Finite Group Databasebeta

← #13168699

Subgroups — #13168699

Order 1728.

1156 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723679C211yesmaximal, characteristic, complemented
2#11723491C212yesmaximal, complemented
3Dic6×C3⋊Dic3C211yesmaximal, characteristic, complemented
4C3×#63902C211yesmaximal, characteristic
5C3×#63887C212yesmaximal, complemented
6#11723037C211yesmaximal, characteristic, complemented
7#11723015C211yesmaximal, characteristic, complemented
8#11722971C212yesmaximal, complemented
9#11722925C211yesmaximal, characteristic
10Dic3×#625C211yesmaximal, characteristic
11D3×Dic3⋊Dic3C212yesmaximal, complemented
12#10595407—31nomaximal
13#10594133—32nomaximal
14C2×C32×Dic6C2211yescharacteristic
15C2×#6293C2212yes—
16C2×D3×C3⋊Dic3C2212yes—
17C2×C3×#625C2211yescharacteristic
18C2×C3×D3×Dic3C2212yes—
19C2×C3×D3×Dic3C2212yes—
20C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
21C3×#3750C2212yes—
22C3×C4×C3⋊Dic3C2211yescharacteristic
23C32×(C2×D3)⋊C4C2212yes—
24#88815C2211yescharacteristic
25#88815C2211yescharacteristic
26#88815C2212yes—
27#88807C2212yes—
28Dic3⋊(C3⋊Dic3)C2211yescharacteristic
29Dic3⋊(C3⋊Dic3)C2211yescharacteristic
30(C2×D3)⋊(C3⋊Dic3)C2212yes—
31Dic3×C3⋊Dic3C2211yescharacteristic
32Dic3×C3⋊Dic3C2211yescharacteristic
33C3×#3724C2211yescharacteristic
34C3×Dic3⋊Dic3C2212yes—
35C3×Dic3⋊Dic3C2212yes—
36C3×(C2×D3)⋊Dic3C2212yes—
37Q8×C3⋊Dic3—31no—
38#64087—32no—
39C3×#946—32no—
40C3×#67⋊C4—32no—
41C3×#878—32no—
42#67⋊Dic3—32no—
43Dic3×#67—32no—
44C2×Dic3⋊Dic3—32no—
45#63964—32no—
46#63919—31no—
47#63902—31no—
48#63902D311yescharacteristic
49C4×#625—31no—
50#63893—32no—
51#63893—32no—
52#63893—32no—
53#63893—62no—
54D3×C4⋊Dic3—32no—
55#63887D312yescomplemented
56D3×Dic3⋊C4—32no—
57#63870—31no—
58#63870—31no—
59#63867—32no—
60#63857—32no—
61#63846—32no—
62Dic3×Dic6—32no—
63Dic3×Dic6—61no—
64#63840—32no—
65C22×C32×D3C2312yes—
66C2×C33×C4C2311yescharacteristic
67C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
68C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
69C2×C3×C3⋊Dic3C2311yescharacteristic
70C2×C3×C3⋊Dic3C2312yes—
71C2×C3×C3⋊Dic3C2312yes—
72C2×C3×C3⋊Dic3C2311yescharacteristic
73C2×C32×Dic3C2312yes—
74C2×C32×Dic3C2311yescharacteristic
75C2×C32×Dic3C2311yescharacteristic
76C32×Dic6—22no—
77#6293—42no—
78D3×C3⋊Dic3—42no—
79C3×#625Q814yescomplemented
80C3×D3×Dic3—24no—
81C3×D3×Dic3—42no—
82#5635—92no—
83#5539—91no—
84C2×C32×Q8—31no—
85C22×C3⋊Dic3—32no—
86C2×#634—32no—
87C2×C3×#67—32no—
88C22×C3×Dic3—32no—
89C22×C3×Dic3—32no—
90C2×C3×C4×D3—32no—
91C2×C3×C4×D3—32no—
92C2×C3×Dic6—21no—
93C2×C3×Dic6C2×D312yes—
94C2×#627—32no—
95C2×#627—32no—
96C2×#627—62no—
97C2×#625C2×D311yescharacteristic
98C2×D3×Dic3—32no—
99C2×D3×Dic3—32no—
100C2×D3×Dic3—62no—
101C2×D3×Dic3C2×D312yes—
102C2×D3×Dic3C2×D312yes—
103C32×C22⋊C4—32no—
104#3757—32no—
105C4⋊(C3⋊Dic3)—32no—
106C4⋊(C3⋊Dic3)—31no—
107C4⋊(C3⋊Dic3)—31no—
108C4⋊(C3⋊Dic3)—31no—
109C4⋊(C3⋊Dic3)—61no—
110C4⋊(C3⋊Dic3)—62no—
111C4⋊(C3⋊Dic3)—121no—
112#3750—32no—
113#3750C2×D312yes—
114C4×C3⋊Dic3—31no—
115C4×C3⋊Dic3—31no—
116C4×C3⋊Dic3C2×D311yescharacteristic
117C3×#217—32no—
118C3×(C2×D3)⋊C4—22no—
119C3×(C2×D3)⋊C4—32no—
120C3×(C2×D3)⋊C4C2×D312yes—
121C3×(C2×D3)⋊C4C2×D312yes—
122C3×C4⋊Dic3—32no—
123C3×C4⋊Dic3—32no—
124C3×Dic3⋊C4—32no—
125C3×Dic3⋊C4—32no—
126C3×Dic3⋊C4—32no—
127C3×Dic3⋊C4—32no—
128C3×Dic3⋊C4—32no—
129C3×Dic3⋊C4—62no—
130C3×C4×Dic3—32no—
131C3×C4×Dic3—61no—
132#3724—32no—
133#3724—31no—
134#3724—31no—
135#3724C2×D311yescharacteristic
136Dic3⋊Dic3—32no—
137Dic3⋊Dic3—32no—
138Dic3⋊Dic3—32no—
139Dic3⋊Dic3—32no—
140Dic3⋊Dic3—32no—
141Dic3⋊Dic3—32no—
142Dic3⋊Dic3—32no—
143Dic3⋊Dic3—32no—
144Dic3⋊Dic3—61no—
145Dic3⋊Dic3—61no—
146Dic3⋊Dic3—62no—
147Dic3⋊Dic3C2×D312yes—
148Dic3⋊Dic3C2×D312yes—
149(C2×D3)⋊Dic3—32no—
150(C2×D3)⋊Dic3—32no—
151(C2×D3)⋊Dic3—62no—
152(C2×D3)⋊Dic3C2×D312yes—
153Dic32—32no—
154Dic32—32no—
155Dic32—61no—
156Dic32—61no—
157C22×C33C2411yescharacteristic
158C2×C32×D3—22no—
159C2×C32×D3—22no—
160C2×C32×D3C2×Q814yes—
161C33×C4—21no—
162C3⋊(C3⋊Dic3)—21no—
163C3⋊(C3⋊Dic3)—21no—
164C3×C3⋊Dic3—22no—
165C3×C3⋊Dic3—22no—
166C3×C3⋊Dic3—21no—
167C3×C3⋊Dic3C2×Q812yes—
168C32×Dic3—21no—
169C32×Dic3—22no—
170C32×Dic3#4112yes—
171C3×C22⋊Q8—92no—
172C3×C4×D4—91no—
173#946—32no—
174Q8×Dic3—92no—
175Q8×Dic3—181no—
176D4×Dic3—91no—
177D4×Dic3—91no—
178C4×#67—92no—
179C2×C4⋊Dic3—92no—
180#924—92no—
181C2×Dic3⋊C4—92no—
182#897—92no—
183#897—92no—
184D3×C4⋊C4—92no—
185#890—91no—
186#889—92no—
187#889—92no—
188#888—91no—
189#888—92no—
190#67⋊C4—32no—
191#67⋊C4—92no—
192#878—32no—
193#878—92no—
194#878—92no—
195#878—92no—
196#878—182no—
197#870—91no—
198C4×Dic6—91no—
199C23×C32—32no—
200C22×C3×D3—22no—
201C22×C3×D3C22×D312yes—
202C22×C3×D3C22×D312yes—
203C32×Q8—62no—
204C2×C32×C4—21no—
205C2×C32×C4—22no—
206C2×C32×C4—31no—
207C2×C32×C4—32no—
208C2×C32×C4—31no—
209C2×C32×C4—41no—
210C2×C32×C4C22×D312yes—
211C2×C32×C4C22×D311yescharacteristic
212C2×C3⋊Dic3—32no—
213C2×C3⋊Dic3—32no—
214C2×C3⋊Dic3—31no—
215C2×C3⋊Dic3—31no—
216C2×C3⋊Dic3—61no—
217C2×C3⋊Dic3—61no—
218C2×C3⋊Dic3—62no—
219C2×C3⋊Dic3—62no—
220C2×C3⋊Dic3—121no—
221C2×C3⋊Dic3—121no—
222C2×C3⋊Dic3—122no—
223C2×C3⋊Dic3C22×D312yes—
224C2×C3⋊Dic3C22×D311yescharacteristic
225C2×C3⋊Dic3C22×D312yes—
226C2×C3⋊Dic3C22×D311yescharacteristic
227#634—122no—
228C3×#67—38no—
229C2×C3×Dic3—21no—
230C2×C3×Dic3—21no—
231C2×C3×Dic3—22no—
232C2×C3×Dic3—32no—
233C2×C3×Dic3—32no—
234C2×C3×Dic3—32no—
235C2×C3×Dic3—32no—
236C2×C3×Dic3—32no—
237C2×C3×Dic3—32no—
238C2×C3×Dic3—61no—
239C2×C3×Dic3—62no—
240C2×C3×Dic3—62no—
241C2×C3×Dic3—61no—
242C2×C3×Dic3—64no—
243C2×C3×Dic3—122no—
244C2×C3×Dic3C22×D312yes—
245C2×C3×Dic3C22×D312yes—
246C2×C3×Dic3C22×D312yes—
247C2×C3×Dic3C22×D312yes—
248C3×C4×D3—64no—
249C3×C4×D3—122no—
250C3×Dic6—24no—
251C3×Dic6—24no—
252#627—122no—
253#627—122no—
254#627—124no—
255#625Dic614yescomplemented
256D3×Dic3—24no—
257D3×Dic3—42no—
258D3×Dic3—122no—
259D3×Dic3—122no—
260D3×Dic3—124no—
261#554—271noSylow
262C2×C33C22×Q811yescharacteristic
263C2×C33C2×#4111yescharacteristic
264C2×C33H(5,𝔽2)11yescharacteristic
265C32×D3—24no—
266C2×C3×Q8—32no—
267C2×C3×Q8—61no—
268C2×C3×D4—91no—
269C22×C3×C4—92no—
270C22×C3×C4—92no—
271C2×#67—32no—
272C22×Dic3—32no—
273C22×Dic3—32no—
274C22×Dic3—92no—
275C22×Dic3—92no—
276C22×Dic3—182no—
277C2×C4×D3—32no—
278C2×C4×D3—32no—
279C2×C4×D3—92no—
280C2×Dic6—92no—
281C2×Dic6—92no—
282C2×Dic6—92no—
283C2×Dic6—182no—
284C2×Dic6D3211yescharacteristic
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—91no—
294#217—32no—
295#217—92no—
296#217—92no—
297#217—182no—
298(C2×D3)⋊C4—32no—
299(C2×D3)⋊C4—92no—
300(C2×D3)⋊C4D3212yes—
301C4⋊Dic3—32no—
302C4⋊Dic3—32no—
303C4⋊Dic3—91no—
304C4⋊Dic3—91no—
305C4⋊Dic3—92no—
306C4⋊Dic3—92no—
307C4⋊Dic3—92no—
308C4⋊Dic3—91no—
309C4⋊Dic3—92no—
310C4⋊Dic3—92no—
311C4⋊Dic3—181no—
312C4⋊Dic3—181no—
313C4⋊Dic3—182no—
314C4⋊Dic3—181no—
315C4⋊Dic3—361no—
316Dic3⋊C4—32no—
317Dic3⋊C4—32no—
318Dic3⋊C4—32no—
319Dic3⋊C4—32no—
320Dic3⋊C4—32no—
321Dic3⋊C4—62no—
322Dic3⋊C4—92no—
323Dic3⋊C4—92no—
324Dic3⋊C4—92no—
325Dic3⋊C4—92no—
326Dic3⋊C4—92no—
327Dic3⋊C4—92no—
328Dic3⋊C4—91no—
329Dic3⋊C4—91no—
330Dic3⋊C4—182no—
331C4×Dic3—32no—
332C4×Dic3—61no—
333C4×Dic3—92no—
334C4×Dic3—92no—
335C4×Dic3—91no—
336C4×Dic3—91no—
337C4×Dic3—181no—
338C4×Dic3—181no—
339C22×C32—21no—
340C22×C32—22no—
341C22×C32—34no—
342C22×C32—41no—
343C22×C32—62no—
344C22×C32—62no—
345C22×C32C23×D311yescharacteristic
346C22×C32C23×D312yes—
347C2×C3×D3—22no—
348C2×C3×D3—22no—
349C2×C3×D3—22no—
350C2×C3×D3—22no—
351C2×C3×D3—24no—
352C2×C3×D3—42no—
353C2×C3×D3—42no—
354C2×C3×D3C2×Dic614yes—
355C2×C3×D3D3×Q814yes—
356C32×C4—21no—
357C32×C4—22no—
358C32×C4—22no—
359C32×C4—32no—
360C32×C4—42no—
361C32×C4—42no—
362C32×C4—61no—
363C32×C4—62no—
364C3⋊Dic3—22no—
365C3⋊Dic3—22no—
366C3⋊Dic3—21no—
367C3⋊Dic3—61no—
368C3⋊Dic3—61no—
369C3⋊Dic3—62no—
370C3⋊Dic3—62no—
371C3⋊Dic3—122no—
372C3⋊Dic3—122no—
373C3⋊Dic3—121no—
374C3⋊Dic3—121no—
375C3⋊Dic3—241no—
376C3⋊Dic3—241no—
377C3⋊Dic3C2×Dic612yes—
378C3×Dic3—22no—
379C3×Dic3—22no—
380C3×Dic3—22no—
381C3×Dic3—22no—
382C3×Dic3—22no—
383C3×Dic3—34no—
384C3×Dic3—42no—
385C3×Dic3—62no—
386C3×Dic3—64no—
387C3×Dic3—62no—
388C3×Dic3—62no—
389C3×Dic3—62no—
390C3×Dic3—62no—
391C3×Dic3—62no—
392C3×Dic3—122no—
393C3×Dic3—121no—
394C3×Dic3#23714yes—
395C4⋊Q8—271no—
396#125—271no—
397#125—271no—
398C22⋊Q8—92no—
399C22⋊Q8—272no—
400C22⋊Q8—272no—
401C4×Q8—271no—
402C4×D4—91no—
403C4×D4—271no—
404C4×D4—271no—
405C2×C4⋊C4—272no—
406C33#55411yescharacteristic, complemented, Sylow
407C23×C3—32no—
408C23×C3—32no—
409C23×C3—62no—
410C22×D3C2×D3212yes—
411C3×Q8—64no—
412C3×Q8—64no—
413C3×D4—94no—
414C2×C3×C4—21no—
415C2×C3×C4—22no—
416C2×C3×C4—32no—
417C2×C3×C4—32no—
418C2×C3×C4—32no—
419C2×C3×C4—32no—
420C2×C3×C4—41no—
421C2×C3×C4—61no—
422C2×C3×C4—61no—
423C2×C3×C4—62no—
424C2×C3×C4—91no—
425C2×C3×C4—91no—
426C2×C3×C4—92no—
427C2×C3×C4—92no—
428C2×C3×C4—184no—
429C2×C3×C4—362no—
430C2×C3×C4C2×D3212yes—
431C2×C3×C4C2×D3211yescharacteristic
432#67—38no—
433C2×Dic3—32no—
434C2×Dic3—32no—
435C2×Dic3—32no—
436C2×Dic3—32no—
437C2×Dic3—32no—
438C2×Dic3—32no—
439C2×Dic3—61no—
440C2×Dic3—62no—
441C2×Dic3—61no—
442C2×Dic3—62no—
443C2×Dic3—64no—
444C2×Dic3—91no—
445C2×Dic3—92no—
446C2×Dic3—91no—
447C2×Dic3—92no—
448C2×Dic3—122no—
449C2×Dic3—181no—
450C2×Dic3—181no—
451C2×Dic3—182no—
452C2×Dic3—182no—
453C2×Dic3—361no—
454C2×Dic3—361no—
455C2×Dic3—362no—
456C2×Dic3—362no—
457C2×Dic3—364no—
458C2×Dic3C2×D3211yescharacteristic
459C2×Dic3C2×D3212yes—
460C2×Dic3C2×D3211yescharacteristic
461C4×D3—64no—
462C4×D3—122no—
463C4×D3—362no—
464Dic6—22no—
465Dic6—362no—
466Dic6—362no—
467Dic6—362no—
468Dic6—364no—
469C2×C32—22no—
470C2×C32—22no—
471C2×C32—22no—
472C2×C32—21no—
473C2×C32—21no—
474C2×C32—21no—
475C2×C32—41no—
476C2×C32—41no—
477C2×C32—41no—
478C2×C32—64no—
479C2×C32C22×Dic611yescharacteristic
480C2×C32C2×#23712yes—
481C2×C32#100411yescharacteristic
482C2×C32C2×D3×Q812yes—
483C2×C32D3×#4111yescharacteristic
484C2×C32#100912yes—
485C3×D3—24no—
486C3×D3—24no—
487C3×D3—44no—
488C2×Q8—31no—
489C2×Q8—272no—
490C2×D4—91no—
491C22×C4—92no—
492C22×C4—92no—
493C22×C4—272no—
494C4⋊C4—92no—
495C4⋊C4—91no—
496C4⋊C4—92no—
497C4⋊C4—92no—
498C4⋊C4—271no—
499C4⋊C4—272no—
500C4⋊C4—271no—
501C4⋊C4—272no—
502C4⋊C4—271no—
503C4⋊C4—271no—
504C4⋊C4—271no—
505C22⋊C4—32no—
506C22⋊C4—92no—
507C22⋊C4—272no—
508C42—91no—
509C42—271no—
510C42—271no—
511C22×C3—21no—
512C22×C3—22no—
513C22×C3—34no—
514C22×C3—34no—
515C22×C3—41no—
516C22×C3—64no—
517C22×C3—62no—
518C22×C3—62no—
519C22×C3—62no—
520C22×C3—62no—
521C22×C3—122no—
522C22×C3—122no—
523C22×C3C22×D3212yes—
524C22×C3C22×D3211yescharacteristic
525C2×D3—22no—
526C2×D3—22no—
527C2×D3D3×Dic614yes—
528C3×C4—22no—
529C3×C4—22no—
530C3×C4—21no—
531C3×C4—34no—
532C3×C4—42no—
533C3×C4—42no—
534C3×C4—62no—
535C3×C4—62no—
536C3×C4—62no—
537C3×C4—62no—
538C3×C4—62no—
539C3×C4—92no—
540C3×C4—122no—
541C3×C4—182no—
542C3×C4—182no—
543C3×C4—181no—
544Dic3—22no—
545Dic3—21no—
546Dic3—34no—
547Dic3—62no—
548Dic3—64no—
549Dic3—62no—
550Dic3—62no—
551Dic3—62no—
552Dic3—62no—
553Dic3—62no—
554Dic3—121no—
555Dic3—122no—
556Dic3—182no—
557Dic3—182no—
558Dic3—181no—
559Dic3—181no—
560Dic3—362no—
561Dic3—362no—
562Dic3—361no—
563Dic3—361no—
564Dic3—721no—
565Dic3—721no—
566Dic3#380512yes—
567C32—21no—
568C32—22no—
569C32—41no—
570C32#553911yescharacteristic, complemented
571C32#563512yescomplemented
572C23—32no—
573Q8—62no—
574Q8—1082no—
575D4—94no—
576C2×C4—31no—
577C2×C4—31no—
578C2×C4—32no—
579C2×C4—91no—
580C2×C4—91no—
581C2×C4—92no—
582C2×C4—92no—
583C2×C4—184no—
584C2×C4—271no—
585C2×C4—271no—
586C2×C4—362no—
587C2×C4—1082no—
588C2×C4D3311yescharacteristic
589C2×C3—21no—
590C2×C3—22no—
591C2×C3—22no—
592C2×C3—21no—
593C2×C3—22no—
594C2×C3—21no—
595C2×C3—41no—
596C2×C3—41no—
597C2×C3—41no—
598C2×C3—64no—
599C2×C3—64no—
600C2×C3—124no—
601C2×C3C2×D3×Dic612yes—
602C2×C3D3×#23712yes—
603C2×C3#6431412yes—
604C2×C3D32×Q811yescharacteristic
605C2×C3#6432011yescharacteristic
606C2×C3C2×#380511yescharacteristic
607D3—24no—
608C22—34no—
609C22—62no—
610C22—62no—
611C22C2×D3311yescharacteristic
612C4—21no—
613C4—32no—
614C4—62no—
615C4—61no—
616C4—92no—
617C4—181no—
618C4—182no—
619C4—182no—
620C4—541no—
621C4—541no—
622C3—22no—
623C3—21no—
624C3—41no—
625C3#1059413312yescomplemented
626C3#1059540711yescharacteristic, complemented
627C2—64no—
628C2D32×Dic611yescharacteristic
629C2#1172438611yescharacteristic
630C2D3×#380511yescharacteristic
subgroupG/H
center, FrattiniC22C2×D33
commutator, socle, cosocleC22×C33C24
FittingC2×C33×C4C23
prime radicalC2×C3×#625C22
hypercenterC2×C4D33
hypocenterC33#554
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. #11723679C2semidirect
2. #11723491C2semidirect
3. Dic6×C3⋊Dic3C2semidirect
5. C3×#63887C2semidirect
6. #11723037C2semidirect
7. #11723015C2semidirect
8. #11722971C2semidirect
11. D3×Dic3⋊Dic3C2semidirect
55. #63887D3semidirect
79. C3×#625Q8semidirect
255. #625Dic6semidirect
406. C33#554semidirect
570. C32#5539semidirect
571. C32#5635semidirect
625. C3#10594133semidirect
626. C3#10595407semidirect

Maximal subgroups

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

subgroupindex
#117236792
#117234912
Dic6×C3⋊Dic32
C3×#639022
C3×#638872
#117230372
#117230152
#117229712
#117229252
Dic3×#6252
D3×Dic3⋊Dic32
#105954073
#105941333