Finite Group Databasebeta

← #13155883

Subgroups — #13155883

Order 1728.

848 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3796C211yesmaximal, characteristic
2#11723431C211yesmaximal, characteristic, complemented
3C2×#88799C211yesmaximal, characteristic, complemented
4C2×#88794C211yesmaximal, characteristic, complemented
5#11722072C211yesmaximal, characteristic, complemented
6C3×#63565C211yesmaximal, characteristic, complemented
7C3×#63562C211yesmaximal, characteristic, complemented
8#10591599—31nomaximal
9#10591463—31nomaximal
10#10590618—31nomaximal
11C2×#6308C2211yescharacteristic
12C2×C3×C4×C3⋊D3C2211yescharacteristic
13C2×C32×D12C2211yescharacteristic
14C2×C32×Dic6C2211yescharacteristic
15C2×C3×#626—21no—
16C2×C3×D3×Dic3—21no—
17C3×#3796—22no—
18C3×#3796—22no—
19C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
20C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
21C32×C4⋊Dic3C2211yescharacteristic
22Dic3⋊(C3⋊Dic3)—21no—
23#88803—21no—
24#88799—22no—
25#88794—22no—
26C2×C3×#239—31no—
27C2×C3×#237—31no—
28C2×#3796D311yescharacteristic
29C2×#3756—31no—
30C2×#3754—31no—
31#64106—31no—
32#63857—31no—
33#63857—31no—
34#63857—61no—
35C2×#3719—31no—
36C2×#3719—31no—
37C2×#3719—61no—
38C2×#3716—31no—
39C2×#3716—31no—
40C2×#3716—61no—
41C3×#835—31no—
42C3×#832—31no—
43C3×#810—31no—
44C3×#809—31no—
45#63565D311yescharacteristic, complemented
46#63562D311yescharacteristic, complemented
47#63557—31no—
48(C2×D3)⋊(C3⋊C8)—31no—
49(C2×D3)⋊(C3⋊C8)—31no—
50C22×C3×C3⋊D3D411yescharacteristic
51C22×C32×D3—21no—
52C2×C33×C4C2311yescharacteristic
53C2×C3⋊(C3⋊Dic3)—21no—
54#6308—21no—
55#6308—21no—
56C2×C3×C3⋊Dic3D411yescharacteristic
57C3×C4×C3⋊D3—21no—
58C3×C4×C3⋊D3—21no—
59C2×C32×Dic3—21no—
60C2×C32×Dic3—21no—
61C32×D12—21no—
62C32×D12D412yes—
63C32×Dic6—21no—
64C32×Dic6D412yes—
65C3×#626—24no—
66C3×D3×Dic3—24no—
67C3×C3⋊(C3⋊C8)—21no—
68C3×C3⋊(C3⋊C8)—21no—
69#5239—91no—
70#4808—91no—
71#4783—91no—
72C2×C32×Q8—31no—
73C2×C32×D4—31no—
74C2×C4×C3⋊D3C2×D311yescharacteristic
75C2×#634—31no—
76C2×#634—31no—
77C2×#634—31no—
78C2×#634—61no—
79C2×#634—61no—
80C2×#634—61no—
81C2×#634—121no—
82C2×C3×#67—61no—
83C22×C3×Dic3—61no—
84C3×#239—62no—
85C3×#239—62no—
86C3×#237—62no—
87C3×#237—62no—
88C2×C3×D12—21no—
89C2×C3×D12—61no—
90C2×C3×D12C2×D311yescharacteristic
91C2×C3×D12C2×D311yescharacteristic
92C2×C3×C4×D3—31no—
93C2×C3×C4×D3—31no—
94C2×C3×C4×D3—61no—
95C2×C3×C4×D3—61no—
96C2×C3×Dic6—21no—
97C2×C3×Dic6C2×D311yescharacteristic
98C2×C3×Dic6C2×D311yescharacteristic
99C2×#626—21no—
100C2×D3×Dic3—21no—
101#3796—22no—
102#3796—22no—
103C32×C4⋊C4—31no—
104#3756—62no—
105#3754—62no—
106#3752—61no—
107C4⋊(C3⋊Dic3)—61no—
108C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
109C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
110C3×C4⋊Dic3—21no—
111C3×C4⋊Dic3C2×D311yescharacteristic
112C3×C4⋊Dic3C2×D311yescharacteristic
113C2×C3×C3⋊C8—31no—
114C2×C3×C3⋊C8—31no—
115C2×C3×C3⋊C8—31no—
116C2×C3×C3⋊C8—31no—
117C2×C3×C3⋊C8—61no—
118C2×C3×C3⋊C8—61no—
119Dic3⋊Dic3—61no—
120Dic3⋊Dic3—61no—
121Dic3⋊Dic3—62no—
122(C2×D3)⋊Dic3—61no—
123(C2×D3)⋊Dic3—61no—
124(C2×D3)⋊Dic3—62no—
125#3719—64no—
126#3719—62no—
127#3719—62no—
128#3716—64no—
129#3716—62no—
130#3716—62no—
131C22×C33C2×D411yescharacteristic
132C2×C3×C3⋊D3—21no—
133C2×C3×C3⋊D3—21no—
134C2×C3×C3⋊D3—21no—
135C2×C32×D3—22no—
136C2×C32×D3—41no—
137C2×C32×D3—41no—
138C33×C4C2×D411yescharacteristic
139C33×C4C2×D411yescharacteristic
140C3⋊(C3⋊Dic3)—41no—
141C3×C3⋊Dic3—21no—
142C32×Dic3—22no—
143C32×Dic3—41no—
144C2×C3×#41—91no—
145C2×#239—31no—
146C2×#237—31no—
147C2×#216—91no—
148C2×#216—91no—
149C2×#216—181no—
150C2×#214—91no—
151C2×#214—91no—
152C2×#214—181no—
153#924—91no—
154C2×Dic12—91no—
155C2×#204—91no—
156#897—91no—
157#897—91no—
158#897—181no—
159C3×#103—91no—
160C3×#102—91no—
161#835—31no—
162#832—31no—
163#830—91no—
164(C2×D3)⋊C8—91no—
165(C2×D3)⋊C8—91no—
166#810—31no—
167#809—31no—
168C23×C32—61no—
169C22×C3⋊D3#6711yescharacteristic
170C22×C3×D3—21no—
171C22×C3×D3—21no—
172C22×C3×D3—31no—
173C22×C3×D3—31no—
174C22×C3×D3—41no—
175C22×C3×D3—61no—
176C32×Q8—32no—
177C32×Q8—61no—
178C32×D4—32no—
179C32×D4—61no—
180C2×C32×C4—21no—
181C2×C32×C4—21no—
182C2×C32×C4—21no—
183C2×C32×C4—41no—
184C2×C32×C4—61no—
185C2×C32×C4—61no—
186C2×C32×C4C22×D311yescharacteristic
187C2×C32×C4C22×D311yescharacteristic
188C2×C32×C4C22×D311yescharacteristic
189C2×C3⋊Dic3—61no—
190C2×C3⋊Dic3—62no—
191C2×C3⋊Dic3—61no—
192C2×C3⋊Dic3—61no—
193C2×C3⋊Dic3—121no—
194C2×C3⋊Dic3—121no—
195C2×C3⋊Dic3—122no—
196C2×C3⋊Dic3#6711yescharacteristic
197C4×C3⋊D3—21no—
198C4×C3⋊D3—21no—
199#634—61no—
200#634—61no—
201#634—62no—
202#634—62no—
203#634—61no—
204#634—61no—
205#634—61no—
206#634—61no—
207#634—121no—
208#634—121no—
209#634—121no—
210#634—121no—
211#634—122no—
212#634—122no—
213C3×#67—64no—
214C2×C3×Dic3—21no—
215C2×C3×Dic3—21no—
216C2×C3×Dic3—22no—
217C2×C3×Dic3—21no—
218C2×C3×Dic3—21no—
219C2×C3×Dic3—31no—
220C2×C3×Dic3—31no—
221C2×C3×Dic3—41no—
222C2×C3×Dic3—61no—
223C2×C3×Dic3—64no—
224C3×D12—21no—
225C3×D12—21no—
226C3×D12—22no—
227C3×D12—41no—
228C3×D12—64no—
229C3×D12D1212yes—
230C3×D12#6712yes—
231C3×C4×D3—64no—
232C3×C4×D3—61no—
233C3×C4×D3—61no—
234C3×C4×D3—61no—
235C3×C4×D3—61no—
236C3×C4×D3—121no—
237C3×C4×D3—121no—
238C3×Dic6—21no—
239C3×Dic6—21no—
240C3×Dic6—22no—
241C3×Dic6—41no—
242C3×Dic6D1212yes—
243C3×Dic6#6712yes—
244#626—24no—
245D3×Dic3—24no—
246C3⋊(C3⋊C8)—21no—
247C3⋊(C3⋊C8)—21no—
248C3×C3⋊C8—62no—
249C3×C3⋊C8—62no—
250C3×C3⋊C8—61no—
251C3×C3⋊C8—61no—
252C3×C3⋊C8—61no—
253C3×C3⋊C8—61no—
254#452—271noSylow
255C2×C33#12011yescharacteristic
256C2×C33#13511yescharacteristic
257C2×C33#13711yescharacteristic
258C3×C3⋊D3—41no—
259C32×D3—42no—
260C3×#41—182no—
261C3×#41—182no—
262C2×C3×Q8—31no—
263C2×C3×Q8—31no—
264C2×C3×Q8—61no—
265C2×C3×D4—31no—
266C2×C3×D4—31no—
267C2×C3×D4—61no—
268C2×C3×D4—181no—
269C22×C3×C4—91no—
270C22×C3×C4—181no—
271C2×#67—61no—
272C22×Dic3—61no—
273#239—62no—
274#239—62no—
275#237—62no—
276#237—62no—
277C2×D12—61no—
278C2×D12D3211yescharacteristic
279C2×C4×D3—31no—
280C2×C4×D3—31no—
281C2×C4×D3—61no—
282C2×C4×D3—61no—
283C2×Dic6—91no—
284C2×Dic6—91no—
285C2×Dic6—91no—
286C2×Dic6—181no—
287C2×Dic6—181no—
288C2×Dic6—181no—
289C2×Dic6—361no—
290C2×Dic6D3211yescharacteristic
291C2×C3×C8—91no—
292C2×C3×C8—91no—
293C3×C4⋊C4—31no—
294C3×C4⋊C4—31no—
295C3×C4⋊C4—61no—
296#217—181no—
297#216—182no—
298#216—182no—
299#216—184no—
300#214—182no—
301#214—182no—
302#214—184no—
303(C2×D3)⋊C4—181no—
304(C2×D3)⋊C4—181no—
305(C2×D3)⋊C4—182no—
306C4⋊Dic3—181no—
307C4⋊Dic3—181no—
308C4⋊Dic3—182no—
309C4⋊Dic3D3211yescharacteristic
310Dic3⋊C4—181no—
311C2×C3⋊C8—31no—
312C2×C3⋊C8—31no—
313C2×C3⋊C8—31no—
314C2×C3⋊C8—31no—
315C2×C3⋊C8—61no—
316C2×C3⋊C8—61no—
317Dic12—182no—
318#204—182no—
319C22×C32—21no—
320C22×C32—21no—
321C22×C32—21no—
322C22×C32—41no—
323C22×C32—62no—
324C22×C32—121no—
325C22×C32—121no—
326C22×C32D3×D411yescharacteristic
327C22×C32D3×D411yescharacteristic
328C22×C32C2×#6711yescharacteristic
329C2×C3⋊D3—21no—
330C2×C3⋊D3—21no—
331C2×C3⋊D3—21no—
332C2×C3×D3—22no—
333C2×C3×D3—22no—
334C2×C3×D3—42no—
335C2×C3×D3—41no—
336C2×C3×D3—41no—
337C2×C3×D3—41no—
338C2×C3×D3—41no—
339C2×C3×D3—62no—
340C2×C3×D3—61no—
341C2×C3×D3—61no—
342C2×C3×D3—61no—
343C2×C3×D3—61no—
344C2×C3×D3—61no—
345C2×C3×D3—61no—
346C2×C3×D3—81no—
347C2×C3×D3—81no—
348C2×C3×D3—121no—
349C2×C3×D3—121no—
350C32×C4—21no—
351C32×C4—21no—
352C32×C4—21no—
353C32×C4—21no—
354C32×C4—21no—
355C32×C4—21no—
356C32×C4—41no—
357C32×C4—41no—
358C32×C4—62no—
359C32×C4—121no—
360C32×C4C2×D1211yescharacteristic
361C32×C4C2×D1211yescharacteristic
362C32×C4D3×D411yescharacteristic
363C32×C4D3×D411yescharacteristic
364C32×C4C2×#6711yescharacteristic
365C32×C4C2×#6711yescharacteristic
366C3⋊Dic3—21no—
367C3⋊Dic3—122no—
368C3⋊Dic3—122no—
369C3⋊Dic3—122no—
370C3⋊Dic3—124no—
371C3⋊Dic3—121no—
372C3⋊Dic3—121no—
373C3⋊Dic3—121no—
374C3×Dic3—22no—
375C3×Dic3—22no—
376C3×Dic3—42no—
377C3×Dic3—42no—
378C3×Dic3—41no—
379C3×Dic3—41no—
380C3×Dic3—61no—
381C3×Dic3—61no—
382C3×Dic3—121no—
383C2×#41—91no—
384C2×Q16—271no—
385C2×SD16—271no—
386C22⋊Q8—271no—
387#103—91no—
388#102—91no—
389C22⋊C8—271no—
390C33#45211yescharacteristic, complemented, Sylow
391C23×C3—61no—
392C23×C3—61no—
393C23×C3—91no—
394C23×C3—121no—
395C22×D3—21no—
396C22×D3—31no—
397C22×D3—31no—
398C22×D3—61no—
399C3×Q8—32no—
400C3×Q8—32no—
401C3×Q8—62no—
402C3×Q8—61no—
403C3×Q8—61no—
404C3×Q8—121no—
405C3×D4—32no—
406C3×D4—32no—
407C3×D4—62no—
408C3×D4—61no—
409C3×D4—61no—
410C3×D4—121no—
411C3×D4—184no—
412C2×C3×C4—21no—
413C2×C3×C4—21no—
414C2×C3×C4—21no—
415C2×C3×C4—41no—
416C2×C3×C4—61no—
417C2×C3×C4—61no—
418C2×C3×C4—62no—
419C2×C3×C4—61no—
420C2×C3×C4—61no—
421C2×C3×C4—91no—
422C2×C3×C4—121no—
423C2×C3×C4—184no—
424C2×C3×C4—181no—
425C2×C3×C4—181no—
426C2×C3×C4C2×D3211yescharacteristic
427C2×C3×C4C2×D3211yescharacteristic
428C2×C3×C4C2×D3211yescharacteristic
429#67—64no—
430C2×Dic3—21no—
431C2×Dic3—21no—
432C2×Dic3—31no—
433C2×Dic3—31no—
434C2×Dic3—61no—
435C2×Dic3—64no—
436C2×Dic3—182no—
437C2×Dic3—181no—
438C2×Dic3—181no—
439C2×Dic3—181no—
440C2×Dic3—361no—
441C2×Dic3—361no—
442C2×Dic3—362no—
443D12—21no—
444D12—64no—
445D12#62612yes—
446C4×D3—64no—
447C4×D3—61no—
448C4×D3—61no—
449C4×D3—61no—
450C4×D3—61no—
451C4×D3—121no—
452C4×D3—121no—
453Dic6—21no—
454Dic6—182no—
455Dic6—182no—
456Dic6—181no—
457Dic6—181no—
458Dic6—181no—
459Dic6—181no—
460Dic6—181no—
461Dic6—181no—
462Dic6—361no—
463Dic6—361no—
464Dic6—361no—
465Dic6—361no—
466Dic6—362no—
467Dic6—362no—
468Dic6#62612yes—
469C3×C8—181no—
470C3×C8—181no—
471C3⋊C8—62no—
472C3⋊C8—62no—
473C3⋊C8—61no—
474C3⋊C8—61no—
475C3⋊C8—61no—
476C3⋊C8—61no—
477C2×C32—21no—
478C2×C32—21no—
479C2×C32—21no—
480C2×C32—21no—
481C2×C32—21no—
482C2×C32—21no—
483C2×C32—21no—
484C2×C32—21no—
485C2×C32—21no—
486C2×C32—41no—
487C2×C32—41no—
488C2×C32—41no—
489C2×C32—122no—
490C2×C32#88211yescharacteristic
491C2×C32#88211yescharacteristic
492C2×C32#91211yescharacteristic
493C2×C32#91511yescharacteristic
494C2×C32#91611yescharacteristic
495C2×C32#91811yescharacteristic
496C2×C32#95011yescharacteristic
497C2×C32#95111yescharacteristic
498C2×C32#95311yescharacteristic
499C3⋊D3—41no—
500C3×D3—42no—
501C3×D3—42no—
502C3×D3—82no—
503C3×D3—122no—
504C3×D3—121no—
505C3×D3—121no—
506#41—182no—
507#41—182no—
508C2×Q8—31no—
509C2×Q8—271no—
510C2×D4—31no—
511C2×D4—181no—
512C22×C4—91no—
513C22×C4—181no—
514Q16—542no—
515SD16—542no—
516C2×C8—91no—
517C2×C8—91no—
518C4⋊C4—31no—
519C4⋊C4—541no—
520C22⋊C4—541no—
521C22×C3—21no—
522C22×C3—21no—
523C22×C3—21no—
524C22×C3—41no—
525C22×C3—62no—
526C22×C3—62no—
527C22×C3—122no—
528C22×C3—121no—
529C22×C3—121no—
530C22×C3—121no—
531C22×C3—121no—
532C22×C3—181no—
533C22×C3—181no—
534C22×C3—181no—
535C22×C3—241no—
536C22×C3—241no—
537C22×C3D3×#6711yescharacteristic
538C22×C3D3×#6711yescharacteristic
539C22×C3#381111yescharacteristic
540C2×D3—22no—
541C2×D3—41no—
542C2×D3—41no—
543C2×D3—62no—
544C2×D3—61no—
545C2×D3—61no—
546C2×D3—61no—
547C2×D3—61no—
548C2×D3—61no—
549C2×D3—61no—
550C2×D3—121no—
551C2×D3—121no—
552C3×C4—21no—
553C3×C4—21no—
554C3×C4—21no—
555C3×C4—21no—
556C3×C4—21no—
557C3×C4—21no—
558C3×C4—41no—
559C3×C4—41no—
560C3×C4—62no—
561C3×C4—62no—
562C3×C4—122no—
563C3×C4—122no—
564C3×C4—121no—
565C3×C4—121no—
566C3×C4—181no—
567C3×C4D3×D1211yescharacteristic
568C3×C4D3×D1211yescharacteristic
569C3×C4C2×#62611yescharacteristic
570C3×C4C2×#62611yescharacteristic
571C3×C4D3×#6711yescharacteristic
572C3×C4D3×#6711yescharacteristic
573Dic3—22no—
574Dic3—41no—
575Dic3—61no—
576Dic3—61no—
577Dic3—121no—
578Dic3—362no—
579Dic3—362no—
580Dic3—362no—
581Dic3—364no—
582Dic3—361no—
583Dic3—361no—
584Dic3—361no—
585C32—21no—
586C32—21no—
587C32—21no—
588C32—41no—
589C32#478311yescharacteristic, complemented
590C32#480811yescharacteristic, complemented
591C32#523911yescharacteristic, complemented
592C23—61no—
593C23—91no—
594Q8—32no—
595Q8—61no—
596Q8—541no—
597Q8—541no—
598D4—32no—
599D4—61no—
600D4—184no—
601C2×C4—61no—
602C2×C4—61no—
603C2×C4—91no—
604C2×C4—184no—
605C2×C4—181no—
606C2×C4—181no—
607C2×C4—541no—
608C2×C4D3311yescharacteristic
609C8—181no—
610C8—181no—
611C2×C3—21no—
612C2×C3—21no—
613C2×C3—21no—
614C2×C3—21no—
615C2×C3—21no—
616C2×C3—21no—
617C2×C3—21no—
618C2×C3—21no—
619C2×C3—21no—
620C2×C3—41no—
621C2×C3—41no—
622C2×C3—41no—
623C2×C3—122no—
624C2×C3—122no—
625C2×C3—242no—
626C2×C3—361no—
627C2×C3#6392311yescharacteristic
628C2×C3#6392911yescharacteristic
629C2×C3#6393311yescharacteristic
630C2×C3#6393611yescharacteristic
631C2×C3#6394311yescharacteristic
632C2×C3#6394711yescharacteristic
633C2×C3#6395111yescharacteristic
634C2×C3#6397711yescharacteristic
635C2×C3#6397711yescharacteristic
636D3—42no—
637D3—122no—
638D3—121no—
639D3—121no—
640C22—62no—
641C22—121no—
642C22—121no—
643C22—181no—
644C22—181no—
645C22—181no—
646C22#8895211yescharacteristic
647C4—62no—
648C4—121no—
649C4—181no—
650C4—1081no—
651C4D3×#62611yescharacteristic
652C4D3×#62611yescharacteristic
653C3—21no—
654C3—21no—
655C3—21no—
656C3—41no—
657C3#1059061811yescharacteristic, complemented
658C3#1059146311yescharacteristic, complemented
659C3#1059159911yescharacteristic, complemented
660C2—122no—
661C2—361no—
662C2#1172285711yescharacteristic
663C2#1172288811yescharacteristic
664C2#1172294811yescharacteristic
subgroupG/H
centerC22#88952
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radicalC2×C3×#3796C2
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. #11723431C2semidirect
3. C2×#88799C2semidirect
4. C2×#88794C2semidirect
5. #11722072C2semidirect
6. C3×#63565C2semidirect
7. C3×#63562C2semidirect
45. #63565D3semidirect
46. #63562D3semidirect
390. C33#452semidirect
589. C32#4783semidirect
590. C32#4808semidirect
591. C32#5239semidirect
657. C3#10590618semidirect
658. C3#10591463semidirect
659. C3#10591599semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37962
#117234312
C2×#887992
C2×#887942
#117220722
C3×#635652
C3×#635622
#105915993
#105914633
#105906183