22×C3—21no—
503C22×C3—21no—
504C22×C3—22no—
505C22×C3—21no—
506C22×C3—22no—
507C22×C3—42no—
508C22×C3—42no—
509C22×C3D4×D912yes—
510C22×C3C23×D911yescharacteristic
511C22×C3#381112yes—
512C22×C3C22×D3211yescharacteristic
513C2×D3—94no—
514C2×D3—92no—
515C2×D3—92no—
516C2×D3—94no—
517C2×D3—92no—
518C2×D3—92no—
519C2×D3—182no—
520C2×D3—184no—
521C2×D3—182no—
522C2×D3—188no—
523C2×D3—188no—
524C2×D3—182no—
525C2×D3—182no—
526C2×D3—182no—
527C2×D3—182no—
528C2×D3—181no—
529C2×D3—181no—
530C2×D3—182no—
531C2×D3—368no—
532C2×D3—362no—
533C2×D3—362no—
534C2×D3—364no—
535C2×D3—364no—
536C2×D3—368no—
537C3×C4—22no—
538C3×C4—21no—
539C3×C4—21no—
540C3×C4—62no—
541C3×C4—62no—
542C3×C4—182no—
543C3×C4—182no—
544Dic3—22no—
545Dic3—22no—
546Dic3—62no—
547Dic3—62no—
548Dic3—182no—
549Dic3—181no—
550Dic3—181no—
551C32D3×#12311yescharacteristic
552C9—21no—
553C9D3×#12311yescharacteristic, complemented
554C23—271no—
555C23—271no—
556C23—274no—
557C23—271no—
558C23—544no—
559C23—542no—
560C23—544no—
561C23C2×D3×D911yescharacteristic
562C23C2×D3×D911yescharacteristic
563D4—22no—
564D4—542no—
565D4—542no—
566D4—542no—
567C2×C4—32no—
568C2×C4—32no—
569C2×C4—62no—
570C2×C4—92no—
571C2×C4—92no—
572C2×C4—182no—
573C2×C4—271no—
574C2×C4—542no—
575C2×C4—544no—
576C2×C4—544no—
577C2×C4—544no—
578C2×C4C2×D3×D911yescharacteristic
579C2×C3—22no—
580C2×C3—22no—
581C2×C3—22no—
582C2×C3—21no—
583C2×C3—42no—
584C2×C3—42no—
585C2×C3—41no—
586C2×C3—41no—
587C2×C3C2×D4×D911yescharacteristic
588C2×C3#41×D912yes—
589C2×C3#6431212yes—
590C2×C3C2×#381111yescharacteristic
591D3—94no—
592D3—94no—
593D3—184no—
594D3—182no—
595D3—182no—
596D3—362no—
597D3—362no—
598D3—361no—
599D3—361no—
600C22—22no—
601C22—22no—
602C22—21no—
603C22—274no—
604C22—272no—
605C22—272no—
606C22—548no—
607C22—542no—
608C22—542no—
609C22—541no—
610C22—1084no—
611C22#8866812yes—
612C22C22×D3×D911yescharacteristic
613C4—21no—
614C4—62no—
615C4—62no—
616C4—182no—
617C4—182no—
618C4—541no—
619C3—21no—
620C3D9×#12311yescharacteristic, complemented
621C3#1059516911yescharacteristic
622C2—22no—
623C2—41no—
624C2—274no—
625C2—542no—
626C2—1081no—
627C2#1172249812yes—
628C2C2×#8866811yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC2×C3×D4×C9C22
socleC22×C32C23×D3
prime radicalC2×D4×C3⋊D9C2
hypercenterC2×D4D3×D9
hypocenterC3×C9C2×#123
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. C22×(C3⋊D9)⋊C4C2semidirect
3. C9×#935C2semidirect
4. C3×#63497C2semidirect
5. #11721275C2semidirect
6. #11721274C2semidirect
7. #11721274C2semidirect
8. #11721255C2semidirect
9. #11721251C2semidirect
10. #11720997C2semidirect
11. #11720980C2semidirect
22. C22×C9×Dic3C22semidirect
23. C22×C3×Dic9C22semidirect
27. C2×(C3⋊D9)⋊C4C22semidirect
28. C9×#217C22semidirect
29. C9×#217C22semidirect
30. C3×#3676C22semidirect
31. C3×#3676C22semidirect
32. C9×Dic3⋊C4C22semidirect
33. C3×Dic9⋊C4C22semidirect
34. #88445C22semidirect
35. #88445C22semidirect
36. #88445C22semidirect
37. Dic3⋊Dic9C22semidirect
38. Dic9⋊Dic3C22semidirect
54. #63497D3semidirect
125. C22×Dic9C2×D3semidirect
139. #3676C2×D3semidirect
140. #3676C2×D3semidirect
146. Dic9⋊C4C2×D3semidirect
174. #935D9semidirect
262. C3⋊D9#123semidirect
281. C22×Dic3C2×D9semidirect
307. #217C2×D9semidirect
308. #217C2×D9semidirect
320. Dic3⋊C4C2×D9semidirect
385. C3×C9C2×#123semidirect
553. C9D3×#123semidirect
620. C3D9×#123semidirect

Maximal subgroups

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

subgroupindex
C2×D4×C3⋊D92
C22×(C3⋊D9)⋊C42
C9×#9352
C3×#634972
#117212752
#117212742
#117212552
#117212512
#117209972
#117209802
#105951693
D9×#1233