Finite Group Databasebeta

← #13172255

Subgroups — #13172255

Order 1728.

1188 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88815C211yesmaximal, characteristic, complemented
2#11723641C212yesmaximal, complemented
3#67×C3⋊Dic3C211yesmaximal, characteristic
4C3×#63982C211yesmaximal, characteristic, complemented
5C3×#67⋊Dic3C212yesmaximal
6#11723037C211yesmaximal, characteristic, complemented
7#11723012C212yesmaximal, complemented
8#11722976C211yesmaximal, characteristic, complemented
9#11722971C212yesmaximal, complemented
10Dic3×#627C211yesmaximal, characteristic, complemented
11D3×#3724C211yesmaximal, characteristic, complemented
12#10595274—31nomaximal
13#10594133—32nomaximal
14C22×C3×C3⋊Dic3C2211yescharacteristic
15C22×C3×C3⋊Dic3C2212yes—
16C2×C32×#67C2211yescharacteristic
17C2×#6293C2212yes—
18C2×D3×C3⋊Dic3C2211yescharacteristic
19C2×C3×#627C2211yescharacteristic
20C2×C3×D3×Dic3C2212yes—
21#88864C2211yescharacteristic
22C32×#217C2212yes—
23#88815C2212yes—
24#88815C2211yescharacteristic
25#88815C2211yescharacteristic
26#88807C2212yes—
27#88806C2212yes—
28Dic3⋊(C3⋊Dic3)C2211yescharacteristic
29Dic3⋊(C3⋊Dic3)C2212yes—
30(C2×D3)⋊(C3⋊Dic3)C2211yescharacteristic
31Dic3×C3⋊Dic3C2211yescharacteristic
32C3×#3724C2211yescharacteristic
33C3×Dic3⋊Dic3C2212yes—
34C3×Dic3⋊Dic3C2212yes—
35C3×(C2×D3)⋊Dic3C2212yes—
36C3×Dic32C2212yes—
37D4×C3⋊Dic3—31no—
38#64087—32no—
39C3×D4×Dic3—32no—
40C3×#67⋊C4—32no—
41C3×#878—32no—
42#63982—31no—
43#63982D311yescharacteristic, complemented
44#67⋊Dic3D312yes—
45Dic3×#67—32no—
46Dic3×#67—61no—
47C2×#3724—31no—
48C2×#3724—31no—
49C2×Dic3⋊Dic3—32no—
50#63964—32no—
51#63964—32no—
52#63964—32no—
53#63964—62no—
54#63919—31no—
55C4×#627—31no—
56#63893—32no—
57#63887—32no—
58D3×Dic3⋊C4—32no—
59#63869—32no—
60#63867—32no—
61#63857—32no—
62#63846—32no—
63Dic3×Dic6—32no—
64#63840—32no—
65C23×C33C2311yescharacteristic
66C22×C32×D3C2311yescharacteristic
67C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
68C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
69C2×C3×C3⋊Dic3—22no—
70C2×C3×C3⋊Dic3—42no—
71C2×C3×C3⋊Dic3C2312yes—
72C2×C3×C3⋊Dic3C2312yes—
73C2×C3×C3⋊Dic3C2311yescharacteristic
74C2×C3×C3⋊Dic3C2311yescharacteristic
75C32×#67Q814yescomplemented
76C2×C32×Dic3C2312yes—
77C2×C32×Dic3C2312yes—
78C2×C32×Dic3C2311yescharacteristic
79#6293—42no—
80D3×C3⋊Dic3—22no—
81C3×#627—22no—
82C3×D3×Dic3—42no—
83#5635—91no—
84#5539—92no—
85C2×C32×D4—31no—
86C22×C3⋊Dic3—31no—
87C22×C3⋊Dic3C2×D311yescharacteristic
88C22×C3⋊Dic3C2×D312yes—
89C2×#634—32no—
90C2×C3×#67—21no—
91C2×C3×#67C2×D312yes—
92C22×C3×Dic3—32no—
93C22×C3×Dic3—32no—
94C22×C3×Dic3—32no—
95C22×C3×Dic3—32no—
96C22×C3×Dic3—62no—
97C22×C3×Dic3—61no—
98C2×C3×C4×D3—32no—
99C2×C3×Dic6—32no—
100C2×#627—32no—
101C2×#627—32no—
102C2×#627—62no—
103C2×#627C2×D311yescharacteristic
104C2×D3×Dic3—32no—
105C2×D3×Dic3—61no—
106C2×D3×Dic3C2×D312yes—
107C32×C22⋊C4—32no—
108#3757—32no—
109#3757—31no—
110#3757—31no—
111#3757—62no—
112#3757—61no—
113#3757—121no—
114C4⋊(C3⋊Dic3)—32no—
115C4⋊(C3⋊Dic3)—31no—
116#3750—32no—
117#3750—32no—
118C4×C3⋊Dic3—31no—
119C3×#217—22no—
120C3×#217—32no—
121C3×#217C2×D312yes—
122C3×#217C2×D312yes—
123C3×(C2×D3)⋊C4—32no—
124C3×C4⋊Dic3—32no—
125C3×C4⋊Dic3—32no—
126C3×Dic3⋊C4—32no—
127C3×Dic3⋊C4—32no—
128C3×Dic3⋊C4—32no—
129C3×C4×Dic3—32no—
130C3×C4×Dic3—32no—
131#3724—32no—
132#3724—31no—
133#3724—31no—
134#3724—32no—
135#3724—32no—
136#3724—62no—
137#3724C2×D311yescharacteristic
138Dic3⋊Dic3—32no—
139Dic3⋊Dic3—32no—
140Dic3⋊Dic3—32no—
141Dic3⋊Dic3—32no—
142Dic3⋊Dic3—32no—
143Dic3⋊Dic3—32no—
144Dic3⋊Dic3—32no—
145Dic3⋊Dic3—32no—
146Dic3⋊Dic3—32no—
147Dic3⋊Dic3—62no—
148Dic3⋊Dic3—61no—
149Dic3⋊Dic3—62no—
150Dic3⋊Dic3C2×D312yes—
151Dic3⋊Dic3C2×D312yes—
152(C2×D3)⋊Dic3—32no—
153(C2×D3)⋊Dic3—61no—
154(C2×D3)⋊Dic3C2×D312yes—
155Dic32—32no—
156Dic32—61no—
157Dic32C2×D312yes—
158C22×C33—21no—
159C22×C33—21no—
160C22×C33C2×Q812yes—
161C22×C33C2411yescharacteristic
162C2×C32×D3—21no—
163C2×C32×D3—21no—
164C2×C32×D3C2×Q812yes—
165C3⋊(C3⋊Dic3)—21no—
166C3⋊(C3⋊Dic3)—21no—
167C3×C3⋊Dic3—21no—
168C3×C3⋊Dic3—22no—
169C3×C3⋊Dic3—22no—
170C3×C3⋊Dic3#4112yes—
171C32×Dic3—22no—
172C32×Dic3—22no—
173C32×Dic3C2×Q812yes—
174C3×C22⋊Q8—91no—
175C3×C4×D4—92no—
176#946—91no—
177Q8×Dic3—91no—
178D4×Dic3—32no—
179D4×Dic3—92no—
180D4×Dic3—181no—
181C4×#67—91no—
182C2×C4⋊Dic3—91no—
183#924—92no—
184#924—92no—
185C2×Dic3⋊C4—92no—
186C2×Dic3⋊C4—92no—
187#897—92no—
188D3×C4⋊C4—91no—
189#890—92no—
190#889—92no—
191#888—91no—
192#888—92no—
193#67⋊C4—32no—
194#878—32no—
195#878—92no—
196#878—92no—
197#878—92no—
198#878—92no—
199#878—182no—
200#870—92no—
201C4×Dic6—92no—
202C23×C32—22no—
203C23×C32—21no—
204C23×C32—31no—
205C23×C32—41no—
206C23×C32C22×D312yes—
207C23×C32C22×D311yescharacteristic
208C22×C3×D3—21no—
209C22×C3×D3C22×D312yes—
210C32×D4—34no—
211C2×C32×C4—31no—
212C2×C32×C4—32no—
213C2×C32×C4—32no—
214C2×C3⋊Dic3—22no—
215C2×C3⋊Dic3—32no—
216C2×C3⋊Dic3—31no—
217C2×C3⋊Dic3—32no—
218C2×C3⋊Dic3—31no—
219C2×C3⋊Dic3—42no—
220C2×C3⋊Dic3—61no—
221C2×C3⋊Dic3—62no—
222C2×C3⋊Dic3—61no—
223C2×C3⋊Dic3—62no—
224C2×C3⋊Dic3—62no—
225C2×C3⋊Dic3—121no—
226C2×C3⋊Dic3—121no—
227C2×C3⋊Dic3C22×D312yes—
228C2×C3⋊Dic3C22×D312yes—
229C2×C3⋊Dic3C22×D311yescharacteristic
230C2×C3⋊Dic3C22×D311yescharacteristic
231#634—122no—
232C3×#67—24no—
233C3×#67Dic618yescomplemented
234C2×C3×Dic3—21no—
235C2×C3×Dic3—22no—
236C2×C3×Dic3—22no—
237C2×C3×Dic3—32no—
238C2×C3×Dic3—32no—
239C2×C3×Dic3—32no—
240C2×C3×Dic3—32no—
241C2×C3×Dic3—32no—
242C2×C3×Dic3—32no—
243C2×C3×Dic3—61no—
244C2×C3×Dic3—62no—
245C2×C3×Dic3—61no—
246C2×C3×Dic3—62no—
247C2×C3×Dic3—64no—
248C2×C3×Dic3—122no—
249C2×C3×Dic3—122no—
250C2×C3×Dic3—122no—
251C2×C3×Dic3—122no—
252C2×C3×Dic3—124no—
253C2×C3×Dic3C22×D312yes—
254C2×C3×Dic3C22×D312yes—
255C2×C3×Dic3C22×D312yes—
256C2×C3×Dic3C22×D312yes—
257C2×C3×Dic3C22×D312yes—
258C3×C4×D3—122no—
259C3×Dic6—64no—
260#627—22no—
261#627—122no—
262#627—122no—
263#627—124no—
264D3×Dic3—42no—
265D3×Dic3—64no—
266D3×Dic3—122no—
267#554—271noSylow
268C2×C33—22no—
269C2×C33C22×Q811yescharacteristic
270C2×C33C2×#4111yescharacteristic
271C2×C33H(5,𝔽2)11yescharacteristic
272C32×D3—22no—
273C2×C3×Q8—91no—
274C2×C3×D4—32no—
275C2×C3×D4—61no—
276C22×C3×C4—92no—
277C22×C3×C4—91no—
278C22×C3×C4—92no—
279C2×#67D3211yescharacteristic
280C22×Dic3—32no—
281C22×Dic3—32no—
282C22×Dic3—32no—
283C22×Dic3—32no—
284C22×Dic3—62no—
285C22×Dic3—61no—
286C22×Dic3—92no—
287C22×Dic3—181no—
288C2×C4×D3—32no—
289C2×C4×D3—91no—
290C2×Dic6—32no—
291C2×Dic6—92no—
292C2×Dic6—92no—
293C2×Dic6—92no—
294C2×Dic6—182no—
295C3×C4⋊C4—92no—
296C3×C4⋊C4—91no—
297C3×C4⋊C4—92no—
298C3×C22⋊C4—32no—
299C3×C22⋊C4—32no—
300C3×C22⋊C4—62no—
301C3×C22⋊C4—92no—
302C3×C42—92no—
303#217—32no—
304#217—92no—
305#217—91no—
306#217—92no—
307#217—181no—
308#217—182no—
309#217—181no—
310#217—361no—
311#217D3212yes—
312(C2×D3)⋊C4—32no—
313(C2×D3)⋊C4—91no—
314C4⋊Dic3—32no—
315C4⋊Dic3—32no—
316C4⋊Dic3—91no—
317C4⋊Dic3—92no—
318C4⋊Dic3—91no—
319C4⋊Dic3—92no—
320C4⋊Dic3—92no—
321C4⋊Dic3—92no—
322C4⋊Dic3—92no—
323C4⋊Dic3—182no—
324C4⋊Dic3—181no—
325Dic3⋊C4—32no—
326Dic3⋊C4—32no—
327Dic3⋊C4—32no—
328Dic3⋊C4—92no—
329Dic3⋊C4—92no—
330Dic3⋊C4—92no—
331Dic3⋊C4—92no—
332Dic3⋊C4—92no—
333Dic3⋊C4—92no—
334Dic3⋊C4—92no—
335Dic3⋊C4—92no—
336Dic3⋊C4—92no—
337Dic3⋊C4—92no—
338Dic3⋊C4—91no—
339Dic3⋊C4—182no—
340Dic3⋊C4—182no—
341C4×Dic3—32no—
342C4×Dic3—32no—
343C4×Dic3—92no—
344C4×Dic3—91no—
345C4×Dic3—181no—
346C22×C32—21no—
347C22×C32—22no—
348C22×C32—22no—
349C22×C32—21no—
350C22×C32—22no—
351C22×C32—21no—
352C22×C32—24no—
353C22×C32—22no—
354C22×C32—32no—
355C22×C32—41no—
356C22×C32—42no—
357C22×C32—41no—
358C22×C32—42no—
359C22×C32—42no—
360C22×C32—41no—
361C22×C32—61no—
362C22×C32—61no—
363C22×C32—81no—
364C22×C32—81no—
365C22×C32C2×Dic614yes—
366C22×C32D3×Q812yes—
367C22×C32C23×D312yes—
368C22×C32C23×D311yescharacteristic
369C2×C3×D3—22no—
370C2×C3×D3—22no—
371C2×C3×D3—22no—
372C2×C3×D3—41no—
373C2×C3×D3—41no—
374C2×C3×D3C2×Dic614yes—
375C32×C4—32no—
376C32×C4—62no—
377C32×C4—62no—
378C3⋊Dic3—22no—
379C3⋊Dic3—21no—
380C3⋊Dic3—22no—
381C3⋊Dic3—61no—
382C3⋊Dic3—62no—
383C3⋊Dic3—62no—
384C3⋊Dic3—61no—
385C3⋊Dic3—121no—
386C3⋊Dic3—122no—
387C3⋊Dic3—122no—
388C3⋊Dic3—121no—
389C3⋊Dic3—241no—
390C3⋊Dic3—241no—
391C3⋊Dic3#23712yes—
392C3×Dic3—22no—
393C3×Dic3—22no—
394C3×Dic3—22no—
395C3×Dic3—22no—
396C3×Dic3—22no—
397C3×Dic3—34no—
398C3×Dic3—42no—
399C3×Dic3—42no—
400C3×Dic3—62no—
401C3×Dic3—64no—
402C3×Dic3—62no—
403C3×Dic3—62no—
404C3×Dic3—62no—
405C3×Dic3—62no—
406C3×Dic3—62no—
407C3×Dic3—121no—
408C3×Dic3—122no—
409C3×Dic3C2×Dic614yes—
410C4⋊Q8—271no—
411#125—272no—
412C22⋊Q8—91no—
413C22⋊Q8—272no—
414C22⋊Q8—271no—
415C22⋊Q8—272no—
416C4×Q8—271no—
417C4×D4—92no—
418C4×D4—271no—
419C2×C4⋊C4—271no—
420C2×C4⋊C4—271no—
421C33#55411yescharacteristic, complemented, Sylow
422C23×C3—22no—
423C23×C3—21no—
424C23×C3—32no—
425C23×C3—41no—
426C23×C3—61no—
427C23×C3C2×D3212yes—
428C23×C3C2×D3211yescharacteristic
429C22×D3C2×D3211yescharacteristic
430C3×Q8—182no—
431C3×D4—38no—
432C3×D4—64no—
433C2×C3×C4—32no—
434C2×C3×C4—32no—
435C2×C3×C4—32no—
436C2×C3×C4—32no—
437C2×C3×C4—32no—
438C2×C3×C4—61no—
439C2×C3×C4—62no—
440C2×C3×C4—62no—
441C2×C3×C4—91no—
442C2×C3×C4—92no—
443C2×C3×C4—92no—
444C2×C3×C4—91no—
445C2×C3×C4—182no—
446C2×C3×C4—362no—
447C2×C3×C4—362no—
448#67#62714yescomplemented
449C2×Dic3—32no—
450C2×Dic3—32no—
451C2×Dic3—32no—
452C2×Dic3—32no—
453C2×Dic3—32no—
454C2×Dic3—32no—
455C2×Dic3—61no—
456C2×Dic3—62no—
457C2×Dic3—62no—
458C2×Dic3—61no—
459C2×Dic3—64no—
460C2×Dic3—92no—
461C2×Dic3—91no—
462C2×Dic3—91no—
463C2×Dic3—92no—
464C2×Dic3—122no—
465C2×Dic3—122no—
466C2×Dic3—122no—
467C2×Dic3—122no—
468C2×Dic3—124no—
469C2×Dic3—181no—
470C2×Dic3—182no—
471C2×Dic3—182no—
472C2×Dic3—181no—
473C2×Dic3—184no—
474C2×Dic3—361no—
475C2×Dic3—361no—
476C2×Dic3—362no—
477C2×Dic3C2×D3212yes—
478C2×Dic3C2×D3212yes—
479C2×Dic3C2×D3211yescharacteristic
480C4×D3—122no—
481C4×D3—182no—
482Dic6—64no—
483Dic6—362no—
484Dic6—362no—
485Dic6—362no—
486Dic6—364no—
487C2×C32—22no—
488C2×C32—24no—
489C2×C32—22no—
490C2×C32—22no—
491C2×C32—21no—
492C2×C32—21no—
493C2×C32—22no—
494C2×C32—21no—
495C2×C32—41no—
496C2×C32—41no—
497C2×C32—41no—
498C2×C32—44no—
499C2×C32—42no—
500C2×C32—62no—
501C2×C32—82no—
502C2×C32C22×Dic612yes—
503C2×C32C2×#23711yescharacteristic
504C2×C32#100412yes—
505C2×C32C2×D3×Q811yescharacteristic
506C2×C32D3×#4112yes—
507C2×C32#100911yescharacteristic
508C3×D3—24no—
509C3×D3—42no—
510C2×Q8—91no—
511C2×Q8—272no—
512C2×D4—31no—
513C22×C4—92no—
514C22×C4—91no—
515C22×C4—92no—
516C22×C4—271no—
517C4⋊C4—91no—
518C4⋊C4—92no—
519C4⋊C4—92no—
520C4⋊C4—272no—
521C4⋊C4—271no—
522C4⋊C4—272no—
523C4⋊C4—272no—
524C4⋊C4—271no—
525C4⋊C4—272no—
526C4⋊C4—271no—
527C22⋊C4—32no—
528C22⋊C4—92no—
529C22⋊C4—271no—
530C22⋊C4—271no—
531C42—92no—
532C42—271no—
533C22×C3—22no—
534C22×C3—21no—
535C22×C3—21no—
536C22×C3—22no—
537C22×C3—22no—
538C22×C3—21no—
539C22×C3—22no—
540C22×C3—24no—
541C22×C3—34no—
542C22×C3—41no—
543C22×C3—42no—
544C22×C3—42no—
545C22×C3—41no—
546C22×C3—41no—
547C22×C3—42no—
548C22×C3—62no—
549C22×C3—62no—
550C22×C3—62no—
551C22×C3—81no—
552C22×C3—81no—
553C22×C3—121no—
554C22×C3—121no—
555C22×C3D3×Dic614yes—
556C22×C3C2×#62712yes—
557C22×C3C22×D3212yes—
558C22×C3C22×D3211yescharacteristic
559C2×D3—21no—
560C2×D3—21no—
561C2×D3C2×#62712yes—
562C3×C4—34no—
563C3×C4—62no—
564C3×C4—62no—
565C3×C4—62no—
566C3×C4—62no—
567C3×C4—62no—
568C3×C4—92no—
569C3×C4—122no—
570C3×C4—122no—
571C3×C4—181no—
572C3×C4—182no—
573C3×C4—182no—
574Dic3—22no—
575Dic3—22no—
576Dic3—34no—
577Dic3—62no—
578Dic3—64no—
579Dic3—62no—
580Dic3—62no—
581Dic3—62no—
582Dic3—62no—
583Dic3—62no—
584Dic3—121no—
585Dic3—122no—
586Dic3—182no—
587Dic3—181no—
588Dic3—181no—
589Dic3—182no—
590Dic3—362no—
591Dic3—361no—
592Dic3—362no—
593Dic3—361no—
594Dic3—721no—
595Dic3—721no—
596Dic3C2×#62712yes—
597C32—22no—
598C32—21no—
599C32—41no—
600C32#553912yescomplemented
601C32#563511yescharacteristic, complemented
602C23—31no—
603C23D3311yescharacteristic
604Q8—182no—
605Q8—1082no—
606D4—34no—
607C2×C4—31no—
608C2×C4—32no—
609C2×C4—32no—
610C2×C4—91no—
611C2×C4—92no—
612C2×C4—92no—
613C2×C4—91no—
614C2×C4—182no—
615C2×C4—271no—
616C2×C4—271no—
617C2×C4—362no—
618C2×C4—362no—
619C2×C4—542no—
620C2×C3—24no—
621C2×C3—22no—
622C2×C3—21no—
623C2×C3—22no—
624C2×C3—22no—
625C2×C3—21no—
626C2×C3—21no—
627C2×C3—22no—
628C2×C3—42no—
629C2×C3—44no—
630C2×C3—41no—
631C2×C3—41no—
632C2×C3—41no—
633C2×C3—64no—
634C2×C3—82no—
635C2×C3—122no—
636C2×C3C2×D3×Dic612yes—
637C2×C3D3×#23712yes—
638C2×C3#6431211yescharacteristic
639C2×C3#6431311yescharacteristic
640C2×C3#6431412yes—
641C2×C3C22×#62711yescharacteristic
642D3—22no—
643C22—21no—
644C22—21no—
645C22—32no—
646C22—61no—
647C22—61no—
648C22D3×#62712yes—
649C22C2×D3311yescharacteristic
650C4—32no—
651C4—62no—
652C4—62no—
653C4—92no—
654C4—181no—
655C4—182no—
656C4—182no—
657C4—541no—
658C4—541no—
659C3—21no—
660C3—22no—
661C3—41no—
662C3#1059413312yescomplemented
663C3#1059527411yescharacteristic, complemented
664C2—22no—
665C2—62no—
666C2#1172440811yescharacteristic
667C2#1172441611yescharacteristic
668C2C2×D3×#62711yescharacteristic
subgroupG/H
center, FrattiniC22C2×D33
commutator, socle, cosocleC22×C33C24
FittingC23×C33C23
prime radicalC2×C32×#67C22
hypercenterC23D33
hypocenterC33#554
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
1. C2×#88815C2semidirect
2. #11723641C2semidirect
4. C3×#63982C2semidirect
6. #11723037C2semidirect
7. #11723012C2semidirect
8. #11722976C2semidirect
9. #11722971C2semidirect
10. Dic3×#627C2semidirect
11. D3×#3724C2semidirect
43. #63982D3semidirect
75. C32×#67Q8semidirect
233. C3×#67Dic6semidirect
421. C33#554semidirect
448. #67#627semidirect
600. C32#5539semidirect
601. C32#5635semidirect
662. C3#10594133semidirect
663. C3#10595274semidirect

Maximal subgroups

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

subgroupindex
C2×#888152
#117236412
#67×C3⋊Dic32
C3×#639822
C3×#67⋊Dic32
#117230372
#117230122
#117229762
#117229712
Dic3×#6272
D3×#37242
#105952743
#105941333