Finite Group Databasebeta

← #13178759

Subgroups — #13178759

Order 1728.

1670 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3832C211yesmaximal, characteristic, complemented
2C2×C4×#1113C211yesmaximal, characteristic, complemented
3C3×#64155C212yesmaximal, complemented
4#11723701C212yesmaximal, complemented
5#11723696C212yesmaximal, complemented
6#11723686C212yesmaximal, complemented
7#11723684C214yesmaximal, complemented
8#11723667C211yesmaximal, characteristic, complemented
9#10595430—31nomaximal
10#10595424—32nomaximal
11C22×#1113C2211yescharacteristic
12C2×C33×Q8C2211yescharacteristic, complemented
13C3×#3832—24no—
14C2×C3×C4⋊(C3⋊D3)C2212yes—
15C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic, complemented
16C2×C3×C4×C3⋊D3C2212yes—
17C2×C3×C4×C3⋊D3C2211yescharacteristic
18C2×C3×C4×C3⋊D3C2212yescomplemented
19C2×#6295C2214yes—
20C2×#6294C2212yes—
21C2×#6294C2211yescharacteristic
22C4×#1113—24no—
23C3×#3752C2214yes—
24C3×C4⋊(C3⋊Dic3)C2212yescomplemented
25C3×#3750C2214yescomplemented
26#88814C2212yes—
27#88814C2214yescomplemented
28#88813C2212yescomplemented
29C2×#3832D311yescharacteristic, complemented
30C2×C3×#239—32no—
31C2×C3×#239—61no—
32C2×C4×D32—32no—
33C2×C4×D32—31no—
34#64155D312yescomplemented
35C3×#946—32no—
36C3×#946—32no—
37C3×#946—62no—
38#63923—32no—
39#63922—34no—
40#63916—32no—
41#63909—32no—
42#63908—32no—
43#63904—34no—
44#63897—32no—
45#63884—34no—
46#63881—34no—
47#63879—34no—
48#63849—31no—
49#63841—32no—
50C22×C3×C3⋊D3C2312yes—
51C22×C3×C3⋊D3C2311yescharacteristic
52C22×C3×C3⋊D3C2312yes—
53C2×#1113—22no—
54C2×#1113—24no—
55C33×Q8—22no—
56C2×C33×C4C2312yes—
57C2×C33×C4C2311yescharacteristic
58C2×C3×C3⋊Dic3C2312yes—
59C2×C3×C3⋊Dic3C2314yes—
60C2×C3×C3⋊Dic3C2311yescharacteristic
61C3×C4⋊(C3⋊D3)—24no—
62C3×C4⋊(C3⋊D3)—22no—
63C3×C4×C3⋊D3—24no—
64C3×C4×C3⋊D3—24no—
65C3×C4×C3⋊D3D414yescomplemented
66#6295—28no—
67#6294—22no—
68#6294—24no—
69#5828—91no—
70#5627—92no—
71C22×D32—32no—
72C22×D32—31no—
73C2×C32×Q8—21no—
74C2×C32×Q8—22no—
75C2×C32×Q8—41no—
76C2×C32×Q8C2×D311yescharacteristic, complemented
77C2×C32×Q8C2×D312yescomplemented
78#3832—24no—
79C2×C4⋊(C3⋊D3)C2×D312yes—
80C2×C4⋊(C3⋊D3)C2×D311yescharacteristic, complemented
81C2×C4×C3⋊D3C2×D312yes—
82C2×C4×C3⋊D3C2×D311yescharacteristic
83C2×C4×C3⋊D3C2×D312yescomplemented
84C3×#239—68no—
85C3×#239—68no—
86C2×C3×D12—34no—
87C2×C3×D12—32no—
88C2×C3×D12—62no—
89C2×C3×D12—61no—
90C2×C3×C4×D3—32no—
91C2×C3×C4×D3—32no—
92C2×C3×C4×D3—32no—
93C2×C3×C4×D3—34no—
94C2×C3×C4×D3—61no—
95C2×C3×C4×D3—62no—
96C2×C3×C4×D3—62no—
97C2×#626—34no—
98C2×#626—34no—
99C2×#625—34no—
100C2×#624—31no—
101C2×#624—32no—
102C2×D3×Dic3—32no—
103C2×D3×Dic3—32no—
104C2×D3×Dic3—32no—
105C4×D32—68no—
106C4×D32—64no—
107#3752C2×D314yes—
108C4⋊(C3⋊Dic3)C2×D312yescomplemented
109#3750C2×D314yescomplemented
110C3×(C2×D3)⋊C4—34no—
111C3×(C2×D3)⋊C4—34no—
112C3×(C2×D3)⋊C4—64no—
113C3×C4⋊Dic3—32no—
114C3×C4⋊Dic3—32no—
115C3×C4⋊Dic3—62no—
116C3×Dic3⋊C4—34no—
117C3×Dic3⋊C4—34no—
118C3×Dic3⋊C4—64no—
119#3722—32no—
120#3722—34no—
121(C2×D3)⋊Dic3—34no—
122(C2×D3)⋊Dic3—34no—
123(C2×D3)⋊Dic3—34no—
124Dic32—34no—
125Dic32—32no—
126C22×C33C2411yescharacteristic
127C2×C3×C3⋊D3—22no—
128C2×C3×C3⋊D3—22no—
129C2×C3×C3⋊D3—22no—
130C2×C3×C3⋊D3—22no—
131C2×C3×C3⋊D3—24no—
132C2×C3×C3⋊D3C2×D412yes—
133C2×C3×C3⋊D3#4114yescomplemented
134#1113—28no—
135C33×C4—22no—
136C33×C4C2×D412yes—
137C3×C3⋊Dic3—22no—
138C3×C3⋊Dic3—24no—
139C3×C3⋊Dic3C2×D412yes—
140C2×C3×#41—91no—
141C2×#239—32no—
142C2×#239—61no—
143C22×C4×D3—91no—
144C22×C4×D3—92no—
145C3×C22⋊Q8—92no—
146#946—32no—
147#946—32no—
148#946—62no—
149#938—92no—
150#937—92no—
151#935—92no—
152#930—92no—
153C4×#67—94no—
154#926—91no—
155#895—92no—
156#894—94no—
157#893—94no—
158#892—92no—
159#882—94no—
160#67⋊C4—94no—
161C22×C3⋊D3C22×D312yes—
162C22×C3⋊D3C22×D312yes—
163C22×C3⋊D3C22×D311yescharacteristic
164C22×C3×D3—32no—
165C22×C3×D3—32no—
166C22×C3×D3—32no—
167C22×C3×D3—34no—
168C22×C3×D3—62no—
169C22×C3×D3—61no—
170C22×C3×D3—62no—
171C2×D32—64no—
172C2×D32—64no—
173C2×D32—64no—
174C2×D32—64no—
175C2×D32—62no—
176C32×Q8—22no—
177C32×Q8—24no—
178C32×Q8—24no—
179C32×Q8—44no—
180C32×Q8—44no—
181C2×C32×C4—22no—
182C2×C32×C4—22no—
183C2×C32×C4—21no—
184C2×C32×C4—24no—
185C2×C32×C4—41no—
186C2×C32×C4—42no—
187C2×C32×C4C22×D312yes—
188C2×C32×C4C22×D314yes—
189C2×C32×C4C22×D312yes—
190C2×C32×C4C22×D311yescharacteristic
191C2×C3⋊Dic3C22×D314yes—
192C2×C3⋊Dic3C22×D312yes—
193C2×C3⋊Dic3C22×D311yescharacteristic
194C4⋊(C3⋊D3)—24no—
195C4⋊(C3⋊D3)—22no—
196C4×C3⋊D3—24no—
197C4×C3⋊D3—24no—
198C4×C3⋊D3#6714yescomplemented
199C2×C3×Dic3—32no—
200C2×C3×Dic3—34no—
201C2×C3×Dic3—34no—
202C2×C3×Dic3—32no—
203C2×C3×Dic3—32no—
204C2×C3×Dic3—61no—
205C2×C3×Dic3—62no—
206C2×C3×Dic3—64no—
207C3×D12—68no—
208C3×D12—68no—
209C3×D12—64no—
210C3×D12—64no—
211C3×C4×D3—38no—
212C3×C4×D3—64no—
213C3×C4×D3—68no—
214C3×C4×D3—68no—
215C3×C4×D3—64no—
216C3×C4×D3—64no—
217C3×C4×D3—124no—
218#626—68no—
219#626—68no—
220#625—68no—
221#624—64no—
222#624—62no—
223D3×Dic3—64no—
224D3×Dic3—64no—
225D3×Dic3—64no—
226#525—271noSylow
227C2×C33C22×D411yescharacteristic
228C2×C33C2×#4112yes—
229C3×C3⋊D3—22no—
230C3×C3⋊D3—24no—
231C3×C3⋊D3—42no—
232C23×D3—92no—
233C23×D3—91no—
234C3×#41—184no—
235C2×C3×Q8—21no—
236C2×C3×Q8—22no—
237C2×C3×Q8—41no—
238C2×C3×Q8D3211yescharacteristic, complemented
239C2×C3×Q8D3212yescomplemented
240C2×C3×D4—92no—
241C2×C3×D4—91no—
242C22×C3×C4—91no—
243C22×C3×C4—92no—
244C22×C3×C4—92no—
245C2×#67—94no—
246C2×#67—94no—
247C22×Dic3—92no—
248C22×Dic3—91no—
249#239—68no—
250#239—68no—
251C2×D12—34no—
252C2×D12—32no—
253C2×D12—62no—
254C2×D12—61no—
255C2×D12—94no—
256C2×C4×D3—32no—
257C2×C4×D3—34no—
258C2×C4×D3—32no—
259C2×C4×D3—32no—
260C2×C4×D3—61no—
261C2×C4×D3—62no—
262C2×C4×D3—62no—
263C2×C4×D3—92no—
264C2×C4×D3—92no—
265C2×C4×D3—92no—
266C2×C4×D3—188no—
267C2×C4×D3—184no—
268C3×C4⋊C4—92no—
269C3×C4⋊C4—94no—
270C3×C22⋊C4—94no—
271#217—92no—
272#217—94no—
273(C2×D3)⋊C4—34no—
274(C2×D3)⋊C4—34no—
275(C2×D3)⋊C4—64no—
276(C2×D3)⋊C4—94no—
277(C2×D3)⋊C4—94no—
278(C2×D3)⋊C4—94no—
279C4⋊Dic3—32no—
280C4⋊Dic3—32no—
281C4⋊Dic3—62no—
282Dic3⋊C4—34no—
283Dic3⋊C4—34no—
284Dic3⋊C4—64no—
285C4×Dic3—94no—
286C4×Dic3—92no—
287C22×C32—22no—
288C22×C32—21no—
289C22×C32—41no—
290C22×C32C23×D312yes—
291C22×C32C23×D311yescharacteristic
292C2×C3⋊D3—22no—
293C2×C3⋊D3—22no—
294C2×C3⋊D3—22no—
295C2×C3⋊D3—22no—
296C2×C3⋊D3—24no—
297C2×C3⋊D3#23914yescomplemented
298C2×C3⋊D3C2×#6712yes—
299C2×C3×D3—34no—
300C2×C3×D3—34no—
301C2×C3×D3—34no—
302C2×C3×D3—62no—
303C2×C3×D3—64no—
304C2×C3×D3—62no—
305C2×C3×D3—62no—
306C2×C3×D3—62no—
307C2×C3×D3—62no—
308C2×C3×D3—62no—
309C2×C3×D3—62no—
310C2×C3×D3—64no—
311C2×C3×D3—64no—
312C2×C3×D3—64no—
313C2×C3×D3—64no—
314C2×C3×D3—122no—
315C2×C3×D3—122no—
316C2×C3×D3—122no—
317C2×C3×D3—124no—
318D32—68no—
319D32—616no—
320C32×C4—22no—
321C32×C4—24no—
322C32×C4—24no—
323C32×C4—24no—
324C32×C4—22no—
325C32×C4—42no—
326C32×C4—44no—
327C32×C4—44no—
328C32×C4D3×D414yes—
329C32×C4C2×#6712yes—
330C3⋊Dic3—22no—
331C3⋊Dic3—24no—
332C3⋊Dic3C2×#6712yes—
333C3×Dic3—34no—
334C3×Dic3—62no—
335C3×Dic3—62no—
336C3×Dic3—64no—
337C3×Dic3—64no—
338C3×Dic3—62no—
339C3×Dic3—122no—
340C3×Dic3—124no—
341C2×#41—91no—
342C23×C4—271no—
343#123—272no—
344C22⋊Q8—92no—
345#121—272no—
346#120—272no—
347C4×D4—274no—
348#117—271no—
349C33#52511yescharacteristic, complemented, Sylow
350C23×C3—91no—
351C23×C3—92no—
352C23×C3—92no—
353C22×D3—32no—
354C22×D3—32no—
355C22×D3—32no—
356C22×D3—34no—
357C22×D3—62no—
358C22×D3—61no—
359C22×D3—62no—
360C22×D3—184no—
361C22×D3—184no—
362C22×D3—184no—
363C22×D3—182no—
364C22×D3—184no—
365C3×Q8—24no—
366C3×Q8—24no—
367C3×Q8—22no—
368C3×Q8—44no—
369C3×Q8—44no—
370C3×D4—184no—
371C3×D4—182no—
372C2×C3×C4—22no—
373C2×C3×C4—22no—
374C2×C3×C4—21no—
375C2×C3×C4—24no—
376C2×C3×C4—42no—
377C2×C3×C4—41no—
378C2×C3×C4—91no—
379C2×C3×C4—94no—
380C2×C3×C4—94no—
381C2×C3×C4—92no—
382C2×C3×C4—184no—
383C2×C3×C4—184no—
384C2×C3×C4C2×D3212yes—
385C2×C3×C4C2×D3214yes—
386C2×C3×C4C2×D3212yes—
387C2×C3×C4C2×D3211yescharacteristic
388#67—188no—
389#67—188no—
390C2×Dic3—32no—
391C2×Dic3—34no—
392C2×Dic3—34no—
393C2×Dic3—32no—
394C2×Dic3—32no—
395C2×Dic3—61no—
396C2×Dic3—62no—
397C2×Dic3—64no—
398C2×Dic3—184no—
399C2×Dic3—182no—
400D12—68no—
401D12—68no—
402D12—64no—
403D12—64no—
404D12—188no—
405C4×D3—38no—
406C4×D3—64no—
407C4×D3—68no—
408C4×D3—68no—
409C4×D3—64no—
410C4×D3—64no—
411C4×D3—124no—
412C4×D3—184no—
413C4×D3—184no—
414C4×D3—184no—
415C2×C32—22no—
416C2×C32—22no—
417C2×C32—22no—
418C2×C32—22no—
419C2×C32—21no—
420C2×C32—41no—
421C2×C32—42no—
422C2×C32C2×D3×D412yes—
423C2×C32C2×#23912yes—
424C2×C32D3×#4112yes—
425C2×C32D3×#4112yes—
426C2×C32C22×#6711yescharacteristic
427C3⋊D3—22no—
428C3⋊D3—24no—
429C3⋊D3—42no—
430C3×D3—64no—
431C3×D3—64no—
432C3×D3—64no—
433C3×D3—124no—
434C3×D3—122no—
435C3×D3—124no—
436C3×D3—124no—
437C24—271no—
438#41—184no—
439C2×Q8#111311yescharacteristic, complemented
440C2×D4—92no—
441C2×D4—91no—
442C2×D4—274no—
443C22×C4—91no—
444C22×C4—92no—
445C22×C4—92no—
446C22×C4—272no—
447C22×C4—271no—
448C22×C4—544no—
449C4⋊C4—92no—
450C4⋊C4—94no—
451C22⋊C4—94no—
452C22⋊C4—272no—
453C22⋊C4—274no—
454C42—272no—
455C22×C3—22no—
456C22×C3—21no—
457C22×C3—41no—
458C22×C3—94no—
459C22×C3—92no—
460C22×C3—182no—
461C22×C3—182no—
462C22×C3—182no—
463C22×C3—184no—
464C22×C3—182no—
465C22×C3C22×D3211yescharacteristic
466C22×C3C22×D3212yes—
467C2×D3—34no—
468C2×D3—34no—
469C2×D3—34no—
470C2×D3—62no—
471C2×D3—64no—
472C2×D3—62no—
473C2×D3—62no—
474C2×D3—62no—
475C2×D3—62no—
476C2×D3—62no—
477C2×D3—62no—
478C2×D3—64no—
479C2×D3—64no—
480C2×D3—64no—
481C2×D3—64no—
482C2×D3—122no—
483C2×D3—122no—
484C2×D3—122no—
485C2×D3—124no—
486C2×D3—1816no—
487C2×D3—188no—
488C3×C4—24no—
489C3×C4—24no—
490C3×C4—22no—
491C3×C4—22no—
492C3×C4—24no—
493C3×C4—44no—
494C3×C4—44no—
495C3×C4—42no—
496C3×C4—92no—
497C3×C4—184no—
498C3×C4—182no—
499C3×C4D3×#6714yes—
500C3×C4#381112yes—
501Dic3—34no—
502Dic3—62no—
503Dic3—64no—
504Dic3—64no—
505Dic3—62no—
506Dic3—62no—
507Dic3—122no—
508Dic3—124no—
509C32—22no—
510C32—21no—
511C32—41no—
512C32#562712yescomplemented
513C32#582811yescharacteristic, complemented
514C23—92no—
515C23—91no—
516C23—92no—
517C23—542no—
518C23—544no—
519Q8—22no—
520D4—184no—
521D4—182no—
522D4—548no—
523C2×C4—91no—
524C2×C4—94no—
525C2×C4—94no—
526C2×C4—92no—
527C2×C4—184no—
528C2×C4—184no—
529C2×C4—544no—
530C2×C4—542no—
531C2×C4C2×#111312yes—
532C2×C4C2×#111311yescharacteristic
533C2×C3—22no—
534C2×C3—22no—
535C2×C3—22no—
536C2×C3—21no—
537C2×C3—22no—
538C2×C3—42no—
539C2×C3—41no—
540C2×C3—184no—
541C2×C3—182no—
542C2×C3—362no—
543C2×C3D3×#23912yes—
544C2×C3D3×#23912yes—
545C2×C3#6431912yes—
546C2×C3C2×D3×#6712yes—
547C2×C3C2×#381111yescharacteristic
548D3—64no—
549D3—64no—
550D3—64no—
551D3—122no—
552D3—124no—
553D3—124no—
554D3—124no—
555C22—94no—
556C22—92no—
557C22—182no—
558C22—182no—
559C22—182no—
560C22—184no—
561C22—182no—
562C22—548no—
563C22C22×#111311yescharacteristic
564C4—22no—
565C4—92no—
566C4—184no—
567C4—182no—
568C4#8904912yes—
569C3—21no—
570C3—22no—
571C3—41no—
572C3#1059542412yescomplemented
573C3#1059543011yescharacteristic, complemented
574C2—184no—
575C2—182no—
576C2—362no—
577C2#1172462612yes—
578C2C2×#8904911yescharacteristic
subgroupG/H
center, FrattiniC22C22×#1113
commutator, socle, cosocleC22×C33C24
FittingC2×C33×Q8C22
hypercenterC2×Q8#1113
hypocenterC33#525
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. C2×C3×#3832C2semidirect
2. C2×C4×#1113C2semidirect
3. C3×#64155C2semidirect
4. #11723701C2semidirect
5. #11723696C2semidirect
6. #11723686C2semidirect
7. #11723684C2semidirect
8. #11723667C2semidirect
12. C2×C33×Q8C22semidirect
15. C2×C3×C4⋊(C3⋊D3)C22semidirect
18. C2×C3×C4×C3⋊D3C22semidirect
24. C3×C4⋊(C3⋊Dic3)C22semidirect
25. C3×#3750C22semidirect
27. #88814C22semidirect
28. #88813C22semidirect
29. C2×#3832D3semidirect
34. #64155D3semidirect
65. C3×C4×C3⋊D3D4semidirect
76. C2×C32×Q8C2×D3semidirect
77. C2×C32×Q8C2×D3semidirect
80. C2×C4⋊(C3⋊D3)C2×D3semidirect
83. C2×C4×C3⋊D3C2×D3semidirect
108. C4⋊(C3⋊Dic3)C2×D3semidirect
109. #3750C2×D3semidirect
133. C2×C3×C3⋊D3#41semidirect
198. C4×C3⋊D3#67semidirect
238. C2×C3×Q8D32semidirect
239. C2×C3×Q8D32semidirect
297. C2×C3⋊D3#239semidirect
349. C33#525semidirect
439. C2×Q8#1113semidirect
512. C32#5627semidirect
513. C32#5828semidirect
572. C3#10595424semidirect
573. C3#10595430semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#38322
C2×C4×#11132
C3×#641552
#117237012
#117236962
#117236862
#117236842
#117236672
#105954303
#105954243