Finite Group Databasebeta

← #13176541

Subgroups — #13176541

Order 1728.

2356 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×D3×C3⋊D3C211yesmaximal, characteristic, complemented
2C2×#89018C211yesmaximal, characteristic, complemented
3#11723936C211yesmaximal, characteristic, complemented
4C3×#64101C211yesmaximal, characteristic, complemented
5C32×#897C211yesmaximal, characteristic, complemented
6#11723542C212yesmaximal, complemented
7#11723516C211yesmaximal, characteristic, complemented
8#11723494C212yesmaximal, complemented
9#11723493C212yesmaximal, complemented
10#11723461C212yesmaximal, complemented
11#11723456C211yesmaximal, characteristic, complemented
12#10596098—31nomaximal
13#10594627—34nomaximal
14C22×D3×C3⋊D3C2211yescharacteristic
15C2×C4⋊(C3⋊(C3⋊D3))C2212yes—
16C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic, complemented
17C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
18C2×C3×C4×C3⋊D3C2211yescharacteristic
19C2×C32×C4×D3C2211yescharacteristic
20C2×C32×Dic6C2211yescharacteristic, complemented
21C2×#6292C2212yes—
22C2×#6291C2212yes—
23C2×#6289C2211yescharacteristic
24C2×#6289C2212yescomplemented
25C2×Dic3×C3⋊D3C2211yescharacteristic
26C2×D3×C3⋊Dic3C2211yescharacteristic
27C4×D3×C3⋊D3—24no—
28#89018—24no—
29C33×C4⋊C4C2211yescharacteristic, complemented
30#88859C2212yescomplemented
31C3×#3752C2212yes—
32C3×C4⋊(C3⋊Dic3)C2211yescharacteristic, complemented
33C3×C4×C3⋊Dic3C2212yescomplemented
34C32×(C2×D3)⋊C4C2212yes—
35C32×C4⋊Dic3C2212yescomplemented
36#88807C2212yescomplemented
37#88804C2212yes—
38#88804C2212yescomplemented
39C2×#3832—31no—
40C22×C4×C3⋊D3—31no—
41C2×C4×D32—34no—
42C2×#3799—34no—
43C32×C22⋊Q8—31no—
44#64155—31no—
45#64139—31no—
46#64104—31no—
47#64104—34no—
48#64104—64no—
49#64103—32no—
50#64102—32no—
51#64101D311yescharacteristic, complemented
52#64091—32no—
53#64090—32no—
54C3×#897D314yescomplemented
55C3×#892—34no—
56#63923—38no—
57#63909—34no—
58#63894—38no—
59#63890—38no—
60#63880—38no—
61#63871—34no—
62C22×C3⋊(C3⋊D3)C2312yes—
63C22×C3⋊(C3⋊D3)C2311yescharacteristic
64C22×C3×C3⋊D3C2311yescharacteristic
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
72C4⋊(C3⋊(C3⋊D3))—24no—
73C4⋊(C3⋊(C3⋊D3))—22no—
74C4×C3⋊(C3⋊D3)—22no—
75C2×C3×C3⋊Dic3C2312yes—
76C2×C3×C3⋊Dic3C2311yescharacteristic
77C3×C4×C3⋊D3D414yescomplemented
78C2×C32×Dic3C2311yescharacteristic
79C2×C32×Dic3C2312yes—
80C32×C4×D3—22no—
81C32×Dic6—22no—
82#6292—24no—
83#6291—24no—
84#6289—24no—
85#6289—22no—
86Dic3×C3⋊D3—22no—
87D3×C3⋊Dic3—22no—
88#5627—94no—
89#5524—91no—
90C23×C3⋊D3—31no—
91C22×D32—34no—
92C2×C32×Q8—31no—
93C22×C32×C4—31no—
94C2×#638—32no—
95C22×C3⋊Dic3—31no—
96#3832—64no—
97C2×C4⋊(C3⋊D3)—32no—
98C2×C4⋊(C3⋊D3)—32no—
99C2×C4⋊(C3⋊D3)—31no—
100C2×C4⋊(C3⋊D3)—34no—
101C2×C4⋊(C3⋊D3)—38no—
102C2×C4⋊(C3⋊D3)—68no—
103C2×C4⋊(C3⋊D3)—64no—
104C2×C4×C3⋊D3—31no—
105C2×C4×C3⋊D3—31no—
106C2×C4×C3⋊D3—34no—
107C2×C4×C3⋊D3—32no—
108C2×C4×C3⋊D3—31no—
109C2×C4×C3⋊D3—64no—
110C2×C4×C3⋊D3—64no—
111C2×C4×C3⋊D3C2×D311yescharacteristic
112C2×C3×C4×D3—34no—
113C2×C3×C4×D3C2×D314yes—
114C2×C3×Dic6C2×D314yescomplemented
115C2×#626—38no—
116C2×#626—38no—
117C2×#624—38no—
118C2×#624—34no—
119C2×D3×Dic3—34no—
120C2×D3×Dic3—34no—
121C4×D32—616no—
122#3799—616no—
123C32×C4⋊C4—24no—
124C32×C4⋊C4—32no—
125C32×C4⋊C4C2×D314yescomplemented
126C32×C4⋊C4C2×D311yescharacteristic, complemented
127C32×C22⋊C4—32no—
128#3752—32no—
129#3752—38no—
130#3752—32no—
131#3752—32no—
132#3752—68no—
133#3752C2×D312yes—
134C4⋊(C3⋊Dic3)C2×D311yescharacteristic, complemented
135#3750—32no—
136C4×C3⋊Dic3C2×D312yescomplemented
137C3×(C2×D3)⋊C4—38no—
138C3×(C2×D3)⋊C4C2×D318yes—
139C3×C4⋊Dic3—34no—
140C3×C4⋊Dic3C2×D318yescomplemented
141C3×C4×Dic3—38no—
142Dic3⋊Dic3—38no—
143#3722—38no—
144#3722—38no—
145C22×C33C2411yescharacteristic
146C2×C3⋊(C3⋊D3)—21no—
147C2×C3⋊(C3⋊D3)—21no—
148C2×C3⋊(C3⋊D3)—22no—
149C2×C3⋊(C3⋊D3)—22no—
150C2×C3⋊(C3⋊D3)—22no—
151C2×C3⋊(C3⋊D3)#4112yescomplemented
152C2×C3×C3⋊D3—21no—
153C2×C3×C3⋊D3—21no—
154C2×C3×C3⋊D3C2×D412yes—
155C2×C32×D3—21no—
156C2×C32×D3—21no—
157C2×C32×D3#4112yescomplemented
158D3×C3⋊D3—28no—
159C33×C4—22no—
160C33×C4C2×D412yes—
161C3⋊(C3⋊Dic3)—21no—
162C3×C3⋊Dic3—22no—
163C3×C3⋊Dic3C2×D412yes—
164C32×Dic3—21no—
165C32×Dic3—22no—
166C2×#239—94no—
167C2×#235—91no—
168C22×C4×D3—94no—
169C22×C4×D3—91no—
170C3×C22⋊Q8—34no—
171C3×#117—91no—
172#946—94no—
173#930—94no—
174#897—91no—
175#897C3⋊D311yescharacteristic, complemented
176#895—91no—
177#895—94no—
178#895—91no—
179#895—184no—
180#894—98no—
181#893—98no—
182#892—34no—
183#886—92no—
184#882—98no—
185#882—92no—
186#67⋊C4—98no—
187C4×D12—92no—
188C4×D12—92no—
189C23×C32—31no—
190C22×C3⋊D3—31no—
191C22×C3⋊D3—32no—
192C22×C3⋊D3—34no—
193C22×C3⋊D3—38no—
194C22×C3⋊D3—64no—
195C22×C3⋊D3—68no—
196C22×C3⋊D3—62no—
197C22×C3⋊D3—62no—
198C22×C3⋊D3—62no—
199C22×C3⋊D3C22×D311yescharacteristic
200C22×C3×D3—34no—
201C22×C3×D3C22×D314yes—
202C2×D32—68no—
203C2×D32—68no—
204C2×D32—68no—
205C32×Q8—62no—
206C2×C32×C4—24no—
207C2×C32×C4—28no—
208C2×C32×C4—32no—
209C2×C32×C4—31no—
210C2×C32×C4—62no—
211C2×C32×C4C22×D318yes—
212C2×C32×C4C22×D314yes—
213C2×C32×C4C22×D312yes—
214C2×C32×C4C22×D311yescharacteristic
215#638—64no—
216C2×C3⋊Dic3—31no—
217C2×C3⋊Dic3—34no—
218C2×C3⋊Dic3—64no—
219C2×C3⋊Dic3—62no—
220C2×C3⋊Dic3C22×D312yes—
221C2×C3⋊Dic3C22×D311yescharacteristic
222C4⋊(C3⋊D3)—64no—
223C4⋊(C3⋊D3)—64no—
224C4⋊(C3⋊D3)—62no—
225C4⋊(C3⋊D3)—616no—
226C4⋊(C3⋊D3)—68no—
227C4⋊(C3⋊D3)—128no—
228C4⋊(C3⋊D3)—1216no—
229C4×C3⋊D3—62no—
230C4×C3⋊D3—62no—
231C4×C3⋊D3—62no—
232C4×C3⋊D3—64no—
233C4×C3⋊D3—68no—
234C4×C3⋊D3—128no—
235C4×C3⋊D3D1214yescomplemented
236C2×C3×Dic3—34no—
237C2×C3×Dic3—38no—
238C2×C3×Dic3C22×D314yes—
239C2×C3×Dic3C22×D318yes—
240C3×C4×D3—28no—
241C3×C4×D3—316no—
242C3×Dic6—28no—
243#626—616no—
244#626—616no—
245#624—616no—
246#624—68no—
247D3×Dic3—68no—
248D3×Dic3—68no—
249#525—271noSylow
250C2×C33C22×D411yescharacteristic
251C2×C33C2×#4111yescharacteristic
252C2×C33C2×#4111yescharacteristic
253C3⋊(C3⋊D3)—22no—
254C3⋊(C3⋊D3)—42no—
255C3×C3⋊D3—22no—
256C32×D3—22no—
257C23×D3—94no—
258C23×D3—91no—
259C2×C3×Q8—34no—
260C22×C3×C4—34no—
261C22×C3×C4—91no—
262C2×#67—98no—
263C2×#67—92no—
264C22×Dic3—94no—
265C22×Dic3—91no—
266#239—1816no—
267#235—184no—
268C2×D12—98no—
269C2×D12—92no—
270C2×D12—98no—
271C2×D12—94no—
272C2×D12—92no—
273C2×D12—91no—
274C2×D12—188no—
275C2×D12—184no—
276C2×C4×D3—34no—
277C2×C4×D3—98no—
278C2×C4×D3—94no—
279C2×C4×D3—94no—
280C2×C4×D3—91no—
281C2×C4×D3—94no—
282C2×C4×D3—91no—
283C2×C4×D3—91no—
284C2×C4×D3—92no—
285C2×C4×D3—184no—
286C2×C4×D3—184no—
287C2×C4×D3—1816no—
288C2×C4×D3C2×C3⋊D311yescharacteristic
289C2×Dic6C2×C3⋊D311yescharacteristic, complemented
290C3×C4⋊C4—24no—
291C3×C4⋊C4—38no—
292C3×C4⋊C4—91no—
293C3×C4⋊C4D3214yescomplemented
294C3×C4⋊C4C2×C3⋊D311yescharacteristic, complemented
295C3×C22⋊C4—38no—
296C3×C22⋊C4—92no—
297C3×C42—92no—
298(C2×D3)⋊C4—38no—
299(C2×D3)⋊C4—98no—
300(C2×D3)⋊C4—98no—
301(C2×D3)⋊C4—98no—
302(C2×D3)⋊C4—92no—
303(C2×D3)⋊C4—92no—
304(C2×D3)⋊C4—92no—
305(C2×D3)⋊C4—188no—
306(C2×D3)⋊C4C2×C3⋊D312yes—
307C4⋊Dic3—34no—
308C4⋊Dic3—92no—
309C4⋊Dic3C2×C3⋊D312yescomplemented
310Dic3⋊C4—98no—
311C4×Dic3—38no—
312C22×C32—24no—
313C22×C32—32no—
314C22×C32—61no—
315C22×C32—61no—
316C22×C32C23×D314yes—
317C22×C32C23×D311yescharacteristic
318C2×C3⋊D3—21no—
319C2×C3⋊D3—21no—
320C2×C3⋊D3—32no—
321C2×C3⋊D3—38no—
322C2×C3⋊D3—68no—
323C2×C3⋊D3—68no—
324C2×C3⋊D3—61no—
325C2×C3⋊D3—61no—
326C2×C3⋊D3—62no—
327C2×C3⋊D3—62no—
328C2×C3⋊D3—62no—
329C2×C3⋊D3—64no—
330C2×C3⋊D3—64no—
331C2×C3⋊D3—68no—
332C2×C3⋊D3—68no—
333C2×C3⋊D3—68no—
334C2×C3⋊D3—124no—
335C2×C3⋊D3—124no—
336C2×C3⋊D3—128no—
337C2×C3⋊D3—128no—
338C2×C3⋊D3—128no—
339C2×C3⋊D3C2×D1212yes—
340C2×C3×D3—24no—
341C2×C3×D3—24no—
342C2×C3×D3—38no—
343C2×C3×D3—64no—
344C2×C3×D3—64no—
345C2×C3×D3#23918yescomplemented
346D32—632no—
347C32×C4—22no—
348C32×C4—28no—
349C32×C4—28no—
350C32×C4—48no—
351C32×C4—61no—
352C32×C4—62no—
353C32×C4C2×D1212yes—
354C32×C4D3×D418yes—
355C3⋊Dic3—22no—
356C3⋊Dic3—61no—
357C3⋊Dic3—64no—
358C3⋊Dic3—124no—
359C3⋊Dic3C2×D1212yes—
360C3×Dic3—24no—
361C3×Dic3—28no—
362C3×Dic3—38no—
363C3×Dic3—68no—
364C2×#41—271no—
365C23×C4—271no—
366#123—272no—
367C22⋊Q8—31no—
368C22⋊Q8—271no—
369#121—271no—
370#121—271no—
371#120—272no—
372C4×D4—272no—
373C4×D4—272no—
374#117—91no—
375C33#52511yescharacteristic, complemented, Sylow
376C23×C3—34no—
377C23×C3—91no—
378C22×D3—34no—
379C22×D3—94no—
380C22×D3—91no—
381C22×D3—98no—
382C22×D3—92no—
383C22×D3—184no—
384C22×D3—188no—
385C22×D3—182no—
386C22×D3—182no—
387C22×D3—188no—
388C22×D3—188no—
389C22×D3—188no—
390C22×D3—182no—
391C22×D3C22×C3⋊D311yescharacteristic
392C3×Q8—68no—
393C2×C3×C4—24no—
394C2×C3×C4—28no—
395C2×C3×C4—38no—
396C2×C3×C4—34no—
397C2×C3×C4—68no—
398C2×C3×C4—91no—
399C2×C3×C4—94no—
400C2×C3×C4—92no—
401C2×C3×C4C2×D3218yes—
402C2×C3×C4C2×D3214yes—
403C2×C3×C4C22×C3⋊D312yes—
404C2×C3×C4C22×C3⋊D311yescharacteristic
405#67—184no—
406#67—1816no—
407C2×Dic3—34no—
408C2×Dic3—38no—
409C2×Dic3—94no—
410C2×Dic3—91no—
411C2×Dic3—184no—
412C2×Dic3—182no—
413C2×Dic3—188no—
414C2×Dic3C22×C3⋊D312yes—
415C2×Dic3C22×C3⋊D311yescharacteristic
416D12—184no—
417D12—1816no—
418D12—1816no—
419D12—184no—
420D12—188no—
421D12—182no—
422D12—368no—
423D12—3616no—
424C4×D3—22no—
425C4×D3—316no—
426C4×D3—182no—
427C4×D3—184no—
428C4×D3—188no—
429C4×D3—1816no—
430C4×D3—188no—
431C4×D3—182no—
432C4×D3—188no—
433C4×D3—182no—
434C4×D3—368no—
435Dic6—22no—
436C2×C32—24no—
437C2×C32—24no—
438C2×C32—24no—
439C2×C32—62no—
440C2×C32C22×D1211yescharacteristic
441C2×C32C2×D3×D414yes—
442C2×C32C2×#23914yes—
443C2×C32D3×#4114yes—
444C2×C32D3×#4111yescharacteristic
445C2×C32D3×#4111yescharacteristic
446C3⋊D3—22no—
447C3⋊D3—62no—
448C3⋊D3—68no—
449C3⋊D3—128no—
450C3⋊D3—122no—
451C3⋊D3—128no—
452C3⋊D3—248no—
453C3×D3—28no—
454C3×D3—68no—
455C24—271no—
456#41—544no—
457C2×Q8—31no—
458C2×D4—272no—
459C2×D4—272no—
460C2×D4—272no—
461C2×D4—271no—
462C22×C4—31no—
463C22×C4—91no—
464C22×C4—271no—
465C22×C4—272no—
466C22×C4—271no—
467C22×C4—271no—
468C22×C4—271no—
469C22×C4—544no—
470C4⋊C4—32no—
471C4⋊C4—91no—
472C4⋊C4—272no—
473C4⋊C4D3×C3⋊D311yescharacteristic, complemented
474C22⋊C4—32no—
475C22⋊C4—92no—
476C22⋊C4—272no—
477C22⋊C4—272no—
478C22⋊C4—272no—
479C42—92no—
480C22×C3—24no—
481C22×C3—38no—
482C22×C3—64no—
483C22×C3—64no—
484C22×C3—92no—
485C22×C3—181no—
486C22×C3—181no—
487C22×C3C22×D3214yes—
488C22×C3C23×C3⋊D311yescharacteristic
489C2×D3—21no—
490C2×D3—21no—
491C2×D3—38no—
492C2×D3—64no—
493C2×D3—64no—
494C2×D3—98no—
495C2×D3—92no—
496C2×D3—188no—
497C2×D3—188no—
498C2×D3—1832no—
499C2×D3—184no—
500C2×D3—184no—
501C2×D3—181no—
502C2×D3—181no—
503C2×D3—188no—
504C2×D3—188no—
505C2×D3—188no—
506C2×D3—182no—
507C2×D3—182no—
508C2×D3—182no—
509C2×D3—364no—
510C2×D3—364no—
511C2×D3—368no—
512C2×D3—368no—
513C2×D3—368no—
514C2×D3#383212yescomplemented
515C3×C4—28no—
516C3×C4—22no—
517C3×C4—28no—
518C3×C4—48no—
519C3×C4—64no—
520C3×C4—68no—
521C3×C4—92no—
522C3×C4—182no—
523C3×C4D3×D1218yes—
524C3×C4D4×C3⋊D312yes—
525Dic3—22no—
526Dic3—21no—
527Dic3—38no—
528Dic3—68no—
529Dic3—184no—
530Dic3—181no—
531Dic3—364no—
532C32—24no—
533C32#552411yescharacteristic, complemented
534C32#562714yescomplemented
535C23—31no—
536C23—91no—
537C23—271no—
538C23—272no—
539C23—542no—
540C23—542no—
541C23—542no—
542Q8—62no—
543D4—544no—
544D4—544no—
545D4—544no—
546D4—542no—
547C2×C4—32no—
548C2×C4—31no—
549C2×C4—62no—
550C2×C4—91no—
551C2×C4—94no—
552C2×C4—92no—
553C2×C4—271no—
554C2×C4—542no—
555C2×C4—544no—
556C2×C4—542no—
557C2×C4—542no—
558C2×C4—542no—
559C2×C4C2×D3×C3⋊D312yes—
560C2×C4C2×D3×C3⋊D311yescharacteristic
561C2×C3—24no—
562C2×C3—24no—
563C2×C3—24no—
564C2×C3—68no—
565C2×C3—182no—
566C2×C3C2×D3×D1214yes—
567C2×C3#6431214yes—
568C2×C3D3×#23914yes—
569C2×C3C2×D4×C3⋊D311yescharacteristic
570C2×C3C2×#383211yescharacteristic
571C2×C3#41×C3⋊D311yescharacteristic
572D3—22no—
573D3—68no—
574D3—182no—
575D3—188no—
576D3—368no—
577D3—362no—
578D3—368no—
579D3—728no—
580C22—32no—
581C22—61no—
582C22—61no—
583C22—92no—
584C22—181no—
585C22—181no—
586C22—272no—
587C22—548no—
588C22—541no—
589C22—541no—
590C22—542no—
591C22—542no—
592C22—542no—
593C22C22×D3×C3⋊D311yescharacteristic
594C4—22no—
595C4—61no—
596C4—62no—
597C4—92no—
598C4—182no—
599C4—541no—
600C4C3⋊D3×D1212yes—
601C3—24no—
602C3#1059462714yescomplemented
603C3#1059609811yescharacteristic, complemented
604C2—62no—
605C2—182no—
606C2—542no—
607C2—1082no—
608C2C2×C3⋊D3×D1211yescharacteristic
609C2#1172458611yescharacteristic
610C2D3×#383211yescharacteristic
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×#89018C2semidirect
3. #11723936C2semidirect
4. C3×#64101C2semidirect
5. C32×#897C2semidirect
6. #11723542C2semidirect
7. #11723516C2semidirect
8. #11723494C2semidirect
9. #11723493C2semidirect
10. #11723461C2semidirect
11. #11723456C2semidirect
16. C2×C4⋊(C3⋊(C3⋊D3))C22semidirect
20. C2×C32×Dic6C22semidirect
24. C2×#6289C22semidirect
29. C33×C4⋊C4C22semidirect
30. #88859C22semidirect
32. C3×C4⋊(C3⋊Dic3)C22semidirect
33. C3×C4×C3⋊Dic3C22semidirect
35. C32×C4⋊Dic3C22semidirect
36. #88807C22semidirect
38. #88804C22semidirect
51. #64101D3semidirect
54. C3×#897D3semidirect
77. C3×C4×C3⋊D3D4semidirect
114. C2×C3×Dic6C2×D3semidirect
125. C32×C4⋊C4C2×D3semidirect
126. C32×C4⋊C4C2×D3semidirect
134. C4⋊(C3⋊Dic3)C2×D3semidirect
136. C4×C3⋊Dic3C2×D3semidirect
140. C3×C4⋊Dic3C2×D3semidirect
151. C2×C3⋊(C3⋊D3)#41semidirect
157. C2×C32×D3#41semidirect
175. #897C3⋊D3semidirect
235. C4×C3⋊D3D12semidirect
289. C2×Dic6C2×C3⋊D3semidirect
293. C3×C4⋊C4D32semidirect
294. C3×C4⋊C4C2×C3⋊D3semidirect
309. C4⋊Dic3C2×C3⋊D3semidirect
345. C2×C3×D3#239semidirect
375. C33#525semidirect
473. C4⋊C4D3×C3⋊D3semidirect
514. C2×D3#3832semidirect
533. C32#5524semidirect
534. C32#5627semidirect
602. C3#10594627semidirect
603. C3#10596098semidirect

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×#890182
#117239362
C3×#641012
C32×#8972
#117235422
#117235162
#117234942
#117234932
#117234612
#117234562
#105960983
#105946273