Finite Group Databasebeta

← #13172155

Subgroups — #13172155

Order 1728.

1474 conjugacy classes of subgroups.

The subgroup lattice has 690 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89046C211yesmaximal, characteristic
2Dic3×#638C212yesmaximal, complemented
3#11723595C211yesmaximal, characteristic, complemented
4C2×C3×#3724C211yesmaximal, characteristic, complemented
5C3×#63952C212yesmaximal, complemented
6#11722945C211yesmaximal, characteristic, complemented
7#11722944C211yesmaximal, characteristic, complemented
8#11722938C211yesmaximal, characteristic, complemented
9#11722931C211yesmaximal, characteristic, complemented
10#11722922C212yesmaximal, complemented
11#11722910C212yesmaximal, complemented
12#10594874—32nomaximal
13#10594051—31nomaximal
14C2×C3×#638C2212yes—
15C22×C3×C3⋊Dic3C2211yescharacteristic
16C22×C32×Dic3C2212yes—
17C2×#6296C2211yescharacteristic
18C2×#6295C2211yescharacteristic
19#89046C418yescomplemented
20C2×#6294C2212yes—
21C2×Dic3×C3⋊D3C2212yes—
22C2×C3×#624C2212yes—
23#88864C2211yescharacteristic
24C32×#217C2211yescharacteristic
25#88807C2211yescharacteristic
26#88807C2211yescharacteristic
27#88807C2212yes—
28#88803C2212yes—
29Dic3×C3⋊Dic3C2211yescharacteristic
30Dic3×C3⋊Dic3C2211yescharacteristic
31Dic3×C3⋊Dic3C2212yes—
32C3×#3724C2212yes—
33C3×#3724C2212yes—
34C3×#3724C2211yescharacteristic
35C3×#3724C2211yescharacteristic
36C3×#3722C2212yes—
37C3×Dic32C2212yes—
38C2×#3805—31no—
39C2×#3804—32no—
40C4×#638—32no—
41#64086—31no—
42C3×C4×#67—32no—
43C2×C3×Dic3⋊C4—32no—
44C3×#67⋊C4—32no—
45Dic3×#67—32no—
46Dic3×#67—32no—
47Dic3×#67—62no—
48C2×#3724D311yescharacteristic, complemented
49#63955—32no—
50#63955—61no—
51#63952D312yescomplemented
52C4×#627—31no—
53#63904—32no—
54#63902—31no—
55C4×#625—31no—
56#63883—32no—
57#63881—32no—
58#63878—32no—
59#63877—32no—
60D3×Dic3⋊C4—32no—
61D3×Dic3⋊C4—32no—
62#63838—31no—
63#63838—32no—
64#63837—32no—
65#63836—32no—
66C23×C33C2311yescharacteristic
67C22×C3×C3⋊D3C2312yes—
68C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
69C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
70C3×#638C2×C418yescomplemented
71C2×C3×C3⋊Dic3C2×C414yescomplemented
72C2×C3×C3⋊Dic3C2312yes—
73C2×C3×C3⋊Dic3C2311yescharacteristic
74C2×C3×C3⋊Dic3C2311yescharacteristic
75C2×C32×Dic3—42no—
76C2×C32×Dic3C2311yescharacteristic
77C2×C32×Dic3C2311yescharacteristic
78C2×C32×Dic3C2312yes—
79C2×C32×Dic3C2312yes—
80#6296C2×C414yescomplemented
81#6295C2×C414yescomplemented
82#6294C2×C418yescomplemented
83Dic3×C3⋊D3—42no—
84C3×#624—42no—
85#5546—92no—
86#5504—91no—
87C22×C32×C4—32no—
88C2×#638C2×D312yes—
89C22×C3⋊Dic3C2×D311yescharacteristic
90C2×C4×C3⋊D3—32no—
91C2×C3×#67—32no—
92C2×C3×#67—32no—
93C2×C3×#67—62no—
94C22×C3×Dic3—22no—
95C22×C3×Dic3—32no—
96C22×C3×Dic3—61no—
97C22×C3×Dic3C2×D312yes—
98C22×C3×Dic3C2×D312yes—
99C2×C3×C4×D3—32no—
100C2×C3×C4×D3—32no—
101C2×#627—32no—
102C2×#627—31no—
103C2×#626—32no—
104C2×#625—31no—
105C2×#624—32no—
106C2×#624C2×D312yes—
107#3805—38no—
108#3804—316no—
109C2×D3×Dic3—32no—
110C2×D3×Dic3—32no—
111C2×D3×Dic3—32no—
112C2×D3×Dic3—32no—
113C2×D3×Dic3—62no—
114C32×C22⋊C4—31no—
115#3757—31no—
116#3757—32no—
117#3757—61no—
118#3757—62no—
119#3757—121no—
120#3752—32no—
121#3750—31no—
122#3750—31no—
123#3750—32no—
124C4×C3⋊Dic3—31no—
125C4×C3⋊Dic3—31no—
126C4×C3⋊Dic3—32no—
127C3×#217—21no—
128C3×#217C2×D312yes—
129C3×(C2×D3)⋊C4—32no—
130C3×(C2×D3)⋊C4—32no—
131C3×Dic3⋊C4—32no—
132C3×Dic3⋊C4—32no—
133C3×Dic3⋊C4—32no—
134C3×Dic3⋊C4—32no—
135C3×Dic3⋊C4—32no—
136C3×Dic3⋊C4—32no—
137C3×C4×Dic3—32no—
138C3×C4×Dic3—32no—
139#3724C2×D312yes—
140#3724C2×D311yescharacteristic
141#3724C2×D312yes—
142#3724C2×D311yescharacteristic
143Dic3⋊Dic3—32no—
144Dic3⋊Dic3—32no—
145Dic3⋊Dic3—32no—
146Dic3⋊Dic3—32no—
147Dic3⋊Dic3—61no—
148Dic3⋊Dic3—61no—
149Dic3⋊Dic3—62no—
150#3722C2×D312yes—
151(C2×D3)⋊Dic3—32no—
152(C2×D3)⋊Dic3—32no—
153(C2×D3)⋊Dic3—62no—
154Dic32—32no—
155Dic32—32no—
156Dic32—32no—
157Dic32—32no—
158Dic32—61no—
159Dic32—61no—
160Dic32—62no—
161Dic32C2×D312yes—
162C22×C33—21no—
163C22×C33—21no—
164C22×C33C22×C412yes—
165C22×C33C2411yescharacteristic
166C2×C3×C3⋊D3—22no—
167C2×C3×C3⋊D3—22no—
168C2×C3×C3⋊D3C22×C414yes—
169C3⋊(C3⋊Dic3)—21no—
170C3⋊(C3⋊Dic3)—21no—
171C3×C3⋊Dic3C22×C412yes—
172C3×C3⋊Dic3C22×C414yes—
173C3×C3⋊Dic3C22×C412yes—
174C32×Dic3—22no—
175C32×Dic3—21no—
176C32×Dic3—22no—
177C32×Dic3—21no—
178C2×#237—92no—
179C2×#235—91no—
180C3×C4×D4—92no—
181C2×C3×C4⋊C4—91no—
182D4×Dic3—92no—
183C4×#67—32no—
184C4×#67—92no—
185C4×#67—92no—
186C4×#67—92no—
187C4×#67—182no—
188#926—91no—
189C2×Dic3⋊C4—32no—
190C2×Dic3⋊C4—92no—
191#893—91no—
192#893—91no—
193#892—92no—
194D3×C4⋊C4—92no—
195D3×C4⋊C4—92no—
196#887—91no—
197#887—91no—
198#887—92no—
199#67⋊C4—32no—
200#67⋊C4—92no—
201#877—92no—
202#877—92no—
203#877—181no—
204#872—92no—
205C4×Dic6—92no—
206C23×C32—21no—
207C23×C32—22no—
208C23×C32—41no—
209C23×C32C22×D312yes—
210C23×C32C22×D311yescharacteristic
211C22×C3⋊D3C22×D312yes—
212C22×C3×D3—32no—
213C22×C3×D3—32no—
214C22×C3×D3—62no—
215C2×C32×C4—31no—
216C2×C32×C4—32no—
217C2×C32×C4—32no—
218C2×C32×C4—31no—
219C2×C32×C4—122no—
220#638C4×D318yescomplemented
221C2×C3⋊Dic3—31no—
222C2×C3⋊Dic3—32no—
223C2×C3⋊Dic3—32no—
224C2×C3⋊Dic3—31no—
225C2×C3⋊Dic3—62no—
226C2×C3⋊Dic3—61no—
227C2×C3⋊Dic3—61no—
228C2×C3⋊Dic3—62no—
229C2×C3⋊Dic3—121no—
230C2×C3⋊Dic3—121no—
231C2×C3⋊Dic3C4×D314yescomplemented
232C2×C3⋊Dic3C22×D312yes—
233C2×C3⋊Dic3C22×D311yescharacteristic
234C2×C3⋊Dic3C22×D311yescharacteristic
235C4×C3⋊D3—122no—
236C3×#67—38no—
237C3×#67—38no—
238C3×#67—68no—
239C2×C3×Dic3—22no—
240C2×C3×Dic3—21no—
241C2×C3×Dic3—21no—
242C2×C3×Dic3—22no—
243C2×C3×Dic3—38no—
244C2×C3×Dic3—32no—
245C2×C3×Dic3—32no—
246C2×C3×Dic3—32no—
247C2×C3×Dic3—32no—
248C2×C3×Dic3—42no—
249C2×C3×Dic3—42no—
250C2×C3×Dic3—44no—
251C2×C3×Dic3—61no—
252C2×C3×Dic3—62no—
253C2×C3×Dic3—61no—
254C2×C3×Dic3—64no—
255C2×C3×Dic3C22×D312yes—
256C2×C3×Dic3C22×D312yes—
257C2×C3×Dic3C22×D312yes—
258C2×C3×Dic3C22×D312yes—
259C2×C3×Dic3C22×D312yes—
260C2×C3×Dic3C22×D312yes—
261C3×C4×D3—122no—
262C3×C4×D3—122no—
263#627—38no—
264#627—34no—
265#626—38no—
266#625—34no—
267#624—38no—
268#624—42no—
269D3×Dic3—38no—
270D3×Dic3—38no—
271D3×Dic3—122no—
272D3×Dic3—122no—
273D3×Dic3—124no—
274#520—271noSylow
275C2×C33—22no—
276C2×C33C23×C411yescharacteristic
277C2×C33H(5,𝔽2)11yescharacteristic
278C2×C332−1+411yescharacteristic
279C3×C3⋊D3—24no—
280C2×C3×D4—92no—
281C22×C3×C4—32no—
282C22×C3×C4—32no—
283C22×C3×C4—62no—
284C22×C3×C4—92no—
285C22×C3×C4—91no—
286C2×#67—32no—
287C2×#67—32no—
288C2×#67—62no—
289C2×#67—92no—
290C22×Dic3—32no—
291C22×Dic3—61no—
292C22×Dic3—92no—
293C22×Dic3—92no—
294C22×Dic3D3212yes—
295#237—916no—
296#235—98no—
297C2×D12—91no—
298C2×C4×D3—32no—
299C2×C4×D3—32no—
300C2×C4×D3—92no—
301C2×C4×D3—92no—
302C2×C4×D3—92no—
303C2×C4×D3—92no—
304C2×C4×D3—182no—
305C2×Dic6—92no—
306C2×Dic6—91no—
307C3×C4⋊C4—92no—
308C3×C4⋊C4—91no—
309C3×C4⋊C4—91no—
310C3×C4⋊C4—92no—
311C3×C22⋊C4—32no—
312C3×C22⋊C4—61no—
313C3×C22⋊C4—92no—
314C3×C42—92no—
315#217—92no—
316#217—91no—
317#217—92no—
318#217—181no—
319#217—182no—
320#217—361no—
321#217D3211yescharacteristic
322(C2×D3)⋊C4—32no—
323(C2×D3)⋊C4—32no—
324(C2×D3)⋊C4—92no—
325(C2×D3)⋊C4—92no—
326(C2×D3)⋊C4—182no—
327C4⋊Dic3—92no—
328C4⋊Dic3—91no—
329C4⋊Dic3—91no—
330Dic3⋊C4—32no—
331Dic3⋊C4—32no—
332Dic3⋊C4—32no—
333Dic3⋊C4—32no—
334Dic3⋊C4—32no—
335Dic3⋊C4—32no—
336Dic3⋊C4—92no—
337Dic3⋊C4—92no—
338Dic3⋊C4—92no—
339Dic3⋊C4—92no—
340Dic3⋊C4—181no—
341Dic3⋊C4—181no—
342Dic3⋊C4—182no—
343C4×Dic3—32no—
344C4×Dic3—32no—
345C4×Dic3—92no—
346C4×Dic3—92no—
347C4×Dic3—92no—
348C4×Dic3—92no—
349C4×Dic3—92no—
350C4×Dic3—91no—
351C4×Dic3—91no—
352C4×Dic3—181no—
353C4×Dic3—181no—
354C4×Dic3—182no—
355C22×C32—21no—
356C22×C32—22no—
357C22×C32—21no—
358C22×C32—22no—
359C22×C32—21no—
360C22×C32—22no—
361C22×C32—22no—
362C22×C32—24no—
363C22×C32—41no—
364C22×C32—42no—
365C22×C32—42no—
366C22×C32—42no—
367C22×C32—41no—
368C22×C32—41no—
369C22×C32—81no—
370C22×C32—81no—
371C22×C32C2×C4×D314yes—
372C22×C32C2×C4×D312yes—
373C22×C32C23×D311yescharacteristic
374C22×C32C23×D312yes—
375C2×C3⋊D3—22no—
376C2×C3⋊D3—22no—
377C2×C3⋊D3C2×C4×D314yes—
378C2×C3×D3—34no—
379C2×C3×D3—34no—
380C2×C3×D3—64no—
381C2×C3×D3—62no—
382C2×C3×D3—62no—
383C2×C3×D3—62no—
384C2×C3×D3—62no—
385C2×C3×D3—122no—
386C2×C3×D3—122no—
387C32×C4—62no—
388C32×C4—61no—
389C32×C4—62no—
390C32×C4—61no—
391C3⋊Dic3—61no—
392C3⋊Dic3—62no—
393C3⋊Dic3—61no—
394C3⋊Dic3—62no—
395C3⋊Dic3—122no—
396C3⋊Dic3—122no—
397C3⋊Dic3—121no—
398C3⋊Dic3—121no—
399C3⋊Dic3—241no—
400C3⋊Dic3—241no—
401C3⋊Dic3C2×C4×D312yes—
402C3⋊Dic3C2×C4×D314yes—
403C3⋊Dic3C2×C4×D312yes—
404C3×Dic3—24no—
405C3×Dic3—22no—
406C3×Dic3—22no—
407C3×Dic3—22no—
408C3×Dic3—22no—
409C3×Dic3—22no—
410C3×Dic3—22no—
411C3×Dic3—34no—
412C3×Dic3—34no—
413C3×Dic3—34no—
414C3×Dic3—34no—
415C3×Dic3—41no—
416C3×Dic3—41no—
417C3×Dic3—42no—
418C3×Dic3—62no—
419C3×Dic3—62no—
420C3×Dic3—64no—
421C2×#41—271no—
422C4×Q8—271no—
423C4×Q8—271no—
424C4×D4—92no—
425C4×D4—271no—
426C4×D4—271no—
427C4×D4—272no—
428#117—272no—
429#117—271no—
430C2×C4⋊C4—91no—
431C2×C4⋊C4—272no—
432C33#52011yescharacteristic, complemented, Sylow
433C23×C3—21no—
434C23×C3—22no—
435C23×C3—41no—
436C23×C3—92no—
437C23×C3C2×D3211yescharacteristic
438C23×C3C2×D3212yes—
439C22×D3—32no—
440C22×D3—32no—
441C22×D3—62no—
442C3×D4—98no—
443C2×C3×C4—32no—
444C2×C3×C4—32no—
445C2×C3×C4—32no—
446C2×C3×C4—32no—
447C2×C3×C4—32no—
448C2×C3×C4—32no—
449C2×C3×C4—62no—
450C2×C3×C4—61no—
451C2×C3×C4—61no—
452C2×C3×C4—62no—
453C2×C3×C4—91no—
454C2×C3×C4—92no—
455C2×C3×C4—91no—
456C2×C3×C4—94no—
457C2×C3×C4—122no—
458C2×C3×C4—122no—
459C2×C3×C4—124no—
460C2×C3×C4—362no—
461#67—38no—
462#67—38no—
463#67—68no—
464#67—98no—
465C2×Dic3—32no—
466C2×Dic3—32no—
467C2×Dic3—32no—
468C2×Dic3—32no—
469C2×Dic3—38no—
470C2×Dic3—42no—
471C2×Dic3—64no—
472C2×Dic3—61no—
473C2×Dic3—62no—
474C2×Dic3—61no—
475C2×Dic3—98no—
476C2×Dic3—92no—
477C2×Dic3—91no—
478C2×Dic3—91no—
479C2×Dic3—92no—
480C2×Dic3—181no—
481C2×Dic3—181no—
482C2×Dic3—182no—
483C2×Dic3—182no—
484C2×Dic3—361no—
485C2×Dic3—361no—
486C2×Dic3—362no—
487C2×Dic3C2×D3211yescharacteristic
488C2×Dic3C2×D3212yes—
489C2×Dic3C2×D3212yes—
490C2×Dic3C2×D3211yescharacteristic
491D12—94no—
492C4×D3—98no—
493C4×D3—98no—
494C4×D3—122no—
495C4×D3—122no—
496C4×D3—362no—
497C4×D3—362no—
498C4×D3—364no—
499Dic6—94no—
500Dic6—98no—
501C2×C32—22no—
502C2×C32—24no—
503C2×C32—22no—
504C2×C32—21no—
505C2×C32—22no—
506C2×C32—21no—
507C2×C32—21no—
508C2×C32—22no—
509C2×C32—41no—
510C2×C32—41no—
511C2×C32—41no—
512C2×C32—42no—
513C2×C32—44no—
514C2×C32—82no—
515C2×C32C22×C4×D311yescharacteristic
516C2×C32C22×C4×D312yes—
517C2×C32#100412yes—
518C2×C32#100411yescharacteristic
519C2×C32#100711yescharacteristic
520C2×C32#101012yes—
521C3⋊D3—24no—
522C3×D3—64no—
523C3×D3—64no—
524C3×D3—124no—
525#41—278no—
526C2×Q8—271no—
527C2×D4—92no—
528C2×D4—271no—
529C22×C4—32no—
530C22×C4—92no—
531C22×C4—91no—
532C22×C4—272no—
533C22×C4—272no—
534C4⋊C4—92no—
535C4⋊C4—91no—
536C4⋊C4—91no—
537C4⋊C4—92no—
538C4⋊C4—271no—
539C4⋊C4—272no—
540C4⋊C4—271no—
541C22⋊C4—31no—
542C22⋊C4—92no—
543C22⋊C4—272no—
544C22⋊C4—271no—
545C42—92no—
546C42—271no—
547C42—271no—
548C42—272no—
549C22×C3—22no—
550C22×C3—21no—
551C22×C3—21no—
552C22×C3—22no—
553C22×C3—21no—
554C22×C3—22no—
555C22×C3—22no—
556C22×C3—24no—
557C22×C3—42no—
558C22×C3—41no—
559C22×C3—41no—
560C22×C3—42no—
561C22×C3—41no—
562C22×C3—42no—
563C22×C3—81no—
564C22×C3—81no—
565C22×C3—94no—
566C22×C3—182no—
567C22×C3—182no—
568C22×C3C4×D3214yes—
569C22×C3C4×D3212yes—
570C22×C3C22×D3212yes—
571C22×C3C22×D3211yescharacteristic
572C2×D3—34no—
573C2×D3—34no—
574C2×D3—64no—
575C2×D3—62no—
576C2×D3—62no—
577C2×D3—62no—
578C2×D3—62no—
579C2×D3—122no—
580C2×D3—122no—
581C3×C4—64no—
582C3×C4—62no—
583C3×C4—62no—
584C3×C4—62no—
585C3×C4—62no—
586C3×C4—62no—
587C3×C4—62no—
588C3×C4—94no—
589C3×C4—92no—
590C3×C4—92no—
591C3×C4—122no—
592C3×C4—121no—
593C3×C4—121no—
594Dic3—21no—
595Dic3—21no—
596Dic3—22no—
597Dic3—22no—
598Dic3—34no—
599Dic3—34no—
600Dic3—34no—
601Dic3—34no—
602Dic3—62no—
603Dic3—62no—
604Dic3—64no—
605Dic3—182no—
606Dic3—181no—
607Dic3—181no—
608Dic3—182no—
609Dic3—362no—
610Dic3—361no—
611Dic3—361no—
612Dic3—362no—
613Dic3—721no—
614Dic3—721no—
615C32—21no—
616C32—22no—
617C32—41no—
618C32#550411yescharacteristic, complemented
619C32#554612yescomplemented
620C23—92no—
621C23D3311yescharacteristic
622Q8—274no—
623D4—98no—
624D4—274no—
625C2×C4—31no—
626C2×C4—32no—
627C2×C4—31no—
628C2×C4—32no—
629C2×C4—92no—
630C2×C4—91no—
631C2×C4—91no—
632C2×C4—94no—
633C2×C4—122no—
634C2×C4—278no—
635C2×C4—271no—
636C2×C4—271no—
637C2×C4—362no—
638C2×C4—1082no—
639C2×C3—22no—
640C2×C3—24no—
641C2×C3—21no—
642C2×C3—22no—
643C2×C3—22no—
644C2×C3—21no—
645C2×C3—21no—
646C2×C3—22no—
647C2×C3—44no—
648C2×C3—42no—
649C2×C3—41no—
650C2×C3—41no—
651C2×C3—41no—
652C2×C3—82no—
653C2×C3—184no—
654C2×C3#6429712yes—
655C2×C3C2×C4×D3211yescharacteristic
656C2×C3C2×C4×D3212yes—
657C2×C3#6430911yescharacteristic
658C2×C3#6431311yescharacteristic
659C2×C3#6433012yes—
660D3—64no—
661D3—64no—
662D3—124no—
663C22—21no—
664C22—21no—
665C22—94no—
666C22—182no—
667C22—182no—
668C22#8894912yes—
669C22C2×D3311yescharacteristic
670C4—62no—
671C4—62no—
672C4—61no—
673C4—61no—
674C4—92no—
675C4—92no—
676C4—94no—
677C4—541no—
678C4—541no—
679C3—22no—
680C3—21no—
681C3—41no—
682C3#1059405111yescharacteristic, complemented
683C3#1059487412yescomplemented
684C2—22no—
685C2—184no—
686C2C2×#8894911yescharacteristic
687C2#1172441411yescharacteristic
688C2#1172442611yescharacteristic
subgroupG/H
center, FrattiniC22C2×D33
commutatorC2×C33C23×C4
FittingC23×C33C23
socle, cosocleC22×C33C24
prime radicalC2×#89046C2
hypercenterC23D33
hypocenterC33#520
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
2. Dic3×#638C2semidirect
3. #11723595C2semidirect
4. C2×C3×#3724C2semidirect
5. C3×#63952C2semidirect
6. #11722945C2semidirect
7. #11722944C2semidirect
8. #11722938C2semidirect
9. #11722931C2semidirect
10. #11722922C2semidirect
11. #11722910C2semidirect
19. #89046C4semidirect
48. C2×#3724D3semidirect
51. #63952D3semidirect
70. C3×#638C2×C4semidirect
71. C2×C3×C3⋊Dic3C2×C4semidirect
80. #6296C2×C4semidirect
81. #6295C2×C4semidirect
82. #6294C2×C4semidirect
220. #638C4×D3semidirect
231. C2×C3⋊Dic3C4×D3semidirect
432. C33#520semidirect
618. C32#5504semidirect
619. C32#5546semidirect
682. C3#10594051semidirect
683. C3#10594874semidirect

Maximal subgroups

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

subgroupindex
C2×#890462
Dic3×#6382
#117235952
C2×C3×#37242
C3×#639522
#117229452
#117229442
#117229382
#117229312
#117229222
#117229102
#105948743
#105940513