Finite Group Databasebeta

← #13178665

Subgroups — #13178665

Order 1728.

1814 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×D4×C3⋊D3C211yesmaximal, characteristic, complemented
2C2×#89042C211yesmaximal, characteristic, complemented
3C3×#64147C212yesmaximal, complemented
4#11723728C212yesmaximal, complemented
5#11723727C214yesmaximal, complemented
6#11723718C212yesmaximal, complemented
7#11723696C212yesmaximal, complemented
8#11723667C211yesmaximal, characteristic, complemented
9#10595241—31nomaximal
10#10595231—32nomaximal
11C23×C3×C3⋊D3C2212yes—
12C2×C33×D4C2211yescharacteristic
13C2×C3×#638C2212yes—
14C2×C3×#638C2214yescomplemented
15C3×D4×C3⋊D3—24no—
16C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
17C2×C3×C4×C3⋊D3C2211yescharacteristic
18C2×C3×C4×C3⋊D3C2212yescomplemented
19C2×#6296C2211yescharacteristic, complemented
20C2×#6295C2212yes—
21C2×#6295C2211yescharacteristic, complemented
22C2×#6295C2214yescomplemented
23#89042—24no—
24C3×#3757C2214yescomplemented
25C3×C4⋊(C3⋊Dic3)C2212yescomplemented
26#88814C2212yes—
27#88814C2214yescomplemented
28#88813C2212yescomplemented
29C2×D4×C3⋊D3D311yescharacteristic, complemented
30C2×C3×D3×D4—32no—
31C2×C3×D3×D4—61no—
32C2×#3798—31no—
33C2×#3798—32no—
34#64147D312yescomplemented
35C3×#938—32no—
36C3×#938—32no—
37C3×#938—62no—
38#63981—34no—
39#63979—32no—
40#63979—34no—
41#63977—34no—
42#63976—34no—
43#63959—34no—
44#63953—32no—
45#63916—32no—
46#63909—32no—
47#63908—32no—
48#63849—31no—
49#63841—32no—
50C23×C33C2312yes—
51C22×C3×C3⋊D3—24no—
52C22×C3×C3⋊D3—44no—
53C22×C3×C3⋊D3C2312yes—
54C22×C3×C3⋊D3C2312yes—
55C22×C3×C3⋊D3C2311yescharacteristic
56C33×D4—22no—
57C2×C33×C4C2311yescharacteristic
58C3×#638—24no—
59C3×#638—44no—
60C2×C3×C3⋊Dic3C2314yes—
61C2×C3×C3⋊Dic3C2312yes—
62C2×C3×C3⋊Dic3C2311yescharacteristic
63C3×C4⋊(C3⋊D3)—22no—
64C3×C4×C3⋊D3—24no—
65C3×C4×C3⋊D3D414yescomplemented
66#6296—22no—
67#6295—24no—
68#6295—22no—
69#6295—44no—
70#5805—91no—
71#5607—92no—
72C23×C3⋊D3C2×D312yes—
73C23×C3×D3—34no—
74C23×C3×D3—62no—
75C2×C32×D4—21no—
76C2×C32×D4—22no—
77C2×C32×D4—41no—
78C2×C32×D4C2×D311yescharacteristic
79C2×C32×D4C2×D312yes—
80C2×#638C2×D312yes—
81C2×#638C2×D314yescomplemented
82D4×C3⋊D3—24no—
83C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
84C2×C4×C3⋊D3C2×D311yescharacteristic
85C2×C4×C3⋊D3C2×D312yescomplemented
86C2×C3×#67—34no—
87C2×C3×#67—34no—
88C2×C3×#67—34no—
89C2×C3×#67—62no—
90C2×C3×#67—64no—
91C3×D3×D4—68no—
92C3×D3×D4—68no—
93C2×C3×D12—32no—
94C2×C3×D12—61no—
95C2×C3×C4×D3—32no—
96C2×C3×C4×D3—32no—
97C2×C3×C4×D3—32no—
98C2×C3×C4×D3—61no—
99C2×C3×C4×D3—62no—
100C2×#627—31no—
101C2×#627—32no—
102C2×#626—32no—
103C2×#626—32no—
104C2×#626—32no—
105C2×#626—34no—
106C2×#626—34no—
107C2×#625—32no—
108C2×#625—31no—
109C2×#625—34no—
110#3798—68no—
111#3798—64no—
112#3757C2×D314yescomplemented
113C4⋊(C3⋊Dic3)C2×D312yescomplemented
114C3×#217—34no—
115C3×#217—34no—
116C3×#217—64no—
117C3×C4⋊Dic3—32no—
118C3×C4⋊Dic3—32no—
119C3×C4⋊Dic3—62no—
120#3722—32no—
121#3722—34no—
122(C2×D3)⋊Dic3—34no—
123(C2×D3)⋊Dic3—34no—
124(C2×D3)⋊Dic3—34no—
125Dic32—34no—
126Dic32—32no—
127C22×C33—22no—
128C22×C33—24no—
129C22×C33C2411yescharacteristic
130C2×C3×C3⋊D3—22no—
131C2×C3×C3⋊D3—22no—
132C2×C3×C3⋊D3—22no—
133C2×C3×C3⋊D3—22no—
134C2×C3×C3⋊D3—22no—
135C2×C3×C3⋊D3—24no—
136C2×C3×C3⋊D3—48no—
137C2×C3×C3⋊D3C2×D412yes—
138C33×C4C2×D412yes—
139C3×C3⋊Dic3—22no—
140C3×C3⋊Dic3—24no—
141C3×C3⋊Dic3C2×D412yes—
142C22×C3×D4—91no—
143C2×D3×D4—32no—
144C2×D3×D4—61no—
145C2×#235—91no—
146C2×#235—92no—
147C3×#121—92no—
148#953—94no—
149#940—92no—
150#940—94no—
151#938—32no—
152#938—32no—
153#938—62no—
154#938—92no—
155#937—94no—
156#936—92no—
157#930—92no—
158#926—91no—
159#895—92no—
160#892—92no—
161#885—94no—
162#882—94no—
163C23×C32—22no—
164C23×C32—24no—
165C23×C32—42no—
166C23×C32C22×D314yes—
167C23×C32C22×D312yes—
168C22×C3⋊D3—24no—
169C22×C3⋊D3—44no—
170C22×C3⋊D3C22×D312yes—
171C22×C3⋊D3C22×D312yes—
172C22×C3⋊D3C22×D311yescharacteristic
173C22×C3×D3—32no—
174C22×C3×D3—34no—
175C22×C3×D3—32no—
176C22×C3×D3—32no—
177C22×C3×D3—61no—
178C22×C3×D3—62no—
179C22×C3×D3—62no—
180C22×C3×D3—68no—
181C22×C3×D3—68no—
182C22×C3×D3—124no—
183C22×C3×D3—124no—
184C22×C3×D3—128no—
185C32×D4—22no—
186C32×D4—24no—
187C32×D4—24no—
188C32×D4—44no—
189C32×D4—44no—
190C2×C32×C4—22no—
191C2×C32×C4—21no—
192C2×C32×C4—41no—
193C2×C32×C4C22×D312yes—
194C2×C32×C4C22×D311yescharacteristic
195#638—24no—
196#638—44no—
197C2×C3⋊Dic3C22×D314yes—
198C2×C3⋊Dic3C22×D312yes—
199C2×C3⋊Dic3C22×D311yescharacteristic
200C4⋊(C3⋊D3)—22no—
201C4×C3⋊D3—24no—
202C4×C3⋊D3#6714yescomplemented
203C3×#67—68no—
204C3×#67—68no—
205C3×#67—124no—
206C3×#67—124no—
207C3×#67—128no—
208C2×C3×Dic3—32no—
209C2×C3×Dic3—34no—
210C2×C3×Dic3—34no—
211C2×C3×Dic3—32no—
212C2×C3×Dic3—32no—
213C2×C3×Dic3—61no—
214C2×C3×Dic3—64no—
215C2×C3×Dic3—62no—
216C3×D12—64no—
217C3×D12—64no—
218C3×C4×D3—38no—
219C3×C4×D3—64no—
220C3×C4×D3—64no—
221C3×C4×D3—64no—
222C3×C4×D3—124no—
223#627—64no—
224#627—62no—
225#626—64no—
226#626—64no—
227#626—64no—
228#626—124no—
229#626—124no—
230#625—64no—
231#625—62no—
232#625—124no—
233#535—271noSylow
234C2×C33—42no—
235C2×C33C22×D411yescharacteristic
236C2×C33H(5,𝔽2)12yes—
237C3×C3⋊D3—22no—
238C3×C3⋊D3—42no—
239C3×C3⋊D3—42no—
240C24×C3—92no—
241C23×D3—34no—
242C23×D3—62no—
243C2×C3×D4—21no—
244C2×C3×D4—22no—
245C2×C3×D4—41no—
246C2×C3×D4—92no—
247C2×C3×D4—94no—
248C2×C3×D4—91no—
249C2×C3×D4—184no—
250C2×C3×D4D3211yescharacteristic
251C2×C3×D4D3212yes—
252C22×C3×C4—91no—
253C22×C3×C4—92no—
254C2×#67—34no—
255C2×#67—34no—
256C2×#67—34no—
257C2×#67—62no—
258C2×#67—64no—
259C2×#67—92no—
260C2×#67—92no—
261C2×#67—92no—
262C2×#67—94no—
263C2×#67—94no—
264D3×D4—68no—
265D3×D4—68no—
266#235—188no—
267#235—184no—
268C2×D12—32no—
269C2×D12—61no—
270C2×D12—92no—
271C2×D12—91no—
272C2×D12—94no—
273C2×C4×D3—32no—
274C2×C4×D3—32no—
275C2×C4×D3—32no—
276C2×C4×D3—61no—
277C2×C4×D3—62no—
278C2×Dic6—92no—
279C2×Dic6—91no—
280C3×C4⋊C4—92no—
281C3×C22⋊C4—94no—
282#217—34no—
283#217—34no—
284#217—64no—
285#217—92no—
286#217—94no—
287(C2×D3)⋊C4—94no—
288(C2×D3)⋊C4—94no—
289(C2×D3)⋊C4—94no—
290C4⋊Dic3—32no—
291C4⋊Dic3—32no—
292C4⋊Dic3—62no—
293C4×Dic3—94no—
294C4×Dic3—92no—
295C22×C32—22no—
296C22×C32—24no—
297C22×C32—24no—
298C22×C32—24no—
299C22×C32—24no—
300C22×C32—24no—
301C22×C32—22no—
302C22×C32—21no—
303C22×C32—41no—
304C22×C32—44no—
305C22×C32—44no—
306C22×C32—44no—
307C22×C32—44no—
308C22×C32—44no—
309C22×C32—84no—
310C22×C32C23×D312yes—
311C22×C32C23×D311yescharacteristic
312C2×C3⋊D3—22no—
313C2×C3⋊D3—22no—
314C2×C3⋊D3—22no—
315C2×C3⋊D3—22no—
316C2×C3⋊D3—24no—
317C2×C3⋊D3—22no—
318C2×C3⋊D3—48no—
319C2×C3⋊D3C2×#6712yes—
320C2×C3×D3—34no—
321C2×C3×D3—62no—
322C2×C3×D3—64no—
323C2×C3×D3—62no—
324C2×C3×D3—62no—
325C2×C3×D3—64no—
326C2×C3×D3—64no—
327C2×C3×D3—62no—
328C2×C3×D3—64no—
329C2×C3×D3—62no—
330C2×C3×D3—64no—
331C2×C3×D3—62no—
332C2×C3×D3—62no—
333C2×C3×D3—64no—
334C2×C3×D3—62no—
335C2×C3×D3—62no—
336C2×C3×D3—122no—
337C2×C3×D3—122no—
338C2×C3×D3—124no—
339C2×C3×D3—1216no—
340C2×C3×D3—1216no—
341C32×C4—24no—
342C32×C4—22no—
343C32×C4—42no—
344C32×C4D3×D414yes—
345C32×C4C2×#6712yes—
346C3⋊Dic3—22no—
347C3⋊Dic3—24no—
348C3⋊Dic3C2×#6712yes—
349C3×Dic3—34no—
350C3×Dic3—62no—
351C3×Dic3—62no—
352C3×Dic3—64no—
353C3×Dic3—64no—
354C3×Dic3—62no—
355C3×Dic3—122no—
356C3×Dic3—124no—
357C2×#41—271no—
358C22×D4—91no—
359#127—272no—
360#124—272no—
361#121—92no—
362#121—272no—
363#120—274no—
364#117—271no—
365C33#53511yescharacteristic, complemented, Sylow
366C23×C3—22no—
367C23×C3—24no—
368C23×C3—42no—
369C23×C3—91no—
370C23×C3—92no—
371C23×C3—92no—
372C23×C3—184no—
373C23×C3—364no—
374C23×C3C2×D3212yes—
375C23×C3C2×D3214yes—
376C22×D3—32no—
377C22×D3—34no—
378C22×D3—32no—
379C22×D3—32no—
380C22×D3—61no—
381C22×D3—62no—
382C22×D3—62no—
383C22×D3—68no—
384C22×D3—68no—
385C22×D3—124no—
386C22×D3—124no—
387C22×D3—128no—
388C3×D4—24no—
389C3×D4—24no—
390C3×D4—22no—
391C3×D4—44no—
392C3×D4—44no—
393C3×D4—184no—
394C3×D4—182no—
395C3×D4—364no—
396C2×C3×C4—22no—
397C2×C3×C4—21no—
398C2×C3×C4—41no—
399C2×C3×C4—91no—
400C2×C3×C4—94no—
401C2×C3×C4—94no—
402C2×C3×C4—92no—
403C2×C3×C4—184no—
404C2×C3×C4C2×D3212yes—
405C2×C3×C4C2×D3211yescharacteristic
406#67—68no—
407#67—68no—
408#67—124no—
409#67—124no—
410#67—128no—
411#67—184no—
412#67—184no—
413#67—184no—
414#67—364no—
415#67—364no—
416C2×Dic3—32no—
417C2×Dic3—34no—
418C2×Dic3—34no—
419C2×Dic3—32no—
420C2×Dic3—32no—
421C2×Dic3—61no—
422C2×Dic3—64no—
423C2×Dic3—62no—
424D12—64no—
425D12—64no—
426D12—182no—
427D12—184no—
428D12—364no—
429C4×D3—38no—
430C4×D3—64no—
431C4×D3—64no—
432C4×D3—64no—
433C4×D3—124no—
434Dic6—184no—
435Dic6—182no—
436C2×C32—22no—
437C2×C32—22no—
438C2×C32—22no—
439C2×C32—22no—
440C2×C32—21no—
441C2×C32—41no—
442C2×C32—42no—
443C2×C32—42no—
444C2×C32—44no—
445C2×C32—44no—
446C2×C32—84no—
447C2×C32—84no—
448C2×C32C2×D3×D412yes—
449C2×C32#100412yes—
450C2×C32#100412yes—
451C2×C32#100912yes—
452C2×C32C22×#6711yescharacteristic
453C3⋊D3—22no—
454C3⋊D3—42no—
455C3⋊D3—42no—
456C3×D3—64no—
457C3×D3—64no—
458C3×D3—124no—
459C3×D3—124no—
460C3×D3—122no—
461C3×D3—122no—
462C3×D3—124no—
463C24—92no—
464#41—544no—
465C2×Q8—271no—
466C2×D4—92no—
467C2×D4—94no—
468C2×D4—91no—
469C2×D4—184no—
470C2×D4—272no—
471C2×D4—271no—
472C2×D4—274no—
473C2×D4#111311yescharacteristic
474C22×C4—91no—
475C22×C4—92no—
476C4⋊C4—92no—
477C22⋊C4—94no—
478C22⋊C4—272no—
479C22⋊C4—274no—
480C42—272no—
481C22×C3—24no—
482C22×C3—24no—
483C22×C3—24no—
484C22×C3—24no—
485C22×C3—24no—
486C22×C3—22no—
487C22×C3—22no—
488C22×C3—21no—
489C22×C3—44no—
490C22×C3—44no—
491C22×C3—44no—
492C22×C3—44no—
493C22×C3—44no—
494C22×C3—41no—
495C22×C3—84no—
496C22×C3—92no—
497C22×C3—182no—
498C22×C3—182no—
499C22×C3—184no—
500C22×C3—182no—
501C22×C3—182no—
502C22×C3—182no—
503C22×C3—368no—
504C22×C3C22×D3212yes—
505C22×C3C22×D3211yescharacteristic
506C2×D3—34no—
507C2×D3—62no—
508C2×D3—64no—
509C2×D3—62no—
510C2×D3—62no—
511C2×D3—64no—
512C2×D3—64no—
513C2×D3—62no—
514C2×D3—64no—
515C2×D3—62no—
516C2×D3—64no—
517C2×D3—62no—
518C2×D3—62no—
519C2×D3—62no—
520C2×D3—64no—
521C2×D3—62no—
522C2×D3—122no—
523C2×D3—124no—
524C2×D3—122no—
525C2×D3—1216no—
526C2×D3—1216no—
527C3×C4—22no—
528C3×C4—24no—
529C3×C4—42no—
530C3×C4—92no—
531C3×C4—184no—
532C3×C4—182no—
533C3×C4D3×#6714yes—
534C3×C4#381112yes—
535Dic3—34no—
536Dic3—62no—
537Dic3—64no—
538Dic3—64no—
539Dic3—62no—
540Dic3—62no—
541Dic3—122no—
542Dic3—124no—
543C32—22no—
544C32—21no—
545C32—41no—
546C32#560712yescomplemented
547C32#580511yescharacteristic, complemented
548C23—91no—
549C23—92no—
550C23—92no—
551C23—184no—
552C23—364no—
553C23C2×#111312yes—
554Q8—542no—
555D4—22no—
556D4—184no—
557D4—182no—
558D4—364no—
559D4—542no—
560D4—544no—
561D4—1084no—
562C2×C4—91no—
563C2×C4—94no—
564C2×C4—94no—
565C2×C4—92no—
566C2×C4—184no—
567C2×C4C2×#111311yescharacteristic
568C2×C3—22no—
569C2×C3—22no—
570C2×C3—22no—
571C2×C3—21no—
572C2×C3—22no—
573C2×C3—44no—
574C2×C3—44no—
575C2×C3—42no—
576C2×C3—42no—
577C2×C3—41no—
578C2×C3—84no—
579C2×C3—84no—
580C2×C3—182no—
581C2×C3—362no—
582C2×C3—362no—
583C2×C3#6431312yes—
584C2×C3#6431412yes—
585C2×C3#6431412yes—
586C2×C3C2×D3×#6712yes—
587C2×C3C2×#381111yescharacteristic
588D3—64no—
589D3—64no—
590D3—124no—
591D3—124no—
592D3—124no—
593D3—122no—
594D3—122no—
595C22—24no—
596C22—22no—
597C22—92no—
598C22—182no—
599C22—182no—
600C22—182no—
601C22—184no—
602C22—182no—
603C22—182no—
604C22—368no—
605C22C22×#111311yescharacteristic
606C4—92no—
607C4—184no—
608C4—182no—
609C4#8904912yes—
610C3—21no—
611C3—22no—
612C3—41no—
613C3#1059523112yescomplemented
614C3#1059524111yescharacteristic, complemented
615C2—42no—
616C2—182no—
617C2—362no—
618C2—362no—
619C2#1172462312yes—
620C2C2×#8904911yescharacteristic
subgroupG/H
center, FrattiniC22C22×#1113
commutator, socle, cosocleC22×C33C24
FittingC2×C33×D4C22
hypercenterC2×D4#1113
hypocenterC33#535
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×C3×D4×C3⋊D3C2semidirect
2. C2×#89042C2semidirect
3. C3×#64147C2semidirect
4. #11723728C2semidirect
5. #11723727C2semidirect
6. #11723718C2semidirect
7. #11723696C2semidirect
8. #11723667C2semidirect
14. C2×C3×#638C22semidirect
18. C2×C3×C4×C3⋊D3C22semidirect
19. C2×#6296C22semidirect
21. C2×#6295C22semidirect
22. C2×#6295C22semidirect
24. C3×#3757C22semidirect
25. C3×C4⋊(C3⋊Dic3)C22semidirect
27. #88814C22semidirect
28. #88813C22semidirect
29. C2×D4×C3⋊D3D3semidirect
34. #64147D3semidirect
65. C3×C4×C3⋊D3D4semidirect
81. C2×#638C2×D3semidirect
85. C2×C4×C3⋊D3C2×D3semidirect
112. #3757C2×D3semidirect
113. C4⋊(C3⋊Dic3)C2×D3semidirect
202. C4×C3⋊D3#67semidirect
365. C33#535semidirect
546. C32#5607semidirect
547. C32#5805semidirect
613. C3#10595231semidirect
614. C3#10595241semidirect

Maximal subgroups

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

subgroupindex
C2×C3×D4×C3⋊D32
C2×#890422
C3×#641472
#117237282
#117237272
#117237182
#117236962
#117236672
#105952413
#105952313