Finite Group Databasebeta

← #13162003

Subgroups — #13162003

Order 1728.

888 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×Q8×C3⋊D3C211yesmaximal, characteristic
2C2×C3×#3756C211yesmaximal, characteristic, complemented
3#11723674C211yesmaximal, characteristic
4C2×#88812C211yesmaximal, characteristic, complemented
5C3×#63662C211yesmaximal, characteristic, complemented
6#11722076C211yesmaximal, characteristic, complemented
7#11722072C211yesmaximal, characteristic
8#10591603—31nomaximal
9#10591595—31nomaximal
10#10591594—31nomaximal
11C2×C33×Q8C2211yescharacteristic
12C3×Q8×C3⋊D3—22no—
13C3×Q8×C3⋊D3—22no—
14C2×C3×C4×C3⋊D3—21no—
15C2×C3×C4×C3⋊D3C2211yescharacteristic
16C2×C3×#634—21no—
17C2×C3×#634C2211yescharacteristic
18C2×C3×#634C2211yescharacteristic
19C3×#3756—22no—
20C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
21C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
22C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
23#88815—21no—
24#88814—21no—
25#88812—22no—
26C2×Q8×C3⋊D3D311yescharacteristic
27C2×C3×D3×Q8—31no—
28C2×C3×D3×Q8—31no—
29C2×C3×D3×Q8—61no—
30C2×#3756D311yescharacteristic, complemented
31C2×C3×#216—31no—
32C2×C3×#216—31no—
33C2×C3×#216—61no—
34#63871—31no—
35#63857—31no—
36#63856—31no—
37C2×#3719—31no—
38C2×#3719—31no—
39C2×#3718—31no—
40#63662D311yescharacteristic, complemented
41C3×#835—31no—
42C3×#835—31no—
43C3×#835—61no—
44#63567—31no—
45#63566—31no—
46#63565—31no—
47#63557—31no—
48(C2×D3)⋊(C3⋊C8)—31no—
49(C2×D3)⋊(C3⋊C8)—31no—
50C22×C3×C3⋊D3D411yescharacteristic
51C33×Q8—21no—
52C33×Q8D412yes—
53C2×C33×C4—21no—
54C2×C33×C4C2311yescharacteristic
55C2×C3×C3⋊Dic3—21no—
56C2×C3×C3⋊Dic3—21no—
57C2×C3×C3⋊Dic3—21no—
58C2×C3×C3⋊Dic3D411yescharacteristic
59C3×C4×C3⋊D3—24no—
60C3×C4×C3⋊D3—21no—
61C3×C4×C3⋊D3—21no—
62C3×#634—24no—
63C3×#634—21no—
64C3×#634—21no—
65C3×#634—21no—
66C3×#634D412yes—
67C3×C3⋊(C3⋊C8)—21no—
68C3×C3⋊(C3⋊C8)—21no—
69#5228—91no—
70#5186—91no—
71#4809—91no—
72C2×C32×Q8—21no—
73C2×C32×Q8—21no—
74C2×C32×Q8—21no—
75C2×C32×Q8—41no—
76C2×C32×Q8C2×D311yescharacteristic
77C2×C32×Q8C2×D311yescharacteristic
78C2×C32×Q8C2×D311yescharacteristic
79Q8×C3⋊D3—22no—
80Q8×C3⋊D3—22no—
81C2×C4×C3⋊D3—21no—
82C2×C4×C3⋊D3C2×D311yescharacteristic
83C2×#634—21no—
84C2×#634C2×D311yescharacteristic
85C2×#634C2×D311yescharacteristic
86C3×D3×Q8—62no—
87C3×D3×Q8—62no—
88C3×D3×Q8—64no—
89C3×D3×Q8—64no—
90C3×D3×Q8—62no—
91C3×D3×Q8—62no—
92C2×C3×C4×D3—31no—
93C2×C3×C4×D3—31no—
94C2×C3×C4×D3—61no—
95C2×C3×C4×D3—61no—
96C2×C3×C4×D3—61no—
97C2×C3×C4×D3—62no—
98C2×C3×Dic6—31no—
99C2×C3×Dic6—31no—
100C2×C3×Dic6—31no—
101C2×C3×Dic6—31no—
102C2×C3×Dic6—61no—
103C2×C3×Dic6—61no—
104C2×C3×Dic6—61no—
105C2×C3×Dic6—62no—
106C2×C3×Dic6—61no—
107#3756—22no—
108C4⋊(C3⋊Dic3)C2×D311yescharacteristic
109C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
110C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
111C3×#216—64no—
112C3×#216—62no—
113C3×#216—62no—
114C3×C4⋊Dic3—31no—
115C3×C4⋊Dic3—31no—
116C3×C4⋊Dic3—61no—
117C2×C3×C3⋊C8—31no—
118C2×C3×C3⋊C8—31no—
119C2×C3×C3⋊C8—31no—
120C2×C3×C3⋊C8—31no—
121C2×C3×C3⋊C8—61no—
122C2×C3×C3⋊C8—61no—
123#3724—61no—
124Dic3⋊Dic3—61no—
125Dic3⋊Dic3—61no—
126#3722—61no—
127(C2×D3)⋊Dic3—61no—
128(C2×D3)⋊Dic3—61no—
129#3719—62no—
130#3719—62no—
131#3718—62no—
132C22×C33C2×D411yescharacteristic
133C2×C3×C3⋊D3—21no—
134C2×C3×C3⋊D3—21no—
135C2×C3×C3⋊D3Q1612yescomplemented
136C33×C4—22no—
137C33×C4C2×D411yescharacteristic
138C33×C4C2×D411yescharacteristic
139C3×C3⋊Dic3—21no—
140C3×C3⋊Dic3—22no—
141C3×C3⋊Dic3—41no—
142C3×C3⋊Dic3—41no—
143C22×C3×Q8—91no—
144C2×D3×Q8—31no—
145C2×D3×Q8—31no—
146C2×D3×Q8—61no—
147C2×C3×Q16—91no—
148#946—91no—
149C2×#216—31no—
150C2×#216—31no—
151C2×#216—61no—
152C2×#216—91no—
153C2×#216—91no—
154#924—91no—
155C2×Dic12—91no—
156#897—91no—
157C3×#103—91no—
158#835—31no—
159#835—31no—
160#835—61no—
161#835—91no—
162#830—91no—
163(C2×D3)⋊C8—91no—
164(C2×D3)⋊C8—91no—
165#816—91no—
166#810—91no—
167C22×C3⋊D3#6711yescharacteristic
168C22×C3×D3—31no—
169C22×C3×D3—31no—
170C22×C3×D3—61no—
171C32×Q8—21no—
172C32×Q8—22no—
173C32×Q8—21no—
174C32×Q8—21no—
175C32×Q8—22no—
176C32×Q8—22no—
177C32×Q8—22no—
178C32×Q8—42no—
179C32×Q8—41no—
180C32×Q8—41no—
181C32×Q8—42no—
182C32×Q8D1212yes—
183C32×Q8#6712yes—
184C32×Q8#6712yes—
185C2×C32×C4—21no—
186C2×C32×C4—22no—
187C2×C32×C4—21no—
188C2×C32×C4—21no—
189C2×C32×C4—21no—
190C2×C32×C4—21no—
191C2×C32×C4—21no—
192C2×C32×C4—41no—
193C2×C32×C4—41no—
194C2×C32×C4—41no—
195C2×C32×C4—42no—
196C2×C32×C4C22×D311yescharacteristic
197C2×C32×C4C22×D311yescharacteristic
198C2×C32×C4C22×D311yescharacteristic
199C2×C3⋊Dic3—21no—
200C2×C3⋊Dic3—21no—
201C2×C3⋊Dic3—21no—
202C2×C3⋊Dic3#6711yescharacteristic
203C4×C3⋊D3—24no—
204C4×C3⋊D3—21no—
205C4×C3⋊D3—21no—
206#634—24no—
207#634—21no—
208#634—21no—
209#634—21no—
210#634#6712yes—
211C2×C3×Dic3—31no—
212C2×C3×Dic3—31no—
213C2×C3×Dic3—61no—
214C2×C3×Dic3—62no—
215C2×C3×Dic3—61no—
216C2×C3×Dic3—61no—
217C2×C3×Dic3—61no—
218C2×C3×Dic3—61no—
219C2×C3×Dic3—61no—
220C2×C3×Dic3—61no—
221C2×C3×Dic3—121no—
222C2×C3×Dic3—121no—
223C3×C4×D3—68no—
224C3×C4×D3—64no—
225C3×C4×D3—64no—
226C3×C4×D3—62no—
227C3×C4×D3—62no—
228C3×C4×D3—61no—
229C3×C4×D3—61no—
230C3×C4×D3—61no—
231C3×C4×D3—61no—
232C3×Dic6—32no—
233C3×Dic6—32no—
234C3×Dic6—62no—
235C3×Dic6—68no—
236C3×Dic6—64no—
237C3×Dic6—64no—
238C3×Dic6—62no—
239C3×Dic6—61no—
240C3×Dic6—61no—
241C3×Dic6—62no—
242C3×Dic6—61no—
243C3×Dic6—61no—
244C3×Dic6—61no—
245C3×Dic6—61no—
246C3×Dic6—121no—
247C3⋊(C3⋊C8)—21no—
248C3⋊(C3⋊C8)—21no—
249C3×C3⋊C8—62no—
250C3×C3⋊C8—61no—
251C3×C3⋊C8—61no—
252C3×C3⋊C8—61no—
253C3×C3⋊C8—61no—
254C3×C3⋊C8—121no—
255#451—271noSylow
256C2×C33#12011yescharacteristic
257C2×C33C2×Q1611yescharacteristic
258C2×C33#13711yescharacteristic
259C3×C3⋊D3—22no—
260C2×C3×Q8—21no—
261C2×C3×Q8—21no—
262C2×C3×Q8—21no—
263C2×C3×Q8—41no—
264C2×C3×Q8—91no—
265C2×C3×Q8—91no—
266C2×C3×Q8—181no—
267C2×C3×Q8—182no—
268C2×C3×Q8—182no—
269C2×C3×Q8D3211yescharacteristic
270C2×C3×Q8D3211yescharacteristic
271C2×C3×Q8D3211yescharacteristic
272C22×C3×C4—91no—
273C22×C3×C4—181no—
274D3×Q8—62no—
275D3×Q8—62no—
276D3×Q8—64no—
277D3×Q8—64no—
278D3×Q8—62no—
279D3×Q8—62no—
280C2×C4×D3—31no—
281C2×C4×D3—31no—
282C2×C4×D3—61no—
283C2×C4×D3—61no—
284C2×C4×D3—61no—
285C2×C4×D3—62no—
286C2×Dic6—31no—
287C2×Dic6—31no—
288C2×Dic6—31no—
289C2×Dic6—31no—
290C2×Dic6—61no—
291C2×Dic6—61no—
292C2×Dic6—61no—
293C2×Dic6—61no—
294C2×Dic6—62no—
295C3×Q16—182no—
296C2×C3×C8—91no—
297C2×C3×C8—91no—
298C3×C4⋊C4—91no—
299#217—181no—
300#216—64no—
301#216—62no—
302#216—62no—
303#216—182no—
304#216—182no—
305(C2×D3)⋊C4—181no—
306(C2×D3)⋊C4—181no—
307C4⋊Dic3—31no—
308C4⋊Dic3—31no—
309C4⋊Dic3—61no—
310C4⋊Dic3—181no—
311Dic3⋊C4—181no—
312Dic3⋊C4—181no—
313C2×C3⋊C8—31no—
314C2×C3⋊C8—31no—
315C2×C3⋊C8—31no—
316C2×C3⋊C8—31no—
317C2×C3⋊C8—61no—
318C2×C3⋊C8—61no—
319Dic12—182no—
320C22×C32—21no—
321C22×C32—21no—
322C22×C32—21no—
323C22×C32—41no—
324C22×C32D3×D411yescharacteristic
325C22×C32D3×D411yescharacteristic
326C22×C32C2×#6711yescharacteristic
327C2×C3⋊D3—21no—
328C2×C3⋊D3—21no—
329C2×C3⋊D3#21612yescomplemented
330C2×C3×D3—32no—
331C2×C3×D3—32no—
332C2×C3×D3—62no—
333C2×C3×D3—62no—
334C2×C3×D3—62no—
335C2×C3×D3—61no—
336C2×C3×D3—61no—
337C2×C3×D3—61no—
338C2×C3×D3—61no—
339C32×C4—22no—
340C32×C4—24no—
341C32×C4—22no—
342C32×C4—22no—
343C32×C4—21no—
344C32×C4—21no—
345C32×C4—21no—
346C32×C4—21no—
347C32×C4—21no—
348C32×C4—21no—
349C32×C4—41no—
350C32×C4—41no—
351C32×C4—42no—
352C32×C4—42no—
353C32×C4—44no—
354C32×C4C2×D1211yescharacteristic
355C32×C4D3×D411yescharacteristic
356C32×C4D3×D411yescharacteristic
357C32×C4C2×#6711yescharacteristic
358C32×C4C2×#6711yescharacteristic
359C32×C4C2×#6711yescharacteristic
360C3⋊Dic3—21no—
361C3⋊Dic3—22no—
362C3⋊Dic3—41no—
363C3⋊Dic3—41no—
364C3×Dic3—62no—
365C3×Dic3—64no—
366C3×Dic3—62no—
367C3×Dic3—61no—
368C3×Dic3—62no—
369C3×Dic3—61no—
370C3×Dic3—122no—
371C3×Dic3—121no—
372C3×Dic3—121no—
373C3×Dic3—121no—
374C3×Dic3—121no—
375C3×Dic3—241no—
376C22×Q8—91no—
377C2×Q16—91no—
378C2×Q16—271no—
379C22⋊Q8—271no—
380#103—91no—
381#103—271no—
382C22⋊C8—271no—
383C33#45111yescharacteristic, complemented, Sylow
384C23×C3—91no—
385C22×D3—31no—
386C22×D3—31no—
387C22×D3—61no—
388C3×Q8—22no—
389C3×Q8—21no—
390C3×Q8—21no—
391C3×Q8—21no—
392C3×Q8—22no—
393C3×Q8—22no—
394C3×Q8—22no—
395C3×Q8—41no—
396C3×Q8—41no—
397C3×Q8—42no—
398C3×Q8—42no—
399C3×Q8—92no—
400C3×Q8—184no—
401C3×Q8—181no—
402C3×Q8—181no—
403C3×Q8—181no—
404C3×Q8#62512yes—
405C3×Q8#62612yes—
406C3×Q8#62612yes—
407C2×C3×C4—22no—
408C2×C3×C4—21no—
409C2×C3×C4—21no—
410C2×C3×C4—21no—
411C2×C3×C4—21no—
412C2×C3×C4—21no—
413C2×C3×C4—21no—
414C2×C3×C4—41no—
415C2×C3×C4—41no—
416C2×C3×C4—42no—
417C2×C3×C4—41no—
418C2×C3×C4—91no—
419C2×C3×C4—184no—
420C2×C3×C4—181no—
421C2×C3×C4—181no—
422C2×C3×C4—181no—
423C2×C3×C4—181no—
424C2×C3×C4—181no—
425C2×C3×C4C2×D3211yescharacteristic
426C2×C3×C4C2×D3211yescharacteristic
427C2×C3×C4C2×D3211yescharacteristic
428C2×Dic3—31no—
429C2×Dic3—31no—
430C2×Dic3—61no—
431C2×Dic3—62no—
432C2×Dic3—61no—
433C2×Dic3—61no—
434C2×Dic3—61no—
435C2×Dic3—61no—
436C2×Dic3—61no—
437C2×Dic3—61no—
438C2×Dic3—121no—
439C2×Dic3—121no—
440C4×D3—68no—
441C4×D3—64no—
442C4×D3—64no—
443C4×D3—62no—
444C4×D3—62no—
445C4×D3—61no—
446C4×D3—61no—
447C4×D3—61no—
448C4×D3—61no—
449Dic6—32no—
450Dic6—32no—
451Dic6—62no—
452Dic6—68no—
453Dic6—64no—
454Dic6—64no—
455Dic6—62no—
456Dic6—61no—
457Dic6—61no—
458Dic6—62no—
459Dic6—61no—
460Dic6—61no—
461Dic6—61no—
462Dic6—61no—
463Dic6—121no—
464C3×C8—181no—
465C3×C8—181no—
466C3⋊C8—62no—
467C3⋊C8—61no—
468C3⋊C8—61no—
469C3⋊C8—61no—
470C3⋊C8—61no—
471C3⋊C8—121no—
472C2×C32—21no—
473C2×C32—21no—
474C2×C32—21no—
475C2×C32—21no—
476C2×C32—21no—
477C2×C32—21no—
478C2×C32—21no—
479C2×C32—21no—
480C2×C32—21no—
481C2×C32—41no—
482C2×C32—41no—
483C2×C32—41no—
484C2×C32#88211yescharacteristic
485C2×C32#91511yescharacteristic
486C2×C32D3×Q1611yescharacteristic
487C2×C32D3×Q1611yescharacteristic
488C2×C32#91811yescharacteristic
489C2×C32#93711yescharacteristic
490C2×C32#94211yescharacteristic
491C2×C32C2×#21611yescharacteristic
492C2×C32#95311yescharacteristic
493C3⋊D3—22no—
494C3×D3—64no—
495C3×D3—62no—
496C3×D3—62no—
497C2×Q8—91no—
498C2×Q8—91no—
499C2×Q8—181no—
500C2×Q8—182no—
501C2×Q8—182no—
502C2×Q8#111311yescharacteristic
503C22×C4—91no—
504C22×C4—181no—
505Q16—182no—
506Q16—542no—
507C2×C8—91no—
508C2×C8—91no—
509C4⋊C4—91no—
510C4⋊C4—541no—
511C22⋊C4—541no—
512C22×C3—21no—
513C22×C3—21no—
514C22×C3—21no—
515C22×C3—41no—
516C22×C3—92no—
517C22×C3—181no—
518C22×C3—181no—
519C22×C3D3×#6711yescharacteristic
520C22×C3D3×#6711yescharacteristic
521C22×C3#381111yescharacteristic
522C2×D3—32no—
523C2×D3—32no—
524C2×D3—62no—
525C2×D3—62no—
526C2×D3—62no—
527C2×D3—61no—
528C2×D3—61no—
529C2×D3—61no—
530C2×D3—61no—
531C3×C4—24no—
532C3×C4—22no—
533C3×C4—22no—
534C3×C4—22no—
535C3×C4—21no—
536C3×C4—21no—
537C3×C4—21no—
538C3×C4—21no—
539C3×C4—21no—
540C3×C4—21no—
541C3×C4—42no—
542C3×C4—42no—
543C3×C4—44no—
544C3×C4—41no—
545C3×C4—41no—
546C3×C4—181no—
547C3×C4—182no—
548C3×C4—361no—
549C3×C4—361no—
550C3×C4C2×#62511yescharacteristic
551C3×C4C2×#62611yescharacteristic
552C3×C4C2×#62611yescharacteristic
553C3×C4D3×#6711yescharacteristic
554C3×C4D3×#6711yescharacteristic
555C3×C4#381111yescharacteristic
556Dic3—62no—
557Dic3—64no—
558Dic3—62no—
559Dic3—61no—
560Dic3—62no—
561Dic3—61no—
562Dic3—122no—
563Dic3—121no—
564Dic3—121no—
565Dic3—121no—
566Dic3—121no—
567Dic3—241no—
568C32—21no—
569C32—21no—
570C32—21no—
571C32—41no—
572C32#480911yescharacteristic, complemented
573C32#518611yescharacteristic, complemented
574C32#522811yescharacteristic, complemented
575C23—91no—
576Q8—21no—
577Q8—92no—
578Q8—184no—
579Q8—181no—
580Q8—181no—
581Q8—181no—
582Q8#629512yes—
583C2×C4—21no—
584C2×C4—91no—
585C2×C4—184no—
586C2×C4—181no—
587C2×C4—181no—
588C2×C4—181no—
589C2×C4—181no—
590C2×C4—181no—
591C2×C4C2×#111311yescharacteristic
592C8—181no—
593C8—181no—
594C2×C3—21no—
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—41no—
604C2×C3—41no—
605C2×C3—41no—
606C2×C3—182no—
607C2×C3D3×#21611yescharacteristic
608C2×C3D3×#21611yescharacteristic
609C2×C3#6394411yescharacteristic
610C2×C3#6394611yescharacteristic
611C2×C3#6394811yescharacteristic
612C2×C3#6395111yescharacteristic
613C2×C3#6397611yescharacteristic
614C2×C3#6397711yescharacteristic
615C2×C3#6398111yescharacteristic
616D3—64no—
617D3—62no—
618D3—62no—
619C22—92no—
620C22—181no—
621C22—181no—
622C22#8904911yescharacteristic
623C4—22no—
624C4—181no—
625C4—182no—
626C4—361no—
627C4—361no—
628C4C2×#629511yescharacteristic
629C4#8904911yescharacteristic
630C3—21no—
631C3—21no—
632C3—21no—
633C3—41no—
634C3#1059159411yescharacteristic, complemented
635C3#1059159511yescharacteristic, complemented
636C3#1059160311yescharacteristic, complemented
637C2—182no—
638C2#1172371011yescharacteristic
639C2#1172371211yescharacteristic
640C2#1172372711yescharacteristic
subgroupG/H
centerC22#89049
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×#1113
FittingC2×C33×Q8C22
socleC22×C33C2×D4
prime radicalC22×C3×C3⋊D3D4
hypercenterC2×Q8#1113
hypocenterC33#451
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
2. C2×C3×#3756C2semidirect
4. C2×#88812C2semidirect
5. C3×#63662C2semidirect
6. #11722076C2semidirect
30. C2×#3756D3semidirect
40. #63662D3semidirect
135. C2×C3×C3⋊D3Q16semidirect
329. C2×C3⋊D3#216semidirect
383. C33#451semidirect
572. C32#4809semidirect
573. C32#5186semidirect
574. C32#5228semidirect
634. C3#10591594semidirect
635. C3#10591595semidirect
636. C3#10591603semidirect

Maximal subgroups

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

subgroupindex
C2×C3×Q8×C3⋊D32
C2×C3×#37562
#117236742
C2×#888122
C3×#636622
#117220762
#117220722
#105916033
#105915953
#105915943