Finite Group Databasebeta

← #13155862

Subgroups — #13155862

Order 1728.

766 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3797C211yesmaximal, characteristic
2#11723674C211yesmaximal, characteristic
3C2×#88798C211yesmaximal, characteristic, complemented
4C2×C3×#3719C211yesmaximal, characteristic, complemented
5#11722037C211yesmaximal, characteristic, complemented
6#11722023C211yesmaximal, characteristic
7C3×#63566C211yesmaximal, characteristic, complemented
8#10591595—31nomaximal
9#10591594—31nomaximal
10#10590950—31nomaximal
11C2×C3×C4×C3⋊D3C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C2×C32×Dic6C2211yescharacteristic
14C2×C32×Dic6C2211yescharacteristic
15C2×C3×#627—21no—
16C2×C3×#624—21no—
17C3×#3797—22no—
18C3×#3797—22no—
19C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
20C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
21C2×C32×C3⋊C8C2211yescharacteristic
22#88815—21no—
23#88814—21no—
24#88798—22no—
25C3×#3719—22no—
26C2×C3×D3×Q8—31no—
27C2×C3×D3×Q8—31no—
28C2×#3797D311yescharacteristic
29C2×#3756—31no—
30C2×C3×#216—31no—
31C2×C3×Dic12—31no—
32#63871—31no—
33#63857—31no—
34#63856—31no—
35C2×#3719D311yescharacteristic, complemented
36C2×#3718—31no—
37C2×#3718—31no—
38C2×#3718—61no—
39#63662—31no—
40#63647—31no—
41C3×#835—31no—
42C3×#816—31no—
43#63566D311yescharacteristic, complemented
44#63565—31no—
45#63565—31no—
46#63565—61no—
47(C2×D3)⋊(C3⋊C8)—31no—
48(C2×D3)⋊(C3⋊C8)—31no—
49(C2×D3)⋊(C3⋊C8)—61no—
50C22×C3×C3⋊D3D411yescharacteristic
51C2×C33×C4C2311yescharacteristic
52C2×C3×C3⋊Dic3—21no—
53C2×C3×C3⋊Dic3—21no—
54C2×C3×C3⋊Dic3D411yescharacteristic
55C3×C4×C3⋊D3—21no—
56C3×C4×C3⋊D3—21no—
57C3×#634—21no—
58C3×#634—21no—
59C2×C32×Dic3—21no—
60C2×C32×Dic3—21no—
61C32×Dic6—21no—
62C32×Dic6—21no—
63C32×Dic6D412yes—
64C32×Dic6D412yes—
65C3×#627—24no—
66C3×#624—24no—
67C3⋊(C3⋊(C3⋊C8))—21no—
68C32×C3⋊C8—21no—
69#5228—91no—
70#5186—91no—
71#4809—91no—
72C2×C32×Q8—31no—
73C2×C32×Q8—31no—
74C2×C4×C3⋊D3C2×D311yescharacteristic
75C2×#634C2×D311yescharacteristic
76C3×D3×Q8—62no—
77C3×D3×Q8—62no—
78C3×D3×Q8—62no—
79C3×D3×Q8—62no—
80C2×C3×C4×D3—31no—
81C2×C3×C4×D3—31no—
82C2×C3×C4×D3—61no—
83C2×C3×C4×D3—61no—
84C2×C3×C4×D3—61no—
85C2×C3×Dic6—21no—
86C2×C3×Dic6—21no—
87C2×C3×Dic6—31no—
88C2×C3×Dic6—31no—
89C2×C3×Dic6—61no—
90C2×C3×Dic6—61no—
91C2×C3×Dic6—61no—
92C2×C3×Dic6C2×D311yescharacteristic
93C2×C3×Dic6C2×D311yescharacteristic
94C2×C3×Dic6C2×D311yescharacteristic
95C2×C3×Dic6C2×D311yescharacteristic
96C2×#627—21no—
97C2×#624—21no—
98#3797—22no—
99#3797—22no—
100C2×C32×C8—31no—
101#3756—62no—
102C4⋊(C3⋊Dic3)C2×D311yescharacteristic
103C2×C3⋊(C3⋊C8)—31no—
104C2×C3⋊(C3⋊C8)—31no—
105C2×C3⋊(C3⋊C8)—31no—
106C2×C3⋊(C3⋊C8)—61no—
107C2×C3⋊(C3⋊C8)—61no—
108C2×C3⋊(C3⋊C8)—61no—
109C2×C3⋊(C3⋊C8)—121no—
110C3×#216—62no—
111C3×C4⋊Dic3—31no—
112C3×C4⋊Dic3—31no—
113C3×C4⋊Dic3—61no—
114C2×C3×C3⋊C8—21no—
115C2×C3×C3⋊C8C2×D311yescharacteristic
116C2×C3×C3⋊C8C2×D311yescharacteristic
117C3×Dic12—62no—
118#3724—61no—
119Dic3⋊Dic3—61no—
120Dic3⋊Dic3—61no—
121#3722—61no—
122(C2×D3)⋊Dic3—61no—
123(C2×D3)⋊Dic3—61no—
124#3719—22no—
125#3718—64no—
126#3718—62no—
127#3718—62no—
128C22×C33C2×D411yescharacteristic
129C2×C3×C3⋊D3—21no—
130C2×C3×C3⋊D3—21no—
131C2×C3×C3⋊D3Q1612yescomplemented
132C33×C4C2×D411yescharacteristic
133C33×C4C2×D411yescharacteristic
134C3×C3⋊Dic3—21no—
135C3×C3⋊Dic3—41no—
136C3×C3⋊Dic3—41no—
137C32×Dic3—22no—
138C32×Dic3—22no—
139C22×C3×Q8—91no—
140C2×D3×Q8—31no—
141C2×D3×Q8—31no—
142C2×C3×Q16—91no—
143#946—91no—
144C2×#216—31no—
145C2×#216—91no—
146C2×#216—91no—
147C2×#216—91no—
148C2×#216—181no—
149#924—91no—
150C2×Dic12—31no—
151#897—91no—
152C3×#103—91no—
153#835—31no—
154#835—91no—
155#835—91no—
156#835—181no—
157#830—91no—
158(C2×D3)⋊C8—91no—
159(C2×D3)⋊C8—91no—
160(C2×D3)⋊C8—181no—
161#816—31no—
162#810—91no—
163C22×C3⋊D3#6711yescharacteristic
164C22×C3×D3—31no—
165C22×C3×D3—31no—
166C22×C3×D3—61no—
167C32×Q8—32no—
168C32×Q8—32no—
169C32×Q8—61no—
170C32×Q8—61no—
171C2×C32×C4—21no—
172C2×C32×C4—21no—
173C2×C32×C4—21no—
174C2×C32×C4—41no—
175C2×C32×C4—61no—
176C2×C32×C4—61no—
177C2×C32×C4C22×D311yescharacteristic
178C2×C32×C4C22×D311yescharacteristic
179C2×C32×C4C22×D311yescharacteristic
180C2×C3⋊Dic3—21no—
181C2×C3⋊Dic3—21no—
182C2×C3⋊Dic3#6711yescharacteristic
183C4×C3⋊D3—21no—
184C4×C3⋊D3—21no—
185#634—21no—
186#634—21no—
187C2×C3×Dic3—21no—
188C2×C3×Dic3—21no—
189C2×C3×Dic3—21no—
190C2×C3×Dic3—21no—
191C2×C3×Dic3—31no—
192C2×C3×Dic3—31no—
193C2×C3×Dic3—41no—
194C2×C3×Dic3—41no—
195C2×C3×Dic3—61no—
196C2×C3×Dic3—61no—
197C2×C3×Dic3—61no—
198C2×C3×Dic3—61no—
199C2×C3×Dic3—61no—
200C2×C3×Dic3—121no—
201C2×C3×Dic3—121no—
202C3×C4×D3—64no—
203C3×C4×D3—64no—
204C3×C4×D3—61no—
205C3×C4×D3—61no—
206C3×C4×D3—61no—
207C3×C4×D3—61no—
208C3×C4×D3—121no—
209C3×C4×D3—121no—
210C3×Dic6—21no—
211C3×Dic6—21no—
212C3×Dic6—21no—
213C3×Dic6—21no—
214C3×Dic6—22no—
215C3×Dic6—22no—
216C3×Dic6—41no—
217C3×Dic6—41no—
218C3×Dic6—64no—
219C3×Dic6—64no—
220C3×Dic6—62no—
221C3×Dic6—61no—
222C3×Dic6—61no—
223C3×Dic6—61no—
224C3×Dic6—61no—
225C3×Dic6—121no—
226C3×Dic6D1212yes—
227C3×Dic6#6712yes—
228C3×Dic6#6712yes—
229C3×Dic6#6712yes—
230#627—24no—
231#624—24no—
232C32×C8—61no—
233C3⋊(C3⋊C8)—61no—
234C3⋊(C3⋊C8)—62no—
235C3⋊(C3⋊C8)—61no—
236C3⋊(C3⋊C8)—61no—
237C3⋊(C3⋊C8)—121no—
238C3⋊(C3⋊C8)—121no—
239C3⋊(C3⋊C8)—122no—
240C3×C3⋊C8—21no—
241C3×C3⋊C8—21no—
242C3×C3⋊C8—41no—
243#451—271noSylow
244C2×C33#12011yescharacteristic
245C2×C33C2×Q1611yescharacteristic
246C2×C33#13711yescharacteristic
247C3×C3⋊D3—22no—
248C2×C3×Q8—31no—
249C2×C3×Q8—31no—
250C2×C3×Q8—31no—
251C2×C3×Q8—31no—
252C2×C3×Q8—61no—
253C2×C3×Q8—61no—
254C2×C3×Q8—91no—
255C2×C3×Q8—181no—
256C2×C3×Q8—182no—
257C2×C3×Q8—182no—
258C22×C3×C4—91no—
259C22×C3×C4—181no—
260D3×Q8—62no—
261D3×Q8—62no—
262D3×Q8—62no—
263D3×Q8—62no—
264C2×C4×D3—31no—
265C2×C4×D3—31no—
266C2×C4×D3—61no—
267C2×C4×D3—61no—
268C2×C4×D3—61no—
269C2×Dic6—31no—
270C2×Dic6—31no—
271C2×Dic6—61no—
272C2×Dic6—61no—
273C2×Dic6—61no—
274C2×Dic6D3211yescharacteristic
275C2×Dic6D3211yescharacteristic
276C3×Q16—182no—
277C2×C3×C8—31no—
278C2×C3×C8—31no—
279C2×C3×C8—61no—
280C3×C4⋊C4—91no—
281#217—181no—
282#216—62no—
283#216—182no—
284#216—182no—
285#216—184no—
286#216—182no—
287(C2×D3)⋊C4—181no—
288(C2×D3)⋊C4—181no—
289C4⋊Dic3—31no—
290C4⋊Dic3—31no—
291C4⋊Dic3—61no—
292C4⋊Dic3—181no—
293Dic3⋊C4—181no—
294Dic3⋊C4—181no—
295C2×C3⋊C8—91no—
296C2×C3⋊C8—91no—
297C2×C3⋊C8—91no—
298C2×C3⋊C8—181no—
299C2×C3⋊C8—181no—
300C2×C3⋊C8—181no—
301C2×C3⋊C8—361no—
302C2×C3⋊C8D3211yescharacteristic
303Dic12—62no—
304C22×C32—21no—
305C22×C32—21no—
306C22×C32—21no—
307C22×C32—41no—
308C22×C32D3×D411yescharacteristic
309C22×C32D3×D411yescharacteristic
310C22×C32C2×#6711yescharacteristic
311C2×C3⋊D3—21no—
312C2×C3⋊D3—21no—
313C2×C3⋊D3#21612yescomplemented
314C2×C3×D3—32no—
315C2×C3×D3—32no—
316C2×C3×D3—62no—
317C2×C3×D3—61no—
318C2×C3×D3—61no—
319C2×C3×D3—61no—
320C2×C3×D3—61no—
321C2×C3×D3—121no—
322C2×C3×D3—121no—
323C32×C4—21no—
324C32×C4—21no—
325C32×C4—21no—
326C32×C4—21no—
327C32×C4—21no—
328C32×C4—21no—
329C32×C4—41no—
330C32×C4—41no—
331C32×C4—62no—
332C32×C4—62no—
333C32×C4C2×D1211yescharacteristic
334C32×C4D3×D411yescharacteristic
335C32×C4D3×D411yescharacteristic
336C32×C4C2×#6711yescharacteristic
337C32×C4C2×#6711yescharacteristic
338C32×C4C2×#6711yescharacteristic
339C3⋊Dic3—21no—
340C3⋊Dic3—41no—
341C3⋊Dic3—41no—
342C3×Dic3—22no—
343C3×Dic3—22no—
344C3×Dic3—22no—
345C3×Dic3—22no—
346C3×Dic3—42no—
347C3×Dic3—42no—
348C3×Dic3—61no—
349C3×Dic3—61no—
350C3×Dic3—121no—
351C3×Dic3—122no—
352C3×Dic3—121no—
353C3×Dic3—121no—
354C3×Dic3—121no—
355C3×Dic3—121no—
356C3×Dic3—241no—
357C22×Q8—91no—
358C2×Q16—91no—
359C2×Q16—271no—
360C22⋊Q8—271no—
361#103—91no—
362#103—271no—
363C22⋊C8—271no—
364C33#45111yescharacteristic, complemented, Sylow
365C23×C3—91no—
366C22×D3—31no—
367C22×D3—31no—
368C22×D3—61no—
369C3×Q8—32no—
370C3×Q8—32no—
371C3×Q8—32no—
372C3×Q8—32no—
373C3×Q8—62no—
374C3×Q8—62no—
375C3×Q8—61no—
376C3×Q8—61no—
377C3×Q8—61no—
378C3×Q8—61no—
379C3×Q8—121no—
380C3×Q8—121no—
381C3×Q8—184no—
382C3×Q8—181no—
383C3×Q8—181no—
384C2×C3×C4—21no—
385C2×C3×C4—21no—
386C2×C3×C4—21no—
387C2×C3×C4—41no—
388C2×C3×C4—61no—
389C2×C3×C4—61no—
390C2×C3×C4—61no—
391C2×C3×C4—61no—
392C2×C3×C4—91no—
393C2×C3×C4—121no—
394C2×C3×C4—121no—
395C2×C3×C4—184no—
396C2×C3×C4—181no—
397C2×C3×C4—181no—
398C2×C3×C4—181no—
399C2×C3×C4—181no—
400C2×C3×C4C2×D3211yescharacteristic
401C2×C3×C4C2×D3211yescharacteristic
402C2×C3×C4C2×D3211yescharacteristic
403C2×Dic3—21no—
404C2×Dic3—21no—
405C2×Dic3—31no—
406C2×Dic3—31no—
407C2×Dic3—61no—
408C2×Dic3—61no—
409C2×Dic3—61no—
410C2×Dic3—61no—
411C2×Dic3—61no—
412C2×Dic3—121no—
413C2×Dic3—121no—
414C4×D3—64no—
415C4×D3—64no—
416C4×D3—61no—
417C4×D3—61no—
418C4×D3—61no—
419C4×D3—61no—
420C4×D3—121no—
421C4×D3—121no—
422Dic6—21no—
423Dic6—21no—
424Dic6—64no—
425Dic6—64no—
426Dic6—62no—
427Dic6—61no—
428Dic6—61no—
429Dic6—61no—
430Dic6—61no—
431Dic6—121no—
432Dic6#62512yes—
433Dic6#62612yes—
434C3×C8—61no—
435C3×C8—61no—
436C3×C8—121no—
437C3⋊C8—21no—
438C3⋊C8—182no—
439C3⋊C8—181no—
440C3⋊C8—181no—
441C3⋊C8—181no—
442C3⋊C8—361no—
443C3⋊C8—361no—
444C3⋊C8—362no—
445C2×C32—21no—
446C2×C32—21no—
447C2×C32—21no—
448C2×C32—21no—
449C2×C32—21no—
450C2×C32—21no—
451C2×C32—21no—
452C2×C32—21no—
453C2×C32—21no—
454C2×C32—41no—
455C2×C32—41no—
456C2×C32—41no—
457C2×C32#88211yescharacteristic
458C2×C32#91511yescharacteristic
459C2×C32D3×Q1611yescharacteristic
460C2×C32D3×Q1611yescharacteristic
461C2×C32#91811yescharacteristic
462C2×C32#93711yescharacteristic
463C2×C32#94211yescharacteristic
464C2×C32C2×#21611yescharacteristic
465C2×C32#95311yescharacteristic
466C3⋊D3—22no—
467C3×D3—62no—
468C3×D3—62no—
469C3×D3—122no—
470C2×Q8—31no—
471C2×Q8—31no—
472C2×Q8—91no—
473C2×Q8—181no—
474C2×Q8—182no—
475C2×Q8—182no—
476C22×C4—91no—
477C22×C4—181no—
478Q16—182no—
479Q16—542no—
480C2×C8—31no—
481C2×C8—271no—
482C4⋊C4—91no—
483C4⋊C4—541no—
484C22⋊C4—541no—
485C22×C3—21no—
486C22×C3—21no—
487C22×C3—21no—
488C22×C3—41no—
489C22×C3—92no—
490C22×C3—181no—
491C22×C3—181no—
492C22×C3#380211yescharacteristic
493C22×C3D3×#6711yescharacteristic
494C22×C3D3×#6711yescharacteristic
495C2×D3—32no—
496C2×D3—32no—
497C2×D3—62no—
498C2×D3—61no—
499C2×D3—61no—
500C2×D3—61no—
501C2×D3—61no—
502C2×D3—121no—
503C2×D3—121no—
504C3×C4—21no—
505C3×C4—21no—
506C3×C4—21no—
507C3×C4—21no—
508C3×C4—21no—
509C3×C4—21no—
510C3×C4—41no—
511C3×C4—41no—
512C3×C4—62no—
513C3×C4—62no—
514C3×C4—62no—
515C3×C4—62no—
516C3×C4—122no—
517C3×C4—122no—
518C3×C4—181no—
519C3×C4—361no—
520C3×C4—361no—
521C3×C4D3×D1211yescharacteristic
522C3×C4C2×#62511yescharacteristic
523C3×C4C2×#62611yescharacteristic
524C3×C4D3×#6711yescharacteristic
525C3×C4D3×#6711yescharacteristic
526C3×C4D3×#6711yescharacteristic
527Dic3—22no—
528Dic3—22no—
529Dic3—61no—
530Dic3—61no—
531Dic3—121no—
532Dic3—122no—
533Dic3—121no—
534Dic3—121no—
535Dic3—121no—
536Dic3—121no—
537Dic3—241no—
538C32—21no—
539C32—21no—
540C32—21no—
541C32—41no—
542C32#480911yescharacteristic, complemented
543C32#518611yescharacteristic, complemented
544C32#522811yescharacteristic, complemented
545C23—91no—
546Q8—32no—
547Q8—32no—
548Q8—61no—
549Q8—61no—
550Q8—184no—
551Q8—181no—
552Q8—181no—
553C2×C4—61no—
554C2×C4—61no—
555C2×C4—91no—
556C2×C4—184no—
557C2×C4—181no—
558C2×C4—181no—
559C2×C4—181no—
560C2×C4—181no—
561C2×C4D3311yescharacteristic
562C8—61no—
563C8—541no—
564C2×C3—21no—
565C2×C3—21no—
566C2×C3—21no—
567C2×C3—21no—
568C2×C3—21no—
569C2×C3—21no—
570C2×C3—21no—
571C2×C3—21no—
572C2×C3—21no—
573C2×C3—41no—
574C2×C3—41no—
575C2×C3—41no—
576C2×C3—182no—
577C2×C3#6380211yescharacteristic
578C2×C3#6381111yescharacteristic
579C2×C3#6392211yescharacteristic
580C2×C3D3×#21611yescharacteristic
581C2×C3D3×#21611yescharacteristic
582C2×C3#6394611yescharacteristic
583C2×C3#6394811yescharacteristic
584C2×C3#6397611yescharacteristic
585C2×C3#6397711yescharacteristic
586D3—62no—
587D3—62no—
588D3—122no—
589C22—92no—
590C22—181no—
591C22—181no—
592C22#8895311yescharacteristic
593C4—62no—
594C4—62no—
595C4—181no—
596C4—361no—
597C4—361no—
598C4D3×#62511yescharacteristic
599C4D3×#62611yescharacteristic
600C3—21no—
601C3—21no—
602C3—21no—
603C3—41no—
604C3#1059095011yescharacteristic, complemented
605C3#1059159411yescharacteristic, complemented
606C3#1059159511yescharacteristic, complemented
607C2—182no—
608C2#1172284711yescharacteristic
609C2#1172289211yescharacteristic
610C2#1172294711yescharacteristic
subgroupG/H
centerC22#88953
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radicalC22×C3×C3⋊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×#88798C2semidirect
4. C2×C3×#3719C2semidirect
5. #11722037C2semidirect
7. C3×#63566C2semidirect
35. C2×#3719D3semidirect
43. #63566D3semidirect
131. C2×C3×C3⋊D3Q16semidirect
313. C2×C3⋊D3#216semidirect
364. C33#451semidirect
542. C32#4809semidirect
543. C32#5186semidirect
544. C32#5228semidirect
604. C3#10590950semidirect
605. C3#10591594semidirect
606. C3#10591595semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37972
#117236742
C2×#887982
C2×C3×#37192
#117220372
#117220232
C3×#635662
#105915953
#105915943
#105909503