Finite Group Databasebeta

← #13155874

Subgroups — #13155874

Order 1728.

790 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
1C2×C3×#3795C211yesmaximal, characteristic
2C2×#88798C211yesmaximal, characteristic, complemented
3C3×#63856C211yesmaximal, characteristic, complemented
4C2×C3×#3715C211yesmaximal, characteristic, complemented
5#11722076C211yesmaximal, characteristic, complemented
6#11722031C211yesmaximal, characteristic, complemented
7(C2×D3)⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic, complemented
8#10591655—31nomaximal
9#10591461—31nomaximal
10#10591170—31nomaximal
11C2×C3×#634C2211yescharacteristic
12C2×C32×D12C2211yescharacteristic
13C2×C32×C4×D3C2211yescharacteristic
14C2×C32×Dic6C2211yescharacteristic
15C2×C3×#625—21no—
16C2×C3×D3×Dic3—21no—
17C3×#3795—22no—
18C3×#3795—22no—
19C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
20C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
21C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
22#88798—22no—
23C3×#3724—21no—
24C3×(C2×D3)⋊Dic3—21no—
25C3×#3715—22no—
26C2×C3×#237—31no—
27C2×C3×#235—31no—
28C2×#3795D311yescharacteristic
29C2×#3756—31no—
30C3×#946—31no—
31C2×C3×#215—31no—
32C2×C3×#214—31no—
33C3×#924—31no—
34#63856D311yescharacteristic, complemented
35C2×#3718—31no—
36C2×#3718—31no—
37C2×#3718—61no—
38C2×#3715D311yescharacteristic, complemented
39#63659—31no—
40#63657—31no—
41#63567—31no—
42#63566—31no—
43#63565—31no—
44#63562—31no—
45#63562—31no—
46#63562—61no—
47(C2×D3)⋊(C3⋊C8)—31no—
48(C2×D3)⋊(C3⋊C8)—31no—
49(C2×D3)⋊(C3⋊C8)—61no—
50C22×C32×D3—21no—
51C22×C32×D3D411yescharacteristic
52C2×C33×C4C2311yescharacteristic
53C2×C3×C3⋊Dic3—21no—
54C2×C3×C3⋊Dic3—21no—
55C3×#634—21no—
56C3×#634D412yes—
57C2×C32×Dic3—21no—
58C2×C32×Dic3D411yescharacteristic
59C32×D12—21no—
60C32×D12D412yes—
61C32×C4×D3—21no—
62C32×C4×D3—21no—
63C32×Dic6—21no—
64C32×Dic6—21no—
65C3×#625—24no—
66C3×D3×Dic3—24no—
67C3⋊(C3⋊(C3⋊C8))—21no—
68C3×C3⋊(C3⋊C8)—21no—
69#5239—91no—
70#5173—91no—
71#5047—91no—
72C2×C32×Q8—31no—
73C2×C32×D4—31no—
74C22×C32×C4—31no—
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—21no—
91C2×C3×Dic6—31no—
92C2×C3×Dic6—31no—
93C2×C3×Dic6—61no—
94C2×C3×Dic6C2×D311yescharacteristic
95C2×C3×Dic6C2×D311yescharacteristic
96C2×#625—21no—
97C2×D3×Dic3—21no—
98#3795—22no—
99#3795—22no—
100#3756—62no—
101C4⋊(C3⋊Dic3)C2×D311yescharacteristic
102C2×C3⋊(C3⋊C8)—31no—
103C2×C3⋊(C3⋊C8)—31no—
104C2×C3⋊(C3⋊C8)—31no—
105C2×C3⋊(C3⋊C8)—61no—
106C2×C3⋊(C3⋊C8)—61no—
107C2×C3⋊(C3⋊C8)—61no—
108C2×C3⋊(C3⋊C8)—121no—
109C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
110C3×#217—61no—
111C3×#215—62no—
112C3×#214—62no—
113C3×(C2×D3)⋊C4—61no—
114C3×C4⋊Dic3—31no—
115C3×C4⋊Dic3—31no—
116C3×C4⋊Dic3—61no—
117C3×Dic3⋊C4—61no—
118C3×Dic3⋊C4—61no—
119C2×C3×C3⋊C8—31no—
120C2×C3×C3⋊C8—31no—
121C2×C3×C3⋊C8—61no—
122#3724—21no—
123(C2×D3)⋊Dic3—21no—
124#3718—62no—
125#3718—62no—
126#3718—64no—
127#3715—22no—
128C22×C33C2×D411yescharacteristic
129C2×C32×D3—22no—
130C2×C32×D3—21no—
131C2×C32×D3—21no—
132C2×C32×D3—21no—
133C2×C32×D3—41no—
134C2×C32×D3—41no—
135C33×C4C2×D411yescharacteristic
136C33×C4C2×D411yescharacteristic
137C3×C3⋊Dic3—22no—
138C3×C3⋊Dic3—41no—
139C32×Dic3—21no—
140C32×Dic3—41no—
141C2×C3×#41—91no—
142C2×#237—31no—
143C2×#235—31no—
144C2×C3×SD16—91no—
145C3×C22⋊Q8—91no—
146#946—31no—
147C2×#216—91no—
148C2×#216—91no—
149C2×#216—91no—
150C2×#216—181no—
151C2×#215—31no—
152C2×#214—31no—
153#924—31no—
154#835—91no—
155#832—91no—
156#832—91no—
157#832—181no—
158#830—91no—
159#830—91no—
160#830—181no—
161(C2×D3)⋊C8—91no—
162#816—91no—
163#810—91no—
164#809—91no—
165C23×C32—31no—
166C23×C32—61no—
167C22×C3×D3—21no—
168C22×C3×D3—21no—
169C22×C3×D3—21no—
170C22×C3×D3—41no—
171C22×C3×D3#6711yescharacteristic
172C22×C3×D3#6711yescharacteristic
173C32×Q8—61no—
174C32×Q8—61no—
175C32×D4—32no—
176C32×D4—61no—
177C2×C32×C4—21no—
178C2×C32×C4—21no—
179C2×C32×C4—21no—
180C2×C32×C4—31no—
181C2×C32×C4—41no—
182C2×C32×C4—61no—
183C2×C32×C4—61no—
184C2×C32×C4—61no—
185C2×C32×C4C22×D311yescharacteristic
186C2×C32×C4C22×D311yescharacteristic
187C2×C32×C4C22×D311yescharacteristic
188C2×C3⋊Dic3—21no—
189C2×C3⋊Dic3—21no—
190#634—21no—
191#634#6712yes—
192C3×#67—64no—
193C3×#67—64no—
194C2×C3×Dic3—21no—
195C2×C3×Dic3—21no—
196C2×C3×Dic3—21no—
197C2×C3×Dic3—41no—
198C2×C3×Dic3—64no—
199C2×C3×Dic3—61no—
200C2×C3×Dic3—61no—
201C2×C3×Dic3—61no—
202C2×C3×Dic3—61no—
203C2×C3×Dic3—121no—
204C2×C3×Dic3—121no—
205C2×C3×Dic3#6711yescharacteristic
206C2×C3×Dic3#6711yescharacteristic
207C3×D12—21no—
208C3×D12—21no—
209C3×D12—22no—
210C3×D12—41no—
211C3×D12#6712yes—
212C3×D12#6712yes—
213C3×C4×D3—21no—
214C3×C4×D3—21no—
215C3×C4×D3—21no—
216C3×C4×D3—21no—
217C3×C4×D3—41no—
218C3×C4×D3—41no—
219C3×C4×D3—64no—
220C3×Dic6—22no—
221C3×Dic6—21no—
222C3×Dic6—21no—
223C3×Dic6—21no—
224C3×Dic6—21no—
225C3×Dic6—32no—
226C3×Dic6—32no—
227C3×Dic6—41no—
228C3×Dic6—62no—
229C3×Dic6—61no—
230C3×Dic6—61no—
231C3×Dic6—121no—
232#625—24no—
233D3×Dic3—24no—
234C3⋊(C3⋊C8)—21no—
235C3⋊(C3⋊C8)—61no—
236C3⋊(C3⋊C8)—62no—
237C3⋊(C3⋊C8)—61no—
238C3⋊(C3⋊C8)—61no—
239C3⋊(C3⋊C8)—121no—
240C3⋊(C3⋊C8)—121no—
241C3⋊(C3⋊C8)—122no—
242C3×C3⋊C8—61no—
243C3×C3⋊C8—61no—
244C3×C3⋊C8—121no—
245#452—271noSylow
246C2×C33#12011yescharacteristic
247C2×C33#13511yescharacteristic
248C2×C33#13711yescharacteristic
249C32×D3—41no—
250C32×D3—42no—
251C3×#41—182no—
252C3×#41—182no—
253C2×C3×Q8—31no—
254C2×C3×Q8—31no—
255C2×C3×Q8—61no—
256C2×C3×Q8—91no—
257C2×C3×D4—31no—
258C2×C3×D4—31no—
259C2×C3×D4—61no—
260C2×C3×D4—181no—
261C22×C3×C4—31no—
262C22×C3×C4—31no—
263C22×C3×C4—61no—
264C22×C3×C4—181no—
265C2×#67—61no—
266C2×#67—61no—
267C22×Dic3—61no—
268#237—62no—
269#237—62no—
270#235—62no—
271#235—62no—
272C2×D12D3211yescharacteristic
273C2×C4×D3—61no—
274C2×C4×D3D3211yescharacteristic
275C2×Dic6—31no—
276C2×Dic6—31no—
277C2×Dic6—61no—
278C2×Dic6D3211yescharacteristic
279C3×SD16—182no—
280C2×C3×C8—91no—
281C3×C4⋊C4—91no—
282C3×C4⋊C4—181no—
283C3×C22⋊C4—181no—
284#217—61no—
285#216—182no—
286#216—182no—
287#216—182no—
288#216—184no—
289#215—62no—
290#214—62no—
291(C2×D3)⋊C4—61no—
292C4⋊Dic3—31no—
293C4⋊Dic3—31no—
294C4⋊Dic3—61no—
295Dic3⋊C4—61no—
296Dic3⋊C4—61no—
297C2×C3⋊C8—31no—
298C2×C3⋊C8—31no—
299C2×C3⋊C8—61no—
300C2×C3⋊C8—91no—
301C2×C3⋊C8—91no—
302C2×C3⋊C8—91no—
303C2×C3⋊C8—181no—
304C2×C3⋊C8—181no—
305C2×C3⋊C8—181no—
306C2×C3⋊C8—361no—
307C22×C32—21no—
308C22×C32—21no—
309C22×C32—21no—
310C22×C32—41no—
311C22×C32—61no—
312C22×C32—61no—
313C22×C32—62no—
314C22×C32—61no—
315C22×C32—121no—
316C22×C32—121no—
317C22×C32D3×D411yescharacteristic
318C22×C32C2×#6711yescharacteristic
319C22×C32C2×#6711yescharacteristic
320C2×C3×D3—22no—
321C2×C3×D3—21no—
322C2×C3×D3—22no—
323C2×C3×D3—21no—
324C2×C3×D3—21no—
325C2×C3×D3—21no—
326C2×C3×D3—21no—
327C2×C3×D3—22no—
328C2×C3×D3—21no—
329C2×C3×D3—41no—
330C2×C3×D3—41no—
331C2×C3×D3—42no—
332C2×C3×D3—41no—
333C2×C3×D3—41no—
334C2×C3×D3—41no—
335C2×C3×D3—41no—
336C2×C3×D3—81no—
337C2×C3×D3—81no—
338C32×C4—21no—
339C32×C4—21no—
340C32×C4—21no—
341C32×C4—21no—
342C32×C4—21no—
343C32×C4—21no—
344C32×C4—41no—
345C32×C4—41no—
346C32×C4—61no—
347C32×C4—121no—
348C32×C4D3×D411yescharacteristic
349C32×C4D3×D411yescharacteristic
350C32×C4D3×D411yescharacteristic
351C32×C4C2×#6711yescharacteristic
352C32×C4C2×#6711yescharacteristic
353C32×C4C2×#6711yescharacteristic
354C3⋊Dic3—22no—
355C3⋊Dic3—41no—
356C3×Dic3—21no—
357C3×Dic3—21no—
358C3×Dic3—41no—
359C3×Dic3—42no—
360C3×Dic3—41no—
361C3×Dic3—41no—
362C3×Dic3—62no—
363C3×Dic3—62no—
364C3×Dic3—122no—
365C3×Dic3—121no—
366C3×Dic3—121no—
367C3×Dic3—241no—
368C2×#41—91no—
369C2×Q16—271no—
370C2×SD16—91no—
371C22⋊Q8—91no—
372#103—271no—
373#102—271no—
374C22⋊C8—271no—
375C33#45211yescharacteristic, complemented, Sylow
376C23×C3—31no—
377C23×C3—31no—
378C23×C3—61no—
379C23×C3—61no—
380C23×C3—61no—
381C23×C3—121no—
382C22×D3—21no—
383C22×D3#62511yescharacteristic
384C3×Q8—62no—
385C3×Q8—61no—
386C3×Q8—61no—
387C3×Q8—61no—
388C3×Q8—61no—
389C3×Q8—92no—
390C3×Q8—121no—
391C3×Q8—181no—
392C3×D4—32no—
393C3×D4—32no—
394C3×D4—62no—
395C3×D4—61no—
396C3×D4—61no—
397C3×D4—121no—
398C3×D4—184no—
399C2×C3×C4—21no—
400C2×C3×C4—21no—
401C2×C3×C4—21no—
402C2×C3×C4—31no—
403C2×C3×C4—31no—
404C2×C3×C4—41no—
405C2×C3×C4—61no—
406C2×C3×C4—61no—
407C2×C3×C4—61no—
408C2×C3×C4—61no—
409C2×C3×C4—61no—
410C2×C3×C4—61no—
411C2×C3×C4—61no—
412C2×C3×C4—121no—
413C2×C3×C4—121no—
414C2×C3×C4—121no—
415C2×C3×C4—184no—
416C2×C3×C4—181no—
417C2×C3×C4—181no—
418C2×C3×C4C2×D3211yescharacteristic
419C2×C3×C4C2×D3211yescharacteristic
420C2×C3×C4C2×D3211yescharacteristic
421#67—64no—
422#67—64no—
423C2×Dic3—21no—
424C2×Dic3—64no—
425C2×Dic3—61no—
426C2×Dic3—61no—
427C2×Dic3—61no—
428C2×Dic3—61no—
429C2×Dic3—121no—
430C2×Dic3—121no—
431C2×Dic3#62511yescharacteristic
432D12—21no—
433D12#62512yes—
434C4×D3—21no—
435C4×D3—21no—
436C4×D3—64no—
437Dic6—21no—
438Dic6—21no—
439Dic6—32no—
440Dic6—32no—
441Dic6—62no—
442Dic6—61no—
443Dic6—61no—
444Dic6—121no—
445C3×C8—181no—
446C3⋊C8—61no—
447C3⋊C8—61no—
448C3⋊C8—121no—
449C3⋊C8—182no—
450C3⋊C8—181no—
451C3⋊C8—181no—
452C3⋊C8—181no—
453C3⋊C8—361no—
454C3⋊C8—361no—
455C3⋊C8—362no—
456C2×C32—21no—
457C2×C32—21no—
458C2×C32—21no—
459C2×C32—21no—
460C2×C32—21no—
461C2×C32—21no—
462C2×C32—21no—
463C2×C32—21no—
464C2×C32—21no—
465C2×C32—41no—
466C2×C32—41no—
467C2×C32—41no—
468C2×C32—121no—
469C2×C32—122no—
470C2×C32#91511yescharacteristic
471C2×C32#91611yescharacteristic
472C2×C32#93711yescharacteristic
473C2×C32#93711yescharacteristic
474C2×C32#94211yescharacteristic
475C2×C32#95011yescharacteristic
476C2×C32#95011yescharacteristic
477C2×C32#95111yescharacteristic
478C2×C32#95311yescharacteristic
479C3×D3—42no—
480C3×D3—42no—
481C3×D3—41no—
482C3×D3—42no—
483C3×D3—41no—
484C3×D3—82no—
485#41—182no—
486#41—182no—
487C2×Q8—31no—
488C2×Q8—91no—
489C2×D4—31no—
490C2×D4—181no—
491C22×C4—31no—
492C22×C4—181no—
493Q16—542no—
494SD16—182no—
495C2×C8—91no—
496C2×C8—271no—
497C4⋊C4—91no—
498C4⋊C4—181no—
499C22⋊C4—181no—
500C22×C3—21no—
501C22×C3—21no—
502C22×C3—21no—
503C22×C3—41no—
504C22×C3—62no—
505C22×C3—61no—
506C22×C3—61no—
507C22×C3—61no—
508C22×C3—61no—
509C22×C3—62no—
510C22×C3—61no—
511C22×C3—61no—
512C22×C3—62no—
513C22×C3—121no—
514C22×C3—121no—
515C22×C3—122no—
516C22×C3—121no—
517C22×C3—121no—
518C22×C3—121no—
519C22×C3—121no—
520C22×C3—241no—
521C22×C3—241no—
522C22×C3C2×#62511yescharacteristic
523C22×C3D3×#6711yescharacteristic
524C22×C3D3×#6711yescharacteristic
525C2×D3—21no—
526C2×D3—21no—
527C2×D3—21no—
528C2×D3—22no—
529C2×D3—41no—
530C2×D3—41no—
531C3×C4—21no—
532C3×C4—21no—
533C3×C4—21no—
534C3×C4—21no—
535C3×C4—21no—
536C3×C4—21no—
537C3×C4—41no—
538C3×C4—41no—
539C3×C4—61no—
540C3×C4—61no—
541C3×C4—121no—
542C3×C4—122no—
543C3×C4—121no—
544C3×C4—121no—
545C3×C4—182no—
546C3×C4—361no—
547C3×C4C2×#62511yescharacteristic
548C3×C4D3×#6711yescharacteristic
549C3×C4D3×#6711yescharacteristic
550C3×C4D3×#6711yescharacteristic
551C3×C4D3×#6711yescharacteristic
552C3×C4#381111yescharacteristic
553Dic3—21no—
554Dic3—41no—
555Dic3—62no—
556Dic3—62no—
557Dic3—122no—
558Dic3—121no—
559Dic3—121no—
560Dic3—241no—
561C32—21no—
562C32—21no—
563C32—21no—
564C32—41no—
565C32#504711yescharacteristic, complemented
566C32#517311yescharacteristic, complemented
567C32#523911yescharacteristic, complemented
568C23—31no—
569C23—61no—
570Q8—61no—
571Q8—61no—
572Q8—92no—
573Q8—181no—
574D4—32no—
575D4—61no—
576D4—184no—
577C2×C4—31no—
578C2×C4—61no—
579C2×C4—61no—
580C2×C4—61no—
581C2×C4—184no—
582C2×C4—181no—
583C2×C4—181no—
584C2×C4D3311yescharacteristic
585C8—181no—
586C8—541no—
587C2×C3—21no—
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—41no—
597C2×C3—41no—
598C2×C3—41no—
599C2×C3—122no—
600C2×C3—121no—
601C2×C3—122no—
602C2×C3—122no—
603C2×C3—121no—
604C2×C3—242no—
605C2×C3#6382211yescharacteristic
606C2×C3#6382711yescharacteristic
607C2×C3#6392911yescharacteristic
608C2×C3#6393511yescharacteristic
609C2×C3#6394611yescharacteristic
610C2×C3#6394711yescharacteristic
611C2×C3#6397611yescharacteristic
612C2×C3#6397611yescharacteristic
613C2×C3#6398311yescharacteristic
614D3—41no—
615D3—42no—
616C22—61no—
617C22—61no—
618C22—61no—
619C22—62no—
620C22—121no—
621C22—121no—
622C22D3×#62511yescharacteristic
623C4—61no—
624C4—121no—
625C4—182no—
626C4—361no—
627C4D3×#62511yescharacteristic
628C4#8895211yescharacteristic
629C3—21no—
630C3—21no—
631C3—21no—
632C3—41no—
633C3#1059117011yescharacteristic, complemented
634C3#1059146111yescharacteristic, complemented
635C3#1059165511yescharacteristic, complemented
636C2—121no—
637C2—122no—
638C2#1172285111yescharacteristic
639C2#1172288211yescharacteristic
640C2#1172294611yescharacteristic
subgroupG/H
centerC22D3×#625
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×#88798C2semidirect
3. C3×#63856C2semidirect
4. C2×C3×#3715C2semidirect
5. #11722076C2semidirect
6. #11722031C2semidirect
7. (C2×D3)⋊(C3⋊(C3⋊C8))C2semidirect
34. #63856D3semidirect
38. C2×#3715D3semidirect
375. C33#452semidirect
565. C32#5047semidirect
566. C32#5173semidirect
567. C32#5239semidirect
633. C3#10591170semidirect
634. C3#10591461semidirect
635. C3#10591655semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37952
C2×#887982
C3×#638562
C2×C3×#37152
#117220762
#117220312
(C2×D3)⋊(C3⋊(C3⋊C8))2
#105916553
#105914613
#105911703