Finite Group Databasebeta

← #13155801

Subgroups — #13155801

Order 1728.

836 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×D3×Dic6C211yesmaximal, characteristic, complemented
2C2×#88811C211yesmaximal, characteristic, complemented
3#11723515C211yesmaximal, characteristic
4C2×C3×#3717C211yesmaximal, characteristic, complemented
5#11722040C211yesmaximal, characteristic, complemented
6#11722038C211yesmaximal, characteristic, complemented
7C3×(C2×D3)⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
8#10591583—31nomaximal
9#10591174—31nomaximal
10#10590822—31nomaximal
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C2×C32×C4×D3C2211yescharacteristic
14C2×C32×Dic6C2211yescharacteristic
15C2×#6290—21no—
16C2×C3×#627—21no—
17C2×C3×D3×Dic3—21no—
18C3×D3×Dic6—22no—
19C3×D3×Dic6—22no—
20C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
21C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
22C2×C32×C3⋊C8C2211yescharacteristic
23#88811—22no—
24#88803—21no—
25C3×#3717—22no—
26C2×C3×D3×Q8—31no—
27C22×C3×Dic6—31no—
28C2×D3×Dic6D311yescharacteristic, complemented
29#64139—31no—
30C2×C3×#215—31no—
31C2×C3×#204—31no—
32#63908—31no—
33#63908—31no—
34#63908—61no—
35C2×#3717—31no—
36C2×#3717D311yescharacteristic, complemented
37C2×#3716—31no—
38C2×#3715—31no—
39#63662—31no—
40#63643—31no—
41C3×#830—31no—
42C3×(C2×D3)⋊C8—31no—
43#63566—31no—
44#63566—31no—
45#63566—31no—
46#63566—31no—
47#63566—61no—
48#63566—61no—
49(C2×D3)⋊(C3⋊C8)D311yescharacteristic, complemented
50C22×C3×C3⋊D3—21no—
51C22×C32×D3D411yescharacteristic
52C2×C33×C4C2311yescharacteristic
53C2×C3⋊(C3⋊Dic3)—21no—
54C2×C3×C3⋊Dic3—21no—
55C3×C4⋊(C3⋊D3)—21no—
56C3×C4⋊(C3⋊D3)—21no—
57C3×#634—21no—
58C3×#634D412yescomplemented
59C2×C32×Dic3—21no—
60C2×C32×Dic3D411yescharacteristic
61C32×C4×D3—21no—
62C32×C4×D3—21no—
63C32×Dic6—21no—
64C32×Dic6D412yescomplemented
65#6290—42no—
66C3×#627—24no—
67C3×D3×Dic3—24no—
68C3×C3⋊(C3⋊C8)—21no—
69C32×C3⋊C8—21no—
70#5227—91no—
71#5170—91no—
72#4738—91no—
73C2×C32×Q8—31no—
74C22×C32×C4—31no—
75C2×#638—61no—
76C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
77C2×#634C2×D311yescharacteristic
78C22×C3×Dic3—61no—
79C3×D3×Q8—62no—
80C3×D3×Q8—62no—
81C2×C3×D12—31no—
82C2×C3×D12—31no—
83C2×C3×D12—61no—
84C2×C3×C4×D3—21no—
85C2×C3×C4×D3—61no—
86C2×C3×C4×D3C2×D311yescharacteristic
87C2×C3×C4×D3C2×D311yescharacteristic
88C2×C3×Dic6—21no—
89C2×C3×Dic6—31no—
90C2×C3×Dic6—31no—
91C2×C3×Dic6—62no—
92C2×C3×Dic6—61no—
93C2×C3×Dic6—61no—
94C2×C3×Dic6—62no—
95C2×C3×Dic6—61no—
96C2×C3×Dic6C2×D311yescharacteristic
97C2×C3×Dic6C2×D311yescharacteristic
98C2×#627—21no—
99C2×#625—62no—
100C2×#625—61no—
101C2×#625—61no—
102C2×D3×Dic3—21no—
103D3×Dic6—22no—
104D3×Dic6—22no—
105C2×C32×C8—31no—
106#3752—61no—
107C4⋊(C3⋊Dic3)—31no—
108C4⋊(C3⋊Dic3)—31no—
109C4⋊(C3⋊Dic3)—31no—
110C4⋊(C3⋊Dic3)—61no—
111C4⋊(C3⋊Dic3)—61no—
112C4⋊(C3⋊Dic3)—61no—
113C4⋊(C3⋊Dic3)—121no—
114C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
115C3×#215—62no—
116C2×C3×C3⋊C8—21no—
117C2×C3×C3⋊C8—31no—
118C2×C3×C3⋊C8—31no—
119C2×C3×C3⋊C8—61no—
120C2×C3×C3⋊C8C2×D311yescharacteristic
121C2×C3×C3⋊C8C2×D311yescharacteristic
122C3×#204—62no—
123(C2×D3)⋊Dic3—62no—
124(C2×D3)⋊Dic3—61no—
125(C2×D3)⋊Dic3—61no—
126#3717—22no—
127#3717—62no—
128#3716—62no—
129#3715—62no—
130C22×C33C2×D411yescharacteristic
131C2×C3×C3⋊D3—41no—
132C2×C3×C3⋊D3—41no—
133C2×C3×C3⋊D3—41no—
134C2×C32×D3—21no—
135C2×C32×D3—21no—
136C2×C32×D3SD1612yescomplemented
137C33×C4C2×D411yescharacteristic
138C33×C4C2×D411yescharacteristic
139C3⋊(C3⋊Dic3)—41no—
140C3×C3⋊Dic3—22no—
141C32×Dic3—22no—
142C32×Dic3—21no—
143C22×C3×Q8—91no—
144C2×D3×Q8—31no—
145C22×Dic6—31no—
146C2×C3×SD16—91no—
147C2×#215—31no—
148C2×#215—91no—
149#938—91no—
150C2×#214—91no—
151#930—91no—
152#930—91no—
153#930—181no—
154C2×#204—31no—
155C2×#204—91no—
156C3×C22⋊C8—91no—
157#835—91no—
158#835—91no—
159#835—91no—
160#835—181no—
161#830—31no—
162(C2×D3)⋊C8—31no—
163#816—91no—
164#816—91no—
165#816—91no—
166#816—181no—
167C23×C32—31no—
168C22×C3⋊D3—21no—
169C22×C3×D3—21no—
170C22×C3×D3—62no—
171C22×C3×D3—61no—
172C22×C3×D3—61no—
173C22×C3×D3D1211yescharacteristic
174C22×C3×D3#6711yescharacteristic
175C32×Q8—32no—
176C32×Q8—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
188#638—122no—
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—
197C4⋊(C3⋊D3)—21no—
198C4⋊(C3⋊D3)—21no—
199#634—21no—
200#634#6712yescomplemented
201C2×C3×Dic3—21no—
202C2×C3×Dic3—21no—
203C2×C3×Dic3—21no—
204C2×C3×Dic3—41no—
205C2×C3×Dic3—64no—
206C2×C3×Dic3—61no—
207C2×C3×Dic3—61no—
208C2×C3×Dic3—121no—
209C2×C3×Dic3D1211yescharacteristic
210C2×C3×Dic3#6711yescharacteristic
211C3×D12—61no—
212C3×D12—61no—
213C3×D12—61no—
214C3×D12—61no—
215C3×D12—121no—
216C3×D12—121no—
217C3×C4×D3—21no—
218C3×C4×D3—21no—
219C3×C4×D3—21no—
220C3×C4×D3—21no—
221C3×C4×D3—41no—
222C3×C4×D3—41no—
223C3×C4×D3—64no—
224C3×Dic6—21no—
225C3×Dic6—21no—
226C3×Dic6—22no—
227C3×Dic6—32no—
228C3×Dic6—32no—
229C3×Dic6—41no—
230C3×Dic6—62no—
231C3×Dic6—64no—
232C3×Dic6—64no—
233C3×Dic6—61no—
234C3×Dic6—61no—
235C3×Dic6—121no—
236C3×Dic6#6712yescomplemented
237C3×Dic6#6712yescomplemented
238#627—24no—
239#625—124no—
240#625—122no—
241#625—122no—
242D3×Dic3—24no—
243C32×C8—61no—
244C3⋊(C3⋊C8)—21no—
245C3×C3⋊C8—21no—
246C3×C3⋊C8—21no—
247C3×C3⋊C8—41no—
248C3×C3⋊C8—61no—
249C3×C3⋊C8—61no—
250C3×C3⋊C8—121no—
251#448—271noSylow
252C2×C33#12011yescharacteristic
253C2×C33C2×SD1611yescharacteristic
254C2×C33#13711yescharacteristic
255C3×C3⋊D3—81no—
256C32×D3—22no—
257C2×C3×Q8—31no—
258C2×C3×Q8—31no—
259C2×C3×Q8—61no—
260C2×C3×Q8—91no—
261C2×C3×Q8—181no—
262C2×C3×Q8—182no—
263C2×C3×Q8—182no—
264C2×C3×D4—91no—
265C22×C3×C4—31no—
266C22×C3×C4—31no—
267C22×C3×C4—61no—
268C22×C3×C4—181no—
269C2×#67—181no—
270C2×#67—181no—
271C2×#67—182no—
272C2×#67—181no—
273C22×Dic3—61no—
274D3×Q8—62no—
275D3×Q8—62no—
276C2×D12—31no—
277C2×D12—31no—
278C2×D12—61no—
279C2×C4×D3—61no—
280C2×C4×D3D3211yescharacteristic
281C2×Dic6—31no—
282C2×Dic6—31no—
283C2×Dic6—61no—
284C2×Dic6—62no—
285C2×Dic6—61no—
286C2×Dic6—61no—
287C2×Dic6—62no—
288C2×Dic6D3211yescharacteristic
289C3×SD16—182no—
290C2×C3×C8—31no—
291C2×C3×C8—31no—
292C2×C3×C8—61no—
293C2×C3×C8—91no—
294#217—181no—
295#215—62no—
296#215—182no—
297#214—182no—
298(C2×D3)⋊C4—181no—
299(C2×D3)⋊C4—181no—
300(C2×D3)⋊C4—182no—
301C4⋊Dic3—91no—
302C4⋊Dic3—91no—
303C4⋊Dic3—91no—
304C4⋊Dic3—181no—
305C4⋊Dic3—181no—
306C4⋊Dic3—181no—
307C4⋊Dic3—361no—
308C2×C3⋊C8—31no—
309C2×C3⋊C8—31no—
310C2×C3⋊C8—61no—
311C2×C3⋊C8D3211yescharacteristic
312#204—62no—
313#204—182no—
314C22×C32—21no—
315C22×C32—21no—
316C22×C32—21no—
317C22×C32—32no—
318C22×C32—41no—
319C22×C32—61no—
320C22×C32—61no—
321C22×C32C2×D1211yescharacteristic
322C22×C32D3×D411yescharacteristic
323C22×C32C2×#6711yescharacteristic
324C2×C3⋊D3—41no—
325C2×C3⋊D3—41no—
326C2×C3⋊D3—41no—
327C2×C3×D3—21no—
328C2×C3×D3—21no—
329C2×C3×D3—21no—
330C2×C3×D3—21no—
331C2×C3×D3—22no—
332C2×C3×D3—41no—
333C2×C3×D3—41no—
334C2×C3×D3—122no—
335C2×C3×D3—122no—
336C2×C3×D3—122no—
337C2×C3×D3—121no—
338C2×C3×D3—121no—
339C2×C3×D3—121no—
340C2×C3×D3—121no—
341C2×C3×D3—121no—
342C2×C3×D3—121no—
343C2×C3×D3#20412yescomplemented
344C2×C3×D3#21512yescomplemented
345C32×C4—21no—
346C32×C4—21no—
347C32×C4—21no—
348C32×C4—21no—
349C32×C4—21no—
350C32×C4—21no—
351C32×C4—41no—
352C32×C4—41no—
353C32×C4—62no—
354C32×C4—61no—
355C32×C4D3×D411yescharacteristic
356C32×C4D3×D411yescharacteristic
357C32×C4D3×D411yescharacteristic
358C32×C4C2×#6711yescharacteristic
359C32×C4C2×#6711yescharacteristic
360C32×C4C2×#6711yescharacteristic
361C3⋊Dic3—22no—
362C3⋊Dic3—121no—
363C3⋊Dic3—122no—
364C3⋊Dic3—121no—
365C3⋊Dic3—121no—
366C3⋊Dic3—241no—
367C3⋊Dic3—241no—
368C3⋊Dic3—242no—
369C3×Dic3—21no—
370C3×Dic3—22no—
371C3×Dic3—22no—
372C3×Dic3—21no—
373C3×Dic3—42no—
374C3×Dic3—41no—
375C3×Dic3—62no—
376C3×Dic3—62no—
377C3×Dic3—122no—
378C22×Q8—91no—
379C2×SD16—91no—
380C2×SD16—271no—
381#121—271no—
382#103—271no—
383#103—271no—
384C22⋊C8—91no—
385C33#44811yescharacteristic, complemented, Sylow
386C23×C3—31no—
387C23×C3—31no—
388C23×C3—61no—
389C23×C3—181no—
390C22×D3—62no—
391C22×D3—61no—
392C22×D3—61no—
393C22×D3#62611yescharacteristic
394C3×Q8—32no—
395C3×Q8—32no—
396C3×Q8—62no—
397C3×Q8—61no—
398C3×Q8—61no—
399C3×Q8—92no—
400C3×Q8—121no—
401C3×Q8—184no—
402C3×Q8—181no—
403C3×D4—181no—
404C3×D4—181no—
405C2×C3×C4—21no—
406C2×C3×C4—21no—
407C2×C3×C4—21no—
408C2×C3×C4—31no—
409C2×C3×C4—31no—
410C2×C3×C4—41no—
411C2×C3×C4—61no—
412C2×C3×C4—61no—
413C2×C3×C4—61no—
414C2×C3×C4—61no—
415C2×C3×C4—61no—
416C2×C3×C4—61no—
417C2×C3×C4—61no—
418C2×C3×C4—121no—
419C2×C3×C4—121no—
420C2×C3×C4—121no—
421C2×C3×C4—184no—
422C2×C3×C4—181no—
423C2×C3×C4C2×D3211yescharacteristic
424C2×C3×C4C2×D3211yescharacteristic
425C2×C3×C4C2×D3211yescharacteristic
426#67—362no—
427#67—362no—
428#67—364no—
429#67—362no—
430C2×Dic3—21no—
431C2×Dic3—64no—
432C2×Dic3—61no—
433C2×Dic3—61no—
434C2×Dic3—121no—
435C2×Dic3—182no—
436C2×Dic3—181no—
437C2×Dic3—181no—
438C2×Dic3—181no—
439C2×Dic3—361no—
440C2×Dic3—361no—
441C2×Dic3—362no—
442C2×Dic3#62611yescharacteristic
443D12—61no—
444D12—61no—
445D12—61no—
446D12—61no—
447D12—121no—
448D12—121no—
449C4×D3—21no—
450C4×D3—21no—
451C4×D3—64no—
452Dic6—21no—
453Dic6—32no—
454Dic6—32no—
455Dic6—62no—
456Dic6—64no—
457Dic6—64no—
458Dic6—61no—
459Dic6—61no—
460Dic6—121no—
461Dic6#62512yescomplemented
462C3×C8—61no—
463C3×C8—61no—
464C3×C8—121no—
465C3×C8—181no—
466C3⋊C8—21no—
467C3⋊C8—61no—
468C3⋊C8—61no—
469C3⋊C8—121no—
470C2×C32—21no—
471C2×C32—21no—
472C2×C32—21no—
473C2×C32—21no—
474C2×C32—21no—
475C2×C32—21no—
476C2×C32—21no—
477C2×C32—21no—
478C2×C32—21no—
479C2×C32—41no—
480C2×C32—41no—
481C2×C32—41no—
482C2×C32—62no—
483C2×C32#88211yescharacteristic
484C2×C32C2×#20411yescharacteristic
485C2×C32#90911yescharacteristic
486C2×C32D3×SD1611yescharacteristic
487C2×C32#91511yescharacteristic
488C2×C32#93711yescharacteristic
489C2×C32C2×#21511yescharacteristic
490C2×C32#94211yescharacteristic
491C2×C32#95311yescharacteristic
492C3⋊D3—81no—
493C3×D3—22no—
494C3×D3—22no—
495C3×D3—42no—
496C3×D3—242no—
497C3×D3—241no—
498C3×D3—241no—
499C2×Q8—31no—
500C2×Q8—91no—
501C2×Q8—181no—
502C2×Q8—182no—
503C2×Q8—182no—
504C2×D4—91no—
505C2×D4—541no—
506C22×C4—31no—
507C22×C4—181no—
508SD16—182no—
509SD16—542no—
510C2×C8—31no—
511C2×C8—91no—
512C4⋊C4—271no—
513C22⋊C4—541no—
514C22×C3—21no—
515C22×C3—21no—
516C22×C3—21no—
517C22×C3—32no—
518C22×C3—32no—
519C22×C3—41no—
520C22×C3—62no—
521C22×C3—61no—
522C22×C3—61no—
523C22×C3—61no—
524C22×C3—61no—
525C22×C3—121no—
526C22×C3—121no—
527C22×C3—361no—
528C22×C3—361no—
529C22×C3—361no—
530C22×C3D3×D1211yescharacteristic
531C22×C3C2×#62611yescharacteristic
532C22×C3D3×#6711yescharacteristic
533C2×D3—21no—
534C2×D3—21no—
535C2×D3—122no—
536C2×D3—122no—
537C2×D3—122no—
538C2×D3—121no—
539C2×D3—121no—
540C2×D3—121no—
541C2×D3—121no—
542C2×D3—121no—
543C2×D3—121no—
544C2×D3#371712yescomplemented
545C3×C4—21no—
546C3×C4—21no—
547C3×C4—21no—
548C3×C4—21no—
549C3×C4—21no—
550C3×C4—21no—
551C3×C4—41no—
552C3×C4—41no—
553C3×C4—61no—
554C3×C4—62no—
555C3×C4—61no—
556C3×C4—62no—
557C3×C4—121no—
558C3×C4—122no—
559C3×C4—182no—
560C3×C4#380211yescharacteristic
561C3×C4C2×#62511yescharacteristic
562C3×C4D3×#6711yescharacteristic
563C3×C4D3×#6711yescharacteristic
564C3×C4D3×#6711yescharacteristic
565C3×C4D3×#6711yescharacteristic
566Dic3—22no—
567Dic3—21no—
568Dic3—62no—
569Dic3—62no—
570Dic3—122no—
571Dic3—362no—
572Dic3—361no—
573Dic3—361no—
574Dic3—361no—
575Dic3—721no—
576Dic3—721no—
577Dic3—722no—
578C32—21no—
579C32—21no—
580C32—21no—
581C32—41no—
582C32#473811yescharacteristic, complemented
583C32#517011yescharacteristic, complemented
584C32#522711yescharacteristic, complemented
585C23—31no—
586C23—181no—
587Q8—32no—
588Q8—61no—
589Q8—92no—
590Q8—184no—
591Q8—181no—
592D4—181no—
593D4—181no—
594D4—1082no—
595C2×C4—31no—
596C2×C4—61no—
597C2×C4—61no—
598C2×C4—61no—
599C2×C4—184no—
600C2×C4—181no—
601C2×C4—541no—
602C2×C4D3311yescharacteristic
603C8—61no—
604C8—181no—
605C2×C3—21no—
606C2×C3—21no—
607C2×C3—21no—
608C2×C3—21no—
609C2×C3—21no—
610C2×C3—21no—
611C2×C3—21no—
612C2×C3—21no—
613C2×C3—21no—
614C2×C3—41no—
615C2×C3—41no—
616C2×C3—41no—
617C2×C3—62no—
618C2×C3—62no—
619C2×C3—122no—
620C2×C3—721no—
621C2×C3D3×#20411yescharacteristic
622C2×C3#6380011yescharacteristic
623C2×C3C2×#371711yescharacteristic
624C2×C3#6383311yescharacteristic
625C2×C3#6392211yescharacteristic
626C2×C3D3×#21511yescharacteristic
627C2×C3#6394611yescharacteristic
628C2×C3#6397611yescharacteristic
629C2×C3#6397711yescharacteristic
630D3—22no—
631D3—242no—
632D3—241no—
633D3—241no—
634C22—32no—
635C22—61no—
636C22—61no—
637C22—361no—
638C22—361no—
639C22—361no—
640C22D3×#62611yescharacteristic
641C4—61no—
642C4—62no—
643C4—182no—
644C4—1081no—
645C4D3×#62511yescharacteristic
646C4#8895311yescharacteristic
647C3—21no—
648C3—21no—
649C3—21no—
650C3—41no—
651C3#1059082211yescharacteristic, complemented
652C3#1059117411yescharacteristic, complemented
653C3#1059158311yescharacteristic, complemented
654C2—62no—
655C2—721no—
656C2D3×#371711yescharacteristic
657C2#1172288411yescharacteristic
658C2#1172294711yescharacteristic
subgroupG/H
centerC22D3×#626
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radical#11723515C2
hypocenterC33#448
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
1. C2×C3×D3×Dic6C2semidirect
2. C2×#88811C2semidirect
4. C2×C3×#3717C2semidirect
5. #11722040C2semidirect
6. #11722038C2semidirect
7. C3×(C2×D3)⋊(C3⋊C8)C2semidirect
28. C2×D3×Dic6D3semidirect
36. C2×#3717D3semidirect
49. (C2×D3)⋊(C3⋊C8)D3semidirect
58. C3×#634D4semidirect
64. C32×Dic6D4semidirect
136. C2×C32×D3SD16semidirect
200. #634#67semidirect
236. C3×Dic6#67semidirect
237. C3×Dic6#67semidirect
343. C2×C3×D3#204semidirect
344. C2×C3×D3#215semidirect
385. C33#448semidirect
461. Dic6#625semidirect
544. C2×D3#3717semidirect
582. C32#4738semidirect
583. C32#5170semidirect
584. C32#5227semidirect
651. C3#10590822semidirect
652. C3#10591174semidirect
653. C3#10591583semidirect

Maximal subgroups

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

subgroupindex
C2×C3×D3×Dic62
C2×#888112
#117235152
C2×C3×#37172
#117220402
#117220382
C3×(C2×D3)⋊(C3⋊C8)2
#105915833
#105911743
#105908223