Finite Group Databasebeta

← #13176540

Subgroups — #13176540

Order 1728.

2276 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×D3×C3⋊D3C211yesmaximal, characteristic, complemented
2C2×#89014C211yesmaximal, characteristic, complemented
3#11723938C211yesmaximal, characteristic, complemented
4C3×#64101C211yesmaximal, characteristic, complemented
5C32×#895C211yesmaximal, characteristic, complemented
6#11723540C212yesmaximal, complemented
7#11723531C211yesmaximal, characteristic, complemented
8#11723523C212yesmaximal, complemented
9D12×C3⋊Dic3C212yesmaximal, complemented
10#11723461C212yesmaximal, complemented
11#11723426C211yesmaximal, characteristic, complemented
12#10596077—31nomaximal
13#10594626—34nomaximal
14C22×D3×C3⋊D3C2211yescharacteristic
15C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
16C2×#6308C2211yescharacteristic, complemented
17C2×C3×C4×C3⋊D3C2211yescharacteristic
18C2×C32×D12C2212yes—
19C2×C32×D12C2211yescharacteristic, complemented
20C2×C32×C4×D3C2211yescharacteristic
21C2×#6291C2212yes—
22C2×#6290C2212yes—
23C2×#6289C2211yescharacteristic
24C2×Dic3×C3⋊D3C2211yescharacteristic
25C2×D3×C3⋊Dic3C2211yescharacteristic
26C2×D3×C3⋊Dic3C2212yescomplemented
27C4×D3×C3⋊D3—24no—
28#89014—24no—
29C33×C4⋊C4C2211yescharacteristic, complemented
30#88859C2212yes—
31C4⋊(C3⋊(C3⋊Dic3))C2212yescomplemented
32C3×#3752C2212yes—
33C3×C4⋊(C3⋊Dic3)C2211yescharacteristic, complemented
34C3×C4×C3⋊Dic3C2212yescomplemented
35C32×(C2×D3)⋊C4C2212yescomplemented
36#88807C2212yescomplemented
37(C2×D3)⋊(C3⋊Dic3)C2212yes—
38(C2×D3)⋊(C3⋊Dic3)C2212yescomplemented
39C2×#3830—31no—
40C22×C4×C3⋊D3—31no—
41C2×C4×D32—34no—
42C2×#3795—34no—
43C32×#121—31no—
44#64147—31no—
45#64146—32no—
46D4×C3⋊Dic3—32no—
47#64133—31no—
48#64106—31no—
49#64106—34no—
50#64106—64no—
51#64101D311yescharacteristic, complemented
52#64092—32no—
53#64090—32no—
54C3×#895D314yescomplemented
55C3×#892—34no—
56#63922—38no—
57#63916—34no—
58Dic3×D12—38no—
59#63891—38no—
60#63880—38no—
61#63857—34no—
62C22×C3⋊(C3⋊D3)C2311yescharacteristic
63C22×C3×C3⋊D3C2311yescharacteristic
64C22×C32×D3C2312yes—
65C22×C32×D3C2311yescharacteristic
66C2×D3×C3⋊D3—22no—
67C2×D3×C3⋊D3—22no—
68C2×D3×C3⋊D3—22no—
69C2×C33×C4C2312yes—
70C2×C33×C4C2311yescharacteristic
71C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
72C2×C3⋊(C3⋊Dic3)C2312yes—
73C4×C3⋊(C3⋊D3)—22no—
74#6308—22no—
75C2×C3×C3⋊Dic3C2312yes—
76C2×C3×C3⋊Dic3C2311yescharacteristic
77C3×C4×C3⋊D3D414yescomplemented
78C2×C32×Dic3C2311yescharacteristic
79C32×D12—24no—
80C32×D12—22no—
81C32×C4×D3—22no—
82#6291—24no—
83#6290—24no—
84#6289—22no—
85Dic3×C3⋊D3—22no—
86D3×C3⋊Dic3—24no—
87D3×C3⋊Dic3—22no—
88#5606—94no—
89#5524—91no—
90C23×C3⋊D3—31no—
91C22×D32—34no—
92C2×C32×D4—31no—
93C2×C32×D4—32no—
94C22×C32×C4—31no—
95C2×#638—32no—
96C2×#638—32no—
97C22×C3⋊Dic3—32no—
98C22×C3⋊Dic3—31no—
99#3830—64no—
100C2×C4×C3⋊D3—31no—
101C2×C4×C3⋊D3—31no—
102C2×C4×C3⋊D3—34no—
103C2×C4×C3⋊D3—31no—
104C2×C4×C3⋊D3—64no—
105C2×C4×C3⋊D3—64no—
106C2×C4×C3⋊D3C2×D311yescharacteristic
107C2×#634—31no—
108C2×#634—34no—
109C2×#634—64no—
110C2×C3×D12C2×D318yes—
111C2×C3×D12C2×D314yescomplemented
112C2×C3×C4×D3—34no—
113C2×C3×C4×D3C2×D314yes—
114C2×#626—38no—
115C2×#625—38no—
116C2×#624—34no—
117C2×D3×Dic3—38no—
118C2×D3×Dic3—34no—
119C2×D3×Dic3—34no—
120C4×D32—616no—
121#3795—616no—
122C32×C4⋊C4—24no—
123C32×C4⋊C4C2×D314yescomplemented
124C32×C4⋊C4C2×D311yescharacteristic, complemented
125C32×C22⋊C4—32no—
126#3757—32no—
127#3757—32no—
128#3752—32no—
129#3752—38no—
130#3752—68no—
131#3752C2×D312yes—
132C4⋊(C3⋊Dic3)—32no—
133C4⋊(C3⋊Dic3)—38no—
134C4⋊(C3⋊Dic3)—68no—
135C4⋊(C3⋊Dic3)C2×D311yescharacteristic, complemented
136#3750—32no—
137C4×C3⋊Dic3C2×D312yescomplemented
138C3×(C2×D3)⋊C4—38no—
139C3×(C2×D3)⋊C4C2×D318yescomplemented
140C3×C4⋊Dic3—34no—
141C3×C4×Dic3—38no—
142Dic3⋊Dic3—38no—
143(C2×D3)⋊Dic3—38no—
144(C2×D3)⋊Dic3—38no—
145C22×C33C2411yescharacteristic
146C2×C3⋊(C3⋊D3)—21no—
147C2×C3⋊(C3⋊D3)—21no—
148C2×C3⋊(C3⋊D3)#4112yescomplemented
149C2×C3×C3⋊D3—21no—
150C2×C3×C3⋊D3—21no—
151C2×C3×C3⋊D3C2×D412yes—
152C2×C32×D3—21no—
153C2×C32×D3—21no—
154C2×C32×D3—22no—
155C2×C32×D3—22no—
156C2×C32×D3—22no—
157C2×C32×D3#4112yescomplemented
158D3×C3⋊D3—28no—
159C33×C4—22no—
160C33×C4C2×D412yes—
161C3⋊(C3⋊Dic3)—21no—
162C3⋊(C3⋊Dic3)—22no—
163C3×C3⋊Dic3—22no—
164C3×C3⋊Dic3C2×D412yes—
165C32×Dic3—21no—
166C2×#237—94no—
167C2×#235—91no—
168C22×C4×D3—94no—
169C22×C4×D3—91no—
170C3×#121—34no—
171C3×#117—91no—
172#938—94no—
173#937—98no—
174D4×Dic3—98no—
175#924—94no—
176#897—91no—
177#897—94no—
178#897—91no—
179#897—184no—
180#895—91no—
181#895C3⋊D311yescharacteristic, complemented
182#892—34no—
183#886—92no—
184#883—98no—
185#882—92no—
186#67⋊C4—98no—
187C4×D12—92no—
188C4×D12—92no—
189C23×C32—31no—
190C23×C32—32no—
191C22×C3⋊D3—31no—
192C22×C3⋊D3—34no—
193C22×C3⋊D3—64no—
194C22×C3⋊D3—62no—
195C22×C3⋊D3—62no—
196C22×C3⋊D3—62no—
197C22×C3⋊D3C22×D311yescharacteristic
198C22×C3×D3—34no—
199C22×C3×D3C22×D318yes—
200C22×C3×D3C22×D314yes—
201C2×D32—68no—
202C2×D32—68no—
203C2×D32—68no—
204C32×D4—64no—
205C32×D4—62no—
206C2×C32×C4—24no—
207C2×C32×C4—28no—
208C2×C32×C4—31no—
209C2×C32×C4—62no—
210C2×C32×C4C22×D318yes—
211C2×C32×C4C22×D314yes—
212C2×C32×C4C22×D312yes—
213C2×C32×C4C22×D311yescharacteristic
214#638—64no—
215#638—64no—
216C2×C3⋊Dic3—32no—
217C2×C3⋊Dic3—31no—
218C2×C3⋊Dic3—34no—
219C2×C3⋊Dic3—38no—
220C2×C3⋊Dic3—64no—
221C2×C3⋊Dic3—68no—
222C2×C3⋊Dic3—62no—
223C2×C3⋊Dic3—64no—
224C2×C3⋊Dic3C22×D312yes—
225C2×C3⋊Dic3C22×D311yescharacteristic
226C4×C3⋊D3—62no—
227C4×C3⋊D3—62no—
228C4×C3⋊D3—62no—
229C4×C3⋊D3—68no—
230C4×C3⋊D3—128no—
231C4×C3⋊D3D1214yescomplemented
232#634—62no—
233#634—68no—
234#634—128no—
235C2×C3×Dic3—34no—
236C2×C3×Dic3—38no—
237C2×C3×Dic3C22×D314yes—
238C3×D12—216no—
239C3×D12—28no—
240C3×C4×D3—28no—
241C3×C4×D3—316no—
242#626—616no—
243#625—616no—
244#624—68no—
245D3×Dic3—68no—
246D3×Dic3—616no—
247D3×Dic3—68no—
248#525—271noSylow
249C2×C33C22×D411yescharacteristic
250C2×C33C2×#4111yescharacteristic
251C2×C33C2×#4111yescharacteristic
252C3⋊(C3⋊D3)—22no—
253C3×C3⋊D3—22no—
254C32×D3—22no—
255C32×D3—42no—
256C23×D3—94no—
257C23×D3—91no—
258C2×C3×D4—38no—
259C2×C3×D4—34no—
260C22×C3×C4—34no—
261C22×C3×C4—91no—
262C2×#67—98no—
263C2×#67—98no—
264C2×#67—92no—
265C22×Dic3—98no—
266C22×Dic3—94no—
267C22×Dic3—91no—
268#237—1816no—
269#235—184no—
270C2×D12—92no—
271C2×D12C2×C3⋊D312yes—
272C2×D12C2×C3⋊D311yescharacteristic, complemented
273C2×C4×D3—34no—
274C2×C4×D3—94no—
275C2×C4×D3—94no—
276C2×C4×D3—91no—
277C2×C4×D3—92no—
278C2×C4×D3—94no—
279C2×C4×D3—91no—
280C2×C4×D3—91no—
281C2×C4×D3—184no—
282C2×C4×D3—184no—
283C2×C4×D3—1816no—
284C2×C4×D3C2×C3⋊D311yescharacteristic
285C2×Dic6—94no—
286C2×Dic6—91no—
287C2×Dic6—184no—
288C3×C4⋊C4—24no—
289C3×C4⋊C4—91no—
290C3×C4⋊C4D3214yescomplemented
291C3×C4⋊C4C2×C3⋊D311yescharacteristic, complemented
292C3×C22⋊C4—38no—
293C3×C22⋊C4—92no—
294C3×C42—92no—
295#217—98no—
296#217—98no—
297(C2×D3)⋊C4—38no—
298(C2×D3)⋊C4—92no—
299(C2×D3)⋊C4—92no—
300(C2×D3)⋊C4—98no—
301(C2×D3)⋊C4—92no—
302(C2×D3)⋊C4—188no—
303(C2×D3)⋊C4C2×C3⋊D312yescomplemented
304C4⋊Dic3—34no—
305C4⋊Dic3—98no—
306C4⋊Dic3—92no—
307C4⋊Dic3—92no—
308C4⋊Dic3—188no—
309Dic3⋊C4—98no—
310C4×Dic3—38no—
311C22×C32—24no—
312C22×C32—32no—
313C22×C32—61no—
314C22×C32—61no—
315C22×C32—62no—
316C22×C32—62no—
317C22×C32—62no—
318C22×C32C23×D314yes—
319C22×C32C23×D311yescharacteristic
320C2×C3⋊D3—21no—
321C2×C3⋊D3—21no—
322C2×C3⋊D3—32no—
323C2×C3⋊D3—38no—
324C2×C3⋊D3—68no—
325C2×C3⋊D3—68no—
326C2×C3⋊D3—61no—
327C2×C3⋊D3—61no—
328C2×C3⋊D3—64no—
329C2×C3⋊D3—64no—
330C2×C3⋊D3—124no—
331C2×C3⋊D3—124no—
332C2×C3⋊D3C2×D1212yes—
333C2×C3×D3—24no—
334C2×C3×D3—24no—
335C2×C3×D3—28no—
336C2×C3×D3—28no—
337C2×C3×D3—28no—
338C2×C3×D3—38no—
339C2×C3×D3—64no—
340C2×C3×D3—64no—
341C2×C3×D3#23718yescomplemented
342D32—632no—
343C32×C4—22no—
344C32×C4—28no—
345C32×C4—28no—
346C32×C4—48no—
347C32×C4—61no—
348C32×C4C2×D1212yes—
349C32×C4D3×D418yes—
350C3⋊Dic3—22no—
351C3⋊Dic3—62no—
352C3⋊Dic3—61no—
353C3⋊Dic3—64no—
354C3⋊Dic3—68no—
355C3⋊Dic3—128no—
356C3⋊Dic3—124no—
357C3⋊Dic3C2×D1212yes—
358C3×Dic3—24no—
359C3×Dic3—38no—
360C3×Dic3—68no—
361C2×#41—271no—
362C23×C4—271no—
363#123—272no—
364C22⋊Q8—271no—
365C22⋊Q8—271no—
366#121—31no—
367#121—271no—
368#120—272no—
369C4×D4—272no—
370C4×D4—272no—
371#117—91no—
372C33#52511yescharacteristic, complemented, Sylow
373C23×C3—34no—
374C23×C3—38no—
375C23×C3—91no—
376C22×D3—34no—
377C22×D3—94no—
378C22×D3—91no—
379C22×D3—184no—
380C22×D3—182no—
381C22×D3—182no—
382C22×D3—188no—
383C22×D3—188no—
384C22×D3—188no—
385C22×D3—182no—
386C22×D3C22×C3⋊D312yes—
387C22×D3C22×C3⋊D311yescharacteristic
388C3×D4—616no—
389C3×D4—68no—
390C2×C3×C4—24no—
391C2×C3×C4—28no—
392C2×C3×C4—34no—
393C2×C3×C4—68no—
394C2×C3×C4—91no—
395C2×C3×C4—94no—
396C2×C3×C4—92no—
397C2×C3×C4C2×D3218yes—
398C2×C3×C4C2×D3214yes—
399C2×C3×C4C22×C3⋊D312yes—
400C2×C3×C4C22×C3⋊D311yescharacteristic
401#67—1816no—
402#67—184no—
403#67—1816no—
404C2×Dic3—34no—
405C2×Dic3—38no—
406C2×Dic3—98no—
407C2×Dic3—94no—
408C2×Dic3—92no—
409C2×Dic3—91no—
410C2×Dic3—184no—
411C2×Dic3—188no—
412C2×Dic3—182no—
413C2×Dic3—188no—
414C2×Dic3—1816no—
415C2×Dic3C22×C3⋊D311yescharacteristic
416D12—24no—
417D12—22no—
418D12—184no—
419C4×D3—22no—
420C4×D3—316no—
421C4×D3—182no—
422C4×D3—188no—
423C4×D3—188no—
424C4×D3—182no—
425C4×D3—184no—
426C4×D3—188no—
427C4×D3—182no—
428C4×D3—368no—
429Dic6—188no—
430Dic6—182no—
431Dic6—368no—
432C2×C32—24no—
433C2×C32—24no—
434C2×C32—24no—
435C2×C32—62no—
436C2×C32—122no—
437C2×C32C22×D1211yescharacteristic
438C2×C32C2×D3×D414yes—
439C2×C32C2×#23714yes—
440C2×C32D3×#4114yes—
441C2×C32D3×#4111yescharacteristic
442C2×C32D3×#4111yescharacteristic
443C3⋊D3—22no—
444C3⋊D3—62no—
445C3⋊D3—68no—
446C3⋊D3—128no—
447C3×D3—28no—
448C3×D3—48no—
449C3×D3—68no—
450C24—271no—
451#41—544no—
452C2×Q8—271no—
453C2×D4—32no—
454C2×D4—31no—
455C2×D4—272no—
456C2×D4—272no—
457C22×C4—31no—
458C22×C4—91no—
459C22×C4—271no—
460C22×C4—271no—
461C22×C4—271no—
462C22×C4—272no—
463C22×C4—271no—
464C22×C4—544no—
465C4⋊C4—91no—
466C4⋊C4—272no—
467C4⋊C4—272no—
468C4⋊C4D3×C3⋊D311yescharacteristic, complemented
469C22⋊C4—32no—
470C22⋊C4—92no—
471C22⋊C4—272no—
472C22⋊C4—272no—
473C22⋊C4—272no—
474C42—92no—
475C22×C3—24no—
476C22×C3—38no—
477C22×C3—64no—
478C22×C3—64no—
479C22×C3—68no—
480C22×C3—68no—
481C22×C3—68no—
482C22×C3—92no—
483C22×C3—181no—
484C22×C3—181no—
485C22×C3C22×D3214yes—
486C22×C3C23×C3⋊D311yescharacteristic
487C2×D3—21no—
488C2×D3—21no—
489C2×D3—22no—
490C2×D3—22no—
491C2×D3—22no—
492C2×D3—38no—
493C2×D3—64no—
494C2×D3—64no—
495C2×D3—98no—
496C2×D3—92no—
497C2×D3—188no—
498C2×D3—188no—
499C2×D3—1832no—
500C2×D3—184no—
501C2×D3—184no—
502C2×D3—181no—
503C2×D3—181no—
504C2×D3—364no—
505C2×D3—364no—
506C2×D3#383012yescomplemented
507C3×C4—28no—
508C3×C4—22no—
509C3×C4—28no—
510C3×C4—48no—
511C3×C4—64no—
512C3×C4—92no—
513C3×C4—182no—
514C3×C4D3×D1218yes—
515C3×C4D4×C3⋊D312yes—
516Dic3—21no—
517Dic3—38no—
518Dic3—68no—
519Dic3—188no—
520Dic3—184no—
521Dic3—182no—
522Dic3—181no—
523Dic3—368no—
524Dic3—364no—
525C32—24no—
526C32#552411yescharacteristic, complemented
527C32#560614yescomplemented
528C23—31no—
529C23—32no—
530C23—91no—
531C23—271no—
532C23—542no—
533C23—542no—
534C23—542no—
535Q8—542no—
536D4—64no—
537D4—62no—
538D4—544no—
539D4—544no—
540C2×C4—31no—
541C2×C4—62no—
542C2×C4—91no—
543C2×C4—94no—
544C2×C4—92no—
545C2×C4—271no—
546C2×C4—272no—
547C2×C4—542no—
548C2×C4—542no—
549C2×C4—542no—
550C2×C4—544no—
551C2×C4—542no—
552C2×C4C2×D3×C3⋊D312yes—
553C2×C4C2×D3×C3⋊D311yescharacteristic
554C2×C3—24no—
555C2×C3—24no—
556C2×C3—24no—
557C2×C3—68no—
558C2×C3—128no—
559C2×C3—182no—
560C2×C3C2×D3×D1214yes—
561C2×C3D3×#23714yes—
562C2×C3#6431914yes—
563C2×C3C2×D4×C3⋊D311yescharacteristic
564C2×C3C2×#383011yescharacteristic
565C2×C3#41×C3⋊D311yescharacteristic
566D3—22no—
567D3—42no—
568D3—68no—
569D3—188no—
570D3—182no—
571D3—368no—
572C22—32no—
573C22—61no—
574C22—61no—
575C22—62no—
576C22—62no—
577C22—62no—
578C22—92no—
579C22—181no—
580C22—181no—
581C22—272no—
582C22—548no—
583C22—541no—
584C22—541no—
585C22C22×D3×C3⋊D311yescharacteristic
586C4—22no—
587C4—61no—
588C4—92no—
589C4—182no—
590C4—541no—
591C4—542no—
592C4C3⋊D3×D1212yes—
593C3—24no—
594C3#1059462614yescomplemented
595C3#1059607711yescharacteristic, complemented
596C2—62no—
597C2—122no—
598C2—182no—
599C2—542no—
600C2C2×C3⋊D3×D1211yescharacteristic
601C2D3×#383011yescharacteristic
602C2#1172459611yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC33×C4⋊C4C22
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33#525
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×C4×D3×C3⋊D3C2semidirect
2. C2×#89014C2semidirect
3. #11723938C2semidirect
4. C3×#64101C2semidirect
5. C32×#895C2semidirect
6. #11723540C2semidirect
7. #11723531C2semidirect
8. #11723523C2semidirect
9. D12×C3⋊Dic3C2semidirect
10. #11723461C2semidirect
11. #11723426C2semidirect
16. C2×#6308C22semidirect
19. C2×C32×D12C22semidirect
26. C2×D3×C3⋊Dic3C22semidirect
29. C33×C4⋊C4C22semidirect
31. C4⋊(C3⋊(C3⋊Dic3))C22semidirect
33. C3×C4⋊(C3⋊Dic3)C22semidirect
34. C3×C4×C3⋊Dic3C22semidirect
35. C32×(C2×D3)⋊C4C22semidirect
36. #88807C22semidirect
38. (C2×D3)⋊(C3⋊Dic3)C22semidirect
51. #64101D3semidirect
54. C3×#895D3semidirect
77. C3×C4×C3⋊D3D4semidirect
111. C2×C3×D12C2×D3semidirect
123. C32×C4⋊C4C2×D3semidirect
124. C32×C4⋊C4C2×D3semidirect
135. C4⋊(C3⋊Dic3)C2×D3semidirect
137. C4×C3⋊Dic3C2×D3semidirect
139. C3×(C2×D3)⋊C4C2×D3semidirect
148. C2×C3⋊(C3⋊D3)#41semidirect
157. C2×C32×D3#41semidirect
181. #895C3⋊D3semidirect
231. C4×C3⋊D3D12semidirect
272. C2×D12C2×C3⋊D3semidirect
290. C3×C4⋊C4D32semidirect
291. C3×C4⋊C4C2×C3⋊D3semidirect
303. (C2×D3)⋊C4C2×C3⋊D3semidirect
341. C2×C3×D3#237semidirect
372. C33#525semidirect
468. C4⋊C4D3×C3⋊D3semidirect
506. C2×D3#3830semidirect
526. C32#5524semidirect
527. C32#5606semidirect
594. C3#10594626semidirect
595. C3#10596077semidirect

Maximal subgroups

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

subgroupindex
C2×C4×D3×C3⋊D32
C2×#890142
#117239382
C3×#641012
C32×#8952
#117235402
#117235312
#117235232
D12×C3⋊Dic32
#117234612
#117234262
#105960773
#105946263