Finite Group Databasebeta

← #13139873

Subgroups — #13139873

Order 1728.

1094 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4⋊(C3⋊D9)C211yesmaximal, characteristic
2C22×(C3⋊D9)⋊C4C211yesmaximal, characteristic, complemented
3C2×#88445C211yesmaximal, characteristic, complemented
4C2×#88445C211yesmaximal, characteristic, complemented
5C2×#88445C211yesmaximal, characteristic, complemented
6C9×C3⋊#95C211yesmaximal, characteristic, complemented
7C3×C9⋊#95C211yesmaximal, characteristic, complemented
8#10591380—31nomaximal
9#10588345—31nomaximal
10C23×C3⋊D9C2211yescharacteristic
11C23×C3⋊D9C2211yescharacteristic
12C22×C3×C4×C9C2211yescharacteristic
13C2×C4⋊(C3⋊D9)—22no—
14C2×C4⋊(C3⋊D9)—22no—
15C2×C4⋊(C3⋊D9)C414yescomplemented
16C22×C9×Dic3C2211yescharacteristic
17C22×C9×Dic3C2211yescharacteristic
18C22×C3×Dic9C2211yescharacteristic
19C22×C3×Dic9C2211yescharacteristic
20C2×(C3⋊D9)⋊C4—22no—
21C2×(C3⋊D9)⋊C4—22no—
22C2×(C3⋊D9)⋊C4—22no—
23#88445—22no—
24#88445—22no—
25#88445—22no—
26C22×C4⋊(C3⋊D3)—31no—
27C22×#624—31no—
28C2×#3722—31no—
29C2×#3722—31no—
30C2×#3722—31no—
31C22×D36—31no—
32C22×D36—61no—
33C22×C4×D9—31no—
34C3×C3⋊#95—31no—
35C3×C3⋊#95D311yescharacteristic
36C2×(C2×D9)⋊C4—31no—
37C2×(C2×D9)⋊C4—31no—
38C2×(C2×D9)⋊C4—31no—
39C9×#95—31no—
40C9⋊#95D311yescharacteristic, complemented
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⋊D9—21no—
49C22×C3⋊D9—21no—
50C22×C3⋊D9—21no—
51C22×C3⋊D9—21no—
52C22×C3⋊D9C2×C412yes—
53C22×C3⋊D9C2×C412yescomplemented
54C22×C3⋊D9D412yes—
55C22×C3⋊D9D412yes—
56C2×C3×C4×C9—21no—
57C2×C3×C4×C9—21no—
58C2×C3×C4×C9C2×C412yescomplemented
59C4⋊(C3⋊D9)—28no—
60C2×C9×Dic3—21no—
61C2×C9×Dic3—21no—
62C2×C9×Dic3—21no—
63C2×C9×Dic3—21no—
64C2×C9×Dic3—21no—
65C2×C9×Dic3D412yescomplemented
66C2×C3×Dic9—21no—
67C2×C3×Dic9—21no—
68C2×C3×Dic9—21no—
69C2×C3×Dic9—21no—
70C2×C3×Dic9—21no—
71C2×C3×Dic9D412yescomplemented
72(C3⋊D9)⋊C4—28no—
73#4665—91no—
74#4665—91no—
75C23×C3⋊D3—31no—
76C23×C3⋊D3—31no—
77C22×C32×C4C2×D311yescharacteristic
78C2×C4⋊(C3⋊D3)—34no—
79C2×C4⋊(C3⋊D3)—62no—
80C2×C4⋊(C3⋊D3)—62no—
81C22×C3×Dic3—31no—
82C22×C3×Dic3—31no—
83C22×C3×Dic3C2×D311yescharacteristic
84C22×C3×Dic3C2×D311yescharacteristic
85C2×#624—62no—
86C2×#624—62no—
87C2×#624—62no—
88C23×D9—31no—
89C23×D9—31no—
90C23×D9—61no—
91C23×D9—61no—
92#3722—62no—
93#3722—62no—
94#3722—62no—
95C22×C4×C9—21no—
96C22×C4×C9—31no—
97C22×C4×C9—31no—
98C22×C4×C9C2×D311yescharacteristic
99C22×Dic9C2×D311yescharacteristic
100C22×Dic9C2×D311yescharacteristic
101C2×D36—34no—
102C2×D36—62no—
103C2×D36—62no—
104C2×D36—64no—
105C2×D36—64no—
106C2×D36—64no—
107C2×C4×D9—62no—
108C2×C4×D9—62no—
109C2×C4×D9—62no—
110(C2×D9)⋊C4—62no—
111(C2×D9)⋊C4—62no—
112(C2×D9)⋊C4—62no—
113C22×C3×C9C22×C411yescharacteristic
114C22×C3×C9C2×D411yescharacteristic
115C22×C3×C9C2×D411yescharacteristic
116C22×C3×C9C2×D411yescharacteristic
117C22×C3×C9C2×D411yescharacteristic
118C22×C3×C9#4111yescharacteristic
119C22×C3×C9#4111yescharacteristic
120C2×C3⋊D9—22no—
121C2×C3⋊D9—22no—
122C2×C3⋊D9—22no—
123C2×C3⋊D9—22no—
124C2×C3⋊D9—22no—
125C2×C3⋊D9—22no—
126C2×C3⋊D9—22no—
127C2×C3⋊D9—22no—
128C2×C3⋊D9—22no—
129C2×C3⋊D9—41no—
130C2×C3⋊D9—41no—
131C2×C3⋊D9—41no—
132C2×C3⋊D9—41no—
133C2×C3⋊D9C22⋊C414yescomplemented
134C3×C4×C9—22no—
135C9×Dic3—22no—
136C9×Dic3—41no—
137C3×Dic9—22no—
138C3×Dic9—41no—
139C22×D12—91no—
140C22×D12—91no—
141C22×D12—181no—
142C22×C4×D3—91no—
143C22×C4×D3—91no—
144C2×(C2×D3)⋊C4—91no—
145C2×(C2×D3)⋊C4—91no—
146C2×(C2×D3)⋊C4—91no—
147C2×(C2×D3)⋊C4—91no—
148C2×(C2×D3)⋊C4—91no—
149C2×(C2×D3)⋊C4—91no—
150C3×#95—31no—
151C3×#95—91no—
152C3⋊#95—31no—
153C3⋊#95D911yescharacteristic, complemented
154C23×C32C22×D311yescharacteristic
155C22×C3⋊D3—32no—
156C22×C3⋊D3—32no—
157C22×C3⋊D3—32no—
158C22×C3⋊D3—32no—
159C22×C3⋊D3—61no—
160C22×C3⋊D3—61no—
161C22×C3⋊D3—61no—
162C22×C3⋊D3—61no—
163C22×C3⋊D3—61no—
164C22×C3⋊D3—61no—
165C22×C3⋊D3—61no—
166C22×C3⋊D3—61no—
167C22×C3⋊D3—61no—
168C22×C3⋊D3—61no—
169C2×C32×C4—21no—
170C2×C32×C4—21no—
171C2×C32×C4C4×D312yes—
172C4⋊(C3⋊D3)—68no—
173C2×C3×Dic3—21no—
174C2×C3×Dic3—21no—
175C2×C3×Dic3—21no—
176C2×C3×Dic3—21no—
177C2×C3×Dic3—21no—
178C2×C3×Dic3—32no—
179C2×C3×Dic3—61no—
180C2×C3×Dic3—61no—
181C2×C3×Dic3—61no—
182C2×C3×Dic3—61no—
183C2×C3×Dic3—61no—
184C2×C3×Dic3D1212yes—
185#624—68no—
186C23×C9—21no—
187C23×C9C22×D311yescharacteristic
188C22×D9—32no—
189C22×D9—32no—
190C22×D9—32no—
191C22×D9—32no—
192C22×D9—62no—
193C22×D9—62no—
194C22×D9—62no—
195C22×D9—62no—
196C22×D9—61no—
197C22×D9—61no—
198C22×D9—61no—
199C22×D9—61no—
200C22×D9—62no—
201C22×D9—62no—
202C22×D9—61no—
203C22×D9—61no—
204C22×D9—61no—
205C22×D9—61no—
206C22×D9—61no—
207C22×D9—61no—
208C22×D9—121no—
209C22×D9—121no—
210C22×D9—121no—
211C22×D9—121no—
212C22×D9—121no—
213C22×D9—121no—
214C22×D9—121no—
215C22×D9—121no—
216C2×C4×C9—22no—
217C2×C4×C9—22no—
218C2×C4×C9—21no—
219C2×C4×C9—21no—
220C2×C4×C9—22no—
221C2×C4×C9—32no—
222C2×C4×C9—61no—
223C2×C4×C9—61no—
224C2×C4×C9—61no—
225C2×C4×C9—61no—
226C2×C4×C9—61no—
227C2×C4×C9C4×D312yescomplemented
228C2×Dic9—21no—
229C2×Dic9—21no—
230C2×Dic9—21no—
231C2×Dic9—21no—
232C2×Dic9—21no—
233C2×Dic9D1212yescomplemented
234D36—68no—
235D36—616no—
236C4×D9—68no—
237#386—271noSylow
238C2×C3×C9C2×C22⋊C411yescharacteristic
239C2×C3×C9C4×D411yescharacteristic
240C2×C3×C9C4×D411yescharacteristic
241C2×C3×C9#12011yescharacteristic
242C2×C3×C9#12111yescharacteristic
243C2×C3×C9#12111yescharacteristic
244C2×C3×C9#12311yescharacteristic
245C3⋊D9—24no—
246C3⋊D9—42no—
247C23×D3—91no—
248C23×D3—91no—
249C23×D3—91no—
250C23×D3—91no—
251C23×D3—181no—
252C23×D3—181no—
253C22×C3×C4—21no—
254C22×C3×C4—31no—
255C22×C3×C4—31no—
256C22×C3×C4—91no—
257C22×C3×C4—91no—
258C22×C3×C4C2×D911yescharacteristic
259C22×C3×C4D3211yescharacteristic
260C22×Dic3—31no—
261C22×Dic3—31no—
262C22×Dic3C2×D911yescharacteristic
263C22×Dic3C2×D911yescharacteristic
264C2×D12—94no—
265C2×D12—94no—
266C2×D12—184no—
267C2×D12—182no—
268C2×D12—182no—
269C2×D12—182no—
270C2×D12—182no—
271C2×D12—184no—
272C2×D12—184no—
273C2×C4×D3—182no—
274C2×C4×D3—182no—
275C2×C4×D3—182no—
276C2×C4×D3—182no—
277C2×C4×D3—182no—
278C2×C4×D3—182no—
279(C2×D3)⋊C4—182no—
280(C2×D3)⋊C4—182no—
281(C2×D3)⋊C4—182no—
282(C2×D3)⋊C4—182no—
283(C2×D3)⋊C4—182no—
284(C2×D3)⋊C4—182no—
285C22×C32C2×C4×D311yescharacteristic
286C22×C32C2×D1211yescharacteristic
287C22×C32#23511yescharacteristic
288C22×C32D3×D411yescharacteristic
289C22×C32D3×D411yescharacteristic
290C22×C32D3×D411yescharacteristic
291C22×C32#23911yescharacteristic
292C2×C3⋊D3—34no—
293C2×C3⋊D3—62no—
294C2×C3⋊D3—62no—
295C2×C3⋊D3—62no—
296C2×C3⋊D3—62no—
297C2×C3⋊D3—62no—
298C2×C3⋊D3—62no—
299C2×C3⋊D3—62no—
300C2×C3⋊D3—62no—
301C2×C3⋊D3—62no—
302C2×C3⋊D3—121no—
303C2×C3⋊D3—121no—
304C2×C3⋊D3—121no—
305C2×C3⋊D3—121no—
306C32×C4—22no—
307C3×Dic3—22no—
308C3×Dic3—41no—
309C3×Dic3—62no—
310C3×Dic3—121no—
311C22×C9—21no—
312C22×C9—21no—
313C22×C9—21no—
314C22×C9—21no—
315C22×C9—21no—
316C22×C9—21no—
317C22×C9—21no—
318C22×C9C2×C4×D311yescharacteristic
319C22×C9C2×D1211yescharacteristic
320C22×C9#23511yescharacteristic
321C22×C9D3×D411yescharacteristic
322C22×C9D3×D411yescharacteristic
323C22×C9D3×D411yescharacteristic
324C22×C9#23911yescharacteristic
325C2×D9—34no—
326C2×D9—64no—
327C2×D9—62no—
328C2×D9—62no—
329C2×D9—62no—
330C2×D9—62no—
331C2×D9—62no—
332C2×D9—62no—
333C2×D9—64no—
334C2×D9—62no—
335C2×D9—62no—
336C2×D9—62no—
337C2×D9—122no—
338C2×D9—122no—
339C2×D9—122no—
340C2×D9—122no—
341C2×D9—122no—
342C2×D9—122no—
343C2×D9—122no—
344C2×D9—122no—
345C2×D9—122no—
346C2×D9—122no—
347C2×D9—122no—
348C2×D9—122no—
349C2×D9—121no—
350C2×D9—121no—
351C2×D9—121no—
352C2×D9—121no—
353C4×C9—24no—
354C4×C9—22no—
355C4×C9—62no—
356C4×C9—121no—
357Dic9—22no—
358Dic9—41no—
359C22×D4—271no—
360C23×C4—271no—
361C2×C22⋊C4—271no—
362C2×C22⋊C4—271no—
363C2×C22⋊C4—271no—
364#95—31no—
365#95—91no—
366C3×C9#38611yescharacteristic, complemented, Sylow
367C23×C3—21no—
368C23×C3C22×D911yescharacteristic
369C23×C3C2×D3211yescharacteristic
370C22×D3—92no—
371C22×D3—92no—
372C22×D3—92no—
373C22×D3—92no—
374C22×D3—92no—
375C22×D3—92no—
376C22×D3—92no—
377C22×D3—92no—
378C22×D3—182no—
379C22×D3—182no—
380C22×D3—182no—
381C22×D3—182no—
382C22×D3—181no—
383C22×D3—181no—
384C22×D3—181no—
385C22×D3—181no—
386C22×D3—181no—
387C22×D3—181no—
388C22×D3—181no—
389C22×D3—181no—
390C22×D3—182no—
391C22×D3—182no—
392C22×D3—181no—
393C22×D3—181no—
394C22×D3—181no—
395C22×D3—181no—
396C22×D3—181no—
397C22×D3—181no—
398C22×D3—181no—
399C22×D3—181no—
400C22×D3—181no—
401C22×D3—181no—
402C22×D3—181no—
403C22×D3—181no—
404C22×D3—361no—
405C22×D3—361no—
406C22×D3—361no—
407C22×D3—361no—
408C22×D3—361no—
409C22×D3—361no—
410C22×D3—361no—
411C22×D3—361no—
412C2×C3×C4—22no—
413C2×C3×C4—22no—
414C2×C3×C4—21no—
415C2×C3×C4—21no—
416C2×C3×C4—21no—
417C2×C3×C4—21no—
418C2×C3×C4—22no—
419C2×C3×C4—32no—
420C2×C3×C4—61no—
421C2×C3×C4—61no—
422C2×C3×C4—61no—
423C2×C3×C4—61no—
424C2×C3×C4—61no—
425C2×C3×C4—92no—
426C2×C3×C4—181no—
427C2×C3×C4—181no—
428C2×C3×C4—181no—
429C2×C3×C4—181no—
430C2×C3×C4—181no—
431C2×C3×C4C4×D912yescomplemented
432C2×C3×C4#62412yes—
433C2×Dic3—21no—
434C2×Dic3—21no—
435C2×Dic3—21no—
436C2×Dic3—21no—
437C2×Dic3—21no—
438C2×Dic3—32no—
439C2×Dic3—61no—
440C2×Dic3—61no—
441C2×Dic3—61no—
442C2×Dic3—61no—
443C2×Dic3—61no—
444C2×Dic3D3612yescomplemented
445D12—188no—
446D12—188no—
447D12—1816no—
448C4×D3—188no—
449C4×D3—188no—
450C2×C32C4×D1211yescharacteristic
451C2×C32D3×C22⋊C411yescharacteristic
452C2×C32#88211yescharacteristic
453C2×C32#88411yescharacteristic
454C2×C32#89311yescharacteristic
455C2×C32#89411yescharacteristic
456C2×C32#89511yescharacteristic
457C3⋊D3—64no—
458C3⋊D3—122no—
459C2×C9—21no—
460C2×C9—21no—
461C2×C9—21no—
462C2×C9—21no—
463C2×C9—21no—
464C2×C9—21no—
465C2×C9—21no—
466C2×C9C4×D1211yescharacteristic
467C2×C9D3×C22⋊C411yescharacteristic
468C2×C9#88211yescharacteristic
469C2×C9#88411yescharacteristic
470C2×C9#89311yescharacteristic
471C2×C9#89411yescharacteristic
472C2×C9#89511yescharacteristic
473D9—64no—
474D9—124no—
475D9—124no—
476D9—122no—
477C24—271no—
478C24—271no—
479C2×D4—274no—
480C2×D4—542no—
481C2×D4—542no—
482C22×C4—31no—
483C22×C4—31no—
484C22×C4—91no—
485C22×C4—91no—
486C22×C4—542no—
487C22×C4—542no—
488C22×C4—542no—
489C22×C4D3×D911yescharacteristic
490C22⋊C4—542no—
491C22⋊C4—542no—
492C22⋊C4—542no—
493C22×C3—21no—
494C22×C3—21no—
495C22×C3—21no—
496C22×C3—21no—
497C22×C3—21no—
498C22×C3—21no—
499C22×C3—21no—
500C22×C3C2×C4×D911yescharacteristic
501C22×C3C2×D3611yescharacteristic
502C22×C3#369711yescharacteristic
503C22×C3D4×D911yescharacteristic
504C22×C3D4×D911yescharacteristic
505C22×C3D4×D911yescharacteristic
506C22×C3#370111yescharacteristic
507C22×C3#379911yescharacteristic
508C22×C3#379911yescharacteristic
509C22×C3D3×D1211yescharacteristic
510C22×C3D3×D1211yescharacteristic
511C22×C3C2×#62411yescharacteristic
512C22×C3#381111yescharacteristic
513C22×C3#381111yescharacteristic
514C2×D3—94no—
515C2×D3—94no—
516C2×D3—184no—
517C2×D3—182no—
518C2×D3—182no—
519C2×D3—182no—
520C2×D3—182no—
521C2×D3—182no—
522C2×D3—182no—
523C2×D3—182no—
524C2×D3—182no—
525C2×D3—182no—
526C2×D3—182no—
527C2×D3—182no—
528C2×D3—182no—
529C2×D3—184no—
530C2×D3—182no—
531C2×D3—182no—
532C2×D3—182no—
533C2×D3—182no—
534C2×D3—182no—
535C2×D3—182no—
536C2×D3—362no—
537C2×D3—362no—
538C2×D3—362no—
539C2×D3—362no—
540C2×D3—362no—
541C2×D3—362no—
542C2×D3—362no—
543C2×D3—362no—
544C2×D3—362no—
545C2×D3—362no—
546C2×D3—362no—
547C2×D3—362no—
548C2×D3—361no—
549C2×D3—361no—
550C2×D3—361no—
551C2×D3—361no—
552C2×D3—361no—
553C2×D3—361no—
554C2×D3—361no—
555C2×D3—361no—
556C3×C4—24no—
557C3×C4—22no—
558C3×C4—22no—
559C3×C4—62no—
560C3×C4—121no—
561C3×C4—182no—
562C3×C4—361no—
563Dic3—22no—
564Dic3—41no—
565Dic3—62no—
566Dic3—121no—
567C32#466511yescharacteristic
568C9—21no—
569C9#466511yescharacteristic, complemented
570C23—272no—
571C23—272no—
572C23—272no—
573C23—272no—
574C23—541no—
575C23—541no—
576C23—541no—
577C23—541no—
578C23—541no—
579C23—541no—
580C23—541no—
581C23—541no—
582C23—541no—
583C23—541no—
584C23C2×D3×D911yescharacteristic
585D4—548no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—32no—
589C2×C4—61no—
590C2×C4—61no—
591C2×C4—61no—
592C2×C4—61no—
593C2×C4—61no—
594C2×C4—92no—
595C2×C4—181no—
596C2×C4—181no—
597C2×C4—181no—
598C2×C4—181no—
599C2×C4—181no—
600C2×C4—548no—
601C2×C4(C3⋊D9)⋊C412yescomplemented
602C2×C3—21no—
603C2×C3—21no—
604C2×C3—21no—
605C2×C3—21no—
606C2×C3—21no—
607C2×C3—21no—
608C2×C3—21no—
609C2×C3C4×D3611yescharacteristic
610C2×C3C22⋊C4×D911yescharacteristic
611C2×C3#6344411yescharacteristic
612C2×C3#6344611yescharacteristic
613C2×C3#6345511yescharacteristic
614C2×C3#6345611yescharacteristic
615C2×C3#6345711yescharacteristic
616C2×C3#6389411yescharacteristic
617C2×C3#6389411yescharacteristic
618C2×C3#6389711yescharacteristic
619C2×C3#6391011yescharacteristic
620C2×C3#6391011yescharacteristic
621C2×C3#6392311yescharacteristic
622C2×C3#6397411yescharacteristic
623D3—184no—
624D3—184no—
625D3—364no—
626D3—364no—
627D3—362no—
628D3—362no—
629C22—274no—
630C22—542no—
631C22—542no—
632C22—542no—
633C22—542no—
634C22—542no—
635C22—542no—
636C22—542no—
637C22—542no—
638C22—542no—
639C22—1081no—
640C22—1081no—
641C22—1081no—
642C22—1081no—
643C22#8863511yescharacteristic
644C22#8864211yescharacteristic
645C22D3×D3611yescharacteristic
646C22D9×D1211yescharacteristic
647C22C2×(C3⋊D9)⋊C411yescharacteristic
648C22#8866811yescharacteristic
649C22#8866811yescharacteristic
650C4—22no—
651C4—62no—
652C4—121no—
653C4—182no—
654C4—361no—
655C3—21no—
656C3#1058834511yescharacteristic, complemented
657C3#1059138011yescharacteristic
658C2—544no—
659C2—1082no—
660C2#1172099211yescharacteristic
661C2#1172099311yescharacteristic
662C2#1172099711yescharacteristic
663C2#1172100511yescharacteristic
664C2#1172103911yescharacteristic
665C2#1172105111yescharacteristic
666C2#1172127411yescharacteristic
subgroupG/H
centerC23C2×D3×D9
commutatorC22×C3×C9C22×C4
FrattiniC23×C3C2×D32
FittingC22×C3×C4×C9C22
socleC23×C32C22×D3
prime radicalC22×C4⋊(C3⋊D9)C2
cosocleC23×C3×C9C23
hypercenterC22×C4D3×D9
hypocenterC3×C9#386
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. C22×(C3⋊D9)⋊C4C2semidirect
3. C2×#88445C2semidirect
4. C2×#88445C2semidirect
5. C2×#88445C2semidirect
6. C9×C3⋊#95C2semidirect
7. C3×C9⋊#95C2semidirect
15. C2×C4⋊(C3⋊D9)C4semidirect
40. C9⋊#95D3semidirect
53. C22×C3⋊D9C2×C4semidirect
58. C2×C3×C4×C9C2×C4semidirect
65. C2×C9×Dic3D4semidirect
71. C2×C3×Dic9D4semidirect
133. C2×C3⋊D9C22⋊C4semidirect
153. C3⋊#95D9semidirect
227. C2×C4×C9C4×D3semidirect
233. C2×Dic9D12semidirect
366. C3×C9#386semidirect
431. C2×C3×C4C4×D9semidirect
444. C2×Dic3D36semidirect
569. C9#4665semidirect
601. C2×C4(C3⋊D9)⋊C4semidirect
656. C3#10588345semidirect

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
C22×(C3⋊D9)⋊C42
C2×#884452
C9×C3⋊#952
C3×C9⋊#952
#105913803
#105883453