Finite Group Databasebeta

← #13155865

Subgroups — #13155865

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×D3×#634C211yesmaximal, characteristic
2C2×#88799C211yesmaximal, characteristic, complemented
3C3×#63856C211yesmaximal, characteristic
4C2×C3×#3719C211yesmaximal, characteristic, complemented
5#11722076C211yesmaximal, characteristic, complemented
6#11722042C211yesmaximal, characteristic, complemented
7C3×(C2×D3)⋊(C3⋊C8)C211yesmaximal, characteristic
8#10591653—31nomaximal
9#10591190—31nomaximal
10#10590949—31nomaximal
11C2×#6308C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C2×C32×C4×D3C2211yescharacteristic
14C2×C32×Dic6C2211yescharacteristic
15C2×#6293—21no—
16C2×D3×C3⋊Dic3—21no—
17D3×#634—22no—
18D3×#634—22no—
19C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
20C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
21C2×C32×C3⋊C8C2211yescharacteristic
22#88799—22no—
23C3×#3724—21no—
24C3×(C2×D3)⋊Dic3—21no—
25C3×#3719—22no—
26C22×#634—31no—
27C2×D3×Dic6—31no—
28C2×D3×Dic6—31no—
29C2×D3×Dic6—61no—
30C2×#3756—31no—
31C3×#946—31no—
32C2×C3×#216—31no—
33C3×#924—31no—
34C2×C3×Dic12—31no—
35#63856D311yescharacteristic
36C2×#3719—31no—
37C2×#3719—31no—
38C2×#3719—61no—
39C2×#3719D311yescharacteristic, complemented
40#63643—31no—
41C3×#830—31no—
42C3×(C2×D3)⋊C8—31no—
43#63567—31no—
44#63567—31no—
45#63567—31no—
46#63567—61no—
47#63566—31no—
48#63565—31no—
49(C2×D3)⋊(C3⋊C8)D311yescharacteristic
50C22×C32×D3D411yescharacteristic
51C2×C33×C4C2311yescharacteristic
52C2×C3⋊(C3⋊Dic3)—21no—
53#6308—21no—
54#6308D412yes—
55C2×C3×C3⋊Dic3—21no—
56C2×C3×C3⋊Dic3—21no—
57C3×#634—21no—
58C3×#634D412yes—
59C2×C32×Dic3—21no—
60C2×C32×Dic3D411yescharacteristic
61C32×C4×D3—21no—
62C32×C4×D3—21no—
63C32×Dic6—21no—
64C32×Dic6—21no—
65#6293—24no—
66D3×C3⋊Dic3—24no—
67C3×C3⋊(C3⋊C8)—21no—
68C32×C3⋊C8—21no—
69#5186—91no—
70#5050—91no—
71#4739—91no—
72C2×C32×Q8—31no—
73C22×C32×C4—31no—
74C22×C3⋊Dic3—61no—
75C2×#634—31no—
76C2×#634—31no—
77C2×#634—31no—
78C2×#634—62no—
79C2×#634—61no—
80C2×#634—61no—
81C2×#634—62no—
82C2×#634—61no—
83C2×#634—61no—
84C2×#634—121no—
85C2×#634C2×D311yescharacteristic
86C2×C3×C4×D3—21no—
87C2×C3×C4×D3C2×D311yescharacteristic
88C2×C3×C4×D3C2×D311yescharacteristic
89C2×C3×Dic6—21no—
90C2×C3×Dic6—31no—
91C2×C3×Dic6—31no—
92C2×C3×Dic6—61no—
93C2×C3×Dic6C2×D311yescharacteristic
94C2×C3×Dic6C2×D311yescharacteristic
95C2×#627—62no—
96C2×#627—61no—
97C2×#627—61no—
98C2×D3×Dic3—62no—
99C2×D3×Dic3—61no—
100C2×D3×Dic3—61no—
101D3×Dic6—62no—
102D3×Dic6—62no—
103D3×Dic6—62no—
104D3×Dic6—62no—
105D3×Dic6—64no—
106D3×Dic6—64no—
107C2×C32×C8—31no—
108#3756—62no—
109C4⋊(C3⋊Dic3)C2×D311yescharacteristic
110C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
111C3×#217—61no—
112C3×#216—62no—
113C3×(C2×D3)⋊C4—61no—
114C3×C4⋊Dic3—31no—
115C3×C4⋊Dic3—31no—
116C3×C4⋊Dic3—61no—
117C3×Dic3⋊C4—61no—
118C3×Dic3⋊C4—61no—
119C2×C3×C3⋊C8—21no—
120C2×C3×C3⋊C8—31no—
121C2×C3×C3⋊C8—31no—
122C2×C3×C3⋊C8—61no—
123C2×C3×C3⋊C8C2×D311yescharacteristic
124C2×C3×C3⋊C8C2×D311yescharacteristic
125C3×Dic12—62no—
126#3724—21no—
127(C2×D3)⋊Dic3—21no—
128#3719—22no—
129#3719—62no—
130#3719—62no—
131#3719—64no—
132C22×C33C2×D411yescharacteristic
133C2×C32×D3—21no—
134C2×C32×D3—21no—
135C2×C32×D3Q1612yescomplemented
136C33×C4C2×D411yescharacteristic
137C33×C4C2×D411yescharacteristic
138C3⋊(C3⋊Dic3)—22no—
139C3×C3⋊Dic3—22no—
140C3×C3⋊Dic3—41no—
141C32×Dic3—21no—
142C32×Dic3—41no—
143C2×D3×Q8—91no—
144C22×Dic6—91no—
145C22×Dic6—91no—
146C22×Dic6—181no—
147C2×C3×Q16—91no—
148C3×C22⋊Q8—91no—
149#946—31no—
150C2×#216—31no—
151C2×#216—91no—
152C2×#216—91no—
153C2×#216—181no—
154#924—31no—
155C2×Dic12—31no—
156C2×Dic12—91no—
157C3×C22⋊C8—91no—
158#835—91no—
159#830—31no—
160(C2×D3)⋊C8—31no—
161#816—91no—
162#816—91no—
163#816—91no—
164#816—181no—
165#810—91no—
166#810—91no—
167C23×C32—31no—
168C22×C3×D3—21no—
169C22×C3×D3D1211yescharacteristic
170C22×C3×D3#6711yescharacteristic
171C32×Q8—61no—
172C32×Q8—61no—
173C2×C32×C4—21no—
174C2×C32×C4—21no—
175C2×C32×C4—21no—
176C2×C32×C4—31no—
177C2×C32×C4—41no—
178C2×C32×C4—61no—
179C2×C32×C4—61no—
180C2×C32×C4—61no—
181C2×C32×C4C22×D311yescharacteristic
182C2×C32×C4C22×D311yescharacteristic
183C2×C32×C4C22×D311yescharacteristic
184C2×C3⋊Dic3—21no—
185C2×C3⋊Dic3—21no—
186C2×C3⋊Dic3—64no—
187C2×C3⋊Dic3—61no—
188C2×C3⋊Dic3—62no—
189C2×C3⋊Dic3—61no—
190C2×C3⋊Dic3—61no—
191C2×C3⋊Dic3—121no—
192C2×C3⋊Dic3—121no—
193C2×C3⋊Dic3—122no—
194#634—21no—
195#634—32no—
196#634—32no—
197#634—32no—
198#634—62no—
199#634—62no—
200#634—62no—
201#634—64no—
202#634—61no—
203#634—62no—
204#634—61no—
205#634—61no—
206#634—122no—
207#634—121no—
208#634—121no—
209#634—122no—
210#634#6712yes—
211C2×C3×Dic3—21no—
212C2×C3×Dic3—21no—
213C2×C3×Dic3—21no—
214C2×C3×Dic3—41no—
215C2×C3×Dic3—62no—
216C2×C3×Dic3—61no—
217C2×C3×Dic3—61no—
218C2×C3×Dic3—61no—
219C2×C3×Dic3—61no—
220C2×C3×Dic3—121no—
221C2×C3×Dic3D1211yescharacteristic
222C2×C3×Dic3#6711yescharacteristic
223C3×C4×D3—22no—
224C3×C4×D3—22no—
225C3×C4×D3—21no—
226C3×C4×D3—21no—
227C3×C4×D3—21no—
228C3×C4×D3—21no—
229C3×Dic6—22no—
230C3×Dic6—21no—
231C3×Dic6—21no—
232C3×Dic6—21no—
233C3×Dic6—21no—
234C3×Dic6—32no—
235C3×Dic6—32no—
236C3×Dic6—41no—
237C3×Dic6—62no—
238C3×Dic6—62no—
239C3×Dic6—61no—
240C3×Dic6—61no—
241#627—68no—
242#627—64no—
243#627—64no—
244D3×Dic3—68no—
245D3×Dic3—64no—
246D3×Dic3—64no—
247C32×C8—61no—
248C3⋊(C3⋊C8)—21no—
249C3×C3⋊C8—21no—
250C3×C3⋊C8—21no—
251C3×C3⋊C8—41no—
252C3×C3⋊C8—62no—
253C3×C3⋊C8—61no—
254C3×C3⋊C8—61no—
255#451—271noSylow
256C2×C33#12011yescharacteristic
257C2×C33C2×Q1611yescharacteristic
258C2×C33#13711yescharacteristic
259C32×D3—22no—
260C2×C3×Q8—31no—
261C2×C3×Q8—31no—
262C2×C3×Q8—61no—
263C2×C3×Q8—91no—
264C22×C3×C4—31no—
265C22×C3×C4—31no—
266C22×C3×C4—61no—
267C22×Dic3—181no—
268C22×Dic3—181no—
269C22×Dic3—182no—
270D3×Q8—182no—
271D3×Q8—182no—
272C2×C4×D3—181no—
273C2×C4×D3D3211yescharacteristic
274C2×Dic6—31no—
275C2×Dic6—31no—
276C2×Dic6—61no—
277C2×Dic6—91no—
278C2×Dic6—91no—
279C2×Dic6—91no—
280C2×Dic6—181no—
281C2×Dic6—181no—
282C2×Dic6—181no—
283C2×Dic6—182no—
284C2×Dic6—182no—
285C2×Dic6—182no—
286C2×Dic6—182no—
287C2×Dic6—184no—
288C2×Dic6—184no—
289C2×Dic6—181no—
290C2×Dic6—181no—
291C2×Dic6—182no—
292C2×Dic6—181no—
293C2×Dic6—361no—
294C2×Dic6D3211yescharacteristic
295C3×Q16—182no—
296C2×C3×C8—31no—
297C2×C3×C8—31no—
298C2×C3×C8—61no—
299C2×C3×C8—91no—
300C3×C4⋊C4—91no—
301C3×C4⋊C4—181no—
302C3×C22⋊C4—181no—
303#217—61no—
304#216—62no—
305#216—182no—
306#216—182no—
307#216—184no—
308(C2×D3)⋊C4—61no—
309C4⋊Dic3—31no—
310C4⋊Dic3—31no—
311C4⋊Dic3—61no—
312Dic3⋊C4—61no—
313Dic3⋊C4—61no—
314C2×C3⋊C8—31no—
315C2×C3⋊C8—31no—
316C2×C3⋊C8—61no—
317C2×C3⋊C8D3211yescharacteristic
318Dic12—62no—
319Dic12—182no—
320C22×C32—21no—
321C22×C32—21no—
322C22×C32—21no—
323C22×C32—32no—
324C22×C32—41no—
325C22×C32—61no—
326C22×C32—61no—
327C22×C32C2×D1211yescharacteristic
328C22×C32D3×D411yescharacteristic
329C22×C32C2×#6711yescharacteristic
330C2×C3×D3—22no—
331C2×C3×D3—22no—
332C2×C3×D3—21no—
333C2×C3×D3—21no—
334C2×C3×D3—21no—
335C2×C3×D3—21no—
336C2×C3×D3—22no—
337C2×C3×D3Dic1212yescomplemented
338C2×C3×D3#21612yescomplemented
339C32×C4—21no—
340C32×C4—21no—
341C32×C4—21no—
342C32×C4—21no—
343C32×C4—21no—
344C32×C4—21no—
345C32×C4—41no—
346C32×C4—41no—
347C32×C4—61no—
348C32×C4—121no—
349C32×C4D3×D411yescharacteristic
350C32×C4D3×D411yescharacteristic
351C32×C4D3×D411yescharacteristic
352C32×C4D3×D411yescharacteristic
353C32×C4D3×D411yescharacteristic
354C32×C4C2×#6711yescharacteristic
355C3⋊Dic3—22no—
356C3⋊Dic3—41no—
357C3⋊Dic3—62no—
358C3⋊Dic3—64no—
359C3⋊Dic3—62no—
360C3⋊Dic3—62no—
361C3⋊Dic3—122no—
362C3⋊Dic3—122no—
363C3⋊Dic3—124no—
364C3×Dic3—22no—
365C3×Dic3—21no—
366C3×Dic3—21no—
367C3×Dic3—42no—
368C3×Dic3—41no—
369C3×Dic3—41no—
370C3×Dic3—64no—
371C3×Dic3—62no—
372C3×Dic3—62no—
373C3×Dic3—121no—
374C3×Dic3—121no—
375C3×Dic3—241no—
376C22×Q8—271no—
377C2×Q16—91no—
378C2×Q16—271no—
379C22⋊Q8—91no—
380#103—271no—
381#103—271no—
382C22⋊C8—91no—
383C33#45111yescharacteristic, complemented, Sylow
384C23×C3—31no—
385C23×C3—31no—
386C23×C3—61no—
387C22×D3#62611yescharacteristic
388C3×Q8—62no—
389C3×Q8—61no—
390C3×Q8—61no—
391C3×Q8—61no—
392C3×Q8—61no—
393C3×Q8—92no—
394C3×Q8—121no—
395C3×Q8—181no—
396C2×C3×C4—21no—
397C2×C3×C4—21no—
398C2×C3×C4—21no—
399C2×C3×C4—31no—
400C2×C3×C4—31no—
401C2×C3×C4—41no—
402C2×C3×C4—61no—
403C2×C3×C4—62no—
404C2×C3×C4—62no—
405C2×C3×C4—61no—
406C2×C3×C4—61no—
407C2×C3×C4—61no—
408C2×C3×C4—61no—
409C2×C3×C4—61no—
410C2×C3×C4—61no—
411C2×C3×C4—121no—
412C2×C3×C4—181no—
413C2×C3×C4—181no—
414C2×C3×C4C2×D3211yescharacteristic
415C2×C3×C4C2×D3211yescharacteristic
416C2×C3×C4C2×D3211yescharacteristic
417C2×Dic3—21no—
418C2×Dic3—62no—
419C2×Dic3—61no—
420C2×Dic3—61no—
421C2×Dic3—61no—
422C2×Dic3—61no—
423C2×Dic3—121no—
424C2×Dic3—188no—
425C2×Dic3—184no—
426C2×Dic3—184no—
427C2×Dic3—182no—
428C2×Dic3—181no—
429C2×Dic3—181no—
430C2×Dic3—181no—
431C2×Dic3—361no—
432C2×Dic3—361no—
433C2×Dic3—362no—
434C2×Dic3#62611yescharacteristic
435C4×D3—21no—
436C4×D3—21no—
437C4×D3—184no—
438Dic6—21no—
439Dic6—21no—
440Dic6—32no—
441Dic6—32no—
442Dic6—62no—
443Dic6—62no—
444Dic6—61no—
445Dic6—61no—
446Dic6—92no—
447Dic6—92no—
448Dic6—92no—
449Dic6—182no—
450Dic6—182no—
451Dic6—182no—
452Dic6—184no—
453Dic6—188no—
454Dic6—184no—
455Dic6—184no—
456Dic6—182no—
457Dic6—181no—
458Dic6—181no—
459Dic6—181no—
460Dic6—362no—
461Dic6—361no—
462Dic6—361no—
463Dic6—362no—
464C3×C8—61no—
465C3×C8—61no—
466C3×C8—121no—
467C3×C8—181no—
468C3⋊C8—21no—
469C3⋊C8—62no—
470C3⋊C8—61no—
471C3⋊C8—61no—
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—62no—
485C2×C32#88211yescharacteristic
486C2×C32C2×Dic1211yescharacteristic
487C2×C32#90911yescharacteristic
488C2×C32D3×Q1611yescharacteristic
489C2×C32#91811yescharacteristic
490C2×C32#93711yescharacteristic
491C2×C32#93711yescharacteristic
492C2×C32C2×#21611yescharacteristic
493C2×C32#95111yescharacteristic
494C3×D3—24no—
495C3×D3—22no—
496C3×D3—22no—
497C2×Q8—31no—
498C2×Q8—91no—
499C2×Q8—271no—
500C2×Q8—541no—
501C2×Q8—542no—
502C2×Q8—542no—
503C22×C4—31no—
504C22×C4—541no—
505Q16—182no—
506Q16—542no—
507C2×C8—31no—
508C2×C8—91no—
509C4⋊C4—91no—
510C4⋊C4—181no—
511C22⋊C4—181no—
512C22×C3—21no—
513C22×C3—21no—
514C22×C3—21no—
515C22×C3—32no—
516C22×C3—32no—
517C22×C3—41no—
518C22×C3—62no—
519C22×C3—62no—
520C22×C3—62no—
521C22×C3—61no—
522C22×C3—61no—
523C22×C3—61no—
524C22×C3—61no—
525C22×C3D3×D1211yescharacteristic
526C22×C3C2×#62611yescharacteristic
527C22×C3D3×#6711yescharacteristic
528C2×D3—21no—
529C2×D3—21no—
530C2×D3#371912yescomplemented
531C3×C4—21no—
532C3×C4—21no—
533C3×C4—21no—
534C3×C4—21no—
535C3×C4—21no—
536C3×C4—21no—
537C3×C4—41no—
538C3×C4—41no—
539C3×C4—62no—
540C3×C4—61no—
541C3×C4—61no—
542C3×C4—122no—
543C3×C4—121no—
544C3×C4—121no—
545C3×C4—182no—
546C3×C4—361no—
547C3×C4#380211yescharacteristic
548C3×C4D3×#6711yescharacteristic
549C3×C4D3×#6711yescharacteristic
550C3×C4#381111yescharacteristic
551C3×C4#381111yescharacteristic
552C3×C4#381111yescharacteristic
553Dic3—21no—
554Dic3—41no—
555Dic3—64no—
556Dic3—62no—
557Dic3—62no—
558Dic3—121no—
559Dic3—121no—
560Dic3—184no—
561Dic3—182no—
562Dic3—182no—
563Dic3—182no—
564Dic3—241no—
565Dic3—362no—
566Dic3—362no—
567Dic3—364no—
568C32—21no—
569C32—21no—
570C32—21no—
571C32—41no—
572C32#473911yescharacteristic, complemented
573C32#505011yescharacteristic, complemented
574C32#518611yescharacteristic, complemented
575C23—31no—
576Q8—61no—
577Q8—61no—
578Q8—92no—
579Q8—181no—
580Q8—272no—
581Q8—544no—
582Q8—541no—
583C2×C4—31no—
584C2×C4—61no—
585C2×C4—61no—
586C2×C4—61no—
587C2×C4—181no—
588C2×C4—181no—
589C2×C4—544no—
590C2×C4—541no—
591C2×C4D3311yescharacteristic
592C8—61no—
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—64no—
607C2×C3—62no—
608C2×C3—62no—
609C2×C3D3×Dic1211yescharacteristic
610C2×C3#6380111yescharacteristic
611C2×C3#6383111yescharacteristic
612C2×C3C2×#371911yescharacteristic
613C2×C3#6392211yescharacteristic
614C2×C3D3×#21611yescharacteristic
615C2×C3#6394311yescharacteristic
616C2×C3#6398111yescharacteristic
617C2×C3#6398311yescharacteristic
618D3—22no—
619C22—32no—
620C22—61no—
621C22—61no—
622C22D3×#62611yescharacteristic
623C4—61no—
624C4—121no—
625C4—182no—
626C4—361no—
627C4—542no—
628C4#8895211yescharacteristic
629C4#8895411yescharacteristic
630C3—21no—
631C3—21no—
632C3—21no—
633C3—41no—
634C3#1059094911yescharacteristic, complemented
635C3#1059119011yescharacteristic, complemented
636C3#1059165311yescharacteristic, complemented
637C2—62no—
638C2D3×#371911yescharacteristic
639C2#1172288711yescharacteristic
640C2#1172294911yescharacteristic
subgroupG/H
centerC22D3×#626
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radicalC22×C32×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
2. C2×#88799C2semidirect
4. C2×C3×#3719C2semidirect
5. #11722076C2semidirect
6. #11722042C2semidirect
39. C2×#3719D3semidirect
135. C2×C32×D3Q16semidirect
337. C2×C3×D3Dic12semidirect
338. C2×C3×D3#216semidirect
383. C33#451semidirect
530. C2×D3#3719semidirect
572. C32#4739semidirect
573. C32#5050semidirect
574. C32#5186semidirect
634. C3#10590949semidirect
635. C3#10591190semidirect
636. C3#10591653semidirect

Maximal subgroups

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

subgroupindex
C2×D3×#6342
C2×#887992
C3×#638562
C2×C3×#37192
#117220762
#117220422
C3×(C2×D3)⋊(C3⋊C8)2
#105916533
#105911903
#105909493