Finite Group Databasebeta

← #13155884

Subgroups — #13155884

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×#3795C211yesmaximal, characteristic
2#11723426C211yesmaximal, characteristic, complemented
3C2×#88800C211yesmaximal, characteristic, complemented
4C2×#88794C211yesmaximal, characteristic, complemented
5#11722076C211yesmaximal, characteristic, complemented
6C3×#63563C211yesmaximal, characteristic, complemented
7C3×(C2×D3)⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
8#10591463—31nomaximal
9#10591173—31nomaximal
10#10590969—31nomaximal
11C2×#6308C2211yescharacteristic
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—
19C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
20C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
21C2×C32×C3⋊C8C2211yescharacteristic
22#88807—21no—
23(C2×D3)⋊(C3⋊Dic3)—21no—
24#88800—22no—
25#88794—22no—
26C2×C3×#237—31no—
27C2×C3×#235—31no—
28C2×#3795D311yescharacteristic
29C2×#3754—31no—
30#64133—31no—
31C2×#3746—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×#832—31no—
42C3×#830—31no—
43C3×#821—31no—
44C3×(C2×D3)⋊C8—31no—
45#63567—31no—
46#63566—31no—
47#63565—31no—
48#63563D311yescharacteristic, complemented
49(C2×D3)⋊(C3⋊C8)D311yescharacteristic, complemented
50C22×C32×D3—21no—
51C22×C32×D3D411yescharacteristic
52C2×C33×C4C2311yescharacteristic
53C2×C3⋊(C3⋊Dic3)—21no—
54#6308—21no—
55#6308—21no—
56C2×C3×C3⋊Dic3—21no—
57C2×C3×C3⋊Dic3—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#4783—91no—
71#4733—91no—
72C2×C32×D4—31no—
73C22×C32×C4—31no—
74C2×#634—31no—
75C2×#634—31no—
76C2×#634—31no—
77C2×#634—61no—
78C2×#634—61no—
79C2×#634—61no—
80C2×#634—121no—
81C2×#634C2×D311yescharacteristic
82C2×C3×#67—61no—
83C2×C3×#67—61no—
84C22×C3×Dic3—61no—
85C3×#237—62no—
86C3×#237—62no—
87C3×#235—62no—
88C3×#235—62no—
89C2×C3×D12—21no—
90C2×C3×D12C2×D311yescharacteristic
91C2×C3×D12C2×D311yescharacteristic
92C2×C3×C4×D3—21no—
93C2×C3×C4×D3—61no—
94C2×C3×C4×D3C2×D311yescharacteristic
95C2×C3×C4×D3C2×D311yescharacteristic
96C2×C3×Dic6—31no—
97C2×C3×Dic6—31no—
98C2×C3×Dic6—61no—
99C2×#625—21no—
100C2×D3×Dic3—21no—
101#3795—22no—
102#3795—22no—
103C2×C32×C8—31no—
104#3757—61no—
105#3754—62no—
106C4⋊(C3⋊Dic3)C2×D311yescharacteristic
107#3750—61no—
108C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
109#3746—62no—
110C3×C4⋊Dic3—31no—
111C3×C4⋊Dic3—31no—
112C3×C4⋊Dic3—61no—
113C2×C3×C3⋊C8—21no—
114C2×C3×C3⋊C8—31no—
115C2×C3×C3⋊C8—31no—
116C2×C3×C3⋊C8—61no—
117C2×C3×C3⋊C8C2×D311yescharacteristic
118C2×C3×C3⋊C8C2×D311yescharacteristic
119Dic3⋊Dic3—62no—
120Dic3⋊Dic3—61no—
121Dic3⋊Dic3—61no—
122(C2×D3)⋊Dic3—62no—
123(C2×D3)⋊Dic3—61no—
124(C2×D3)⋊Dic3—61no—
125#3719—62no—
126#3719—62no—
127#3719—64no—
128#3716—64no—
129#3716—62no—
130#3716—62no—
131C22×C33C2×D411yescharacteristic
132C2×C32×D3—22no—
133C2×C32×D3—21no—
134C2×C32×D3—21no—
135C2×C32×D3—21no—
136C2×C32×D3—41no—
137C2×C32×D3—41no—
138C33×C4C2×D411yescharacteristic
139C33×C4C2×D411yescharacteristic
140C3⋊(C3⋊Dic3)—41no—
141C3×C3⋊Dic3—22no—
142C3×C3⋊Dic3—41no—
143C32×Dic3—21no—
144C2×C3×#41—91no—
145C2×#237—31no—
146C2×#235—31no—
147C2×#216—91no—
148C2×#214—91no—
149C2×#214—91no—
150C2×#214—181no—
151#924—91no—
152#924—91no—
153#924—181no—
154C2×Dic12—91no—
155C2×Dic12—91no—
156C2×Dic12—181no—
157C2×#204—91no—
158#897—91no—
159C3×#102—91no—
160C3×C22⋊C8—91no—
161#835—91no—
162#832—31no—
163#830—31no—
164#821—31no—
165(C2×D3)⋊C8—31no—
166#816—91no—
167#810—91no—
168C23×C32—31no—
169C23×C32—61no—
170C22×C3×D3—21no—
171C22×C3×D3—21no—
172C22×C3×D3—21no—
173C22×C3×D3—41no—
174C22×C3×D3D1211yescharacteristic
175C22×C3×D3#6711yescharacteristic
176C32×D4—32no—
177C32×D4—61no—
178C2×C32×C4—21no—
179C2×C32×C4—21no—
180C2×C32×C4—21no—
181C2×C32×C4—31no—
182C2×C32×C4—41no—
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—
190C2×C3⋊Dic3—61no—
191C2×C3⋊Dic3—62no—
192C2×C3⋊Dic3—61no—
193C2×C3⋊Dic3—61no—
194C2×C3⋊Dic3—121no—
195C2×C3⋊Dic3—121no—
196C2×C3⋊Dic3—122no—
197#634—21no—
198#634—61no—
199#634—61no—
200#634—62no—
201#634—62no—
202#634—61no—
203#634—61no—
204#634—61no—
205#634—61no—
206#634—121no—
207#634—121no—
208#634—121no—
209#634—121no—
210#634—122no—
211#634—122no—
212#634#6712yes—
213C3×#67—64no—
214C3×#67—64no—
215C2×C3×Dic3—21no—
216C2×C3×Dic3—64no—
217C2×C3×Dic3—61no—
218C2×C3×Dic3—61no—
219C2×C3×Dic3—62no—
220C2×C3×Dic3—61no—
221C2×C3×Dic3—61no—
222C2×C3×Dic3—121no—
223C2×C3×Dic3D1211yescharacteristic
224C2×C3×Dic3#6711yescharacteristic
225C3×D12—21no—
226C3×D12—21no—
227C3×D12—22no—
228C3×D12—41no—
229C3×D12D1212yes—
230C3×D12#6712yes—
231C3×C4×D3—21no—
232C3×C4×D3—21no—
233C3×C4×D3—21no—
234C3×C4×D3—21no—
235C3×C4×D3—41no—
236C3×C4×D3—41no—
237C3×C4×D3—64no—
238C3×Dic6—32no—
239C3×Dic6—32no—
240C3×Dic6—62no—
241C3×Dic6—61no—
242C3×Dic6—61no—
243C3×Dic6—121no—
244#625—24no—
245D3×Dic3—24no—
246C32×C8—61no—
247C3⋊(C3⋊C8)—21no—
248C3×C3⋊C8—22no—
249C3×C3⋊C8—21no—
250C3×C3⋊C8—21no—
251C3×C3⋊C8—62no—
252C3×C3⋊C8—61no—
253C3×C3⋊C8—61no—
254#452—271noSylow
255C2×C33#12011yescharacteristic
256C2×C33#13511yescharacteristic
257C2×C33#13711yescharacteristic
258C32×D3—41no—
259C32×D3—42no—
260C3×#41—182no—
261C3×#41—182no—
262C2×C3×Q8—91no—
263C2×C3×D4—31no—
264C2×C3×D4—31no—
265C2×C3×D4—61no—
266C2×C3×D4—181no—
267C22×C3×C4—31no—
268C22×C3×C4—31no—
269C22×C3×C4—61no—
270C22×C3×C4—181no—
271C2×#67—61no—
272C2×#67—61no—
273C22×Dic3—61no—
274#237—62no—
275#237—62no—
276#235—62no—
277#235—62no—
278C2×D12D3211yescharacteristic
279C2×C4×D3—61no—
280C2×C4×D3D3211yescharacteristic
281C2×Dic6—31no—
282C2×Dic6—31no—
283C2×Dic6—61no—
284C2×Dic6—91no—
285C2×Dic6—91no—
286C2×Dic6—91no—
287C2×Dic6—181no—
288C2×Dic6—181no—
289C2×Dic6—181no—
290C2×Dic6—361no—
291C2×C3×C8—31no—
292C2×C3×C8—31no—
293C2×C3×C8—61no—
294C2×C3×C8—91no—
295C3×C4⋊C4—91no—
296#217—181no—
297#217—181no—
298#217—182no—
299#216—182no—
300#214—182no—
301#214—182no—
302#214—184no—
303(C2×D3)⋊C4—181no—
304C4⋊Dic3—31no—
305C4⋊Dic3—31no—
306C4⋊Dic3—61no—
307C4⋊Dic3—181no—
308Dic3⋊C4—181no—
309Dic3⋊C4—181no—
310Dic3⋊C4—182no—
311C2×C3⋊C8—31no—
312C2×C3⋊C8—31no—
313C2×C3⋊C8—61no—
314C2×C3⋊C8D3211yescharacteristic
315Dic12—182no—
316Dic12—182no—
317Dic12—184no—
318#204—182no—
319C22×C32—21no—
320C22×C32—21no—
321C22×C32—21no—
322C22×C32—41no—
323C22×C32—61no—
324C22×C32—62no—
325C22×C32—61no—
326C22×C32—61no—
327C22×C32—121no—
328C22×C32—121no—
329C22×C32C2×D1211yescharacteristic
330C22×C32D3×D411yescharacteristic
331C22×C32C2×#6711yescharacteristic
332C2×C3×D3—22no—
333C2×C3×D3—21no—
334C2×C3×D3—22no—
335C2×C3×D3—21no—
336C2×C3×D3—21no—
337C2×C3×D3—21no—
338C2×C3×D3—21no—
339C2×C3×D3—22no—
340C2×C3×D3—21no—
341C2×C3×D3—41no—
342C2×C3×D3—41no—
343C2×C3×D3—42no—
344C2×C3×D3—41no—
345C2×C3×D3—41no—
346C2×C3×D3—41no—
347C2×C3×D3—41no—
348C2×C3×D3—81no—
349C2×C3×D3—81no—
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—61no—
359C32×C4C2×D1211yescharacteristic
360C32×C4D3×D411yescharacteristic
361C32×C4D3×D411yescharacteristic
362C32×C4D3×D411yescharacteristic
363C32×C4C2×#6711yescharacteristic
364C32×C4C2×#6711yescharacteristic
365C3⋊Dic3—22no—
366C3⋊Dic3—41no—
367C3⋊Dic3—122no—
368C3⋊Dic3—122no—
369C3⋊Dic3—122no—
370C3⋊Dic3—124no—
371C3⋊Dic3—121no—
372C3⋊Dic3—121no—
373C3⋊Dic3—121no—
374C3×Dic3—21no—
375C3×Dic3—21no—
376C3×Dic3—41no—
377C3×Dic3—62no—
378C3×Dic3—62no—
379C3×Dic3—122no—
380C3×Dic3—122no—
381C3×Dic3—121no—
382C3×Dic3—121no—
383C2×#41—91no—
384C2×Q16—271no—
385C2×SD16—271no—
386C22⋊Q8—271no—
387#103—271no—
388#102—91no—
389C22⋊C8—91no—
390C33#45211yescharacteristic, complemented, Sylow
391C23×C3—31no—
392C23×C3—31no—
393C23×C3—61no—
394C23×C3—61no—
395C23×C3—61no—
396C23×C3—121no—
397C22×D3—21no—
398C22×D3#62611yescharacteristic
399C3×Q8—92no—
400C3×Q8—181no—
401C3×D4—32no—
402C3×D4—32no—
403C3×D4—62no—
404C3×D4—61no—
405C3×D4—61no—
406C3×D4—121no—
407C3×D4—184no—
408C2×C3×C4—21no—
409C2×C3×C4—21no—
410C2×C3×C4—21no—
411C2×C3×C4—31no—
412C2×C3×C4—31no—
413C2×C3×C4—41no—
414C2×C3×C4—61no—
415C2×C3×C4—61no—
416C2×C3×C4—61no—
417C2×C3×C4—61no—
418C2×C3×C4—61no—
419C2×C3×C4—121no—
420C2×C3×C4—121no—
421C2×C3×C4—184no—
422C2×C3×C4—181no—
423C2×C3×C4—181no—
424C2×C3×C4C2×D3211yescharacteristic
425C2×C3×C4C2×D3211yescharacteristic
426C2×C3×C4C2×D3211yescharacteristic
427#67—64no—
428#67—64no—
429C2×Dic3—64no—
430C2×Dic3—61no—
431C2×Dic3—61no—
432C2×Dic3—62no—
433C2×Dic3—61no—
434C2×Dic3—61no—
435C2×Dic3—121no—
436C2×Dic3—182no—
437C2×Dic3—181no—
438C2×Dic3—181no—
439C2×Dic3—181no—
440C2×Dic3—361no—
441C2×Dic3—361no—
442C2×Dic3—362no—
443C2×Dic3#62611yescharacteristic
444D12—21no—
445D12#62612yes—
446C4×D3—21no—
447C4×D3—21no—
448C4×D3—64no—
449Dic6—32no—
450Dic6—32no—
451Dic6—62no—
452Dic6—61no—
453Dic6—61no—
454Dic6—121no—
455Dic6—182no—
456Dic6—182no—
457Dic6—181no—
458Dic6—181no—
459Dic6—181no—
460Dic6—181no—
461Dic6—181no—
462Dic6—181no—
463Dic6—361no—
464Dic6—361no—
465Dic6—361no—
466Dic6—361no—
467Dic6—362no—
468Dic6—362no—
469C3×C8—62no—
470C3×C8—61no—
471C3×C8—61no—
472C3×C8—181no—
473C3⋊C8—21no—
474C3⋊C8—62no—
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—121no—
490C2×C32—122no—
491C2×C32#88211yescharacteristic
492C2×C32#88211yescharacteristic
493C2×C32#90411yescharacteristic
494C2×C32#90911yescharacteristic
495C2×C32#91211yescharacteristic
496C2×C32#91511yescharacteristic
497C2×C32#95011yescharacteristic
498C2×C32#95111yescharacteristic
499C2×C32#95311yescharacteristic
500C3×D3—42no—
501C3×D3—41no—
502C3×D3—42no—
503C3×D3—41no—
504C3×D3—42no—
505C3×D3—82no—
506#41—182no—
507#41—182no—
508C2×Q8—91no—
509C2×Q8—271no—
510C2×D4—31no—
511C2×D4—181no—
512C22×C4—31no—
513C22×C4—181no—
514Q16—542no—
515SD16—542no—
516C2×C8—31no—
517C2×C8—91no—
518C4⋊C4—91no—
519C4⋊C4—541no—
520C22⋊C4—541no—
521C22×C3—21no—
522C22×C3—21no—
523C22×C3—21no—
524C22×C3—41no—
525C22×C3—62no—
526C22×C3—61no—
527C22×C3—61no—
528C22×C3—61no—
529C22×C3—62no—
530C22×C3—61no—
531C22×C3—61no—
532C22×C3—62no—
533C22×C3—61no—
534C22×C3—121no—
535C22×C3—121no—
536C22×C3—122no—
537C22×C3—121no—
538C22×C3—121no—
539C22×C3—121no—
540C22×C3—121no—
541C22×C3—241no—
542C22×C3—241no—
543C22×C3D3×D1211yescharacteristic
544C22×C3C2×#62611yescharacteristic
545C22×C3D3×#6711yescharacteristic
546C2×D3—21no—
547C2×D3—21no—
548C2×D3—21no—
549C2×D3—22no—
550C2×D3—41no—
551C2×D3—41no—
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—61no—
561C3×C4—61no—
562C3×C4—121no—
563C3×C4—182no—
564C3×C4—361no—
565C3×C4D3×D1211yescharacteristic
566C3×C4C2×#62611yescharacteristic
567C3×C4D3×#6711yescharacteristic
568C3×C4D3×#6711yescharacteristic
569C3×C4D3×#6711yescharacteristic
570C3×C4#381111yescharacteristic
571Dic3—21no—
572Dic3—62no—
573Dic3—62no—
574Dic3—122no—
575Dic3—122no—
576Dic3—121no—
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#473311yescharacteristic, complemented
590C32#478311yescharacteristic, complemented
591C32#523911yescharacteristic, complemented
592C23—31no—
593C23—61no—
594Q8—92no—
595Q8—181no—
596Q8—541no—
597Q8—541no—
598D4—32no—
599D4—61no—
600D4—184no—
601C2×C4—31no—
602C2×C4—61no—
603C2×C4—61no—
604C2×C4—184no—
605C2×C4—181no—
606C2×C4—181no—
607C2×C4—541no—
608C2×C4D3311yescharacteristic
609C8—61no—
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—121no—
626C2×C3—121no—
627C2×C3—122no—
628C2×C3—242no—
629C2×C3#6380011yescharacteristic
630C2×C3#6380711yescharacteristic
631C2×C3#6382911yescharacteristic
632C2×C3#6383111yescharacteristic
633C2×C3#6392311yescharacteristic
634C2×C3#6392911yescharacteristic
635C2×C3#6393311yescharacteristic
636C2×C3#6397711yescharacteristic
637C2×C3#6397711yescharacteristic
638D3—41no—
639D3—42no—
640C22—61no—
641C22—61no—
642C22—61no—
643C22—62no—
644C22—121no—
645C22—121no—
646C22D3×#62611yescharacteristic
647C4—61no—
648C4—182no—
649C4—361no—
650C4—1081no—
651C4D3×#62611yescharacteristic
652C4#8895211yescharacteristic
653C3—21no—
654C3—21no—
655C3—21no—
656C3—41no—
657C3#1059096911yescharacteristic, complemented
658C3#1059117311yescharacteristic, complemented
659C3#1059146311yescharacteristic, complemented
660C2—121no—
661C2—122no—
662C2#1172285411yescharacteristic
663C2#1172288611yescharacteristic
664C2#1172294811yescharacteristic
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. #11723426C2semidirect
3. C2×#88800C2semidirect
4. C2×#88794C2semidirect
5. #11722076C2semidirect
6. C3×#63563C2semidirect
7. C3×(C2×D3)⋊(C3⋊C8)C2semidirect
48. #63563D3semidirect
49. (C2×D3)⋊(C3⋊C8)D3semidirect
390. C33#452semidirect
589. C32#4733semidirect
590. C32#4783semidirect
591. C32#5239semidirect
657. C3#10590969semidirect
658. C3#10591173semidirect
659. C3#10591463semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37952
#117234262
C2×#888002
C2×#887942
#117220762
C3×#635632
C3×(C2×D3)⋊(C3⋊C8)2
#105914633
#105911733
#105909693