Finite Group Databasebeta

← #13155863

Subgroups — #13155863

Order 1728.

824 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×D3×Dic6C211yesmaximal, characteristic
2#11723426C211yesmaximal, characteristic
3C2×#88800C211yesmaximal, characteristic, complemented
4C2×#88799C211yesmaximal, characteristic, complemented
5#11722076C211yesmaximal, characteristic, complemented
6C3×#63566C211yesmaximal, characteristic, complemented
7C3×(C2×D3)⋊(C3⋊C8)C211yesmaximal, characteristic
8#10591595—31nomaximal
9#10591186—31nomaximal
10#10590952—31nomaximal
11C2×#6308C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C2×C32×C4×D3C2211yescharacteristic
14C2×C32×Dic6C2211yescharacteristic
15C2×C3×#627—21no—
16C2×C3×D3×Dic3—21no—
17C3×D3×Dic6—22no—
18C3×D3×Dic6—22no—
19C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
20C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
21C2×C32×C3⋊C8C2211yescharacteristic
22#88807—21no—
23(C2×D3)⋊(C3⋊Dic3)—21no—
24#88800—22no—
25#88799—22no—
26C2×C3×D3×Q8—31no—
27C22×C3×Dic6—31no—
28C2×D3×Dic6D311yescharacteristic
29C2×#3756—31no—
30#64133—31no—
31C2×#3746—31no—
32#63857—31no—
33#63857—31no—
34#63857—61no—
35C2×#3719—31no—
36C2×#3719—31no—
37C2×#3719—31no—
38C2×#3719—31no—
39C2×#3719—61no—
40C2×#3719—61no—
41C3×#835—31no—
42C3×#830—31no—
43C3×(C2×D3)⋊C8—31no—
44C3×#816—31no—
45#63567—31no—
46#63566—31no—
47#63566D311yescharacteristic, complemented
48#63565—31no—
49(C2×D3)⋊(C3⋊C8)D311yescharacteristic
50C22×C32×D3D411yescharacteristic
51C2×C33×C4C2311yescharacteristic
52C2×C3⋊(C3⋊Dic3)—21no—
53#6308—21no—
54#6308—21no—
55C2×C3×C3⋊Dic3—21no—
56C2×C3×C3⋊Dic3—21no—
57C3×#634—21no—
58C3×#634D412yes—
59C2×C32×Dic3—21no—
60C2×C32×Dic3D411yescharacteristic
61C32×C4×D3—21no—
62C32×C4×D3—21no—
63C32×Dic6—21no—
64C32×Dic6D412yes—
65C3×#627—24no—
66C3×D3×Dic3—24no—
67C3×C3⋊(C3⋊C8)—21no—
68C32×C3⋊C8—21no—
69#5228—91no—
70#4809—91no—
71#4739—91no—
72C2×C32×Q8—31no—
73C22×C32×C4—31no—
74C2×#634—31no—
75C2×#634—31no—
76C2×#634—31no—
77C2×#634—61no—
78C2×#634—61no—
79C2×#634—61no—
80C2×#634—121no—
81C2×#634C2×D311yescharacteristic
82C22×C3×Dic3—61no—
83C3×D3×Q8—62no—
84C3×D3×Q8—62no—
85C2×C3×C4×D3—21no—
86C2×C3×C4×D3—61no—
87C2×C3×C4×D3C2×D311yescharacteristic
88C2×C3×C4×D3C2×D311yescharacteristic
89C2×C3×Dic6—21no—
90C2×C3×Dic6—31no—
91C2×C3×Dic6—31no—
92C2×C3×Dic6—62no—
93C2×C3×Dic6—61no—
94C2×C3×Dic6—61no—
95C2×C3×Dic6—62no—
96C2×C3×Dic6—61no—
97C2×C3×Dic6C2×D311yescharacteristic
98C2×C3×Dic6C2×D311yescharacteristic
99C2×#627—21no—
100C2×D3×Dic3—21no—
101D3×Dic6—22no—
102D3×Dic6—22no—
103C2×C32×C8—31no—
104#3757—61no—
105#3756—62no—
106C4⋊(C3⋊Dic3)C2×D311yescharacteristic
107#3750—61no—
108C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
109#3746—62no—
110C3×C4⋊Dic3—31no—
111C3×C4⋊Dic3—31no—
112C3×C4⋊Dic3—61no—
113C2×C3×C3⋊C8—21no—
114C2×C3×C3⋊C8—31no—
115C2×C3×C3⋊C8—31no—
116C2×C3×C3⋊C8—61no—
117C2×C3×C3⋊C8C2×D311yescharacteristic
118C2×C3×C3⋊C8C2×D311yescharacteristic
119Dic3⋊Dic3—62no—
120Dic3⋊Dic3—61no—
121Dic3⋊Dic3—61no—
122(C2×D3)⋊Dic3—62no—
123(C2×D3)⋊Dic3—61no—
124(C2×D3)⋊Dic3—61no—
125#3719—62no—
126#3719—62no—
127#3719—64no—
128#3719—64no—
129#3719—62no—
130#3719—62no—
131C22×C33C2×D411yescharacteristic
132C2×C32×D3—21no—
133C2×C32×D3—21no—
134C2×C32×D3Q1612yescomplemented
135C33×C4C2×D411yescharacteristic
136C33×C4C2×D411yescharacteristic
137C3⋊(C3⋊Dic3)—41no—
138C3×C3⋊Dic3—22no—
139C3×C3⋊Dic3—41no—
140C32×Dic3—22no—
141C32×Dic3—21no—
142C22×C3×Q8—91no—
143C2×D3×Q8—31no—
144C22×Dic6—31no—
145C2×#216—91no—
146C2×#216—91no—
147C2×#216—91no—
148C2×#216—181no—
149#924—91no—
150#924—91no—
151#924—181no—
152C2×Dic12—91no—
153C2×Dic12—91no—
154C2×Dic12—91no—
155C2×Dic12—181no—
156#897—91no—
157C3×#103—91no—
158C3×C22⋊C8—91no—
159#835—31no—
160#835—91no—
161#830—31no—
162(C2×D3)⋊C8—31no—
163#816—31no—
164#816—91no—
165#810—91no—
166C23×C32—31no—
167C22×C3×D3—21no—
168C22×C3×D3D1211yescharacteristic
169C22×C3×D3#6711yescharacteristic
170C32×Q8—32no—
171C32×Q8—61no—
172C2×C32×C4—21no—
173C2×C32×C4—21no—
174C2×C32×C4—21no—
175C2×C32×C4—31no—
176C2×C32×C4—41no—
177C2×C32×C4—61no—
178C2×C32×C4—61no—
179C2×C32×C4—61no—
180C2×C32×C4C22×D311yescharacteristic
181C2×C32×C4C22×D311yescharacteristic
182C2×C32×C4C22×D311yescharacteristic
183C2×C3⋊Dic3—21no—
184C2×C3⋊Dic3—21no—
185C2×C3⋊Dic3—61no—
186C2×C3⋊Dic3—62no—
187C2×C3⋊Dic3—61no—
188C2×C3⋊Dic3—61no—
189C2×C3⋊Dic3—121no—
190C2×C3⋊Dic3—121no—
191C2×C3⋊Dic3—122no—
192#634—21no—
193#634—61no—
194#634—61no—
195#634—62no—
196#634—62no—
197#634—61no—
198#634—61no—
199#634—61no—
200#634—61no—
201#634—121no—
202#634—121no—
203#634—121no—
204#634—121no—
205#634—122no—
206#634—122no—
207#634#6712yes—
208C2×C3×Dic3—21no—
209C2×C3×Dic3—21no—
210C2×C3×Dic3—21no—
211C2×C3×Dic3—41no—
212C2×C3×Dic3—64no—
213C2×C3×Dic3—61no—
214C2×C3×Dic3—61no—
215C2×C3×Dic3—62no—
216C2×C3×Dic3—61no—
217C2×C3×Dic3—61no—
218C2×C3×Dic3—121no—
219C2×C3×Dic3D1211yescharacteristic
220C2×C3×Dic3#6711yescharacteristic
221C3×C4×D3—21no—
222C3×C4×D3—21no—
223C3×C4×D3—21no—
224C3×C4×D3—21no—
225C3×C4×D3—41no—
226C3×C4×D3—41no—
227C3×C4×D3—64no—
228C3×Dic6—21no—
229C3×Dic6—21no—
230C3×Dic6—22no—
231C3×Dic6—32no—
232C3×Dic6—32no—
233C3×Dic6—41no—
234C3×Dic6—62no—
235C3×Dic6—64no—
236C3×Dic6—64no—
237C3×Dic6—61no—
238C3×Dic6—61no—
239C3×Dic6—121no—
240C3×Dic6D1212yes—
241C3×Dic6#6712yes—
242#627—24no—
243D3×Dic3—24no—
244C32×C8—61no—
245C3⋊(C3⋊C8)—21no—
246C3×C3⋊C8—22no—
247C3×C3⋊C8—21no—
248C3×C3⋊C8—21no—
249C3×C3⋊C8—62no—
250C3×C3⋊C8—61no—
251C3×C3⋊C8—61no—
252#451—271noSylow
253C2×C33#12011yescharacteristic
254C2×C33C2×Q1611yescharacteristic
255C2×C33#13711yescharacteristic
256C32×D3—22no—
257C2×C3×Q8—31no—
258C2×C3×Q8—31no—
259C2×C3×Q8—61no—
260C2×C3×Q8—91no—
261C2×C3×Q8—181no—
262C2×C3×Q8—182no—
263C2×C3×Q8—182no—
264C22×C3×C4—31no—
265C22×C3×C4—31no—
266C22×C3×C4—61no—
267C22×C3×C4—181no—
268C22×Dic3—61no—
269D3×Q8—62no—
270D3×Q8—62no—
271C2×C4×D3—61no—
272C2×C4×D3D3211yescharacteristic
273C2×Dic6—31no—
274C2×Dic6—31no—
275C2×Dic6—61no—
276C2×Dic6—62no—
277C2×Dic6—61no—
278C2×Dic6—61no—
279C2×Dic6—62no—
280C2×Dic6—91no—
281C2×Dic6—91no—
282C2×Dic6—91no—
283C2×Dic6—181no—
284C2×Dic6—181no—
285C2×Dic6—181no—
286C2×Dic6—361no—
287C2×Dic6D3211yescharacteristic
288C2×C3×C8—31no—
289C2×C3×C8—31no—
290C2×C3×C8—61no—
291C2×C3×C8—91no—
292C3×C4⋊C4—91no—
293#217—181no—
294#217—181no—
295#217—182no—
296#216—182no—
297#216—182no—
298#216—182no—
299#216—184no—
300(C2×D3)⋊C4—181no—
301C4⋊Dic3—31no—
302C4⋊Dic3—31no—
303C4⋊Dic3—61no—
304C4⋊Dic3—181no—
305Dic3⋊C4—181no—
306Dic3⋊C4—181no—
307Dic3⋊C4—182no—
308C2×C3⋊C8—31no—
309C2×C3⋊C8—31no—
310C2×C3⋊C8—61no—
311C2×C3⋊C8D3211yescharacteristic
312Dic12—182no—
313Dic12—182no—
314Dic12—184no—
315Dic12—182no—
316C22×C32—21no—
317C22×C32—21no—
318C22×C32—21no—
319C22×C32—32no—
320C22×C32—41no—
321C22×C32—61no—
322C22×C32—61no—
323C22×C32C2×D1211yescharacteristic
324C22×C32D3×D411yescharacteristic
325C22×C32C2×#6711yescharacteristic
326C2×C3×D3—21no—
327C2×C3×D3—21no—
328C2×C3×D3—21no—
329C2×C3×D3—21no—
330C2×C3×D3—22no—
331C2×C3×D3—41no—
332C2×C3×D3—41no—
333C2×C3×D3Dic1212yescomplemented
334C2×C3×D3#21612yescomplemented
335C32×C4—21no—
336C32×C4—21no—
337C32×C4—21no—
338C32×C4—21no—
339C32×C4—21no—
340C32×C4—21no—
341C32×C4—41no—
342C32×C4—41no—
343C32×C4—61no—
344C32×C4—62no—
345C32×C4C2×D1211yescharacteristic
346C32×C4D3×D411yescharacteristic
347C32×C4D3×D411yescharacteristic
348C32×C4D3×D411yescharacteristic
349C32×C4C2×#6711yescharacteristic
350C32×C4C2×#6711yescharacteristic
351C3⋊Dic3—22no—
352C3⋊Dic3—41no—
353C3⋊Dic3—122no—
354C3⋊Dic3—122no—
355C3⋊Dic3—122no—
356C3⋊Dic3—124no—
357C3⋊Dic3—121no—
358C3⋊Dic3—121no—
359C3⋊Dic3—121no—
360C3×Dic3—22no—
361C3×Dic3—21no—
362C3×Dic3—22no—
363C3×Dic3—21no—
364C3×Dic3—42no—
365C3×Dic3—41no—
366C3×Dic3—62no—
367C3×Dic3—62no—
368C3×Dic3—122no—
369C3×Dic3—122no—
370C3×Dic3—121no—
371C3×Dic3—121no—
372C22×Q8—91no—
373C2×Q16—271no—
374C2×Q16—271no—
375C22⋊Q8—271no—
376#103—91no—
377#103—271no—
378C22⋊C8—91no—
379C33#45111yescharacteristic, complemented, Sylow
380C23×C3—31no—
381C23×C3—31no—
382C23×C3—61no—
383C22×D3#62611yescharacteristic
384C3×Q8—32no—
385C3×Q8—32no—
386C3×Q8—62no—
387C3×Q8—61no—
388C3×Q8—61no—
389C3×Q8—92no—
390C3×Q8—121no—
391C3×Q8—184no—
392C3×Q8—181no—
393C2×C3×C4—21no—
394C2×C3×C4—21no—
395C2×C3×C4—21no—
396C2×C3×C4—31no—
397C2×C3×C4—31no—
398C2×C3×C4—41no—
399C2×C3×C4—61no—
400C2×C3×C4—61no—
401C2×C3×C4—61no—
402C2×C3×C4—61no—
403C2×C3×C4—61no—
404C2×C3×C4—61no—
405C2×C3×C4—61no—
406C2×C3×C4—121no—
407C2×C3×C4—121no—
408C2×C3×C4—121no—
409C2×C3×C4—184no—
410C2×C3×C4—181no—
411C2×C3×C4—181no—
412C2×C3×C4C2×D3211yescharacteristic
413C2×C3×C4C2×D3211yescharacteristic
414C2×C3×C4C2×D3211yescharacteristic
415C2×Dic3—21no—
416C2×Dic3—64no—
417C2×Dic3—61no—
418C2×Dic3—61no—
419C2×Dic3—62no—
420C2×Dic3—61no—
421C2×Dic3—61no—
422C2×Dic3—121no—
423C2×Dic3—182no—
424C2×Dic3—181no—
425C2×Dic3—181no—
426C2×Dic3—181no—
427C2×Dic3—361no—
428C2×Dic3—361no—
429C2×Dic3—362no—
430C2×Dic3#62611yescharacteristic
431C4×D3—21no—
432C4×D3—21no—
433C4×D3—64no—
434Dic6—21no—
435Dic6—32no—
436Dic6—32no—
437Dic6—62no—
438Dic6—64no—
439Dic6—64no—
440Dic6—61no—
441Dic6—61no—
442Dic6—121no—
443Dic6—182no—
444Dic6—182no—
445Dic6—181no—
446Dic6—181no—
447Dic6—181no—
448Dic6—181no—
449Dic6—181no—
450Dic6—181no—
451Dic6—361no—
452Dic6—361no—
453Dic6—361no—
454Dic6—361no—
455Dic6—362no—
456Dic6—362no—
457Dic6#62612yes—
458C3×C8—62no—
459C3×C8—61no—
460C3×C8—61no—
461C3×C8—181no—
462C3⋊C8—21no—
463C3⋊C8—62no—
464C3⋊C8—61no—
465C3⋊C8—61no—
466C2×C32—21no—
467C2×C32—21no—
468C2×C32—21no—
469C2×C32—21no—
470C2×C32—21no—
471C2×C32—21no—
472C2×C32—21no—
473C2×C32—21no—
474C2×C32—21no—
475C2×C32—41no—
476C2×C32—41no—
477C2×C32—41no—
478C2×C32—62no—
479C2×C32#88211yescharacteristic
480C2×C32#88211yescharacteristic
481C2×C32C2×Dic1211yescharacteristic
482C2×C32#90911yescharacteristic
483C2×C32#91511yescharacteristic
484C2×C32D3×Q1611yescharacteristic
485C2×C32#94211yescharacteristic
486C2×C32C2×#21611yescharacteristic
487C2×C32#95311yescharacteristic
488C3×D3—22no—
489C3×D3—22no—
490C3×D3—42no—
491C2×Q8—31no—
492C2×Q8—91no—
493C2×Q8—181no—
494C2×Q8—182no—
495C2×Q8—182no—
496C2×Q8—271no—
497C22×C4—31no—
498C22×C4—181no—
499Q16—542no—
500Q16—542no—
501C2×C8—31no—
502C2×C8—91no—
503C4⋊C4—91no—
504C4⋊C4—541no—
505C22⋊C4—541no—
506C22×C3—21no—
507C22×C3—21no—
508C22×C3—21no—
509C22×C3—32no—
510C22×C3—32no—
511C22×C3—41no—
512C22×C3—62no—
513C22×C3—61no—
514C22×C3—61no—
515C22×C3—61no—
516C22×C3—61no—
517C22×C3—121no—
518C22×C3—121no—
519C22×C3D3×D1211yescharacteristic
520C22×C3C2×#62611yescharacteristic
521C22×C3D3×#6711yescharacteristic
522C2×D3—21no—
523C2×D3—21no—
524C2×D3#371912yescomplemented
525C3×C4—21no—
526C3×C4—21no—
527C3×C4—21no—
528C3×C4—21no—
529C3×C4—21no—
530C3×C4—21no—
531C3×C4—41no—
532C3×C4—41no—
533C3×C4—61no—
534C3×C4—62no—
535C3×C4—61no—
536C3×C4—62no—
537C3×C4—121no—
538C3×C4—122no—
539C3×C4—182no—
540C3×C4—361no—
541C3×C4D3×D1211yescharacteristic
542C3×C4C2×#62611yescharacteristic
543C3×C4D3×#6711yescharacteristic
544C3×C4D3×#6711yescharacteristic
545C3×C4D3×#6711yescharacteristic
546C3×C4#381111yescharacteristic
547Dic3—22no—
548Dic3—21no—
549Dic3—62no—
550Dic3—62no—
551Dic3—122no—
552Dic3—122no—
553Dic3—121no—
554Dic3—121no—
555Dic3—362no—
556Dic3—362no—
557Dic3—362no—
558Dic3—364no—
559Dic3—361no—
560Dic3—361no—
561Dic3—361no—
562C32—21no—
563C32—21no—
564C32—21no—
565C32—41no—
566C32#473911yescharacteristic, complemented
567C32#480911yescharacteristic, complemented
568C32#522811yescharacteristic, complemented
569C23—31no—
570Q8—32no—
571Q8—61no—
572Q8—92no—
573Q8—184no—
574Q8—181no—
575Q8—541no—
576Q8—541no—
577C2×C4—31no—
578C2×C4—61no—
579C2×C4—61no—
580C2×C4—61no—
581C2×C4—184no—
582C2×C4—181no—
583C2×C4—181no—
584C2×C4—541no—
585C2×C4D3311yescharacteristic
586C8—61no—
587C8—181no—
588C2×C3—21no—
589C2×C3—21no—
590C2×C3—21no—
591C2×C3—21no—
592C2×C3—21no—
593C2×C3—21no—
594C2×C3—21no—
595C2×C3—21no—
596C2×C3—21no—
597C2×C3—41no—
598C2×C3—41no—
599C2×C3—41no—
600C2×C3—62no—
601C2×C3—62no—
602C2×C3—122no—
603C2×C3D3×Dic1211yescharacteristic
604C2×C3#6380011yescharacteristic
605C2×C3#6383311yescharacteristic
606C2×C3C2×#371911yescharacteristic
607C2×C3#6392311yescharacteristic
608C2×C3D3×#21611yescharacteristic
609C2×C3#6394611yescharacteristic
610C2×C3#6397711yescharacteristic
611C2×C3#6397711yescharacteristic
612D3—22no—
613C22—32no—
614C22—61no—
615C22—61no—
616C22D3×#62611yescharacteristic
617C4—61no—
618C4—62no—
619C4—182no—
620C4—361no—
621C4—1081no—
622C4D3×#62611yescharacteristic
623C4#8895211yescharacteristic
624C3—21no—
625C3—21no—
626C3—21no—
627C3—41no—
628C3#1059095211yescharacteristic, complemented
629C3#1059118611yescharacteristic, complemented
630C3#1059159511yescharacteristic, complemented
631C2—62no—
632C2D3×#371911yescharacteristic
633C2#1172288411yescharacteristic
634C2#1172294811yescharacteristic
subgroupG/H
centerC22D3×#626
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radicalC22×C32×D3D4
hypocenterC33#451
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
3. C2×#88800C2semidirect
4. C2×#88799C2semidirect
5. #11722076C2semidirect
6. C3×#63566C2semidirect
47. #63566D3semidirect
134. C2×C32×D3Q16semidirect
333. C2×C3×D3Dic12semidirect
334. C2×C3×D3#216semidirect
379. C33#451semidirect
524. C2×D3#3719semidirect
566. C32#4739semidirect
567. C32#4809semidirect
568. C32#5228semidirect
628. C3#10590952semidirect
629. C3#10591186semidirect
630. C3#10591595semidirect

Maximal subgroups

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

subgroupindex
C2×C3×D3×Dic62
#117234262
C2×#888002
C2×#887992
#117220762
C3×#635662
C3×(C2×D3)⋊(C3⋊C8)2
#105915953
#105911863
#105909523