Finite Group Databasebeta

← #13155877

Subgroups — #13155877

Order 1728.

800 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3795C211yesmaximal, characteristic
2C2×#88812C211yesmaximal, characteristic, complemented
3#11723425C211yesmaximal, characteristic, complemented
4C2×C3×#3716C211yesmaximal, characteristic, complemented
5#11722040C211yesmaximal, characteristic, complemented
6#11722032C211yesmaximal, characteristic, complemented
7C3×(C2×D3)⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
8#10591461—31nomaximal
9#10591173—31nomaximal
10#10590960—31nomaximal
11C2×C3×#634C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C2×C32×D12C2211yescharacteristic
14C2×C32×C4×D3C2211yescharacteristic
15C2×C3×#625—21no—
16C2×C3×D3×Dic3—21no—
17C3×#3795—22no—
18C3×#3795—22no—
19C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
20C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
21C2×C32×C3⋊C8C2211yescharacteristic
22#88812—22no—
23#88806—21no—
24(C2×D3)⋊(C3⋊Dic3)—21no—
25C3×#3716—22no—
26C2×C3×#237—31no—
27C2×C3×#235—31no—
28C2×#3795D311yescharacteristic
29#64133—31no—
30C2×C3×#214—31no—
31C2×C3×#204—31no—
32#63856—31no—
33#63856—31no—
34#63856—61no—
35C2×#3719—31no—
36C2×#3719—31no—
37C2×#3718—31no—
38C2×#3716D311yescharacteristic, complemented
39#63659—31no—
40#63643—31no—
41C3×#830—31no—
42C3×(C2×D3)⋊C8—31no—
43#63566—31no—
44#63566—31no—
45#63566—61no—
46#63563—31no—
47#63563—31no—
48#63563—61no—
49(C2×D3)⋊(C3⋊C8)D311yescharacteristic, complemented
50C22×C32×D3—21no—
51C22×C32×D3D411yescharacteristic
52C2×C33×C4C2311yescharacteristic
53C2×C3⋊(C3⋊Dic3)—21no—
54C2×C3×C3⋊Dic3—21no—
55C2×C3×C3⋊Dic3—21no—
56C3×#634—21no—
57C3×#634—21no—
58C3×#634—21no—
59C3×#634D412yes—
60C2×C32×Dic3D411yescharacteristic
61C32×D12—21no—
62C32×D12D412yes—
63C32×C4×D3—21no—
64C32×C4×D3—21no—
65C3×#625—24no—
66C3×D3×Dic3—24no—
67C3×C3⋊(C3⋊C8)—21no—
68C32×C3⋊C8—21no—
69#5239—91no—
70#5173—91no—
71#4733—91no—
72C2×C32×D4—31no—
73C22×C32×C4—31no—
74C2×#634C2×D311yescharacteristic
75C2×#634C2×D311yescharacteristic
76C2×C3×#67—61no—
77C2×C3×#67—61no—
78C22×C3×Dic3—61no—
79C3×#237—62no—
80C3×#237—62no—
81C3×#235—62no—
82C3×#235—62no—
83C2×C3×D12—21no—
84C2×C3×D12C2×D311yescharacteristic
85C2×C3×D12C2×D311yescharacteristic
86C2×C3×C4×D3—21no—
87C2×C3×C4×D3—61no—
88C2×C3×C4×D3C2×D311yescharacteristic
89C2×C3×C4×D3C2×D311yescharacteristic
90C2×C3×Dic6—31no—
91C2×C3×Dic6—31no—
92C2×C3×Dic6—31no—
93C2×C3×Dic6—31no—
94C2×C3×Dic6—61no—
95C2×C3×Dic6—61no—
96C2×#625—21no—
97C2×D3×Dic3—21no—
98#3795—22no—
99#3795—22no—
100C2×C32×C8—31no—
101#3757—61no—
102C4⋊(C3⋊Dic3)—31no—
103C4⋊(C3⋊Dic3)—31no—
104C4⋊(C3⋊Dic3)—31no—
105C4⋊(C3⋊Dic3)—61no—
106C4⋊(C3⋊Dic3)—61no—
107C4⋊(C3⋊Dic3)—61no—
108C4⋊(C3⋊Dic3)—121no—
109#3750—61no—
110C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
111C3×#214—62no—
112C2×C3×C3⋊C8—21no—
113C2×C3×C3⋊C8—31no—
114C2×C3×C3⋊C8—31no—
115C2×C3×C3⋊C8—61no—
116C2×C3×C3⋊C8C2×D311yescharacteristic
117C2×C3×C3⋊C8C2×D311yescharacteristic
118C3×#204—62no—
119#3724—62no—
120#3724—61no—
121#3724—61no—
122(C2×D3)⋊Dic3—62no—
123(C2×D3)⋊Dic3—61no—
124(C2×D3)⋊Dic3—61no—
125#3719—62no—
126#3719—62no—
127#3718—62no—
128#3716—22no—
129C22×C33C2×D411yescharacteristic
130C2×C32×D3—22no—
131C2×C32×D3—21no—
132C2×C32×D3—21no—
133C2×C32×D3—21no—
134C2×C32×D3—41no—
135C2×C32×D3—41no—
136C33×C4C2×D411yescharacteristic
137C33×C4C2×D411yescharacteristic
138C3⋊(C3⋊Dic3)—41no—
139C3×C3⋊Dic3—22no—
140C3×C3⋊Dic3—41no—
141C32×Dic3—21no—
142C2×C3×#41—91no—
143C2×#237—31no—
144C2×#235—31no—
145C2×C3×SD16—91no—
146#946—91no—
147C2×#216—91no—
148C2×#216—91no—
149C2×#214—31no—
150#924—91no—
151#924—91no—
152#924—181no—
153C2×Dic12—91no—
154C2×#204—31no—
155C3×C22⋊C8—91no—
156#835—91no—
157#832—91no—
158#832—91no—
159#832—181no—
160#830—31no—
161#821—91no—
162(C2×D3)⋊C8—31no—
163#816—91no—
164#816—91no—
165#816—181no—
166C23×C32—31no—
167C23×C32—61no—
168C22×C3×D3—21no—
169C22×C3×D3—21no—
170C22×C3×D3—21no—
171C22×C3×D3—41no—
172C22×C3×D3D1211yescharacteristic
173C22×C3×D3#6711yescharacteristic
174C32×D4—32no—
175C32×D4—61no—
176C2×C32×C4—21no—
177C2×C32×C4—21no—
178C2×C32×C4—21no—
179C2×C32×C4—31no—
180C2×C32×C4—41no—
181C2×C32×C4—61no—
182C2×C32×C4—61no—
183C2×C32×C4C22×D311yescharacteristic
184C2×C32×C4C22×D311yescharacteristic
185C2×C32×C4C22×D311yescharacteristic
186C2×C3⋊Dic3—21no—
187C2×C3⋊Dic3—21no—
188C2×C3⋊Dic3—61no—
189C2×C3⋊Dic3—62no—
190C2×C3⋊Dic3—61no—
191C2×C3⋊Dic3—61no—
192C2×C3⋊Dic3—121no—
193C2×C3⋊Dic3—121no—
194C2×C3⋊Dic3—122no—
195#634—21no—
196#634—21no—
197#634—21no—
198#634#6712yes—
199C3×#67—64no—
200C3×#67—64no—
201C2×C3×Dic3—21no—
202C2×C3×Dic3—64no—
203C2×C3×Dic3—61no—
204C2×C3×Dic3—61no—
205C2×C3×Dic3—62no—
206C2×C3×Dic3—61no—
207C2×C3×Dic3—61no—
208C2×C3×Dic3—121no—
209C2×C3×Dic3D1211yescharacteristic
210C2×C3×Dic3#6711yescharacteristic
211C3×D12—21no—
212C3×D12—21no—
213C3×D12—22no—
214C3×D12—41no—
215C3×D12#6712yes—
216C3×D12#6712yes—
217C3×C4×D3—21no—
218C3×C4×D3—21no—
219C3×C4×D3—21no—
220C3×C4×D3—21no—
221C3×C4×D3—41no—
222C3×C4×D3—41no—
223C3×C4×D3—64no—
224C3×Dic6—32no—
225C3×Dic6—32no—
226C3×Dic6—62no—
227C3×Dic6—61no—
228C3×Dic6—61no—
229C3×Dic6—61no—
230C3×Dic6—61no—
231C3×Dic6—61no—
232C3×Dic6—61no—
233C3×Dic6—121no—
234C3×Dic6—121no—
235C3×Dic6—121no—
236#625—24no—
237D3×Dic3—24no—
238C32×C8—61no—
239C3⋊(C3⋊C8)—21no—
240C3×C3⋊C8—21no—
241C3×C3⋊C8—21no—
242C3×C3⋊C8—41no—
243C3×C3⋊C8—61no—
244C3×C3⋊C8—61no—
245C3×C3⋊C8—121no—
246#452—271noSylow
247C2×C33#12011yescharacteristic
248C2×C33#13511yescharacteristic
249C2×C33#13711yescharacteristic
250C32×D3—41no—
251C32×D3—42no—
252C3×#41—182no—
253C3×#41—182no—
254C2×C3×Q8—91no—
255C2×C3×Q8—91no—
256C2×C3×D4—31no—
257C2×C3×D4—31no—
258C2×C3×D4—61no—
259C2×C3×D4—181no—
260C22×C3×C4—31no—
261C22×C3×C4—31no—
262C22×C3×C4—61no—
263C22×C3×C4—181no—
264C2×#67—61no—
265C2×#67—61no—
266C22×Dic3—61no—
267#237—62no—
268#237—62no—
269#235—62no—
270#235—62no—
271C2×D12D3211yescharacteristic
272C2×C4×D3—61no—
273C2×C4×D3D3211yescharacteristic
274C2×Dic6—31no—
275C2×Dic6—31no—
276C2×Dic6—31no—
277C2×Dic6—31no—
278C2×Dic6—61no—
279C2×Dic6—61no—
280C3×SD16—182no—
281C2×C3×C8—31no—
282C2×C3×C8—31no—
283C2×C3×C8—61no—
284C2×C3×C8—91no—
285#217—181no—
286#217—181no—
287#217—182no—
288#216—182no—
289#216—182no—
290#214—62no—
291(C2×D3)⋊C4—181no—
292C4⋊Dic3—91no—
293C4⋊Dic3—91no—
294C4⋊Dic3—91no—
295C4⋊Dic3—181no—
296C4⋊Dic3—181no—
297C4⋊Dic3—181no—
298C4⋊Dic3—361no—
299Dic3⋊C4—181no—
300Dic3⋊C4—181no—
301Dic3⋊C4—182no—
302Dic3⋊C4—181no—
303C2×C3⋊C8—31no—
304C2×C3⋊C8—31no—
305C2×C3⋊C8—61no—
306C2×C3⋊C8D3211yescharacteristic
307Dic12—182no—
308#204—62no—
309C22×C32—21no—
310C22×C32—21no—
311C22×C32—21no—
312C22×C32—41no—
313C22×C32—61no—
314C22×C32—62no—
315C22×C32—61no—
316C22×C32—61no—
317C22×C32—121no—
318C22×C32—121no—
319C22×C32C2×D1211yescharacteristic
320C22×C32D3×D411yescharacteristic
321C22×C32C2×#6711yescharacteristic
322C2×C3×D3—22no—
323C2×C3×D3—21no—
324C2×C3×D3—21no—
325C2×C3×D3—21no—
326C2×C3×D3—22no—
327C2×C3×D3—21no—
328C2×C3×D3—21no—
329C2×C3×D3—22no—
330C2×C3×D3—21no—
331C2×C3×D3—41no—
332C2×C3×D3—41no—
333C2×C3×D3—42no—
334C2×C3×D3—41no—
335C2×C3×D3—41no—
336C2×C3×D3—41no—
337C2×C3×D3—41no—
338C2×C3×D3—81no—
339C2×C3×D3—81no—
340C32×C4—21no—
341C32×C4—21no—
342C32×C4—21no—
343C32×C4—21no—
344C32×C4—21no—
345C32×C4—21no—
346C32×C4—41no—
347C32×C4—41no—
348C32×C4—61no—
349C32×C4D3×D411yescharacteristic
350C32×C4D3×D411yescharacteristic
351C32×C4D3×D411yescharacteristic
352C32×C4C2×#6711yescharacteristic
353C32×C4C2×#6711yescharacteristic
354C32×C4C2×#6711yescharacteristic
355C3⋊Dic3—22no—
356C3⋊Dic3—41no—
357C3⋊Dic3—121no—
358C3⋊Dic3—122no—
359C3⋊Dic3—121no—
360C3⋊Dic3—121no—
361C3⋊Dic3—241no—
362C3⋊Dic3—241no—
363C3⋊Dic3—242no—
364C3×Dic3—21no—
365C3×Dic3—21no—
366C3×Dic3—41no—
367C3×Dic3—62no—
368C3×Dic3—62no—
369C3×Dic3—122no—
370C3×Dic3—122no—
371C3×Dic3—121no—
372C3×Dic3—121no—
373C2×#41—91no—
374C2×Q16—271no—
375C2×SD16—91no—
376C22⋊Q8—271no—
377#103—271no—
378#102—271no—
379C22⋊C8—91no—
380C33#45211yescharacteristic, complemented, Sylow
381C23×C3—31no—
382C23×C3—31no—
383C23×C3—61no—
384C23×C3—61no—
385C23×C3—61no—
386C23×C3—121no—
387C22×D3—21no—
388C22×D3#62611yescharacteristic
389C3×Q8—92no—
390C3×Q8—181no—
391C3×Q8—181no—
392C3×Q8—181no—
393C3×D4—32no—
394C3×D4—32no—
395C3×D4—62no—
396C3×D4—61no—
397C3×D4—61no—
398C3×D4—121no—
399C3×D4—184no—
400C2×C3×C4—21no—
401C2×C3×C4—21no—
402C2×C3×C4—21no—
403C2×C3×C4—31no—
404C2×C3×C4—31no—
405C2×C3×C4—41no—
406C2×C3×C4—61no—
407C2×C3×C4—61no—
408C2×C3×C4—61no—
409C2×C3×C4—61no—
410C2×C3×C4—61no—
411C2×C3×C4—121no—
412C2×C3×C4—121no—
413C2×C3×C4—184no—
414C2×C3×C4—181no—
415C2×C3×C4—181no—
416C2×C3×C4C2×D3211yescharacteristic
417C2×C3×C4C2×D3211yescharacteristic
418C2×C3×C4C2×D3211yescharacteristic
419#67—64no—
420#67—64no—
421C2×Dic3—64no—
422C2×Dic3—61no—
423C2×Dic3—61no—
424C2×Dic3—62no—
425C2×Dic3—61no—
426C2×Dic3—61no—
427C2×Dic3—121no—
428C2×Dic3—182no—
429C2×Dic3—181no—
430C2×Dic3—181no—
431C2×Dic3—181no—
432C2×Dic3—361no—
433C2×Dic3—361no—
434C2×Dic3—362no—
435C2×Dic3#62611yescharacteristic
436D12—21no—
437D12#62512yes—
438C4×D3—21no—
439C4×D3—21no—
440C4×D3—64no—
441Dic6—32no—
442Dic6—32no—
443Dic6—62no—
444Dic6—61no—
445Dic6—61no—
446Dic6—61no—
447Dic6—61no—
448Dic6—61no—
449Dic6—61no—
450Dic6—121no—
451Dic6—121no—
452Dic6—121no—
453C3×C8—61no—
454C3×C8—61no—
455C3×C8—121no—
456C3×C8—181no—
457C3⋊C8—21no—
458C3⋊C8—61no—
459C3⋊C8—61no—
460C3⋊C8—121no—
461C2×C32—21no—
462C2×C32—21no—
463C2×C32—21no—
464C2×C32—21no—
465C2×C32—21no—
466C2×C32—21no—
467C2×C32—21no—
468C2×C32—21no—
469C2×C32—21no—
470C2×C32—41no—
471C2×C32—41no—
472C2×C32—41no—
473C2×C32—121no—
474C2×C32—122no—
475C2×C32#88211yescharacteristic
476C2×C32#90411yescharacteristic
477C2×C32#90911yescharacteristic
478C2×C32#91511yescharacteristic
479C2×C32#91611yescharacteristic
480C2×C32#93711yescharacteristic
481C2×C32#95011yescharacteristic
482C2×C32#95111yescharacteristic
483C2×C32#95311yescharacteristic
484C3×D3—42no—
485C3×D3—42no—
486C3×D3—41no—
487C3×D3—42no—
488C3×D3—41no—
489C3×D3—82no—
490#41—182no—
491#41—182no—
492C2×Q8—91no—
493C2×Q8—91no—
494C2×D4—31no—
495C2×D4—181no—
496C22×C4—31no—
497C22×C4—181no—
498Q16—542no—
499SD16—182no—
500C2×C8—31no—
501C2×C8—91no—
502C4⋊C4—271no—
503C4⋊C4—541no—
504C22⋊C4—541no—
505C22×C3—21no—
506C22×C3—21no—
507C22×C3—21no—
508C22×C3—41no—
509C22×C3—62no—
510C22×C3—61no—
511C22×C3—61no—
512C22×C3—61no—
513C22×C3—62no—
514C22×C3—61no—
515C22×C3—61no—
516C22×C3—61no—
517C22×C3—62no—
518C22×C3—121no—
519C22×C3—121no—
520C22×C3—122no—
521C22×C3—121no—
522C22×C3—121no—
523C22×C3—121no—
524C22×C3—121no—
525C22×C3—241no—
526C22×C3—241no—
527C22×C3D3×D1211yescharacteristic
528C22×C3C2×#62611yescharacteristic
529C22×C3D3×#6711yescharacteristic
530C2×D3—21no—
531C2×D3—21no—
532C2×D3—22no—
533C2×D3—21no—
534C2×D3—41no—
535C2×D3—41no—
536C3×C4—21no—
537C3×C4—21no—
538C3×C4—21no—
539C3×C4—21no—
540C3×C4—21no—
541C3×C4—21no—
542C3×C4—41no—
543C3×C4—41no—
544C3×C4—61no—
545C3×C4—61no—
546C3×C4—121no—
547C3×C4—182no—
548C3×C4—361no—
549C3×C4#380211yescharacteristic
550C3×C4C2×#62511yescharacteristic
551C3×C4D3×#6711yescharacteristic
552C3×C4D3×#6711yescharacteristic
553C3×C4D3×#6711yescharacteristic
554C3×C4D3×#6711yescharacteristic
555Dic3—21no—
556Dic3—62no—
557Dic3—62no—
558Dic3—122no—
559Dic3—122no—
560Dic3—121no—
561Dic3—121no—
562Dic3—362no—
563Dic3—361no—
564Dic3—361no—
565Dic3—361no—
566Dic3—721no—
567Dic3—721no—
568Dic3—722no—
569C32—21no—
570C32—21no—
571C32—21no—
572C32—41no—
573C32#473311yescharacteristic, complemented
574C32#517311yescharacteristic, complemented
575C32#523911yescharacteristic, complemented
576C23—31no—
577C23—61no—
578Q8—92no—
579Q8—181no—
580Q8—181no—
581Q8—181no—
582D4—32no—
583D4—61no—
584D4—184no—
585C2×C4—31no—
586C2×C4—61no—
587C2×C4—61no—
588C2×C4—184no—
589C2×C4—181no—
590C2×C4—181no—
591C2×C4—541no—
592C2×C4D3311yescharacteristic
593C8—61no—
594C8—181no—
595C2×C3—21no—
596C2×C3—21no—
597C2×C3—21no—
598C2×C3—21no—
599C2×C3—21no—
600C2×C3—21no—
601C2×C3—21no—
602C2×C3—21no—
603C2×C3—21no—
604C2×C3—41no—
605C2×C3—41no—
606C2×C3—41no—
607C2×C3—122no—
608C2×C3—122no—
609C2×C3—121no—
610C2×C3—121no—
611C2×C3—122no—
612C2×C3—242no—
613C2×C3#6380011yescharacteristic
614C2×C3#6380611yescharacteristic
615C2×C3#6382911yescharacteristic
616C2×C3#6383111yescharacteristic
617C2×C3#6392211yescharacteristic
618C2×C3#6392911yescharacteristic
619C2×C3#6393511yescharacteristic
620C2×C3#6397611yescharacteristic
621C2×C3#6397711yescharacteristic
622D3—41no—
623D3—42no—
624C22—61no—
625C22—61no—
626C22—61no—
627C22—62no—
628C22—121no—
629C22—121no—
630C22D3×#62611yescharacteristic
631C4—61no—
632C4—182no—
633C4—361no—
634C4—1081no—
635C4D3×#62511yescharacteristic
636C4#8895311yescharacteristic
637C3—21no—
638C3—21no—
639C3—21no—
640C3—41no—
641C3#1059096011yescharacteristic, complemented
642C3#1059117311yescharacteristic, complemented
643C3#1059146111yescharacteristic, complemented
644C2—121no—
645C2—122no—
646C2#1172285311yescharacteristic
647C2#1172288611yescharacteristic
648C2#1172294711yescharacteristic
subgroupG/H
centerC22D3×#626
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radicalC2×C3×#3795C2
hypocenterC33#452
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
2. C2×#88812C2semidirect
3. #11723425C2semidirect
4. C2×C3×#3716C2semidirect
5. #11722040C2semidirect
6. #11722032C2semidirect
7. C3×(C2×D3)⋊(C3⋊C8)C2semidirect
38. C2×#3716D3semidirect
49. (C2×D3)⋊(C3⋊C8)D3semidirect
380. C33#452semidirect
573. C32#4733semidirect
574. C32#5173semidirect
575. C32#5239semidirect
641. C3#10590960semidirect
642. C3#10591173semidirect
643. C3#10591461semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37952
C2×#888122
#117234252
C2×C3×#37162
#117220402
#117220322
C3×(C2×D3)⋊(C3⋊C8)2
#105914613
#105911733
#105909603