Finite Group Databasebeta

← #13156851

Subgroups — #13156851

Order 1728.

1458 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#625C211yesmaximal, characteristic
2C2×#88815C211yesmaximal, characteristic, complemented
3C2×(C2×D3)⋊(C3⋊Dic3)C212yesmaximal, complemented
4C2×Dic3×C3⋊Dic3C211yesmaximal, characteristic, complemented
5C2×C3×(C2×D3)⋊Dic3C212yesmaximal, complemented
6#10591707—32nomaximal
7#10591385—31nomaximal
8C23×C32×D3C2212yes—
9C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
10C22×C3×C3⋊Dic3C2212yes—
11C22×C3×C3⋊Dic3C2211yescharacteristic
12C22×C32×Dic3C2211yescharacteristic
13C2×C3×#625—22no—
14C2×C3×#625—22no—
15C2×C3×#625C414yescomplemented
16#88815—22no—
17(C2×D3)⋊(C3⋊Dic3)—24no—
18Dic3×C3⋊Dic3—22no—
19C3×(C2×D3)⋊Dic3—24no—
20C22×C3×#67—32no—
21C22×#625D311yescharacteristic
22C2×#3757—32no—
23C2×C4×C3⋊Dic3—31no—
24C2×C3×#217—32no—
25C2×C3×(C2×D3)⋊C4—32no—
26C2×#3724—31no—
27C2×Dic3⋊Dic3—32no—
28C2×(C2×D3)⋊Dic3—32no—
29C2×(C2×D3)⋊Dic3—32no—
30C2×(C2×D3)⋊Dic3—62no—
31C2×(C2×D3)⋊Dic3D312yescomplemented
32C2×Dic32—32no—
33C2×Dic32—61no—
34C23×C33C2311yescharacteristic
35C22×C32×D3—22no—
36C22×C32×D3—22no—
37C22×C32×D3—22no—
38C22×C32×D3—22no—
39C22×C32×D3—22no—
40C22×C32×D3—22no—
41C22×C32×D3C2×C414yes—
42C2×C3⋊(C3⋊Dic3)—21no—
43C2×C3⋊(C3⋊Dic3)—21no—
44C2×C3⋊(C3⋊Dic3)D412yescomplemented
45C2×C3×C3⋊Dic3—22no—
46C2×C3×C3⋊Dic3—22no—
47C2×C3×C3⋊Dic3—22no—
48C2×C3×C3⋊Dic3C2×C412yes—
49C2×C3×C3⋊Dic3D412yes—
50C2×C3×C3⋊Dic3D412yes—
51C2×C32×Dic3—21no—
52C2×C32×Dic3—21no—
53C2×C32×Dic3D412yescomplemented
54C3×#625—28no—
55#5221—91no—
56#4957—92no—
57C24×C32—32no—
58C23×C3×D3—22no—
59C23×C3×D3C2×D312yes—
60C23×C3×D3C2×D312yes—
61C22×C32×C4—31no—
62C22×C3⋊Dic3—31no—
63C22×C3⋊Dic3—32no—
64C22×C3⋊Dic3—61no—
65C22×C3⋊Dic3—62no—
66C22×C3⋊Dic3—121no—
67C22×C3⋊Dic3C2×D312yes—
68C22×C3⋊Dic3C2×D311yescharacteristic
69C2×C3×#67—38no—
70C2×C3×#67—64no—
71C2×C3×#67—64no—
72C22×C3×Dic3—21no—
73C22×C3×Dic3—32no—
74C22×C3×Dic3—32no—
75C22×C3×Dic3—32no—
76C22×C3×Dic3—61no—
77C22×C3×Dic3—62no—
78C22×C3×Dic3C2×D312yes—
79C2×#625—22no—
80C2×#625—22no—
81C2×#625Dic314yescomplemented
82#3757—64no—
83C4×C3⋊Dic3—62no—
84C3×#217—64no—
85C3×(C2×D3)⋊C4—64no—
86#3724—62no—
87Dic3⋊Dic3—64no—
88(C2×D3)⋊Dic3—24no—
89(C2×D3)⋊Dic3—64no—
90(C2×D3)⋊Dic3—64no—
91(C2×D3)⋊Dic3—68no—
92Dic32—64no—
93Dic32—64no—
94C22×C33C22×C411yescharacteristic
95C22×C33C2×D411yescharacteristic
96C22×C33C2×D411yescharacteristic
97C22×C33C2×D411yescharacteristic
98C22×C33C2×D411yescharacteristic
99C22×C33#4111yescharacteristic
100C22×C33#4111yescharacteristic
101C2×C32×D3—24no—
102C2×C32×D3—24no—
103C2×C32×D3—24no—
104C2×C32×D3—42no—
105C2×C32×D3—42no—
106C2×C32×D3—42no—
107C2×C32×D3—42no—
108C3⋊(C3⋊Dic3)—22no—
109C3×C3⋊Dic3—42no—
110C3×C3⋊Dic3C22⋊C414yescomplemented
111C32×Dic3—22no—
112C22×C3×D4—91no—
113C22×#67—32no—
114C2×C3×C22⋊C4—92no—
115C2×#217—32no—
116C2×#217—92no—
117C2×#217—92no—
118C2×#217—182no—
119C2×(C2×D3)⋊C4—32no—
120C2×(C2×D3)⋊C4—92no—
121C2×C4⋊Dic3—91no—
122C2×Dic3⋊C4—92no—
123C2×C4×Dic3—92no—
124C2×C4×Dic3—91no—
125C2×C4×Dic3—181no—
126C23×C32—22no—
127C23×C32—21no—
128C23×C32—34no—
129C23×C32—41no—
130C23×C32—62no—
131C23×C32—62no—
132C23×C32—62no—
133C23×C32—62no—
134C23×C32—62no—
135C23×C32—62no—
136C23×C32C22×D312yes—
137C23×C32C22×D311yescharacteristic
138C22×C3×D3—24no—
139C22×C3×D3—24no—
140C22×C3×D3—22no—
141C22×C3×D3—22no—
142C22×C3×D3—22no—
143C22×C3×D3—22no—
144C22×C3×D3—22no—
145C22×C3×D3—22no—
146C22×C3×D3—22no—
147C22×C3×D3—22no—
148C22×C3×D3—22no—
149C22×C3×D3—22no—
150C22×C3×D3—22no—
151C22×C3×D3—22no—
152C22×C3×D3—24no—
153C22×C3×D3—42no—
154C22×C3×D3—42no—
155C22×C3×D3—42no—
156C22×C3×D3—42no—
157C22×C3×D3C4×D314yes—
158C22×C3×D3C2×Dic314yes—
159C2×C32×C4—32no—
160C2×C32×C4—61no—
161C2×C32×C4—61no—
162C2×C3⋊Dic3—22no—
163C2×C3⋊Dic3—22no—
164C2×C3⋊Dic3—22no—
165C2×C3⋊Dic3—32no—
166C2×C3⋊Dic3—34no—
167C2×C3⋊Dic3—62no—
168C2×C3⋊Dic3—64no—
169C2×C3⋊Dic3—61no—
170C2×C3⋊Dic3—61no—
171C2×C3⋊Dic3—62no—
172C2×C3⋊Dic3—62no—
173C2×C3⋊Dic3—62no—
174C2×C3⋊Dic3—62no—
175C2×C3⋊Dic3—122no—
176C2×C3⋊Dic3—122no—
177C2×C3⋊Dic3—122no—
178C2×C3⋊Dic3—122no—
179C2×C3⋊Dic3—122no—
180C2×C3⋊Dic3C2×Dic312yes—
181C2×C3⋊Dic3#6712yes—
182C2×C3⋊Dic3#6712yes—
183C3×#67—616no—
184C2×C3×Dic3—22no—
185C2×C3×Dic3—22no—
186C2×C3×Dic3—22no—
187C2×C3×Dic3—22no—
188C2×C3×Dic3—22no—
189C2×C3×Dic3—34no—
190C2×C3×Dic3—34no—
191C2×C3×Dic3—34no—
192C2×C3×Dic3—62no—
193C2×C3×Dic3—62no—
194C2×C3×Dic3—62no—
195C2×C3×Dic3—64no—
196C2×C3×Dic3—62no—
197C2×C3×Dic3—62no—
198C2×C3×Dic3—62no—
199C2×C3×Dic3—62no—
200C2×C3×Dic3—62no—
201C2×C3×Dic3—62no—
202C2×C3×Dic3—122no—
203C2×C3×Dic3—122no—
204C2×C3×Dic3#6714yescomplemented
205#625—28no—
206#390—271noSylow
207C2×C33C2×C22⋊C411yescharacteristic
208C2×C33C4×D411yescharacteristic
209C2×C33C4×D411yescharacteristic
210C2×C33#12111yescharacteristic
211C2×C33#12111yescharacteristic
212C2×C33#12411yescharacteristic
213C2×C33#12711yescharacteristic
214C32×D3—44no—
215C24×C3—32no—
216C24×C3—32no—
217C24×C3—62no—
218C23×D3D3212yes—
219C2×C3×D4—94no—
220C2×C3×D4—182no—
221C2×C3×D4—182no—
222C22×C3×C4—32no—
223C22×C3×C4—61no—
224C22×C3×C4—91no—
225C22×C3×C4—92no—
226C2×#67—38no—
227C2×#67—64no—
228C2×#67—64no—
229C22×Dic3—32no—
230C22×Dic3—32no—
231C22×Dic3—32no—
232C22×Dic3—61no—
233C22×Dic3—62no—
234C22×Dic3—92no—
235C22×Dic3—91no—
236C22×Dic3—181no—
237C22×Dic3—182no—
238C22×Dic3—361no—
239C22×Dic3D3211yescharacteristic
240C3×C22⋊C4—184no—
241#217—64no—
242#217—184no—
243#217—184no—
244#217—188no—
245(C2×D3)⋊C4—64no—
246(C2×D3)⋊C4—184no—
247C4⋊Dic3—182no—
248Dic3⋊C4—184no—
249C4×Dic3—182no—
250C4×Dic3—184no—
251C4×Dic3—184no—
252C22×C32—21no—
253C22×C32—21no—
254C22×C32—21no—
255C22×C32—21no—
256C22×C32—22no—
257C22×C32—22no—
258C22×C32—21no—
259C22×C32—22no—
260C22×C32—21no—
261C22×C32—22no—
262C22×C32—22no—
263C22×C32—22no—
264C22×C32—21no—
265C22×C32—22no—
266C22×C32—41no—
267C22×C32—41no—
268C22×C32—41no—
269C22×C32—41no—
270C22×C32—41no—
271C22×C32—41no—
272C22×C32—41no—
273C22×C32—64no—
274C22×C32—64no—
275C22×C32—64no—
276C22×C32—122no—
277C22×C32—122no—
278C22×C32—122no—
279C22×C32—122no—
280C22×C32C2×C4×D312yes—
281C22×C32#23512yes—
282C22×C32D3×D412yes—
283C22×C32D3×D412yes—
284C22×C32D3×D411yescharacteristic
285C22×C32D3×D412yes—
286C22×C32D3×D411yescharacteristic
287C22×C32#23711yescharacteristic
288C22×C32#23712yes—
289C22×C32#23711yescharacteristic
290C22×C32C22×Dic311yescharacteristic
291C22×C32C2×#6712yes—
292C22×C32C2×#6711yescharacteristic
293C22×C32C2×#6711yescharacteristic
294C2×C3×D3—28no—
295C2×C3×D3—24no—
296C2×C3×D3—24no—
297C2×C3×D3—24no—
298C2×C3×D3—24no—
299C2×C3×D3—24no—
300C2×C3×D3—24no—
301C2×C3×D3—44no—
302C2×C3×D3—44no—
303C2×C3×D3—44no—
304C2×C3×D3—44no—
305C2×C3×D3—44no—
306C2×C3×D3—44no—
307C2×C3×D3—42no—
308C2×C3×D3—42no—
309C2×C3×D3—42no—
310C2×C3×D3—42no—
311C2×C3×D3—42no—
312C2×C3×D3—42no—
313C2×C3×D3—42no—
314C2×C3×D3—42no—
315C32×C4—62no—
316C3⋊Dic3—42no—
317C3⋊Dic3—62no—
318C3⋊Dic3—64no—
319C3⋊Dic3—64no—
320C3⋊Dic3—124no—
321C3⋊Dic3—124no—
322C3⋊Dic3#21714yescomplemented
323C3×Dic3—24no—
324C3×Dic3—24no—
325C3×Dic3—38no—
326C3×Dic3—64no—
327C3×Dic3—124no—
328C3×Dic3—122no—
329C3×Dic3—122no—
330C22×D4—91no—
331C2×C4⋊C4—271no—
332C2×C22⋊C4—92no—
333C2×C22⋊C4—272no—
334C2×C42—271no—
335C33#39011yescharacteristic, complemented, Sylow
336C23×C3—21no—
337C23×C3—22no—
338C23×C3—34no—
339C23×C3—34no—
340C23×C3—41no—
341C23×C3—64no—
342C23×C3—64no—
343C23×C3—64no—
344C23×C3—62no—
345C23×C3—62no—
346C23×C3—62no—
347C23×C3—62no—
348C23×C3—62no—
349C23×C3—62no—
350C23×C3—62no—
351C23×C3—62no—
352C23×C3—62no—
353C23×C3—62no—
354C23×C3—62no—
355C23×C3—62no—
356C23×C3—122no—
357C23×C3—122no—
358C23×C3—122no—
359C23×C3—122no—
360C23×C3C2×D3211yescharacteristic
361C23×C3C2×D3212yes—
362C22×D3—22no—
363C22×D3—22no—
364C22×D3—22no—
365C22×D3—22no—
366C22×D3—22no—
367C22×D3—22no—
368C22×D3D3×Dic314yes—
369C3×D4—188no—
370C2×C3×C4—34no—
371C2×C3×C4—62no—
372C2×C3×C4—62no—
373C2×C3×C4—62no—
374C2×C3×C4—62no—
375C2×C3×C4—62no—
376C2×C3×C4—92no—
377C2×C3×C4—92no—
378C2×C3×C4—92no—
379C2×C3×C4—182no—
380C2×C3×C4—182no—
381C2×C3×C4—182no—
382#67—616no—
383C2×Dic3—21no—
384C2×Dic3—21no—
385C2×Dic3—34no—
386C2×Dic3—34no—
387C2×Dic3—34no—
388C2×Dic3—62no—
389C2×Dic3—62no—
390C2×Dic3—62no—
391C2×Dic3—64no—
392C2×Dic3—62no—
393C2×Dic3—62no—
394C2×Dic3—62no—
395C2×Dic3—62no—
396C2×Dic3—62no—
397C2×Dic3—62no—
398C2×Dic3—92no—
399C2×Dic3—94no—
400C2×Dic3—122no—
401C2×Dic3—122no—
402C2×Dic3—182no—
403C2×Dic3—184no—
404C2×Dic3—182no—
405C2×Dic3—182no—
406C2×Dic3—181no—
407C2×Dic3—182no—
408C2×Dic3—182no—
409C2×Dic3—181no—
410C2×Dic3—362no—
411C2×Dic3—362no—
412C2×Dic3—362no—
413C2×Dic3—362no—
414C2×Dic3—362no—
415C2×Dic3#62512yescomplemented
416C2×C32—21no—
417C2×C32—21no—
418C2×C32—21no—
419C2×C32—21no—
420C2×C32—21no—
421C2×C32—21no—
422C2×C32—22no—
423C2×C32—22no—
424C2×C32—22no—
425C2×C32—22no—
426C2×C32—22no—
427C2×C32—22no—
428C2×C32—22no—
429C2×C32—21no—
430C2×C32—41no—
431C2×C32—41no—
432C2×C32—41no—
433C2×C32—41no—
434C2×C32—41no—
435C2×C32—41no—
436C2×C32—41no—
437C2×C32—124no—
438C2×C32D3×C22⋊C412yes—
439C2×C32#67⋊C412yes—
440C2×C32#88412yes—
441C2×C32#88512yes—
442C2×C32C4×#6712yes—
443C2×C32D4×Dic311yescharacteristic
444C2×C32D4×Dic311yescharacteristic
445C2×C32#93611yescharacteristic
446C2×C32#93811yescharacteristic
447C2×C32#93811yescharacteristic
448C2×C32#93912yes—
449C2×C32#94012yes—
450C2×C32#94011yescharacteristic
451C2×C32C2×#21711yescharacteristic
452C3×D3—48no—
453C3×D3—44no—
454C3×D3—44no—
455C24—32no—
456C2×D4—94no—
457C2×D4—182no—
458C2×D4—182no—
459C22×C4—31no—
460C22×C4—91no—
461C22×C4—92no—
462C22×C4—271no—
463C4⋊C4—542no—
464C22⋊C4—184no—
465C22⋊C4—544no—
466C42—542no—
467C22×C3—21no—
468C22×C3—22no—
469C22×C3—22no—
470C22×C3—21no—
471C22×C3—21no—
472C22×C3—22no—
473C22×C3—21no—
474C22×C3—22no—
475C22×C3—22no—
476C22×C3—21no—
477C22×C3—21no—
478C22×C3—22no—
479C22×C3—21no—
480C22×C3—22no—
481C22×C3—41no—
482C22×C3—41no—
483C22×C3—41no—
484C22×C3—41no—
485C22×C3—41no—
486C22×C3—41no—
487C22×C3—41no—
488C22×C3—68no—
489C22×C3—64no—
490C22×C3—64no—
491C22×C3—64no—
492C22×C3—64no—
493C22×C3—64no—
494C22×C3—64no—
495C22×C3—124no—
496C22×C3—124no—
497C22×C3—124no—
498C22×C3—124no—
499C22×C3—124no—
500C22×C3—124no—
501C22×C3—122no—
502C22×C3—122no—
503C22×C3—122no—
504C22×C3—122no—
505C22×C3—122no—
506C22×C3—122no—
507C22×C3—122no—
508C22×C3—122no—
509C22×C3#379811yescharacteristic
510C22×C3C4×D3211yescharacteristic
511C22×C3#380211yescharacteristic
512C22×C3#380211yescharacteristic
513C22×C3C2×D3×Dic312yes—
514C22×C3#380412yes—
515C22×C3#380511yescharacteristic
516C22×C3#380512yes—
517C22×C3C2×#62511yescharacteristic
518C22×C3D3×#6712yes—
519C22×C3D3×#6712yes—
520C22×C3D3×#6712yes—
521C22×C3#381111yescharacteristic
522C22×C3#381112yes—
523C2×D3—24no—
524C2×D3—24no—
525C2×D3—24no—
526C2×D3—42no—
527C2×D3—42no—
528C2×D3—42no—
529C2×D3—42no—
530C3×C4—64no—
531C3×C4—64no—
532C3×C4—94no—
533C3×C4—362no—
534Dic3—22no—
535Dic3—38no—
536Dic3—64no—
537Dic3—124no—
538Dic3—122no—
539Dic3—122no—
540Dic3—184no—
541Dic3—182no—
542Dic3—184no—
543Dic3—364no—
544Dic3—364no—
545C32—21no—
546C32—22no—
547C32—41no—
548C32#495712yescomplemented
549C32#522111yescharacteristic, complemented
550C23—34no—
551C23—62no—
552C23—62no—
553C23—62no—
554C23—62no—
555C23—62no—
556C23—62no—
557C23D3311yescharacteristic
558D4—188no—
559C2×C4—32no—
560C2×C4—61no—
561C2×C4—61no—
562C2×C4—92no—
563C2×C4—92no—
564C2×C4—92no—
565C2×C4—182no—
566C2×C4—182no—
567C2×C4—182no—
568C2×C4—272no—
569C2×C4—541no—
570C2×C4—541no—
571C2×C3—22no—
572C2×C3—21no—
573C2×C3—21no—
574C2×C3—22no—
575C2×C3—22no—
576C2×C3—21no—
577C2×C3—22no—
578C2×C3—21no—
579C2×C3—21no—
580C2×C3—22no—
581C2×C3—21no—
582C2×C3—22no—
583C2×C3—22no—
584C2×C3—21no—
585C2×C3—41no—
586C2×C3—41no—
587C2×C3—41no—
588C2×C3—41no—
589C2×C3—41no—
590C2×C3—41no—
591C2×C3—41no—
592C2×C3—128no—
593C2×C3—124no—
594C2×C3—124no—
595C2×C3C4×#62511yescharacteristic
596C2×C3#6390211yescharacteristic
597C2×C3#6391211yescharacteristic
598C2×C3#6391411yescharacteristic
599C2×C3#6391511yescharacteristic
600C2×C3#6392111yescharacteristic
601C2×C3#6395812yes—
602C2×C3#6395912yes—
603C2×C3D3×#21712yes—
604C2×C3#6397012yes—
605C2×C3Dic3×#6712yes—
606C2×C3#67⋊Dic312yes—
607C2×C3#6397912yes—
608C2×C3#6398011yescharacteristic
609D3—44no—
610C22—64no—
611C22—64no—
612C22—64no—
613C22—122no—
614C22—122no—
615C22—122no—
616C22—122no—
617C22D32×Dic311yescharacteristic
618C22D3×#62511yescharacteristic
619C22#8895211yescharacteristic
620C22#8895311yescharacteristic
621C22#8895411yescharacteristic
622C22#8895911yescharacteristic
623C22#8896311yescharacteristic
624C4—62no—
625C4—94no—
626C4—362no—
627C4—542no—
628C3—22no—
629C3—21no—
630C3—41no—
631C3#1059138511yescharacteristic, complemented
632C3#1059170712yescomplemented
633C2—124no—
634C2#1172290211yescharacteristic
635C2Dic3×#62511yescharacteristic
636C2#1172292511yescharacteristic
637C2#1172295311yescharacteristic
638C2#1172296411yescharacteristic
639C2#1172300611yescharacteristic
640C2#1172302911yescharacteristic
subgroupG/H
center, Frattini, hypercenterC23D33
commutatorC22×C33C22×C4
Fitting, socle, cosocleC23×C33C23
prime radicalC22×C3×#625C2
hypocenterC33#390
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C23

Complements

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

NH ≅ G/Ntype
2. C2×#88815C2semidirect
3. C2×(C2×D3)⋊(C3⋊Dic3)C2semidirect
4. C2×Dic3×C3⋊Dic3C2semidirect
5. C2×C3×(C2×D3)⋊Dic3C2semidirect
15. C2×C3×#625C4semidirect
31. C2×(C2×D3)⋊Dic3D3semidirect
44. C2×C3⋊(C3⋊Dic3)D4semidirect
53. C2×C32×Dic3D4semidirect
81. C2×#625Dic3semidirect
110. C3×C3⋊Dic3C22⋊C4semidirect
204. C2×C3×Dic3#67semidirect
322. C3⋊Dic3#217semidirect
335. C33#390semidirect
415. C2×Dic3#625semidirect
548. C32#4957semidirect
549. C32#5221semidirect
631. C3#10591385semidirect
632. C3#10591707semidirect

Maximal subgroups

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

subgroupindex
C22×C3×#6252
C2×#888152
C2×(C2×D3)⋊(C3⋊Dic3)2
C2×Dic3×C3⋊Dic32
C2×C3×(C2×D3)⋊Dic32
#105917073
#105913853