Finite Group Databasebeta

← #13176543

Subgroups — #13176543

Order 1728.

2380 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D3×C4⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
2C2×#89017C211yesmaximal, characteristic, complemented
3#11723936C211yesmaximal, characteristic, complemented
4C3×#64102C211yesmaximal, characteristic, complemented
5C32×#897C211yesmaximal, characteristic, complemented
6#11723542C212yesmaximal, complemented
7C3⋊D3×(C2×D3)⋊C4C212yesmaximal, complemented
8#11723526C212yesmaximal, complemented
9C4×#6291C211yesmaximal, characteristic, complemented
10#11723493C212yesmaximal, complemented
11#11723440C211yesmaximal, characteristic, complemented
12#10596101—31nomaximal
13#10594629—34nomaximal
14C22×D3×C3⋊D3C2212yes—
15C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
16C2×C4⋊(C3⋊(C3⋊D3))C2212yescomplemented
17C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic, complemented
18C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
19C2×C3×C4×C3⋊D3C2212yescomplemented
20C2×C32×C4×D3C2211yescharacteristic
21C2×C32×Dic6C2211yescharacteristic, complemented
22C2×#6292C2212yes—
23C2×#6292C2212yescomplemented
24C2×#6291C2211yescharacteristic
25C2×Dic3×C3⋊D3C2212yescomplemented
26D3×C4⋊(C3⋊D3)—24no—
27#89017—24no—
28C33×C4⋊C4C2211yescharacteristic, complemented
29#88859C2212yescomplemented
30C3×#3752C2212yescomplemented
31C3×C4×C3⋊Dic3C2211yescharacteristic, complemented
32C32×(C2×D3)⋊C4C2212yes—
33C32×C4⋊Dic3C2212yescomplemented
34#88807C2211yescharacteristic, complemented
35#88804C2211yescharacteristic, complemented
36#88804C2212yescomplemented
37#88804C2212yescomplemented
38(C2×D3)⋊(C3⋊Dic3)C2211yescharacteristic
39C2×#3832—31no—
40C22×C4⋊(C3⋊D3)—31no—
41C2×D3×D12—34no—
42C2×#3796—34no—
43C32×C22⋊Q8—31no—
44#64156—31no—
45C4×#638—31no—
46#64104—32no—
47#64104—31no—
48#64104—34no—
49#64104—64no—
50#64103—32no—
51#64102D311yescharacteristic, complemented
52#64091—32no—
53C22⋊C4×C3⋊D3—32no—
54C3×#897D314yescomplemented
55C3×#893—34no—
56#63923—38no—
57D3×(C2×D3)⋊C4—38no—
58#63909—38no—
59C4×#626—34no—
60#63890—38no—
61#63864—34no—
62C22×C3⋊(C3⋊D3)C2312yes—
63C22×C3⋊(C3⋊D3)C2311yescharacteristic
64C22×C3×C3⋊D3C2312yes—
65C22×C32×D3C2311yescharacteristic
66C2×D3×C3⋊D3—24no—
67C2×D3×C3⋊D3—24no—
68C2×D3×C3⋊D3—42no—
69C2×C33×C4C2312yes—
70C2×C33×C4C2311yescharacteristic
71C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
72C4⋊(C3⋊(C3⋊D3))—22no—
73C4⋊(C3⋊(C3⋊D3))—42no—
74C4×C3⋊(C3⋊D3)—22no—
75C2×C3×C3⋊Dic3C2311yescharacteristic
76C2×C3×C3⋊Dic3C2311yescharacteristic
77C3×C4⋊(C3⋊D3)D414yescomplemented
78C3×C4×C3⋊D3—42no—
79C2×C32×Dic3C2312yes—
80C2×C32×Dic3C2311yescharacteristic
81C32×C4×D3—22no—
82C32×Dic6—22no—
83#6292—24no—
84#6292—24no—
85#6291—22no—
86Dic3×C3⋊D3—24no—
87#5630—94no—
88#5554—91no—
89C23×C3⋊D3—32no—
90C22×D32—38no—
91C2×C32×Q8—31no—
92C22×C32×C4—31no—
93C2×#638—31no—
94#3832—64no—
95C2×C4⋊(C3⋊D3)—32no—
96C2×C4⋊(C3⋊D3)—32no—
97C2×C4⋊(C3⋊D3)—31no—
98C2×C4⋊(C3⋊D3)—34no—
99C2×C4⋊(C3⋊D3)—32no—
100C2×C4⋊(C3⋊D3)—38no—
101C2×C4⋊(C3⋊D3)—64no—
102C2×C4⋊(C3⋊D3)—64no—
103C2×C4⋊(C3⋊D3)—68no—
104C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
105C2×C4×C3⋊D3—32no—
106C2×C4×C3⋊D3—31no—
107C2×C4×C3⋊D3—34no—
108C2×C4×C3⋊D3—64no—
109C2×C4×C3⋊D3C2×D312yescomplemented
110C2×C3×D12—34no—
111C2×C3×C4×D3—38no—
112C2×C3×C4×D3C2×D314yes—
113C2×C3×Dic6C2×D314yescomplemented
114C2×#626—34no—
115C2×#626—38no—
116C2×#626—38no—
117C2×D3×Dic3—38no—
118D3×D12—616no—
119#3796—616no—
120C32×C4⋊C4—24no—
121C32×C4⋊C4—32no—
122C32×C4⋊C4C2×D314yescomplemented
123C32×C4⋊C4C2×D311yescharacteristic, complemented
124C32×C22⋊C4—32no—
125#3757—31no—
126#3752—32no—
127#3752—38no—
128#3752—32no—
129#3752—32no—
130#3752—31no—
131#3752—68no—
132#3752C2×D312yescomplemented
133#3750—31no—
134C4×C3⋊Dic3C2×D311yescharacteristic, complemented
135C3×(C2×D3)⋊C4—38no—
136C3×(C2×D3)⋊C4C2×D318yes—
137C3×C4⋊Dic3C2×D318yescomplemented
138C3×C4×Dic3—34no—
139Dic3⋊Dic3—34no—
140#3722—38no—
141#3722—38no—
142#3722—34no—
143(C2×D3)⋊Dic3—34no—
144C22×C33C2411yescharacteristic
145C2×C3⋊(C3⋊D3)—21no—
146C2×C3⋊(C3⋊D3)—22no—
147C2×C3⋊(C3⋊D3)—21no—
148C2×C3⋊(C3⋊D3)—22no—
149C2×C3⋊(C3⋊D3)—21no—
150C2×C3⋊(C3⋊D3)—22no—
151C2×C3×C3⋊D3—22no—
152C2×C3×C3⋊D3—22no—
153C2×C3×C3⋊D3C2×D414yes—
154C2×C32×D3—21no—
155C2×C32×D3—21no—
156C2×C32×D3#4112yescomplemented
157D3×C3⋊D3—48no—
158C33×C4—22no—
159C33×C4C2×D412yes—
160C3⋊(C3⋊Dic3)—21no—
161C3×C3⋊Dic3—21no—
162C3×C3⋊Dic3—21no—
163C32×Dic3—22no—
164C32×Dic3—21no—
165C2×#239—94no—
166C2×#237—91no—
167C2×D3×D4—91no—
168C22×D12—94no—
169C3×C22⋊Q8—34no—
170C3×C4×D4—91no—
171#947—94no—
172#930—92no—
173C4×#67—94no—
174C2×(C2×D3)⋊C4—92no—
175#897C3⋊D311yescharacteristic, complemented
176#895—98no—
177#895—94no—
178#895—91no—
179#895—184no—
180#894—98no—
181#893—34no—
182#886—92no—
183#882—98no—
184#882—92no—
185D3×C22⋊C4—98no—
186#875—91no—
187C4×D12—91no—
188C23×C32—31no—
189C22×C3⋊D3—32no—
190C22×C3⋊D3—31no—
191C22×C3⋊D3—38no—
192C22×C3⋊D3—34no—
193C22×C3⋊D3—64no—
194C22×C3⋊D3—68no—
195C22×C3⋊D3—64no—
196C22×C3⋊D3—64no—
197C22×C3⋊D3—122no—
198C22×C3⋊D3C22×D312yes—
199C22×C3×D3—38no—
200C22×C3×D3C22×D314yes—
201C2×D32—616no—
202C2×D32—616no—
203C2×D32—128no—
204C32×Q8—62no—
205C2×C32×C4—24no—
206C2×C32×C4—28no—
207C2×C32×C4—31no—
208C2×C32×C4—32no—
209C2×C32×C4—62no—
210C2×C32×C4C22×D318yes—
211C2×C32×C4C22×D314yes—
212C2×C32×C4C22×D312yes—
213C2×C32×C4C22×D311yescharacteristic
214#638—62no—
215C2×C3⋊Dic3—31no—
216C2×C3⋊Dic3—34no—
217C2×C3⋊Dic3—64no—
218C2×C3⋊Dic3C22×D311yescharacteristic
219C2×C3⋊Dic3C22×D311yescharacteristic
220C4⋊(C3⋊D3)—64no—
221C4⋊(C3⋊D3)—64no—
222C4⋊(C3⋊D3)—62no—
223C4⋊(C3⋊D3)—68no—
224C4⋊(C3⋊D3)—128no—
225C4⋊(C3⋊D3)—122no—
226C4⋊(C3⋊D3)—128no—
227C4⋊(C3⋊D3)—1216no—
228C4⋊(C3⋊D3)D1214yescomplemented
229C4×C3⋊D3—42no—
230C4×C3⋊D3—64no—
231C4×C3⋊D3—62no—
232C4×C3⋊D3—68no—
233C4×C3⋊D3—128no—
234C2×C3×Dic3—34no—
235C2×C3×Dic3—34no—
236C2×C3×Dic3C22×D318yes—
237C2×C3×Dic3C22×D314yes—
238C3×D12—316no—
239C3×C4×D3—28no—
240C3×C4×D3—128no—
241C3×Dic6—28no—
242#626—616no—
243#626—616no—
244#626—68no—
245D3×Dic3—616no—
246#546—271noSylow
247C2×C33C22×D411yescharacteristic
248C2×C33C2×#4111yescharacteristic
249C2×C33H(5,𝔽2)11yescharacteristic
250C3⋊(C3⋊D3)—41no—
251C3⋊(C3⋊D3)—42no—
252C3×C3⋊D3—24no—
253C32×D3—22no—
254C23×D3—98no—
255C23×D3—92no—
256C2×C3×Q8—34no—
257C2×C3×D4—91no—
258C22×C3×C4—34no—
259C22×C3×C4—92no—
260C2×#67—94no—
261C2×#67—92no—
262C2×#67—92no—
263C22×Dic3—92no—
264#239—1816no—
265#237—184no—
266D3×D4—184no—
267C2×D12—34no—
268C2×D12—98no—
269C2×D12—98no—
270C2×D12—91no—
271C2×D12—94no—
272C2×D12—91no—
273C2×D12—98no—
274C2×D12—92no—
275C2×D12—184no—
276C2×D12—188no—
277C2×D12—1816no—
278C2×C4×D3—38no—
279C2×C4×D3—98no—
280C2×C4×D3—94no—
281C2×C4×D3—91no—
282C2×C4×D3—184no—
283C2×C4×D3C2×C3⋊D311yescharacteristic
284C2×Dic6C2×C3⋊D311yescharacteristic, complemented
285C3×C4⋊C4—24no—
286C3×C4⋊C4—38no—
287C3×C4⋊C4D3214yescomplemented
288C3×C4⋊C4C2×C3⋊D311yescharacteristic, complemented
289C3×C22⋊C4—38no—
290C3×C22⋊C4—92no—
291C3×C42—91no—
292#217—94no—
293(C2×D3)⋊C4—38no—
294(C2×D3)⋊C4—98no—
295(C2×D3)⋊C4—98no—
296(C2×D3)⋊C4—94no—
297(C2×D3)⋊C4—91no—
298(C2×D3)⋊C4—98no—
299(C2×D3)⋊C4—92no—
300(C2×D3)⋊C4—91no—
301(C2×D3)⋊C4—92no—
302(C2×D3)⋊C4—92no—
303(C2×D3)⋊C4—188no—
304(C2×D3)⋊C4C2×C3⋊D312yes—
305C4⋊Dic3—91no—
306C4⋊Dic3C2×C3⋊D312yescomplemented
307Dic3⋊C4—94no—
308C4×Dic3—34no—
309C22×C32—24no—
310C22×C32—32no—
311C22×C32—61no—
312C22×C32—61no—
313C22×C32C23×D314yes—
314C22×C32C23×D311yescharacteristic
315C2×C3⋊D3—22no—
316C2×C3⋊D3—22no—
317C2×C3⋊D3—68no—
318C2×C3⋊D3—61no—
319C2×C3⋊D3—62no—
320C2×C3⋊D3—61no—
321C2×C3⋊D3—62no—
322C2×C3⋊D3—61no—
323C2×C3⋊D3—62no—
324C2×C3⋊D3—64no—
325C2×C3⋊D3—64no—
326C2×C3⋊D3—68no—
327C2×C3⋊D3—64no—
328C2×C3⋊D3—68no—
329C2×C3⋊D3—68no—
330C2×C3⋊D3—124no—
331C2×C3⋊D3—124no—
332C2×C3⋊D3—128no—
333C2×C3⋊D3—128no—
334C2×C3⋊D3—128no—
335C2×C3⋊D3—128no—
336C2×C3⋊D3C2×D1214yes—
337C2×C3×D3—24no—
338C2×C3×D3—24no—
339C2×C3×D3—316no—
340C2×C3×D3—68no—
341C2×C3×D3—68no—
342C2×C3×D3#23918yescomplemented
343D32—1232no—
344C32×C4—22no—
345C32×C4—28no—
346C32×C4—28no—
347C32×C4—48no—
348C32×C4—62no—
349C32×C4—61no—
350C32×C4C2×D1212yes—
351C32×C4D3×D418yes—
352C3⋊Dic3—21no—
353C3⋊Dic3—21no—
354C3⋊Dic3—61no—
355C3⋊Dic3—64no—
356C3⋊Dic3—124no—
357C3×Dic3—28no—
358C3×Dic3—24no—
359C3×Dic3—64no—
360C3×Dic3—64no—
361C2×#41—271no—
362C22×D4—271no—
363#124—271no—
364#123—272no—
365C22⋊Q8—31no—
366#121—272no—
367#121—271no—
368#120—272no—
369C4×D4—91no—
370C4×D4—271no—
371C2×C22⋊C4—272no—
372C33#54611yescharacteristic, complemented, Sylow
373C23×C3—34no—
374C23×C3—92no—
375C22×D3—38no—
376C22×D3—98no—
377C22×D3—94no—
378C22×D3—92no—
379C22×D3—91no—
380C22×D3—184no—
381C22×D3—188no—
382C22×D3—184no—
383C22×D3—1816no—
384C22×D3—1816no—
385C22×D3—184no—
386C22×D3—362no—
387C22×D3—368no—
388C22×D3C22×C3⋊D311yescharacteristic
389C3×Q8—68no—
390C3×D4—94no—
391C2×C3×C4—24no—
392C2×C3×C4—28no—
393C2×C3×C4—34no—
394C2×C3×C4—38no—
395C2×C3×C4—68no—
396C2×C3×C4—91no—
397C2×C3×C4—91no—
398C2×C3×C4—362no—
399C2×C3×C4C2×D3218yes—
400C2×C3×C4C2×D3214yes—
401C2×C3×C4C22×C3⋊D312yes—
402C2×C3×C4C22×C3⋊D311yescharacteristic
403#67—184no—
404#67—184no—
405#67—188no—
406C2×Dic3—34no—
407C2×Dic3—34no—
408C2×Dic3—94no—
409C2×Dic3—91no—
410C2×Dic3—184no—
411C2×Dic3—184no—
412C2×Dic3C22×C3⋊D311yescharacteristic
413C2×Dic3C22×C3⋊D312yes—
414D12—316no—
415D12—1816no—
416D12—1816no—
417D12—182no—
418D12—188no—
419D12—182no—
420D12—368no—
421D12—368no—
422D12—362no—
423D12—3616no—
424C4×D3—22no—
425C4×D3—128no—
426C4×D3—1816no—
427C4×D3—188no—
428C4×D3—182no—
429C4×D3—368no—
430Dic6—22no—
431C2×C32—24no—
432C2×C32—24no—
433C2×C32—24no—
434C2×C32—62no—
435C2×C32C22×D1211yescharacteristic
436C2×C32C2×D3×D414yes—
437C2×C32#100411yescharacteristic
438C2×C32C2×#23914yes—
439C2×C32D3×#4111yescharacteristic
440C2×C32#100914yes—
441C3⋊D3—24no—
442C3⋊D3—128no—
443C3⋊D3—121no—
444C3⋊D3—122no—
445C3⋊D3—124no—
446C3⋊D3—128no—
447C3⋊D3—248no—
448C3×D3—28no—
449C3×D3—616no—
450C24—272no—
451#41—544no—
452C2×Q8—31no—
453C2×D4—91no—
454C2×D4—272no—
455C2×D4—272no—
456C2×D4—271no—
457C2×D4—271no—
458C2×D4—272no—
459C2×D4—544no—
460C22×C4—31no—
461C22×C4—92no—
462C22×C4—272no—
463C22×C4—271no—
464C4⋊C4—32no—
465C4⋊C4—271no—
466C4⋊C4D3×C3⋊D311yescharacteristic, complemented
467C22⋊C4—32no—
468C22⋊C4—92no—
469C22⋊C4—272no—
470C22⋊C4—271no—
471C22⋊C4—272no—
472C22⋊C4—271no—
473C22⋊C4—272no—
474C42—91no—
475C22×C3—24no—
476C22×C3—38no—
477C22×C3—64no—
478C22×C3—64no—
479C22×C3—94no—
480C22×C3—182no—
481C22×C3—182no—
482C22×C3C22×D3214yes—
483C22×C3C23×C3⋊D311yescharacteristic
484C2×D3—21no—
485C2×D3—21no—
486C2×D3—316no—
487C2×D3—68no—
488C2×D3—68no—
489C2×D3—188no—
490C2×D3—184no—
491C2×D3—188no—
492C2×D3—184no—
493C2×D3—188no—
494C2×D3—184no—
495C2×D3—188no—
496C2×D3—181no—
497C2×D3—182no—
498C2×D3—181no—
499C2×D3—182no—
500C2×D3—181no—
501C2×D3—182no—
502C2×D3—364no—
503C2×D3—364no—
504C2×D3—368no—
505C2×D3—368no—
506C2×D3—368no—
507C2×D3—368no—
508C2×D3—3632no—
509C2×D3#383212yescomplemented
510C3×C4—28no—
511C3×C4—22no—
512C3×C4—28no—
513C3×C4—48no—
514C3×C4—68no—
515C3×C4—64no—
516C3×C4—181no—
517C3×C4—181no—
518C3×C4D3×D1218yes—
519C3×C4D4×C3⋊D312yes—
520Dic3—22no—
521Dic3—21no—
522Dic3—64no—
523Dic3—64no—
524Dic3—184no—
525Dic3—181no—
526Dic3—364no—
527C32—24no—
528C32#555411yescharacteristic, complemented
529C32#563014yescomplemented
530C23—31no—
531C23—92no—
532C23—272no—
533C23—271no—
534C23—544no—
535C23—544no—
536C23—1082no—
537Q8—62no—
538D4—94no—
539D4—544no—
540D4—544no—
541D4—542no—
542D4—542no—
543D4—1082no—
544C2×C4—32no—
545C2×C4—31no—
546C2×C4—62no—
547C2×C4—91no—
548C2×C4—91no—
549C2×C4—271no—
550C2×C4—362no—
551C2×C4—544no—
552C2×C4—542no—
553C2×C4C2×D3×C3⋊D312yes—
554C2×C4C2×D3×C3⋊D311yescharacteristic
555C2×C3—24no—
556C2×C3—24no—
557C2×C3—24no—
558C2×C3—68no—
559C2×C3—184no—
560C2×C3C2×D3×D1214yes—
561C2×C3#6431414yes—
562C2×C3D3×#23914yes—
563C2×C3C2×D4×C3⋊D311yescharacteristic
564C2×C3C2×#383211yescharacteristic
565C2×C3#6436611yescharacteristic
566D3—22no—
567D3—616no—
568D3—368no—
569D3—362no—
570D3—361no—
571D3—368no—
572D3—364no—
573D3—728no—
574C22—32no—
575C22—61no—
576C22—61no—
577C22—94no—
578C22—182no—
579C22—182no—
580C22—541no—
581C22—542no—
582C22—541no—
583C22—542no—
584C22—541no—
585C22—542no—
586C22—1088no—
587C22C22×D3×C3⋊D311yescharacteristic
588C4—22no—
589C4—61no—
590C4—62no—
591C4—181no—
592C4—181no—
593C4—541no—
594C4C3⋊D3×D1212yes—
595C3—24no—
596C3#1059462914yescomplemented
597C3#1059610111yescharacteristic, complemented
598C2—62no—
599C2—184no—
600C2—1082no—
601C2—1081no—
602C2C2×C3⋊D3×D1211yescharacteristic
603C2#1172458911yescharacteristic
604C2D3×#383211yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC33×C4⋊C4C22
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33#546
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×D3×C4⋊(C3⋊D3)C2semidirect
2. C2×#89017C2semidirect
3. #11723936C2semidirect
4. C3×#64102C2semidirect
5. C32×#897C2semidirect
6. #11723542C2semidirect
7. C3⋊D3×(C2×D3)⋊C4C2semidirect
8. #11723526C2semidirect
9. C4×#6291C2semidirect
10. #11723493C2semidirect
11. #11723440C2semidirect
16. C2×C4⋊(C3⋊(C3⋊D3))C22semidirect
17. C2×C4×C3⋊(C3⋊D3)C22semidirect
19. C2×C3×C4×C3⋊D3C22semidirect
21. C2×C32×Dic6C22semidirect
23. C2×#6292C22semidirect
25. C2×Dic3×C3⋊D3C22semidirect
28. C33×C4⋊C4C22semidirect
29. #88859C22semidirect
30. C3×#3752C22semidirect
31. C3×C4×C3⋊Dic3C22semidirect
33. C32×C4⋊Dic3C22semidirect
34. #88807C22semidirect
35. #88804C22semidirect
36. #88804C22semidirect
37. #88804C22semidirect
51. #64102D3semidirect
54. C3×#897D3semidirect
77. C3×C4⋊(C3⋊D3)D4semidirect
109. C2×C4×C3⋊D3C2×D3semidirect
113. C2×C3×Dic6C2×D3semidirect
122. C32×C4⋊C4C2×D3semidirect
123. C32×C4⋊C4C2×D3semidirect
132. #3752C2×D3semidirect
134. C4×C3⋊Dic3C2×D3semidirect
137. C3×C4⋊Dic3C2×D3semidirect
156. C2×C32×D3#41semidirect
175. #897C3⋊D3semidirect
228. C4⋊(C3⋊D3)D12semidirect
284. C2×Dic6C2×C3⋊D3semidirect
287. C3×C4⋊C4D32semidirect
288. C3×C4⋊C4C2×C3⋊D3semidirect
306. C4⋊Dic3C2×C3⋊D3semidirect
342. C2×C3×D3#239semidirect
372. C33#546semidirect
466. C4⋊C4D3×C3⋊D3semidirect
509. C2×D3#3832semidirect
528. C32#5554semidirect
529. C32#5630semidirect
596. C3#10594629semidirect
597. C3#10596101semidirect

Maximal subgroups

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

subgroupindex
C2×D3×C4⋊(C3⋊D3)2
C2×#890172
#117239362
C3×#641022
C32×#8972
#117235422
C3⋊D3×(C2×D3)⋊C42
#117235262
C4×#62912
#117234932
#117234402
#105961013
#105946293