Finite Group Databasebeta

← #13139876

Subgroups — #13139876

Order 1728.

906 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6192C211yesmaximal, characteristic
2C2×#88536C211yesmaximal, characteristic, complemented
3C2×C3×(C2×D9)⋊C4C211yesmaximal, characteristic, complemented
4C2×C4×C9×Dic3C211yesmaximal, characteristic, complemented
5C2×#88445C211yesmaximal, characteristic, complemented
6C2×(C2×D9)⋊Dic3C211yesmaximal, characteristic, complemented
7C2×Dic3⋊Dic9C211yesmaximal, characteristic, complemented
8#10591383—31nomaximal
9#10588619—31nomaximal
10C23×C3⋊D9C2211yescharacteristic
11C23×C3×D9C2211yescharacteristic
12C22×C3×C4×C9C2211yescharacteristic
13C22×C3⋊Dic9C2211yescharacteristic
14C22×C9×Dic3C2211yescharacteristic
15C22×C9×Dic3C2211yescharacteristic
16C22×C3×Dic9C2211yescharacteristic
17C2×#6192—22no—
18C2×#6192—22no—
19C2×#6192C414yescomplemented
20#88536—22no—
21C3×(C2×D9)⋊C4—22no—
22C4×C9×Dic3—22no—
23#88445—22no—
24(C2×D9)⋊Dic3—22no—
25Dic3⋊Dic9—22no—
26C22×#626—31no—
27C2×#3752—31no—
28C2×C3×(C2×D3)⋊C4—31no—
29C2×C3×C4×Dic3D311yescharacteristic
30C2×Dic3⋊Dic3—31no—
31C2×#3722—31no—
32C2×(C2×D3)⋊Dic3—31no—
33C22×D36—31no—
34C2×C42×C9—31no—
35C2×(C2×D9)⋊C4—31no—
36C2×(C2×D9)⋊C4—31no—
37C2×(C2×D9)⋊C4—31no—
38C2×(C2×D9)⋊C4—61no—
39C2×(C2×D9)⋊C4D311yescharacteristic, complemented
40C2×C4⋊Dic9—31no—
41C23×C3×C9C2311yescharacteristic
42C22×C3⋊D9—21no—
43C22×C3⋊D9—21no—
44C22×C3⋊D9—21no—
45C22×C3⋊D9—21no—
46C22×C3⋊D9—21no—
47C22×C3⋊D9—21no—
48C22×C3⋊D9C2×C412yes—
49C22×C3×D9—21no—
50C22×C3×D9—21no—
51C22×C3×D9—21no—
52C22×C3×D9—21no—
53C22×C3×D9—21no—
54C22×C3×D9—21no—
55C22×C3×D9C2×C412yes—
56C2×C3×C4×C9—21no—
57C2×C3×C4×C9—21no—
58C2×C3×C4×C9D412yescomplemented
59C2×C3⋊Dic9—21no—
60C2×C3⋊Dic9—21no—
61C2×C3⋊Dic9—21no—
62C2×C9×Dic3—21no—
63C2×C9×Dic3—21no—
64C2×C9×Dic3C2×C412yes—
65C2×C9×Dic3D412yes—
66C2×C9×Dic3D412yes—
67C2×C9×Dic3D412yescomplemented
68C2×C3×Dic9—21no—
69C2×C3×Dic9—21no—
70C2×C3×Dic9—21no—
71#6192—28no—
72#4957—91no—
73#4939—91no—
74C23×C3⋊D3—31no—
75C23×C3×D3—31no—
76C22×C32×C4C2×D311yescharacteristic
77C22×C3⋊Dic3—31no—
78C22×C3×Dic3—31no—
79C22×C3×Dic3C2×D311yescharacteristic
80C22×C3×Dic3C2×D311yescharacteristic
81C2×#626—34no—
82C2×#626—62no—
83C2×#626—62no—
84C23×D9—31no—
85C23×D9—61no—
86C23×D9C2×D311yescharacteristic
87#3752—62no—
88C3×(C2×D3)⋊C4—62no—
89C3×C4×Dic3—22no—
90Dic3⋊Dic3—62no—
91#3722—62no—
92(C2×D3)⋊Dic3—62no—
93C22×C4×C9—21no—
94C22×C4×C9—31no—
95C22×C4×C9—31no—
96C22×C4×C9C2×D311yescharacteristic
97C22×Dic9—31no—
98C22×Dic9—61no—
99C22×Dic9C2×D311yescharacteristic
100C2×D36—34no—
101C2×D36—62no—
102C2×D36—62no—
103C42×C9—62no—
104(C2×D9)⋊C4—22no—
105(C2×D9)⋊C4—62no—
106(C2×D9)⋊C4—64no—
107(C2×D9)⋊C4—62no—
108(C2×D9)⋊C4—62no—
109C4⋊Dic9—62no—
110C22×C3×C9C22×C411yescharacteristic
111C22×C3×C9C2×D411yescharacteristic
112C22×C3×C9C2×D411yescharacteristic
113C22×C3×C9C2×D411yescharacteristic
114C22×C3×C9C2×D411yescharacteristic
115C22×C3×C9#4111yescharacteristic
116C22×C3×C9#4111yescharacteristic
117C2×C3⋊D9—22no—
118C2×C3⋊D9—22no—
119C2×C3⋊D9—22no—
120C2×C3⋊D9—41no—
121C2×C3⋊D9—41no—
122C2×C3⋊D9—41no—
123C2×C3⋊D9—41no—
124C2×C3×D9—22no—
125C2×C3×D9—22no—
126C2×C3×D9—22no—
127C2×C3×D9—41no—
128C2×C3×D9—41no—
129C2×C3×D9—41no—
130C2×C3×D9—41no—
131C3×C4×C9—22no—
132C3⋊Dic9—41no—
133C9×Dic3—22no—
134C9×Dic3C22⋊C414yescomplemented
135C3×Dic9—41no—
136C22×#67—91no—
137C22×D12—91no—
138C2×C3×C22⋊C4—91no—
139C2×C3×C42—31no—
140C2×#217—91no—
141C2×(C2×D3)⋊C4—31no—
142C2×(C2×D3)⋊C4—91no—
143C2×(C2×D3)⋊C4—91no—
144C2×(C2×D3)⋊C4—91no—
145C2×(C2×D3)⋊C4—91no—
146C2×(C2×D3)⋊C4—91no—
147C2×(C2×D3)⋊C4—181no—
148C2×C4⋊Dic3—91no—
149C2×Dic3⋊C4—91no—
150C2×C4×Dic3D911yescharacteristic, complemented
151C23×C32C22×D311yescharacteristic
152C22×C3⋊D3—32no—
153C22×C3⋊D3—61no—
154C22×C3⋊D3—61no—
155C22×C3⋊D3—61no—
156C22×C3⋊D3—61no—
157C22×C3⋊D3—61no—
158C22×C3⋊D3—61no—
159C22×C3×D3—32no—
160C22×C3×D3—61no—
161C22×C3×D3—61no—
162C22×C3×D3—61no—
163C22×C3×D3—61no—
164C22×C3×D3—61no—
165C22×C3×D3—61no—
166C2×C32×C4—21no—
167C2×C32×C4—21no—
168C2×C32×C4D1212yes—
169C2×C3⋊Dic3—61no—
170C2×C3⋊Dic3—61no—
171C2×C3⋊Dic3—61no—
172C2×C3×Dic3—21no—
173C2×C3×Dic3—21no—
174C2×C3×Dic3—61no—
175C2×C3×Dic3—61no—
176C2×C3×Dic3—61no—
177C2×C3×Dic3C4×D312yes—
178C2×C3×Dic3D1212yes—
179C2×C3×Dic3D1212yes—
180C2×C3×Dic3#6712yes—
181#626—68no—
182C23×C9—21no—
183C23×C9C22×D311yescharacteristic
184C22×D9—21no—
185C22×D9—21no—
186C22×D9—21no—
187C22×D9—21no—
188C22×D9—21no—
189C22×D9—21no—
190C22×D9—32no—
191C22×D9—62no—
192C22×D9—62no—
193C22×D9—62no—
194C22×D9—61no—
195C22×D9—61no—
196C22×D9—61no—
197C22×D9—61no—
198C22×D9—61no—
199C22×D9—61no—
200C22×D9—121no—
201C22×D9—121no—
202C22×D9—121no—
203C22×D9—121no—
204C22×D9C4×D312yes—
205C2×C4×C9—21no—
206C2×C4×C9—21no—
207C2×C4×C9—22no—
208C2×C4×C9—32no—
209C2×C4×C9—32no—
210C2×C4×C9—32no—
211C2×C4×C9—32no—
212C2×C4×C9—41no—
213C2×C4×C9—41no—
214C2×C4×C9—61no—
215C2×C4×C9—61no—
216C2×C4×C9#6712yescomplemented
217C2×Dic9—21no—
218C2×Dic9—21no—
219C2×Dic9—21no—
220C2×Dic9—62no—
221C2×Dic9—61no—
222C2×Dic9—61no—
223C2×Dic9—61no—
224C2×Dic9—121no—
225C2×Dic9—121no—
226D36—68no—
227#390—271noSylow
228C2×C3×C9C2×C22⋊C411yescharacteristic
229C2×C3×C9C4×D411yescharacteristic
230C2×C3×C9C4×D411yescharacteristic
231C2×C3×C9#12111yescharacteristic
232C2×C3×C9#12111yescharacteristic
233C2×C3×C9#12411yescharacteristic
234C2×C3×C9#12711yescharacteristic
235C3⋊D9—42no—
236C3×D9—42no—
237C24×C3—91no—
238C23×D3—31no—
239C23×D3—91no—
240C23×D3—91no—
241C23×D3—181no—
242C22×C3×C4—21no—
243C22×C3×C4—31no—
244C22×C3×C4—31no—
245C22×C3×C4—91no—
246C22×C3×C4C2×D911yescharacteristic
247C22×C3×C4D3211yescharacteristic
248C2×#67—94no—
249C2×#67—182no—
250C2×#67—182no—
251C22×Dic3—31no—
252C22×Dic3—91no—
253C22×Dic3—91no—
254C22×Dic3—181no—
255C22×Dic3C2×D911yescharacteristic
256C22×Dic3C2×D911yescharacteristic
257C2×D12—94no—
258C2×D12—182no—
259C2×D12—182no—
260C3×C22⋊C4—182no—
261C3×C42—62no—
262#217—182no—
263(C2×D3)⋊C4—62no—
264(C2×D3)⋊C4—182no—
265(C2×D3)⋊C4—184no—
266(C2×D3)⋊C4—182no—
267(C2×D3)⋊C4—182no—
268(C2×D3)⋊C4—182no—
269(C2×D3)⋊C4—182no—
270C4⋊Dic3—182no—
271Dic3⋊C4—182no—
272C4×Dic3—22no—
273C22×C32C2×C4×D311yescharacteristic
274C22×C32C2×D1211yescharacteristic
275C22×C32C2×D1211yescharacteristic
276C22×C32C2×D1211yescharacteristic
277C22×C32#23511yescharacteristic
278C22×C32#23511yescharacteristic
279C22×C32C2×#6711yescharacteristic
280C2×C3⋊D3—62no—
281C2×C3⋊D3—62no—
282C2×C3⋊D3—62no—
283C2×C3⋊D3—121no—
284C2×C3⋊D3—121no—
285C2×C3⋊D3—121no—
286C2×C3⋊D3—121no—
287C2×C3×D3—62no—
288C2×C3×D3—62no—
289C2×C3×D3—62no—
290C2×C3×D3—121no—
291C2×C3×D3—121no—
292C2×C3×D3—121no—
293C2×C3×D3—121no—
294C32×C4—22no—
295C3⋊Dic3—121no—
296C3×Dic3—22no—
297C3×Dic3—121no—
298C3×Dic3(C2×D3)⋊C414yes—
299C22×C9—21no—
300C22×C9—21no—
301C22×C9—21no—
302C22×C9—21no—
303C22×C9—21no—
304C22×C9—21no—
305C22×C9—21no—
306C22×C9C2×C4×D311yescharacteristic
307C22×C9#23511yescharacteristic
308C22×C9D3×D411yescharacteristic
309C22×C9D3×D411yescharacteristic
310C22×C9D3×D411yescharacteristic
311C22×C9#23711yescharacteristic
312C22×C9C2×#6711yescharacteristic
313C2×D9—22no—
314C2×D9—22no—
315C2×D9—22no—
316C2×D9—41no—
317C2×D9—41no—
318C2×D9—41no—
319C2×D9—41no—
320C2×D9—64no—
321C2×D9—62no—
322C2×D9—62no—
323C2×D9—62no—
324C2×D9—122no—
325C2×D9—122no—
326C2×D9—122no—
327C2×D9—122no—
328C2×D9—122no—
329C2×D9—122no—
330C2×D9—121no—
331C2×D9—121no—
332C2×D9—121no—
333C2×D9—121no—
334C4×C9—22no—
335C4×C9—34no—
336C4×C9—42no—
337C4×C9—62no—
338Dic9—41no—
339Dic9—122no—
340Dic9—121no—
341C22×D4—271no—
342C2×C4⋊C4—271no—
343C2×C22⋊C4—91no—
344C2×C22⋊C4—271no—
345C2×C22⋊C4—271no—
346C2×C22⋊C4—271no—
347C2×C42—31no—
348C3×C9#39011yescharacteristic, complemented, Sylow
349C23×C3—21no—
350C23×C3—92no—
351C23×C3—181no—
352C23×C3—181no—
353C23×C3—181no—
354C23×C3—181no—
355C23×C3—181no—
356C23×C3—181no—
357C23×C3C22×D911yescharacteristic
358C23×C3C2×D3211yescharacteristic
359C22×D3—32no—
360C22×D3—61no—
361C22×D3—61no—
362C22×D3—61no—
363C22×D3—61no—
364C22×D3—61no—
365C22×D3—61no—
366C22×D3—92no—
367C22×D3—92no—
368C22×D3—182no—
369C22×D3—182no—
370C22×D3—182no—
371C22×D3—181no—
372C22×D3—181no—
373C22×D3—181no—
374C22×D3—181no—
375C22×D3—181no—
376C22×D3—181no—
377C22×D3—181no—
378C22×D3—181no—
379C22×D3—181no—
380C22×D3—181no—
381C22×D3—181no—
382C22×D3—181no—
383C22×D3—361no—
384C22×D3—361no—
385C22×D3—361no—
386C22×D3—361no—
387C2×C3×C4—21no—
388C2×C3×C4—21no—
389C2×C3×C4—21no—
390C2×C3×C4—21no—
391C2×C3×C4—22no—
392C2×C3×C4—32no—
393C2×C3×C4—32no—
394C2×C3×C4—32no—
395C2×C3×C4—32no—
396C2×C3×C4—41no—
397C2×C3×C4—41no—
398C2×C3×C4—61no—
399C2×C3×C4—61no—
400C2×C3×C4—181no—
401C2×C3×C4—181no—
402C2×C3×C4—181no—
403C2×C3×C4D3612yescomplemented
404C2×C3×C4#62612yes—
405#67—188no—
406C2×Dic3—21no—
407C2×Dic3—21no—
408C2×Dic3—61no—
409C2×Dic3—61no—
410C2×Dic3—61no—
411C2×Dic3—182no—
412C2×Dic3—181no—
413C2×Dic3—181no—
414C2×Dic3—181no—
415C2×Dic3—181no—
416C2×Dic3—181no—
417C2×Dic3—181no—
418C2×Dic3—361no—
419C2×Dic3—361no—
420C2×Dic3C4×D912yes—
421C2×Dic3D3612yes—
422C2×Dic3D3612yescomplemented
423C2×Dic3#61112yes—
424D12—188no—
425C2×C32C4×D1211yescharacteristic
426C2×C32C4×D1211yescharacteristic
427C2×C32#87411yescharacteristic
428C2×C32#87511yescharacteristic
429C2×C32C2×(C2×D3)⋊C411yescharacteristic
430C2×C32#93011yescharacteristic
431C2×C32#93011yescharacteristic
432C3⋊D3—122no—
433C3×D3—122no—
434C2×C9—21no—
435C2×C9—21no—
436C2×C9—21no—
437C2×C9—21no—
438C2×C9—21no—
439C2×C9—21no—
440C2×C9—21no—
441C2×C9D3×C22⋊C411yescharacteristic
442C2×C9#67⋊C411yescharacteristic
443C2×C9#88411yescharacteristic
444C2×C9#88511yescharacteristic
445C2×C9C4×#6711yescharacteristic
446C2×C9#93911yescharacteristic
447C2×C9#94011yescharacteristic
448D9—42no—
449D9—124no—
450D9—122no—
451C24—91no—
452C24—271no—
453C2×D4—274no—
454C2×D4—542no—
455C2×D4—542no—
456C22×C4—31no—
457C22×C4—31no—
458C22×C4—91no—
459C22×C4—271no—
460C22×C4D3×D911yescharacteristic
461C4⋊C4—542no—
462C22⋊C4—182no—
463C22⋊C4—542no—
464C22⋊C4—542no—
465C22⋊C4—542no—
466C42—62no—
467C22×C3—21no—
468C22×C3—21no—
469C22×C3—21no—
470C22×C3—21no—
471C22×C3—21no—
472C22×C3—21no—
473C22×C3—21no—
474C22×C3—182no—
475C22×C3—182no—
476C22×C3—182no—
477C22×C3—361no—
478C22×C3—361no—
479C22×C3—361no—
480C22×C3—361no—
481C22×C3C2×C4×D911yescharacteristic
482C22×C3C2×D3611yescharacteristic
483C22×C3C2×D3611yescharacteristic
484C22×C3C2×D3611yescharacteristic
485C22×C3#369711yescharacteristic
486C22×C3#369711yescharacteristic
487C22×C3C2×#61111yescharacteristic
488C22×C3#379811yescharacteristic
489C22×C3C4×D3211yescharacteristic
490C22×C3D3×D1211yescharacteristic
491C22×C3D3×D1211yescharacteristic
492C22×C3#380411yescharacteristic
493C22×C3C2×#62611yescharacteristic
494C22×C3D3×#6711yescharacteristic
495C2×D3—62no—
496C2×D3—62no—
497C2×D3—62no—
498C2×D3—121no—
499C2×D3—121no—
500C2×D3—121no—
501C2×D3—121no—
502C2×D3—184no—
503C2×D3—182no—
504C2×D3—182no—
505C2×D3—182no—
506C2×D3—182no—
507C2×D3—182no—
508C2×D3—182no—
509C2×D3—362no—
510C2×D3—362no—
511C2×D3—362no—
512C2×D3—362no—
513C2×D3—362no—
514C2×D3—362no—
515C2×D3—361no—
516C2×D3—361no—
517C2×D3—361no—
518C2×D3—361no—
519C2×D3—361no—
520C2×D3—361no—
521C2×D3—361no—
522C2×D3—361no—
523C3×C4—22no—
524C3×C4—22no—
525C3×C4—34no—
526C3×C4—42no—
527C3×C4—62no—
528C3×C4—361no—
529Dic3—22no—
530Dic3—121no—
531Dic3—362no—
532Dic3—361no—
533Dic3—361no—
534Dic3(C2×D9)⋊C414yescomplemented
535C32#493911yescharacteristic
536C9—21no—
537C9#495711yescharacteristic, complemented
538C23—92no—
539C23—181no—
540C23—181no—
541C23—181no—
542C23—181no—
543C23—181no—
544C23—181no—
545C23—272no—
546C23—541no—
547C23—541no—
548C23—541no—
549C23—541no—
550C23—541no—
551C23—541no—
552C23C2×D3×D911yescharacteristic
553D4—548no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—32no—
557C2×C4—32no—
558C2×C4—32no—
559C2×C4—32no—
560C2×C4—61no—
561C2×C4—61no—
562C2×C4—181no—
563C2×C4—181no—
564C2×C4—181no—
565C2×C4—541no—
566C2×C4—541no—
567C2×C4—541no—
568C2×C4#619212yescomplemented
569C2×C3—21no—
570C2×C3—21no—
571C2×C3—21no—
572C2×C3—21no—
573C2×C3—21no—
574C2×C3—21no—
575C2×C3—21no—
576C2×C3—362no—
577C2×C3C4×D3611yescharacteristic
578C2×C3C4×D3611yescharacteristic
579C2×C3#6343611yescharacteristic
580C2×C3#6343711yescharacteristic
581C2×C3C2×(C2×D9)⋊C411yescharacteristic
582C2×C3#6349211yescharacteristic
583C2×C3#6349211yescharacteristic
584C2×C3#6385211yescharacteristic
585C2×C3#6388011yescharacteristic
586C2×C3C4×#62611yescharacteristic
587C2×C3#6390611yescharacteristic
588C2×C3#6391111yescharacteristic
589C2×C3D3×(C2×D3)⋊C411yescharacteristic
590C2×C3#6397811yescharacteristic
591D3—122no—
592D3—364no—
593D3—362no—
594D3—362no—
595C22—182no—
596C22—182no—
597C22—182no—
598C22—361no—
599C22—361no—
600C22—361no—
601C22—361no—
602C22—542no—
603C22—542no—
604C22—542no—
605C22—1081no—
606C22—1081no—
607C22—1081no—
608C22—1081no—
609C22#8864011yescharacteristic
610C22C4×D3×D911yescharacteristic
611C22D3×D3611yescharacteristic
612C22D3×D3611yescharacteristic
613C22#8865811yescharacteristic
614C22C2×#619211yescharacteristic
615C22D3×#61111yescharacteristic
616C4—22no—
617C4—34no—
618C4—62no—
619C4—361no—
620C4—1081no—
621C3—21no—
622C3#1058861911yescharacteristic, complemented
623C3#1059138311yescharacteristic
624C2—362no—
625C2—1082no—
626C2#1172094011yescharacteristic
627C2#1172097711yescharacteristic
628C2C4×#619211yescharacteristic
629C2#1172100611yescharacteristic
630C2#1172103411yescharacteristic
631C2D3×(C2×D9)⋊C411yescharacteristic
632C2#1172127711yescharacteristic
subgroupG/H
centerC23C2×D3×D9
commutatorC22×C3×C9C22×C4
FrattiniC23×C3C2×D32
FittingC22×C3×C4×C9C22
socleC23×C32C22×D3
prime radicalC22×#6192C2
cosocleC23×C3×C9C23
hypercenterC22×C4D3×D9
hypocenterC3×C9#390
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C2×#88536C2semidirect
3. C2×C3×(C2×D9)⋊C4C2semidirect
4. C2×C4×C9×Dic3C2semidirect
5. C2×#88445C2semidirect
6. C2×(C2×D9)⋊Dic3C2semidirect
7. C2×Dic3⋊Dic9C2semidirect
19. C2×#6192C4semidirect
39. C2×(C2×D9)⋊C4D3semidirect
58. C2×C3×C4×C9D4semidirect
67. C2×C9×Dic3D4semidirect
134. C9×Dic3C22⋊C4semidirect
150. C2×C4×Dic3D9semidirect
216. C2×C4×C9#67semidirect
348. C3×C9#390semidirect
403. C2×C3×C4D36semidirect
422. C2×Dic3D36semidirect
534. Dic3(C2×D9)⋊C4semidirect
537. C9#4957semidirect
568. C2×C4#6192semidirect
622. C3#10588619semidirect

Maximal subgroups

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

subgroupindex
C22×#61922
C2×#885362
C2×C3×(C2×D9)⋊C42
C2×C4×C9×Dic32
C2×#884452
C2×(C2×D9)⋊Dic32
C2×Dic3⋊Dic92
#105913833
#105886193