Finite Group Databasebeta

← #13178109

Subgroups — #13178109

Order 1728.

3020 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6312C211yesmaximal, characteristic, complemented
2C2×#89039C211yesmaximal, characteristic, complemented
3C3×#64162C211yesmaximal, characteristic, complemented
4C32×#953C211yesmaximal, characteristic, complemented
5#11723643C212yesmaximal, complemented
6#11723628C212yesmaximal, complemented
7#11723627C211yesmaximal, characteristic, complemented
8#11723623C212yesmaximal, complemented
9#11723609C211yesmaximal, characteristic, complemented
10#11723603C211yesmaximal, characteristic, complemented
11#11723599C212yesmaximal, complemented
12#10596060—31nomaximal
13#10595528—34nomaximal
14C24×C33C2211yescharacteristic, complemented
15C23×C3⋊(C3⋊D3)C2211yescharacteristic
16C22×D3×C3⋊D3C2211yescharacteristic, complemented
17C2×#6312—24no—
18C2×#6312C2212yes—
19C2×#6312C2212yescomplemented
20C22×C3⋊(C3⋊Dic3)C2211yescharacteristic, complemented
21C2×C3×#638C2211yescharacteristic
22C2×C3×#638C2212yescomplemented
23C2×C32×#67C2211yescharacteristic
24C2×C32×#67C2212yescomplemented
25#89039—24no—
26C2×#6292C2211yescharacteristic, complemented
27C2×#6291C2211yescharacteristic, complemented
28C2×#6289C2211yescharacteristic, complemented
29C2×#6289C2212yescomplemented
30C3×#3757C2212yescomplemented
31C3×#3757C2211yescharacteristic, complemented
32C32×#217C2211yescharacteristic, complemented
33C32×#217C2212yescomplemented
34#88807C2212yescomplemented
35Dic3⋊(C3⋊Dic3)C2212yescomplemented
36#88804C2212yescomplemented
37#88803C2212yescomplemented
38(C2×D3)⋊(C3⋊Dic3)C2212yescomplemented
39C22×#638—31no—
40C22×#638—34no—
41C22×#638—64no—
42C2×D4×C3⋊D3—31no—
43C2×#3811—34no—
44C32×#120—31no—
45#64162D311yescharacteristic, complemented
46#64147—32no—
47#64146—32no—
48#64144—31no—
49#64095—31no—
50#64093—32no—
51#64092—32no—
52C22⋊C4×C3⋊D3—31no—
53C3×#953—34no—
54C3×#953D314yescomplemented
55#63983—38no—
56#63975—38no—
57#63974—34no—
58#63962—34no—
59#63962—34no—
60#63958—38no—
61#63958—38no—
62C23×C33—22no—
63C23×C33—42no—
64C23×C33C2312yes—
65C23×C33C2311yescharacteristic
66C22×C3⋊(C3⋊D3)—42no—
67C22×C3⋊(C3⋊D3)D414yescomplemented
68C22×C3⋊(C3⋊D3)C2311yescharacteristic
69C22×C3⋊(C3⋊D3)C2311yescharacteristic
70C22×C3×C3⋊D3C2311yescharacteristic
71C22×C32×D3C2311yescharacteristic
72C2×D3×C3⋊D3—22no—
73C2×D3×C3⋊D3—42no—
74#6312—24no—
75#6312—24no—
76C2×C3⋊(C3⋊Dic3)—22no—
77C2×C3⋊(C3⋊Dic3)C2312yes—
78C3×#638—22no—
79C3×#638—42no—
80C2×C3×C3⋊Dic3C2311yescharacteristic
81C2×C3×C3⋊Dic3C2312yes—
82C32×#67—22no—
83C32×#67—42no—
84C2×C32×Dic3C2311yescharacteristic
85C2×C32×Dic3C2312yes—
86#6292—22no—
87#6291—22no—
88#6289—22no—
89#6289—42no—
90#5589—94no—
91#5589—91no—
92C24×C32—24no—
93C24×C32C2×D314yescomplemented
94C24×C32C2×D311yescharacteristic, complemented
95C23×C3⋊D3—31no—
96C23×C3⋊D3—34no—
97C23×C3⋊D3—31no—
98C23×C3⋊D3—64no—
99C22×D32—34no—
100C2×C32×D4—31no—
101C2×C32×D4—32no—
102C2×#638—32no—
103C2×#638—32no—
104C2×#638—38no—
105C2×#638—38no—
106C2×#638—31no—
107C2×#638—64no—
108C2×#638—616no—
109C2×#638—68no—
110C2×#638—68no—
111C2×#638—632no—
112C2×#638C2×D311yescharacteristic
113C2×#638C2×D312yescomplemented
114C22×C3⋊Dic3—31no—
115C22×C3⋊Dic3—34no—
116C22×C3⋊Dic3—64no—
117D4×C3⋊D3—64no—
118C2×C4⋊(C3⋊D3)—31no—
119C2×C4×C3⋊D3—31no—
120C2×C4×C3⋊D3—32no—
121C2×C3×#67—34no—
122C2×C3×#67—38no—
123C2×C3×#67C2×D314yes—
124C2×C3×#67C2×D318yescomplemented
125#3811—616no—
126C2×#626—34no—
127C2×#626—34no—
128C2×#624—34no—
129C2×#624—38no—
130C32×C22⋊C4—31no—
131C32×C22⋊C4—32no—
132#3757—32no—
133#3757C2×D311yescharacteristic, complemented
134#3757C2×D312yescomplemented
135#3752—32no—
136#3752—32no—
137C4⋊(C3⋊Dic3)—32no—
138#3750—32no—
139C3×#217—38no—
140C3×#217—34no—
141C3×#217C2×D318yescomplemented
142C3×#217C2×D314yescomplemented
143Dic3⋊Dic3—38no—
144Dic3⋊Dic3—38no—
145#3722—38no—
146(C2×D3)⋊Dic3—38no—
147(C2×D3)⋊Dic3—38no—
148C22×C33—22no—
149C22×C33—22no—
150C22×C33—22no—
151C22×C33—22no—
152C22×C33—44no—
153C22×C33C2×D412yes—
154C22×C33C2411yescharacteristic
155C2×C3⋊(C3⋊D3)—22no—
156C2×C3⋊(C3⋊D3)—22no—
157C2×C3⋊(C3⋊D3)—44no—
158C2×C3⋊(C3⋊D3)C2×D412yes—
159C2×C3⋊(C3⋊D3)C2×D412yes—
160C2×C3×C3⋊D3—21no—
161C2×C3×C3⋊D3—22no—
162C2×C32×D3—21no—
163C2×C32×D3—22no—
164D3×C3⋊D3—44no—
165C3⋊(C3⋊Dic3)—22no—
166C3×C3⋊Dic3—21no—
167C3×C3⋊Dic3—22no—
168C32×Dic3—21no—
169C32×Dic3—22no—
170C22×#67—94no—
171C22×#67—91no—
172C22×#67—184no—
173C2×D3×D4—94no—
174C2×D3×D4—91no—
175C3×#120—34no—
176C3×#120—91no—
177#953—34no—
178#953C3⋊D311yescharacteristic, complemented
179#938—98no—
180#938—92no—
181#937—98no—
182#937—92no—
183#935—94no—
184#935—91no—
185#886—94no—
186#886—91no—
187#884—98no—
188#884—92no—
189#883—98no—
190#883—92no—
191D3×C22⋊C4—94no—
192D3×C22⋊C4—91no—
193C23×C32—216no—
194C23×C32—28no—
195C23×C32—22no—
196C23×C32—24no—
197C23×C32—28no—
198C23×C32—31no—
199C23×C32—42no—
200C23×C32—48no—
201C23×C32—416no—
202C23×C32C22×D318yes—
203C23×C32C22×D314yes—
204C23×C32C22×D312yes—
205C23×C32C22×D311yescharacteristic
206C22×C3⋊D3—31no—
207C22×C3⋊D3—31no—
208C22×C3⋊D3—34no—
209C22×C3⋊D3—34no—
210C22×C3⋊D3—34no—
211C22×C3⋊D3—316no—
212C22×C3⋊D3—616no—
213C22×C3⋊D3—64no—
214C22×C3⋊D3—64no—
215C22×C3⋊D3—62no—
216C22×C3⋊D3—122no—
217C22×C3⋊D3—122no—
218C22×C3⋊D3—128no—
219C22×C3⋊D3—1216no—
220C22×C3⋊D3C22×D311yescharacteristic
221C22×C3×D3—34no—
222C22×C3×D3C22×D314yes—
223C2×D32—68no—
224C2×D32—128no—
225C32×D4—62no—
226C32×D4—122no—
227C2×C32×C4—31no—
228C2×C32×C4—32no—
229#638—22no—
230#638—42no—
231#638—62no—
232#638—64no—
233#638—64no—
234#638—616no—
235#638—616no—
236#638—632no—
237#638—632no—
238C2×C3⋊Dic3—32no—
239C2×C3⋊Dic3—38no—
240C2×C3⋊Dic3—68no—
241C2×C3⋊Dic3—616no—
242C2×C3⋊Dic3—68no—
243C2×C3⋊Dic3—62no—
244C2×C3⋊Dic3C22×D312yes—
245C2×C3⋊Dic3C22×D311yescharacteristic
246C4⋊(C3⋊D3)—62no—
247C4×C3⋊D3—62no—
248C4×C3⋊D3—122no—
249C3×#67—28no—
250C3×#67—48no—
251C3×#67—68no—
252C3×#67—128no—
253C2×C3×Dic3—38no—
254C2×C3×Dic3—34no—
255C2×C3×Dic3C22×D318yes—
256C2×C3×Dic3C22×D314yes—
257#626—68no—
258#626—68no—
259#624—68no—
260#624—128no—
261#534—271noSylow
262C2×C33—22no—
263C2×C33—42no—
264C2×C33C22×D411yescharacteristic
265C2×C33H(5,𝔽2)12yes—
266C3⋊(C3⋊D3)—22no—
267C3⋊(C3⋊D3)—22no—
268C3×C3⋊D3—41no—
269C32×D3—41no—
270C24×C3—24no—
271C24×C3D3214yescomplemented
272C24×C3C2×C3⋊D311yescharacteristic, complemented
273C23×D3—94no—
274C23×D3—94no—
275C23×D3—91no—
276C23×D3—91no—
277C23×D3—184no—
278C2×C3×D4—34no—
279C2×C3×D4—38no—
280C2×C3×D4—91no—
281C2×C3×D4—92no—
282C2×#67—34no—
283C2×#67—38no—
284C2×#67—94no—
285C2×#67—98no—
286C2×#67—98no—
287C2×#67—92no—
288C2×#67—92no—
289C2×#67—91no—
290C2×#67—188no—
291C2×#67—188no—
292C2×#67—1816no—
293C2×#67—184no—
294C2×#67—1832no—
295C2×#67C2×C3⋊D311yescharacteristic
296C2×#67C2×C3⋊D312yescomplemented
297C22×Dic3—94no—
298C22×Dic3—91no—
299C22×Dic3—184no—
300D3×D4—184no—
301D3×D4—1816no—
302C2×D12—94no—
303C2×D12—91no—
304C2×C4×D3—94no—
305C2×C4×D3—91no—
306C2×C4×D3—98no—
307C2×C4×D3—92no—
308C3×C22⋊C4—34no—
309C3×C22⋊C4—38no—
310C3×C22⋊C4—91no—
311C3×C22⋊C4—92no—
312#217—34no—
313#217—38no—
314#217—98no—
315#217—92no—
316#217C2×C3⋊D311yescharacteristic, complemented
317#217C2×C3⋊D312yescomplemented
318(C2×D3)⋊C4—98no—
319(C2×D3)⋊C4—98no—
320(C2×D3)⋊C4—92no—
321(C2×D3)⋊C4—92no—
322C4⋊Dic3—98no—
323C4⋊Dic3—92no—
324Dic3⋊C4—98no—
325Dic3⋊C4—92no—
326C22×C32—216no—
327C22×C32—28no—
328C22×C32—22no—
329C22×C32—216no—
330C22×C32—28no—
331C22×C32—22no—
332C22×C32—22no—
333C22×C32—22no—
334C22×C32—24no—
335C22×C32—28no—
336C22×C32—28no—
337C22×C32—28no—
338C22×C32—48no—
339C22×C32—48no—
340C22×C32—432no—
341C22×C32—416no—
342C22×C32—44no—
343C22×C32—61no—
344C22×C32—62no—
345C22×C32D3×D418yes—
346C22×C32D3×D412yes—
347C22×C32C23×D314yes—
348C22×C32C23×D311yescharacteristic
349C2×C3⋊D3—21no—
350C2×C3⋊D3—22no—
351C2×C3⋊D3—32no—
352C2×C3⋊D3—32no—
353C2×C3⋊D3—38no—
354C2×C3⋊D3—38no—
355C2×C3⋊D3—68no—
356C2×C3⋊D3—68no—
357C2×C3⋊D3—62no—
358C2×C3⋊D3—62no—
359C2×C3⋊D3—68no—
360C2×C3⋊D3—68no—
361C2×C3⋊D3—128no—
362C2×C3⋊D3—128no—
363C2×C3⋊D3—124no—
364C2×C3⋊D3—124no—
365C2×C3⋊D3—1216no—
366C2×C3⋊D3—1232no—
367C2×C3×D3—24no—
368C2×C3×D3—28no—
369C2×C3×D3—64no—
370C2×C3×D3—68no—
371D32—1216no—
372C32×C4—61no—
373C32×C4—62no—
374C3⋊Dic3—21no—
375C3⋊Dic3—22no—
376C3⋊Dic3—616no—
377C3⋊Dic3—68no—
378C3⋊Dic3—62no—
379C3×Dic3—24no—
380C3×Dic3—28no—
381C3×Dic3—64no—
382C3×Dic3—68no—
383C22×D4—271no—
384C22×D4—271no—
385#123—272no—
386#123—271no—
387#123—271no—
388#121—272no—
389#121—272no—
390#120—31no—
391#120—91no—
392#120—272no—
393C2×C22⋊C4—271no—
394C33#53411yescharacteristic, complemented, Sylow
395C23×C3—216no—
396C23×C3—28no—
397C23×C3—22no—
398C23×C3—24no—
399C23×C3—28no—
400C23×C3—34no—
401C23×C3—48no—
402C23×C3—42no—
403C23×C3—416no—
404C23×C3—91no—
405C23×C3C2×D3218yes—
406C23×C3C2×D3214yes—
407C23×C3C22×C3⋊D312yes—
408C23×C3C22×C3⋊D311yescharacteristic
409C22×D3—34no—
410C22×D3—94no—
411C22×D3—94no—
412C22×D3—91no—
413C22×D3—91no—
414C22×D3—916no—
415C22×D3—94no—
416C22×D3—184no—
417C22×D3—184no—
418C22×D3—1816no—
419C22×D3—182no—
420C22×D3—188no—
421C22×D3—362no—
422C22×D3—368no—
423C22×D3—368no—
424C22×D3—362no—
425C22×D3—3616no—
426C22×D3C22×C3⋊D311yescharacteristic
427C3×D4—68no—
428C3×D4—128no—
429C3×D4—182no—
430C3×D4—362no—
431C2×C3×C4—34no—
432C2×C3×C4—38no—
433C2×C3×C4—91no—
434C2×C3×C4—92no—
435#67—22no—
436#67—42no—
437#67—68no—
438#67—128no—
439#67—182no—
440#67—188no—
441#67—1816no—
442#67—1816no—
443#67—184no—
444#67—184no—
445#67—1832no—
446#67—1832no—
447C2×Dic3—38no—
448C2×Dic3—34no—
449C2×Dic3—98no—
450C2×Dic3—92no—
451C2×Dic3—188no—
452C2×Dic3—1816no—
453C2×Dic3—188no—
454C2×Dic3—182no—
455C2×Dic3C22×C3⋊D312yes—
456C2×Dic3C22×C3⋊D311yescharacteristic
457D12—188no—
458D12—182no—
459C4×D3—182no—
460C4×D3—188no—
461C4×D3—362no—
462C4×D3—368no—
463C2×C32—216no—
464C2×C32—28no—
465C2×C32—22no—
466C2×C32—24no—
467C2×C32—28no—
468C2×C32—416no—
469C2×C32—48no—
470C2×C32—42no—
471C2×C32—121no—
472C2×C32C2×D3×D414yes—
473C2×C32C2×D3×D411yescharacteristic
474C2×C32#100418yes—
475C2×C32#100412yes—
476C3⋊D3—41no—
477C3⋊D3—62no—
478C3⋊D3—62no—
479C3⋊D3—68no—
480C3⋊D3—68no—
481C3⋊D3—128no—
482C3⋊D3—128no—
483C3×D3—44no—
484C3×D3—124no—
485C24—271no—
486C24—271no—
487C24D3×C3⋊D311yescharacteristic, complemented
488C2×D4—31no—
489C2×D4—32no—
490C2×D4—91no—
491C2×D4—92no—
492C2×D4—271no—
493C2×D4—271no—
494C2×D4—272no—
495C2×D4—272no—
496C2×D4—544no—
497C2×D4—544no—
498C22×C4—271no—
499C22×C4—272no—
500C22×C4—271no—
501C4⋊C4—272no—
502C4⋊C4—272no—
503C22⋊C4—31no—
504C22⋊C4—32no—
505C22⋊C4—91no—
506C22⋊C4—92no—
507C22⋊C4—272no—
508C22⋊C4—272no—
509C22⋊C4—272no—
510C22×C3—216no—
511C22×C3—28no—
512C22×C3—22no—
513C22×C3—216no—
514C22×C3—28no—
515C22×C3—28no—
516C22×C3—28no—
517C22×C3—22no—
518C22×C3—22no—
519C22×C3—22no—
520C22×C3—24no—
521C22×C3—28no—
522C22×C3—48no—
523C22×C3—48no—
524C22×C3—432no—
525C22×C3—416no—
526C22×C3—44no—
527C22×C3—64no—
528C22×C3—68no—
529C22×C3—181no—
530C22×C3—182no—
531C22×C3#381118yes—
532C22×C3D4×C3⋊D312yes—
533C22×C3C22×D3214yes—
534C22×C3C23×C3⋊D311yescharacteristic
535C2×D3—21no—
536C2×D3—22no—
537C2×D3—64no—
538C2×D3—68no—
539C2×D3—98no—
540C2×D3—98no—
541C2×D3—92no—
542C2×D3—92no—
543C2×D3—188no—
544C2×D3—188no—
545C2×D3—188no—
546C2×D3—188no—
547C2×D3—182no—
548C2×D3—182no—
549C2×D3—368no—
550C2×D3—368no—
551C2×D3—364no—
552C2×D3—3616no—
553C2×D3—3616no—
554C2×D3—364no—
555C2×D3—3632no—
556C3×C4—64no—
557C3×C4—68no—
558C3×C4—181no—
559C3×C4—182no—
560Dic3—21no—
561Dic3—22no—
562Dic3—64no—
563Dic3—68no—
564Dic3—1816no—
565Dic3—188no—
566Dic3—182no—
567C32—24no—
568C32#558914yescomplemented
569C32#558911yescharacteristic, complemented
570C23—22no—
571C23—31no—
572C23—42no—
573C23—91no—
574C23—271no—
575C23—271no—
576C23—274no—
577C23—542no—
578C23—1082no—
579C23—1082no—
580C23C2×D3×C3⋊D312yes—
581C23C2×D3×C3⋊D311yescharacteristic
582D4—62no—
583D4—122no—
584D4—182no—
585D4—362no—
586D4—542no—
587D4—542no—
588D4—544no—
589D4—544no—
590C2×C4—32no—
591C2×C4—31no—
592C2×C4—92no—
593C2×C4—91no—
594C2×C4—272no—
595C2×C4—542no—
596C2×C4—542no—
597C2×C4—1082no—
598C2×C3—216no—
599C2×C3—28no—
600C2×C3—22no—
601C2×C3—24no—
602C2×C3—28no—
603C2×C3—416no—
604C2×C3—48no—
605C2×C3—42no—
606C2×C3—124no—
607C2×C3—361no—
608C2×C3C2×#381114yes—
609C2×C3#6433018yes—
610C2×C3C2×D4×C3⋊D311yescharacteristic
611C2×C3#6436112yes—
612D3—41no—
613D3—124no—
614D3—188no—
615D3—188no—
616D3—182no—
617D3—182no—
618D3—368no—
619D3—368no—
620C22—22no—
621C22—22no—
622C22—22no—
623C22—22no—
624C22—44no—
625C22—61no—
626C22—62no—
627C22—181no—
628C22—182no—
629C22—272no—
630C22—272no—
631C22—542no—
632C22—542no—
633C22—1084no—
634C22—1084no—
635C22#8903912yes—
636C22C22×D3×C3⋊D311yescharacteristic
637C4—61no—
638C4—62no—
639C4—181no—
640C4—182no—
641C4—542no—
642C3—24no—
643C3#1059552814yescomplemented
644C3#1059606011yescharacteristic, complemented
645C2—22no—
646C2—42no—
647C2—121no—
648C2—361no—
649C2—542no—
650C2—542no—
651C2C2×#8903911yescharacteristic
652C2#1172461112yes—
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC24×C33C22
hypercenterC24D3×C3⋊D3
hypocenterC33#534
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C24

Complements

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

NH ≅ G/Ntype
1. C22×#6312C2semidirect
2. C2×#89039C2semidirect
3. C3×#64162C2semidirect
4. C32×#953C2semidirect
5. #11723643C2semidirect
6. #11723628C2semidirect
7. #11723627C2semidirect
8. #11723623C2semidirect
9. #11723609C2semidirect
10. #11723603C2semidirect
11. #11723599C2semidirect
14. C24×C33C22semidirect
16. C22×D3×C3⋊D3C22semidirect
19. C2×#6312C22semidirect
20. C22×C3⋊(C3⋊Dic3)C22semidirect
22. C2×C3×#638C22semidirect
24. C2×C32×#67C22semidirect
26. C2×#6292C22semidirect
27. C2×#6291C22semidirect
28. C2×#6289C22semidirect
29. C2×#6289C22semidirect
30. C3×#3757C22semidirect
31. C3×#3757C22semidirect
32. C32×#217C22semidirect
33. C32×#217C22semidirect
34. #88807C22semidirect
35. Dic3⋊(C3⋊Dic3)C22semidirect
36. #88804C22semidirect
37. #88803C22semidirect
38. (C2×D3)⋊(C3⋊Dic3)C22semidirect
45. #64162D3semidirect
54. C3×#953D3semidirect
67. C22×C3⋊(C3⋊D3)D4semidirect
93. C24×C32C2×D3semidirect
94. C24×C32C2×D3semidirect
113. C2×#638C2×D3semidirect
124. C2×C3×#67C2×D3semidirect
133. #3757C2×D3semidirect
134. #3757C2×D3semidirect
141. C3×#217C2×D3semidirect
142. C3×#217C2×D3semidirect
178. #953C3⋊D3semidirect
271. C24×C3D32semidirect
272. C24×C3C2×C3⋊D3semidirect
296. C2×#67C2×C3⋊D3semidirect
316. #217C2×C3⋊D3semidirect
317. #217C2×C3⋊D3semidirect
394. C33#534semidirect
487. C24D3×C3⋊D3semidirect
568. C32#5589semidirect
569. C32#5589semidirect
643. C3#10595528semidirect
644. C3#10596060semidirect

Maximal subgroups

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

subgroupindex
C22×#63122
C2×#890392
C3×#641622
C32×#9532
#117236432
#117236282
#117236272
#117236232
#117236092
#117236032
#117235992
#105960603
#105955283