Finite Group Databasebeta

← #13178275

Subgroups — #13178275

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
1C3×#64090C211yesmaximal, characteristic
2C3×#64087C212yesmaximal, complemented
3#11723726C212yesmaximal
4C2×#88815C211yesmaximal, characteristic, complemented
5#11723721C211yesmaximal, characteristic, complemented
6#11723689C211yesmaximal, characteristic, complemented
7#11723688C212yesmaximal, complemented
8#11723678C211yesmaximal, characteristic, complemented
9#11723671C211yesmaximal, characteristic, complemented
10#11723669C212yesmaximal, complemented
11#11723665C211yesmaximal, characteristic, complemented
12#10594168—31nomaximal
13#10594146—32nomaximal
14C2×C3×#638C2211yescharacteristic
15C22×C3×C3⋊Dic3C2211yescharacteristic
16C22×C3×C3⋊Dic3C2212yes—
17C2×C3×C4×C3⋊D3C2211yescharacteristic
18C2×C3×#634C2212yes—
19C2×#6296C2211yescharacteristic
20C2×#6294C2212yes—
21C33×C22⋊C4C2211yescharacteristic
22C3×#3757C2212yes—
23C3×#3752C2211yescharacteristic
24C3×C4⋊(C3⋊Dic3)C2212yes—
25C3×#3750C2212yes—
26C3×#3750C2212yes—
27C3×#3750C2211yescharacteristic
28C3×C4×C3⋊Dic3C2211yescharacteristic
29#88815C2211yescharacteristic
30#88815C2212yes—
31#88815C2212yes—
32#88815C2212yes—
33#88815C2211yescharacteristic
34#88815C2211yescharacteristic
35#88814C2212yes—
36#88813C2212yes—
37#64090D311yescharacteristic
38#64087D312yescomplemented
39C3×#67⋊C4—32no—
40C3×#67⋊C4—61no—
41C3×#878—32no—
42C3×#878—32no—
43C3×#878—62no—
44#63982—31no—
45#67⋊Dic3—32no—
46Dic3×#67—32no—
47C2×#3724—31no—
48C2×Dic3⋊Dic3—32no—
49#63964—32no—
50#63952—32no—
51#63900—31no—
52#63896—32no—
53D3×C4⋊Dic3—32no—
54#63888—32no—
55#63887—32no—
56#63873—31no—
57#63866—32no—
58#63865—31no—
59#63851—32no—
60#63847—32no—
61#63846—32no—
62#63845—32no—
63#63838—31no—
64#63837—32no—
65C23×C33C2311yescharacteristic
66C22×C3×C3⋊D3C2311yescharacteristic
67C2×C33×C4C2311yescharacteristic
68C2×C33×C4C2311yescharacteristic
69C3×#638Q814yescomplemented
70C2×C3×C3⋊Dic3—22no—
71C2×C3×C3⋊Dic3—42no—
72C2×C3×C3⋊Dic3C2312yes—
73C2×C3×C3⋊Dic3C2312yes—
74C2×C3×C3⋊Dic3C2311yescharacteristic
75C2×C3×C3⋊Dic3C2312yes—
76C2×C3×C3⋊Dic3C2312yes—
77C2×C3×C3⋊Dic3C2311yescharacteristic
78C2×C3×C3⋊Dic3C2311yescharacteristic
79C3×C4×C3⋊D3—22no—
80C3×#634—42no—
81#6296—22no—
82#6294—42no—
83#5636—92no—
84#5543—91no—
85C2×#638C2×D311yescharacteristic
86C22×C3⋊Dic3C2×D311yescharacteristic
87C22×C3⋊Dic3C2×D312yes—
88C2×C4×C3⋊D3C2×D311yescharacteristic
89C2×#634C2×D312yes—
90C2×C3×#67—32no—
91C2×C3×#67—61no—
92C22×C3×Dic3—32no—
93C22×C3×Dic3—32no—
94C22×C3×Dic3—32no—
95C22×C3×Dic3—62no—
96C22×C3×Dic3—61no—
97C2×C3×C4×D3—32no—
98C2×C3×C4×D3—61no—
99C2×C3×Dic6—32no—
100C2×C3×Dic6—32no—
101C2×C3×Dic6—62no—
102C2×#627—32no—
103C2×#627—31no—
104C2×#624—32no—
105C2×D3×Dic3—32no—
106C2×D3×Dic3—32no—
107C32×C22⋊C4—22no—
108C32×C22⋊C4—21no—
109C32×C22⋊C4—41no—
110C32×C22⋊C4C2×D311yescharacteristic
111C32×C22⋊C4C2×D312yes—
112#3757C2×D312yes—
113#3752C2×D311yescharacteristic
114C4⋊(C3⋊Dic3)C2×D312yes—
115#3750C2×D312yes—
116#3750C2×D312yes—
117#3750C2×D311yescharacteristic
118C4×C3⋊Dic3C2×D311yescharacteristic
119C3×#217—32no—
120C3×#217—32no—
121C3×#217—62no—
122C3×(C2×D3)⋊C4—32no—
123C3×(C2×D3)⋊C4—61no—
124C3×C4⋊Dic3—32no—
125C3×C4⋊Dic3—32no—
126C3×C4⋊Dic3—62no—
127C3×Dic3⋊C4—32no—
128C3×Dic3⋊C4—32no—
129C3×Dic3⋊C4—32no—
130C3×Dic3⋊C4—32no—
131C3×Dic3⋊C4—32no—
132C3×Dic3⋊C4—62no—
133C3×Dic3⋊C4—62no—
134C3×Dic3⋊C4—61no—
135C3×C4×Dic3—32no—
136C3×C4×Dic3—61no—
137#3724—31no—
138#3724—32no—
139#3724—32no—
140#3724—32no—
141#3724—31no—
142#3724—31no—
143Dic3⋊Dic3—32no—
144Dic3⋊Dic3—32no—
145Dic3⋊Dic3—32no—
146Dic3⋊Dic3—32no—
147Dic3⋊Dic3—32no—
148Dic3⋊Dic3—32no—
149Dic3⋊Dic3—32no—
150Dic3⋊Dic3—32no—
151Dic3⋊Dic3—32no—
152#3722—32no—
153(C2×D3)⋊Dic3—32no—
154(C2×D3)⋊Dic3—32no—
155Dic32—32no—
156Dic32—32no—
157Dic32—32no—
158C22×C33—21no—
159C22×C33—21no—
160C22×C33C2×Q812yes—
161C22×C33C2411yescharacteristic
162C2×C3×C3⋊D3—21no—
163C2×C3×C3⋊D3—21no—
164C2×C3×C3⋊D3C2×Q812yes—
165C33×C4—21no—
166C33×C4—21no—
167C3×C3⋊Dic3—22no—
168C3×C3⋊Dic3—21no—
169C3×C3⋊Dic3—22no—
170C3×C3⋊Dic3—22no—
171C3×C3⋊Dic3—22no—
172C3×C3⋊Dic3C2×Q812yes—
173C3×C3⋊Dic3#4112yes—
174C3×C22⋊Q8—92no—
175C3×C4×D4—91no—
176#946—92no—
177#944—92no—
178D4×Dic3—92no—
179C4×#67—92no—
180C2×C4⋊Dic3—91no—
181C2×C4⋊Dic3—91no—
182#924—91no—
183#924—91no—
184C2×Dic3⋊C4—92no—
185#896—92no—
186#896—92no—
187D3×C4⋊C4—92no—
188#890—92no—
189#889—92no—
190#889—92no—
191#887—92no—
192#67⋊C4—32no—
193#67⋊C4—61no—
194#67⋊C4—92no—
195#878—32no—
196#878—32no—
197#878—62no—
198#878—92no—
199#878—92no—
200#869—91no—
201C4×Dic6—91no—
202C23×C32—22no—
203C23×C32—21no—
204C23×C32—41no—
205C23×C32C22×D311yescharacteristic
206C23×C32C22×D312yes—
207C22×C3⋊D3C22×D311yescharacteristic
208C22×C3×D3—32no—
209C22×C3×D3—61no—
210C2×C32×C4—21no—
211C2×C32×C4—22no—
212C2×C32×C4—22no—
213C2×C32×C4—21no—
214C2×C32×C4—41no—
215C2×C32×C4—41no—
216C2×C32×C4C22×D312yes—
217C2×C32×C4C22×D311yescharacteristic
218C2×C32×C4C22×D312yes—
219C2×C32×C4C22×D311yescharacteristic
220#638Dic614yescomplemented
221C2×C3⋊Dic3—22no—
222C2×C3⋊Dic3—42no—
223C2×C3⋊Dic3C22×D312yes—
224C2×C3⋊Dic3C22×D312yes—
225C2×C3⋊Dic3C22×D312yes—
226C2×C3⋊Dic3C22×D311yescharacteristic
227C2×C3⋊Dic3C22×D312yes—
228C2×C3⋊Dic3C22×D311yescharacteristic
229C2×C3⋊Dic3C22×D311yescharacteristic
230C4×C3⋊D3—22no—
231#634—42no—
232C3×#67—38no—
233C3×#67—64no—
234C2×C3×Dic3—32no—
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—32no—
242C2×C3×Dic3—32no—
243C2×C3×Dic3—32no—
244C2×C3×Dic3—32no—
245C2×C3×Dic3—61no—
246C2×C3×Dic3—61no—
247C2×C3×Dic3—62no—
248C2×C3×Dic3—62no—
249C2×C3×Dic3—62no—
250C2×C3×Dic3—61no—
251C2×C3×Dic3—62no—
252C2×C3×Dic3—64no—
253C2×C3×Dic3—122no—
254C2×C3×Dic3—122no—
255C2×C3×Dic3—122no—
256C2×C3×Dic3—124no—
257C3×C4×D3—64no—
258C3×C4×D3—122no—
259C3×Dic6—122no—
260C3×Dic6—122no—
261C3×Dic6—124no—
262#627—64no—
263#627—62no—
264#624—122no—
265D3×Dic3—122no—
266D3×Dic3—122no—
267#554—271noSylow
268C2×C33—22no—
269C2×C33C22×Q811yescharacteristic
270C2×C33C2×#4111yescharacteristic
271C2×C33H(5,𝔽2)11yescharacteristic
272C3×C3⋊D3—22no—
273C2×C3×Q8—92no—
274C2×C3×D4—91no—
275C22×C3×C4—91no—
276C22×C3×C4—92no—
277C22×C3×C4—91no—
278C2×#67—32no—
279C2×#67—61no—
280C22×Dic3—32no—
281C22×Dic3—32no—
282C22×Dic3—32no—
283C22×Dic3—62no—
284C22×Dic3—61no—
285C22×Dic3—92no—
286C2×C4×D3—32no—
287C2×C4×D3—61no—
288C2×C4×D3—92no—
289C2×C4×D3—92no—
290C2×Dic6—32no—
291C2×Dic6—32no—
292C2×Dic6—62no—
293C2×Dic6—91no—
294C2×Dic6—92no—
295C3×C4⋊C4—92no—
296C3×C4⋊C4—92no—
297C3×C4⋊C4—92no—
298C3×C4⋊C4—91no—
299C3×C22⋊C4—22no—
300C3×C22⋊C4—21no—
301C3×C22⋊C4—41no—
302C3×C22⋊C4—92no—
303C3×C22⋊C4—91no—
304C3×C22⋊C4D3211yescharacteristic
305C3×C22⋊C4D3212yes—
306C3×C42—91no—
307#217—32no—
308#217—32no—
309#217—62no—
310#217—92no—
311(C2×D3)⋊C4—32no—
312(C2×D3)⋊C4—61no—
313(C2×D3)⋊C4—92no—
314(C2×D3)⋊C4—92no—
315C4⋊Dic3—32no—
316C4⋊Dic3—32no—
317C4⋊Dic3—62no—
318C4⋊Dic3—91no—
319C4⋊Dic3—92no—
320C4⋊Dic3—91no—
321C4⋊Dic3—91no—
322C4⋊Dic3—92no—
323C4⋊Dic3—92no—
324Dic3⋊C4—32no—
325Dic3⋊C4—32no—
326Dic3⋊C4—32no—
327Dic3⋊C4—32no—
328Dic3⋊C4—32no—
329Dic3⋊C4—62no—
330Dic3⋊C4—62no—
331Dic3⋊C4—61no—
332Dic3⋊C4—92no—
333Dic3⋊C4—92no—
334Dic3⋊C4—92no—
335Dic3⋊C4—92no—
336Dic3⋊C4—92no—
337Dic3⋊C4—92no—
338Dic3⋊C4—92no—
339Dic3⋊C4—92no—
340Dic3⋊C4—92no—
341C4×Dic3—32no—
342C4×Dic3—61no—
343C4×Dic3—92no—
344C4×Dic3—92no—
345C4×Dic3—92no—
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—41no—
355C22×C32—42no—
356C22×C32—41no—
357C22×C32—42no—
358C22×C32—42no—
359C22×C32—41no—
360C22×C32—81no—
361C22×C32—81no—
362C22×C32C2×Dic612yes—
363C22×C32D3×Q814yes—
364C22×C32C23×D311yescharacteristic
365C22×C32C23×D312yes—
366C2×C3⋊D3—21no—
367C2×C3⋊D3—21no—
368C2×C3⋊D3C2×Dic612yes—
369C2×C3×D3—34no—
370C2×C3×D3—62no—
371C2×C3×D3—62no—
372C2×C3×D3—62no—
373C2×C3×D3—121no—
374C2×C3×D3—121no—
375C32×C4—21no—
376C32×C4—22no—
377C32×C4—22no—
378C32×C4—21no—
379C32×C4—42no—
380C32×C4—41no—
381C32×C4—42no—
382C32×C4—41no—
383C32×C4—81no—
384C32×C4—81no—
385C3⋊Dic3—22no—
386C3⋊Dic3—21no—
387C3⋊Dic3—22no—
388C3⋊Dic3—22no—
389C3⋊Dic3—22no—
390C3⋊Dic3C2×Dic612yes—
391C3⋊Dic3#23712yes—
392C3×Dic3—34no—
393C3×Dic3—34no—
394C3×Dic3—62no—
395C3×Dic3—62no—
396C3×Dic3—64no—
397C3×Dic3—62no—
398C3×Dic3—62no—
399C3×Dic3—62no—
400C3×Dic3—62no—
401C3×Dic3—62no—
402C3×Dic3—62no—
403C3×Dic3—62no—
404C3×Dic3—62no—
405C3×Dic3—62no—
406C3×Dic3—122no—
407C3×Dic3—122no—
408C3×Dic3—121no—
409C3×Dic3—122no—
410C4⋊Q8—271no—
411#125—272no—
412C22⋊Q8—92no—
413C22⋊Q8—271no—
414C22⋊Q8—272no—
415C22⋊Q8—271no—
416C4×Q8—271no—
417C4×D4—91no—
418C4×D4—272no—
419C2×C4⋊C4—271no—
420C2×C4⋊C4—271no—
421C33#55411yescharacteristic, complemented, Sylow
422C23×C3—22no—
423C23×C3—21no—
424C23×C3—41no—
425C23×C3—91no—
426C23×C3C2×D3212yes—
427C23×C3C2×D3211yescharacteristic
428C22×D3—32no—
429C22×D3—61no—
430C3×Q8—362no—
431C3×D4—94no—
432C2×C3×C4—21no—
433C2×C3×C4—22no—
434C2×C3×C4—22no—
435C2×C3×C4—21no—
436C2×C3×C4—41no—
437C2×C3×C4—41no—
438C2×C3×C4—91no—
439C2×C3×C4—91no—
440C2×C3×C4—92no—
441C2×C3×C4—92no—
442C2×C3×C4—92no—
443C2×C3×C4—91no—
444C2×C3×C4—92no—
445C2×C3×C4—182no—
446C2×C3×C4—182no—
447C2×C3×C4—362no—
448C2×C3×C4C2×D3212yes—
449C2×C3×C4C2×D3211yescharacteristic
450C2×C3×C4C2×D3212yes—
451C2×C3×C4C2×D3211yescharacteristic
452#67—38no—
453#67—64no—
454C2×Dic3—32no—
455C2×Dic3—32no—
456C2×Dic3—32no—
457C2×Dic3—32no—
458C2×Dic3—32no—
459C2×Dic3—32no—
460C2×Dic3—32no—
461C2×Dic3—32no—
462C2×Dic3—32no—
463C2×Dic3—32no—
464C2×Dic3—32no—
465C2×Dic3—61no—
466C2×Dic3—61no—
467C2×Dic3—62no—
468C2×Dic3—62no—
469C2×Dic3—62no—
470C2×Dic3—61no—
471C2×Dic3—62no—
472C2×Dic3—64no—
473C2×Dic3—122no—
474C2×Dic3—122no—
475C2×Dic3—122no—
476C2×Dic3—124no—
477C2×Dic3—362no—
478C4×D3—64no—
479C4×D3—122no—
480C4×D3—362no—
481C4×D3—362no—
482Dic6—122no—
483Dic6—122no—
484Dic6—124no—
485Dic6—184no—
486Dic6—182no—
487C2×C32—24no—
488C2×C32—22no—
489C2×C32—22no—
490C2×C32—21no—
491C2×C32—22no—
492C2×C32—21no—
493C2×C32—21no—
494C2×C32—22no—
495C2×C32—41no—
496C2×C32—41no—
497C2×C32—41no—
498C2×C32—44no—
499C2×C32—42no—
500C2×C32—82no—
501C2×C32C22×Dic611yescharacteristic
502C2×C32C2×#23711yescharacteristic
503C2×C32#100412yes—
504C2×C32C2×D3×Q812yes—
505C2×C32D3×#4112yes—
506C2×C32#100911yescharacteristic
507C3⋊D3—22no—
508C3×D3—64no—
509C3×D3—122no—
510C2×Q8—92no—
511C2×Q8—271no—
512C2×D4—91no—
513C22×C4—91no—
514C22×C4—92no—
515C22×C4—91no—
516C22×C4—272no—
517C4⋊C4—92no—
518C4⋊C4—91no—
519C4⋊C4—92no—
520C4⋊C4—92no—
521C4⋊C4—271no—
522C4⋊C4—272no—
523C4⋊C4—271no—
524C4⋊C4—271no—
525C4⋊C4—272no—
526C4⋊C4—272no—
527C22⋊C4—92no—
528C22⋊C4—91no—
529C22⋊C4—272no—
530C22⋊C4#111311yescharacteristic
531C42—91no—
532C42—272no—
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—41no—
542C22×C3—42no—
543C22×C3—42no—
544C22×C3—41no—
545C22×C3—41no—
546C22×C3—42no—
547C22×C3—81no—
548C22×C3—81no—
549C22×C3—92no—
550C22×C3—181no—
551C22×C3—181no—
552C22×C3D3×Dic614yes—
553C22×C3#379712yes—
554C22×C3C22×D3212yes—
555C22×C3C22×D3211yescharacteristic
556C2×D3—34no—
557C2×D3—62no—
558C2×D3—62no—
559C2×D3—62no—
560C2×D3—121no—
561C2×D3—121no—
562C3×C4—22no—
563C3×C4—21no—
564C3×C4—21no—
565C3×C4—22no—
566C3×C4—42no—
567C3×C4—41no—
568C3×C4—42no—
569C3×C4—41no—
570C3×C4—81no—
571C3×C4—81no—
572C3×C4—92no—
573C3×C4—92no—
574C3×C4—182no—
575C3×C4—181no—
576C3×C4—182no—
577C3×C4—182no—
578C3×C4—182no—
579Dic3—34no—
580Dic3—34no—
581Dic3—62no—
582Dic3—62no—
583Dic3—64no—
584Dic3—62no—
585Dic3—62no—
586Dic3—62no—
587Dic3—62no—
588Dic3—62no—
589Dic3—62no—
590Dic3—62no—
591Dic3—62no—
592Dic3—62no—
593Dic3—122no—
594Dic3—121no—
595Dic3—122no—
596Dic3—122no—
597C32—22no—
598C32—21no—
599C32—41no—
600C32#554311yescharacteristic, complemented
601C32#563612yescomplemented
602C23—91no—
603C23C2×#111311yescharacteristic
604Q8—362no—
605Q8—542no—
606D4—94no—
607C2×C4—91no—
608C2×C4—91no—
609C2×C4—92no—
610C2×C4—92no—
611C2×C4—91no—
612C2×C4—92no—
613C2×C4—92no—
614C2×C4—182no—
615C2×C4—182no—
616C2×C4—362no—
617C2×C4—1082no—
618C2×C4C2×#111311yescharacteristic
619C2×C4C2×#111311yescharacteristic
620C2×C3—24no—
621C2×C3—22no—
622C2×C3—21no—
623C2×C3—22no—
624C2×C3—21no—
625C2×C3—22no—
626C2×C3—22no—
627C2×C3—21no—
628C2×C3—42no—
629C2×C3—44no—
630C2×C3—41no—
631C2×C3—41no—
632C2×C3—41no—
633C2×C3—82no—
634C2×C3—182no—
635C2×C3C2×D3×Dic612yes—
636C2×C3C2×#379711yescharacteristic
637C2×C3D3×#23712yes—
638C2×C3#6431211yescharacteristic
639C2×C3#6431311yescharacteristic
640C2×C3#6431412yes—
641D3—64no—
642D3—122no—
643C22—21no—
644C22—21no—
645C22—92no—
646C22—181no—
647C22—181no—
648C22#8904012yes—
649C22C22×#111311yescharacteristic
650C4—21no—
651C4—21no—
652C4—92no—
653C4—92no—
654C4—181no—
655C4—182no—
656C4—182no—
657C4—182no—
658C4—182no—
659C3—21no—
660C3—22no—
661C3—41no—
662C3#1059414612yescomplemented
663C3#1059416811yescharacteristic, complemented
664C2—22no—
665C2—182no—
666C2C2×#8904011yescharacteristic
667C2#1172462211yescharacteristic
668C2#1172462311yescharacteristic
subgroupG/H
center, FrattiniC22C22×#1113
commutator, socle, cosocleC22×C33C24
FittingC33×C22⋊C4C22
prime radicalC2×C3×#638C22
hypercenterC22⋊C4#1113
hypocenterC33#554
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C3×#64087C2semidirect
4. C2×#88815C2semidirect
5. #11723721C2semidirect
6. #11723689C2semidirect
7. #11723688C2semidirect
8. #11723678C2semidirect
9. #11723671C2semidirect
10. #11723669C2semidirect
11. #11723665C2semidirect
38. #64087D3semidirect
69. C3×#638Q8semidirect
220. #638Dic6semidirect
421. C33#554semidirect
600. C32#5543semidirect
601. C32#5636semidirect
662. C3#10594146semidirect
663. C3#10594168semidirect

Maximal subgroups

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

subgroupindex
C3×#640902
C3×#640872
#117237262
C2×#888152
#117237212
#117236892
#117236882
#117236782
#117236712
#117236692
#117236652
#105941683
#105941463