Finite Group Databasebeta

← #13172242

Subgroups — #13172242

Order 1728.

1198 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723721C211yesmaximal, characteristic
2#11723642C212yesmaximal
3C2×Dic3×C3⋊Dic3C211yesmaximal, characteristic
4C3×#63982C211yesmaximal, characteristic
5C3×#63955C212yesmaximal
6#11723037C211yesmaximal, characteristic, complemented
7#11723035C211yesmaximal, characteristic, complemented
8#11723013C212yesmaximal, complemented
9#11722944C211yesmaximal, characteristic, complemented
10#11722942C212yesmaximal, complemented
11Dic3×#627C211yesmaximal, characteristic, complemented
12#10595050—31nomaximal
13#10594029—32nomaximal
14C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
15C22×C3×C3⋊Dic3C2211yescharacteristic
16C22×C32×Dic3C2211yescharacteristic
17C2×#6296C2211yescharacteristic
18C2×#6293C2212yes—
19C2×C3×#627C2211yescharacteristic
20C3×#3757C2212yes—
21C32×#217C2212yes—
22#88815C2212yes—
23#88815C2211yescharacteristic
24#88807C2212yes—
25#88806C2212yes—
26Dic3⋊(C3⋊Dic3)C2212yes—
27Dic3×C3⋊Dic3C2211yescharacteristic
28Dic3×C3⋊Dic3C2211yescharacteristic
29Dic3×C3⋊Dic3C2211yescharacteristic
30Dic3×C3⋊Dic3C2211yescharacteristic
31C3×#3724C2211yescharacteristic
32C3×Dic3⋊Dic3C2212yes—
33C3×Dic3⋊Dic3C2212yes—
34C3×Dic3⋊Dic3C2212yes—
35C3×Dic32C2212yes—
36C3×Dic32C2212yes—
37C2×C4×C3⋊Dic3—31no—
38#64087—32no—
39C3×#926—32no—
40C3×#878—32no—
41C3×#877—32no—
42#63982—31no—
43#63982—32no—
44#63982—32no—
45#63982—62no—
46#63982D311yescharacteristic
47#63964—32no—
48#63955D312yes—
49C2×Dic32—32no—
50C2×Dic32—61no—
51#63919—31no—
52#63919—31no—
53C4×#627—31no—
54C4×#627—31no—
55C4×#627—32no—
56#63872—32no—
57#63846—32no—
58#63845—32no—
59#63844—32no—
60Dic3×Dic6—32no—
61#63840—32no—
62#63839—32no—
63#63837—32no—
64#63837—32no—
65C23×C33C2311yescharacteristic
66C2×C3⋊(C3⋊Dic3)—22no—
67C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
68C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
69C2×C3×C3⋊Dic3—22no—
70C2×C3×C3⋊Dic3C2312yes—
71C2×C3×C3⋊Dic3C2312yes—
72C2×C3×C3⋊Dic3C2311yescharacteristic
73C2×C3×C3⋊Dic3C2311yescharacteristic
74C2×C32×Dic3Q814yescomplemented
75C2×C32×Dic3C2312yes—
76C2×C32×Dic3C2312yes—
77C2×C32×Dic3C2311yescharacteristic
78C2×C32×Dic3C2311yescharacteristic
79#6296—22no—
80#6293—24no—
81C3×#627—22no—
82#5622—91no—
83#5517—92no—
84C22×C32×C4—31no—
85C22×C3⋊Dic3—32no—
86C22×C3⋊Dic3—31no—
87C22×C3⋊Dic3—61no—
88C22×C3⋊Dic3—62no—
89C22×C3⋊Dic3—121no—
90C22×C3⋊Dic3C2×D311yescharacteristic
91C2×#634—32no—
92C22×C3×Dic3—21no—
93C22×C3×Dic3—32no—
94C22×C3×Dic3—61no—
95C22×C3×Dic3C2×D312yes—
96C2×C3×Dic6—32no—
97C2×#627—32no—
98C2×#627—31no—
99C2×#627—32no—
100C2×#627—32no—
101C2×#627—62no—
102C2×#627C2×D311yescharacteristic
103C32×C22⋊C4—32no—
104#3757C2×D312yes—
105C4⋊(C3⋊Dic3)—32no—
106#3750—32no—
107#3750—32no—
108C4×C3⋊Dic3—31no—
109C4×C3⋊Dic3—31no—
110C4×C3⋊Dic3—31no—
111C4×C3⋊Dic3—31no—
112C3×#217—22no—
113C3×#217—32no—
114C3×#217—32no—
115C3×#217—62no—
116C3×#217C2×D312yes—
117C3×#217C2×D312yes—
118C3×C4⋊Dic3—32no—
119C3×C4⋊Dic3—32no—
120C3×C4⋊Dic3—32no—
121C3×Dic3⋊C4—32no—
122C3×Dic3⋊C4—32no—
123C3×Dic3⋊C4—32no—
124C3×Dic3⋊C4—32no—
125C3×C4×Dic3—32no—
126C3×C4×Dic3—32no—
127C3×C4×Dic3—32no—
128C3×C4×Dic3—32no—
129#3724—32no—
130#3724—31no—
131#3724—32no—
132#3724—32no—
133#3724—62no—
134#3724C2×D311yescharacteristic
135Dic3⋊Dic3—32no—
136Dic3⋊Dic3—32no—
137Dic3⋊Dic3—32no—
138Dic3⋊Dic3—32no—
139Dic3⋊Dic3—32no—
140Dic3⋊Dic3—32no—
141Dic3⋊Dic3—32no—
142Dic3⋊Dic3—62no—
143Dic3⋊Dic3—62no—
144Dic3⋊Dic3C2×D312yes—
145Dic3⋊Dic3C2×D312yes—
146Dic3⋊Dic3C2×D312yes—
147Dic32—32no—
148Dic32—32no—
149Dic32—32no—
150Dic32—32no—
151Dic32—61no—
152Dic32—61no—
153Dic32—61no—
154Dic32—61no—
155Dic32C2×D312yes—
156Dic32C2×D312yes—
157C22×C33—21no—
158C22×C33—21no—
159C22×C33C2×Q812yes—
160C22×C33C2411yescharacteristic
161C3⋊(C3⋊Dic3)—21no—
162C3⋊(C3⋊Dic3)#4112yes—
163C3×C3⋊Dic3—21no—
164C3×C3⋊Dic3—22no—
165C3×C3⋊Dic3—22no—
166C3×C3⋊Dic3#4112yes—
167C32×Dic3—22no—
168C32×Dic3—22no—
169C32×Dic3C2×Q812yes—
170C32×Dic3C2×Q812yes—
171C3×C22⋊Q8—91no—
172C3×#117—92no—
173Q8×Dic3—91no—
174#944—91no—
175#926—32no—
176#924—91no—
177C2×C4×Dic3—92no—
178C2×C4×Dic3—91no—
179C2×C4×Dic3—181no—
180#890—92no—
181#890—92no—
182#889—92no—
183#888—91no—
184#888—92no—
185#888—92no—
186#887—91no—
187#887—92no—
188#878—32no—
189#878—92no—
190#878—92no—
191#878—92no—
192#878—92no—
193#878—182no—
194#877—32no—
195C4×Dic6—92no—
196C4×Dic6—92no—
197C4×Dic6—92no—
198C4×Dic6—92no—
199C23×C32—21no—
200C23×C32—22no—
201C23×C32—41no—
202C23×C32C22×D312yes—
203C23×C32C22×D311yescharacteristic
204C2×C32×C4—34no—
205C2×C32×C4—31no—
206C2×C32×C4—31no—
207C2×C32×C4—32no—
208C2×C32×C4—32no—
209C2×C3⋊Dic3—22no—
210C2×C3⋊Dic3—32no—
211C2×C3⋊Dic3—31no—
212C2×C3⋊Dic3—32no—
213C2×C3⋊Dic3—31no—
214C2×C3⋊Dic3—61no—
215C2×C3⋊Dic3—61no—
216C2×C3⋊Dic3—62no—
217C2×C3⋊Dic3—62no—
218C2×C3⋊Dic3—62no—
219C2×C3⋊Dic3—64no—
220C2×C3⋊Dic3—64no—
221C2×C3⋊Dic3—121no—
222C2×C3⋊Dic3—121no—
223C2×C3⋊Dic3—124no—
224C2×C3⋊Dic3—124no—
225C2×C3⋊Dic3C22×D312yes—
226C2×C3⋊Dic3C22×D312yes—
227C2×C3⋊Dic3C22×D311yescharacteristic
228C2×C3⋊Dic3C22×D311yescharacteristic
229#634—64no—
230C2×C3×Dic3—24no—
231C2×C3×Dic3—21no—
232C2×C3×Dic3—21no—
233C2×C3×Dic3—22no—
234C2×C3×Dic3—22no—
235C2×C3×Dic3—32no—
236C2×C3×Dic3—32no—
237C2×C3×Dic3—32no—
238C2×C3×Dic3—32no—
239C2×C3×Dic3—32no—
240C2×C3×Dic3—32no—
241C2×C3×Dic3—61no—
242C2×C3×Dic3—61no—
243C2×C3×Dic3—62no—
244C2×C3×Dic3—62no—
245C2×C3×Dic3—64no—
246C2×C3×Dic3—64no—
247C2×C3×Dic3Dic618yescomplemented
248C2×C3×Dic3C22×D312yes—
249C2×C3×Dic3C22×D312yes—
250C2×C3×Dic3C22×D312yes—
251C2×C3×Dic3C22×D312yes—
252C2×C3×Dic3C22×D312yes—
253C2×C3×Dic3C22×D312yes—
254C3×Dic6—64no—
255#627—22no—
256#627—64no—
257#627—64no—
258#627—64no—
259#627—62no—
260#627—124no—
261#533—271noSylow
262C2×C33—22no—
263C2×C33C22×Q811yescharacteristic
264C2×C33C2×#4111yescharacteristic
265C2×C33C2×#4111yescharacteristic
266C2×C3×Q8—91no—
267C22×C3×C4—32no—
268C22×C3×C4—61no—
269C22×C3×C4—91no—
270C22×Dic3—32no—
271C22×Dic3—61no—
272C22×Dic3—91no—
273C22×Dic3—92no—
274C22×Dic3—181no—
275C22×Dic3—182no—
276C22×Dic3—361no—
277C22×Dic3D3211yescharacteristic
278C2×Dic6—32no—
279C2×Dic6—92no—
280C2×Dic6—91no—
281C2×Dic6—92no—
282C2×Dic6—92no—
283C2×Dic6—92no—
284C2×Dic6—182no—
285C3×C4⋊C4—91no—
286C3×C4⋊C4—92no—
287C3×C4⋊C4—92no—
288C3×C4⋊C4—92no—
289C3×C22⋊C4—32no—
290C3×C22⋊C4—32no—
291C3×C22⋊C4—62no—
292C3×C22⋊C4—92no—
293C3×C42—92no—
294C3×C42—92no—
295#217—32no—
296#217—32no—
297#217—62no—
298#217D3212yes—
299C4⋊Dic3—32no—
300C4⋊Dic3—32no—
301C4⋊Dic3—32no—
302C4⋊Dic3—92no—
303C4⋊Dic3—91no—
304C4⋊Dic3—92no—
305C4⋊Dic3—92no—
306C4⋊Dic3—92no—
307C4⋊Dic3—182no—
308Dic3⋊C4—32no—
309Dic3⋊C4—32no—
310Dic3⋊C4—32no—
311Dic3⋊C4—32no—
312Dic3⋊C4—92no—
313Dic3⋊C4—92no—
314Dic3⋊C4—92no—
315Dic3⋊C4—92no—
316Dic3⋊C4—92no—
317Dic3⋊C4—92no—
318Dic3⋊C4—92no—
319Dic3⋊C4—92no—
320Dic3⋊C4—92no—
321Dic3⋊C4—182no—
322Dic3⋊C4—182no—
323C4×Dic3—32no—
324C4×Dic3—32no—
325C4×Dic3—32no—
326C4×Dic3—32no—
327C4×Dic3—92no—
328C4×Dic3—92no—
329C4×Dic3—92no—
330C4×Dic3—92no—
331C4×Dic3—91no—
332C4×Dic3—91no—
333C4×Dic3—91no—
334C4×Dic3—91no—
335C4×Dic3—181no—
336C4×Dic3—181no—
337C4×Dic3—181no—
338C4×Dic3—181no—
339C22×C32—21no—
340C22×C32—21no—
341C22×C32—22no—
342C22×C32—22no—
343C22×C32—22no—
344C22×C32—22no—
345C22×C32—22no—
346C22×C32—21no—
347C22×C32—24no—
348C22×C32—22no—
349C22×C32—42no—
350C22×C32—41no—
351C22×C32—42no—
352C22×C32—42no—
353C22×C32—42no—
354C22×C32—42no—
355C22×C32C2×Dic614yes—
356C22×C32D3×Q812yes—
357C22×C32C23×D312yes—
358C22×C32C23×D311yescharacteristic
359C32×C4—32no—
360C32×C4—32no—
361C32×C4—62no—
362C32×C4—62no—
363C3⋊Dic3—21no—
364C3⋊Dic3—22no—
365C3⋊Dic3—22no—
366C3⋊Dic3—32no—
367C3⋊Dic3—34no—
368C3⋊Dic3—64no—
369C3⋊Dic3—62no—
370C3⋊Dic3—61no—
371C3⋊Dic3—62no—
372C3⋊Dic3—62no—
373C3⋊Dic3—122no—
374C3⋊Dic3—122no—
375C3⋊Dic3—122no—
376C3⋊Dic3#23712yes—
377C3×Dic3—22no—
378C3×Dic3—22no—
379C3×Dic3—22no—
380C3×Dic3—22no—
381C3×Dic3—22no—
382C3×Dic3—22no—
383C3×Dic3—34no—
384C3×Dic3—42no—
385C3×Dic3—42no—
386C3×Dic3—62no—
387C3×Dic3—62no—
388C3×Dic3—62no—
389C3×Dic3—62no—
390C3×Dic3—62no—
391C3×Dic3—62no—
392C3×Dic3—62no—
393C3×Dic3—122no—
394C3×Dic3—122no—
395C3×Dic3C2×Dic614yes—
396C3×Dic3C2×Dic614yes—
397C4⋊Q8—271no—
398C4⋊Q8—271no—
399#125—272no—
400C22⋊Q8—91no—
401C22⋊Q8—272no—
402C22⋊Q8—271no—
403C4×Q8—272no—
404C4×Q8—271no—
405C4×Q8—271no—
406#117—92no—
407C2×C42—271no—
408C33#53311yescharacteristic, complemented, Sylow
409C23×C3—21no—
410C23×C3—22no—
411C23×C3—41no—
412C23×C3C2×D3212yes—
413C23×C3C2×D3211yescharacteristic
414C3×Q8—182no—
415C2×C3×C4—32no—
416C2×C3×C4—32no—
417C2×C3×C4—38no—
418C2×C3×C4—32no—
419C2×C3×C4—32no—
420C2×C3×C4—32no—
421C2×C3×C4—32no—
422C2×C3×C4—61no—
423C2×C3×C4—61no—
424C2×C3×C4—64no—
425C2×C3×C4—62no—
426C2×C3×C4—62no—
427C2×C3×C4—91no—
428C2×C3×C4—91no—
429C2×C3×C4—92no—
430C2×C3×C4—92no—
431C2×C3×C4—182no—
432C2×Dic3—32no—
433C2×Dic3—32no—
434C2×Dic3—32no—
435C2×Dic3—32no—
436C2×Dic3—32no—
437C2×Dic3—32no—
438C2×Dic3—61no—
439C2×Dic3—61no—
440C2×Dic3—62no—
441C2×Dic3—62no—
442C2×Dic3—64no—
443C2×Dic3—64no—
444C2×Dic3—91no—
445C2×Dic3—92no—
446C2×Dic3—91no—
447C2×Dic3—92no—
448C2×Dic3—181no—
449C2×Dic3—181no—
450C2×Dic3—182no—
451C2×Dic3—182no—
452C2×Dic3—182no—
453C2×Dic3—184no—
454C2×Dic3—184no—
455C2×Dic3—361no—
456C2×Dic3—361no—
457C2×Dic3—364no—
458C2×Dic3—364no—
459C2×Dic3#62714yescomplemented
460C2×Dic3C2×D3212yes—
461C2×Dic3C2×D3212yes—
462C2×Dic3C2×D3211yescharacteristic
463C2×Dic3C2×D3211yescharacteristic
464Dic6—64no—
465Dic6—184no—
466Dic6—184no—
467Dic6—182no—
468Dic6—184no—
469Dic6—184no—
470Dic6—364no—
471C2×C32—22no—
472C2×C32—24no—
473C2×C32—24no—
474C2×C32—22no—
475C2×C32—21no—
476C2×C32—21no—
477C2×C32—22no—
478C2×C32—22no—
479C2×C32—21no—
480C2×C32—41no—
481C2×C32—41no—
482C2×C32—41no—
483C2×C32—44no—
484C2×C32—44no—
485C2×C32C22×Dic612yes—
486C2×C32C2×#23711yescharacteristic
487C2×C32C2×D3×Q811yescharacteristic
488C2×C32D3×#4112yes—
489C2×C32D3×#4112yes—
490C2×C32D3×#4111yescharacteristic
491C2×Q8—91no—
492C2×Q8—272no—
493C2×Q8—271no—
494C22×C4—31no—
495C22×C4—91no—
496C22×C4—271no—
497C4⋊C4—92no—
498C4⋊C4—92no—
499C4⋊C4—92no—
500C4⋊C4—91no—
501C4⋊C4—272no—
502C4⋊C4—272no—
503C4⋊C4—272no—
504C4⋊C4—272no—
505C4⋊C4—271no—
506C22⋊C4—32no—
507C22⋊C4—92no—
508C42—92no—
509C42—92no—
510C42—271no—
511C42—271no—
512C42—271no—
513C42—271no—
514C22×C3—22no—
515C22×C3—22no—
516C22×C3—21no—
517C22×C3—21no—
518C22×C3—22no—
519C22×C3—22no—
520C22×C3—21no—
521C22×C3—22no—
522C22×C3—22no—
523C22×C3—24no—
524C22×C3—42no—
525C22×C3—42no—
526C22×C3—42no—
527C22×C3—42no—
528C22×C3—41no—
529C22×C3—42no—
530C22×C3D3×Dic614yes—
531C22×C3C2×#62712yes—
532C22×C3C22×D3212yes—
533C22×C3C22×D3211yescharacteristic
534C3×C4—34no—
535C3×C4—34no—
536C3×C4—62no—
537C3×C4—62no—
538C3×C4—62no—
539C3×C4—62no—
540C3×C4—62no—
541C3×C4—62no—
542C3×C4—92no—
543C3×C4—122no—
544C3×C4—122no—
545C3×C4—181no—
546C3×C4—182no—
547C3×C4—182no—
548Dic3—22no—
549Dic3—22no—
550Dic3—34no—
551Dic3—62no—
552Dic3—62no—
553Dic3—62no—
554Dic3—62no—
555Dic3—62no—
556Dic3—62no—
557Dic3—62no—
558Dic3—92no—
559Dic3—94no—
560Dic3—122no—
561Dic3—122no—
562Dic3—184no—
563Dic3—182no—
564Dic3—182no—
565Dic3—181no—
566Dic3—182no—
567Dic3—362no—
568Dic3—362no—
569Dic3—362no—
570Dic3C2×#62712yes—
571Dic3C2×#62712yes—
572C32—21no—
573C32—22no—
574C32—41no—
575C32#551712yescomplemented
576C32#562211yescharacteristic, complemented
577C23D3311yescharacteristic
578Q8—182no—
579Q8—544no—
580Q8—542no—
581C2×C4—31no—
582C2×C4—31no—
583C2×C4—34no—
584C2×C4—32no—
585C2×C4—32no—
586C2×C4—91no—
587C2×C4—92no—
588C2×C4—92no—
589C2×C4—91no—
590C2×C4—182no—
591C2×C4—271no—
592C2×C4—271no—
593C2×C4—542no—
594C2×C3—24no—
595C2×C3—22no—
596C2×C3—24no—
597C2×C3—21no—
598C2×C3—22no—
599C2×C3—22no—
600C2×C3—21no—
601C2×C3—21no—
602C2×C3—22no—
603C2×C3—44no—
604C2×C3—44no—
605C2×C3—41no—
606C2×C3—41no—
607C2×C3—41no—
608C2×C3C2×D3×Dic612yes—
609C2×C3#6430611yescharacteristic
610C2×C3#6430611yescharacteristic
611C2×C3D3×#23712yes—
612C2×C3#6431212yes—
613C2×C3C22×#62711yescharacteristic
614C22—21no—
615C22—21no—
616C22D3×#62712yes—
617C22C2×D3311yescharacteristic
618C4—32no—
619C4—32no—
620C4—62no—
621C4—62no—
622C4—92no—
623C4—181no—
624C4—182no—
625C4—182no—
626C4—272no—
627C4—541no—
628C3—21no—
629C3—22no—
630C3—41no—
631C3#1059402912yescomplemented
632C3#1059505011yescharacteristic, complemented
633C2—22no—
634C2#1172440711yescharacteristic
635C2#1172441111yescharacteristic
636C2C2×D3×#62711yescharacteristic
subgroupG/H
center, FrattiniC22C2×D33
commutator, socle, cosocleC22×C33C24
Fitting, prime radicalC23×C33C23
hypercenterC23D33
hypocenterC33#533
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C23

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
6. #11723037C2semidirect
7. #11723035C2semidirect
8. #11723013C2semidirect
9. #11722944C2semidirect
10. #11722942C2semidirect
11. Dic3×#627C2semidirect
74. C2×C32×Dic3Q8semidirect
247. C2×C3×Dic3Dic6semidirect
408. C33#533semidirect
459. C2×Dic3#627semidirect
575. C32#5517semidirect
576. C32#5622semidirect
631. C3#10594029semidirect
632. C3#10595050semidirect

Maximal subgroups

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

subgroupindex
#117237212
#117236422
C2×Dic3×C3⋊Dic32
C3×#639822
C3×#639552
#117230372
#117230352
#117230132
#117229442
#117229422
Dic3×#6272
#105950503
#105940293