Finite Group Databasebeta

← #13177318

Subgroups — #13177318

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×C32×#235C211yesmaximal, characteristic, complemented
2C2×C3⋊D3×D12C211yesmaximal, characteristic, complemented
3#11723971C211yesmaximal, characteristic, complemented
4C3×C4×#638C211yesmaximal, characteristic, complemented
5#11723643C212yesmaximal, complemented
6#11723616C212yesmaximal, complemented
7D3×#3752C212yesmaximal, complemented
8D12×C3⋊Dic3C211yesmaximal, characteristic, complemented
9#11723440C211yesmaximal, characteristic, complemented
10#11723435C211yesmaximal, characteristic, complemented
11#11723414C212yesmaximal, complemented
12#10596300—31nomaximal
13#10595067—34nomaximal
14C22×D3×C3⋊D3C2212yes—
15C22×C33×C4C2211yescharacteristic, complemented
16C2×#6312C2212yescomplemented
17C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
18C2×C3×#638C2211yescharacteristic
19C2×C3×C4×C3⋊D3C2211yescharacteristic
20C2×C32×#67C2212yescomplemented
21C32×#235—24no—
22C2×C32×D12C2211yescharacteristic
23C2×C32×C4×D3C2212yescomplemented
24C2×C32×Dic6C2211yescharacteristic, complemented
25C2×#6291C2212yes—
26C2×D3×C3⋊Dic3C2212yescomplemented
27C3⋊D3×D12—24no—
28#88859C2212yescomplemented
29C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic, complemented
30C3×#3757C2211yescharacteristic, complemented
31C3×#3752C2211yescharacteristic
32C3×#3750C2211yescharacteristic, complemented
33C3×C4×C3⋊Dic3C2211yescharacteristic, complemented
34#88806C2212yescomplemented
35#88804C2212yescomplemented
36#88804C2212yescomplemented
37#88803C2212yes—
38(C2×D3)⋊(C3⋊Dic3)C2212yescomplemented
39C2×C32×#41—31no—
40C2×D4×C3⋊D3—31no—
41C2×C3×#235D314yescomplemented
42C2×D3×D12—34no—
43#64156—31no—
44#64155—31no—
45#64148—32no—
46#64146—32no—
47D4×C3⋊Dic3—31no—
48#64139—31no—
49#64139—34no—
50#64139—64no—
51#64138—32no—
52C4×#638D311yescharacteristic, complemented
53C2×#3752—32no—
54C3×C4×#67—34no—
55#63983—38no—
56#63971—38no—
57D3×(C2×D3)⋊C4—38no—
58Dic3×D12—34no—
59#63864—34no—
60#63856—34no—
61#63850—38no—
62C23×C33C2311yescharacteristic
63C22×C3⋊(C3⋊D3)C2312yes—
64C22×C3×C3⋊D3C2311yescharacteristic
65C22×C32×D3C2312yes—
66C2×D3×C3⋊D3—24no—
67C2×D3×C3⋊D3—24no—
68C2×D3×C3⋊D3—42no—
69C2×C33×C4—22no—
70C2×C33×C4C2311yescharacteristic
71C2×C33×C4C2311yescharacteristic
72#6312—42no—
73C2×C3⋊(C3⋊Dic3)C2312yes—
74C4⋊(C3⋊(C3⋊D3))—22no—
75C3×#638—22no—
76C2×C3×C3⋊Dic3C2311yescharacteristic
77C2×C3×C3⋊Dic3C2311yescharacteristic
78C2×C3×C3⋊Dic3C2311yescharacteristic
79C3×C4×C3⋊D3—22no—
80C32×#67—24no—
81C2×C32×Dic3C2312yes—
82C32×D12D414yescomplemented
83C32×C4×D3—24no—
84C32×Dic6—22no—
85#6291—24no—
86D3×C3⋊Dic3—42no—
87#5829—94no—
88#5550—91no—
89C23×C3⋊D3—32no—
90C22×D32—38no—
91C32×#41—64no—
92C2×C32×Q8—31no—
93C2×C32×D4—32no—
94C2×C32×D4—31no—
95C22×C32×C4—24no—
96C22×C32×C4—32no—
97C22×C32×C4C2×D314yescomplemented
98C22×C32×C4C2×D311yescharacteristic, complemented
99C2×#638—32no—
100C2×#638—38no—
101C2×#638—32no—
102C2×#638—68no—
103C2×#638C2×D311yescharacteristic
104C22×C3⋊Dic3—32no—
105D4×C3⋊D3—64no—
106C2×C4⋊(C3⋊D3)—31no—
107C2×C4⋊(C3⋊D3)—34no—
108C2×C4⋊(C3⋊D3)—64no—
109C2×C4×C3⋊D3C2×D311yescharacteristic
110C2×C3×#67—34no—
111C2×C3×#67C2×D318yescomplemented
112C3×#235—216no—
113C2×C3×D12C2×D314yes—
114C2×C3×C4×D3—34no—
115C2×C3×C4×D3C2×D318yescomplemented
116C2×C3×Dic6C2×D314yescomplemented
117C2×#626—38no—
118C2×D3×Dic3—38no—
119D3×D12—616no—
120#3757—32no—
121#3757C2×D311yescharacteristic, complemented
122#3752—32no—
123#3752—32no—
124#3752—38no—
125#3752—32no—
126#3752—32no—
127#3752—68no—
128#3752C2×D311yescharacteristic
129C4⋊(C3⋊Dic3)—31no—
130C4⋊(C3⋊Dic3)—34no—
131C4⋊(C3⋊Dic3)—64no—
132#3750—32no—
133#3750C2×D311yescharacteristic, complemented
134C4×C3⋊Dic3C2×D311yescharacteristic, complemented
135C3×#217—34no—
136C3×(C2×D3)⋊C4—34no—
137C3×Dic3⋊C4—34no—
138C3×C4×Dic3—34no—
139#3724—38no—
140#3722—38no—
141#3722—38no—
142(C2×D3)⋊Dic3—38no—
143(C2×D3)⋊Dic3—38no—
144C22×C33—21no—
145C22×C33—21no—
146C22×C33—21no—
147C22×C33C2411yescharacteristic
148C2×C3⋊(C3⋊D3)—22no—
149C2×C3⋊(C3⋊D3)—22no—
150C2×C3⋊(C3⋊D3)—22no—
151C2×C3×C3⋊D3—21no—
152C2×C3×C3⋊D3—21no—
153C2×C3×C3⋊D3#4112yescomplemented
154C2×C32×D3—22no—
155C2×C32×D3—22no—
156C2×C32×D3C2×D414yes—
157D3×C3⋊D3—48no—
158C33×C4—21no—
159C33×C4C2×D412yes—
160C3⋊(C3⋊Dic3)—22no—
161C3×C3⋊Dic3—21no—
162C3×C3⋊Dic3—21no—
163C3×C3⋊Dic3—21no—
164C32×Dic3—22no—
165C2×C3×#41—34no—
166C2×D3×D4—94no—
167C2×#235C3⋊D311yescharacteristic, complemented
168C22×D12—91no—
169C3×C4×D4—91no—
170#947—94no—
171#946—94no—
172#939—98no—
173#937—98no—
174#937—92no—
175D4×Dic3—94no—
176#930—94no—
177#930—91no—
178#930—184no—
179#929—98no—
180C4×#67—34no—
181C2×(C2×D3)⋊C4—98no—
182#924—91no—
183#894—92no—
184#884—92no—
185D3×C22⋊C4—92no—
186#875—91no—
187C4×D12—91no—
188C23×C32—24no—
189C23×C32—32no—
190C23×C32C22×D314yes—
191C23×C32C22×D311yescharacteristic
192C22×C3⋊D3—32no—
193C22×C3⋊D3—38no—
194C22×C3⋊D3—68no—
195C22×C3⋊D3—64no—
196C22×C3⋊D3—64no—
197C22×C3⋊D3—122no—
198C22×C3⋊D3C22×D311yescharacteristic
199C22×C3×D3—34no—
200C22×C3×D3C22×D318yes—
201C2×D32—616no—
202C2×D32—616no—
203C2×D32—128no—
204C32×Q8—62no—
205C32×D4—34no—
206C32×D4—64no—
207C2×C32×C4—22no—
208C2×C32×C4—24no—
209C2×C32×C4—24no—
210C2×C32×C4—28no—
211C2×C32×C4—32no—
212C2×C32×C4—48no—
213C2×C32×C4—64no—
214C2×C32×C4C22×D314yes—
215C2×C32×C4C22×D314yes—
216C2×C32×C4C22×D311yescharacteristic
217C2×C32×C4C22×D311yescharacteristic
218#638—22no—
219#638—64no—
220#638—122no—
221#638—128no—
222#638—1216no—
223C2×C3⋊Dic3—32no—
224C2×C3⋊Dic3—38no—
225C2×C3⋊Dic3—68no—
226C2×C3⋊Dic3—122no—
227C2×C3⋊Dic3C22×D311yescharacteristic
228C2×C3⋊Dic3C22×D311yescharacteristic
229C2×C3⋊Dic3C22×D311yescharacteristic
230C4⋊(C3⋊D3)—62no—
231C4⋊(C3⋊D3)—68no—
232C4⋊(C3⋊D3)—128no—
233C4×C3⋊D3—22no—
234C3×#67—216no—
235C3×#67—68no—
236C2×C3×Dic3—34no—
237C2×C3×Dic3—34no—
238C2×C3×Dic3—34no—
239C2×C3×Dic3C22×D318yes—
240C3×D12#67116yescomplemented
241C3×C4×D3—216no—
242C3×C4×D3—68no—
243C3×Dic6—28no—
244#626—616no—
245D3×Dic3—128no—
246#546—271noSylow
247C2×C33—41no—
248C2×C33C22×D411yescharacteristic
249C2×C33C2×#4111yescharacteristic
250C2×C33H(5,𝔽2)11yescharacteristic
251C3⋊(C3⋊D3)—42no—
252C3×C3⋊D3—22no—
253C32×D3—24no—
254C23×D3—98no—
255C23×D3—92no—
256C3×#41—616no—
257C2×C3×Q8—34no—
258C2×C3×D4—38no—
259C2×C3×D4—34no—
260C2×C3×D4—91no—
261C22×C3×C4—24no—
262C22×C3×C4—38no—
263C22×C3×C4—91no—
264C22×C3×C4D3214yescomplemented
265C22×C3×C4C2×C3⋊D311yescharacteristic, complemented
266C2×#67—34no—
267C2×#67—98no—
268C2×#67—98no—
269C2×#67—92no—
270C2×#67—188no—
271C2×#67C2×C3⋊D312yescomplemented
272C22×Dic3—98no—
273D3×D4—1816no—
274#235—24no—
275C2×D12—92no—
276C2×D12—94no—
277C2×D12—91no—
278C2×D12—184no—
279C2×D12—184no—
280C2×D12C2×C3⋊D311yescharacteristic
281C2×C4×D3—34no—
282C2×C4×D3—92no—
283C2×C4×D3C2×C3⋊D312yescomplemented
284C2×Dic6C2×C3⋊D311yescharacteristic, complemented
285C3×C4⋊C4—91no—
286C3×C22⋊C4—91no—
287C3×C22⋊C4—91no—
288C3×C42—91no—
289#217—34no—
290#217—98no—
291#217—92no—
292(C2×D3)⋊C4—34no—
293(C2×D3)⋊C4—98no—
294(C2×D3)⋊C4—98no—
295(C2×D3)⋊C4—98no—
296(C2×D3)⋊C4—92no—
297(C2×D3)⋊C4—98no—
298(C2×D3)⋊C4—92no—
299(C2×D3)⋊C4—92no—
300(C2×D3)⋊C4—92no—
301(C2×D3)⋊C4—188no—
302C4⋊Dic3—94no—
303C4⋊Dic3—91no—
304C4⋊Dic3—184no—
305Dic3⋊C4—34no—
306Dic3⋊C4—98no—
307Dic3⋊C4—92no—
308C4×Dic3—34no—
309C22×C32—28no—
310C22×C32—21no—
311C22×C32—21no—
312C22×C32—24no—
313C22×C32—21no—
314C22×C32—24no—
315C22×C32—24no—
316C22×C32—24no—
317C22×C32—34no—
318C22×C32—44no—
319C22×C32—44no—
320C22×C32—62no—
321C22×C32—62no—
322C22×C32C23×D314yes—
323C22×C32C23×D311yescharacteristic
324C2×C3⋊D3—21no—
325C2×C3⋊D3—21no—
326C2×C3⋊D3—62no—
327C2×C3⋊D3—62no—
328C2×C3⋊D3—62no—
329C2×C3⋊D3—68no—
330C2×C3⋊D3—68no—
331C2×C3⋊D3—68no—
332C2×C3⋊D3—128no—
333C2×C3⋊D3—128no—
334C2×C3⋊D3—128no—
335C2×C3⋊D3—128no—
336C2×C3⋊D3#23512yescomplemented
337C2×C3×D3—28no—
338C2×C3×D3—28no—
339C2×C3×D3—38no—
340C2×C3×D3—64no—
341C2×C3×D3—64no—
342C2×C3×D3C2×#67116yes—
343D32—1232no—
344C32×C4—21no—
345C32×C4—28no—
346C32×C4—24no—
347C32×C4—44no—
348C32×C4—62no—
349C32×C4D3×D412yes—
350C32×C4C2×#6718yes—
351C3⋊Dic3—21no—
352C3⋊Dic3—21no—
353C3⋊Dic3—21no—
354C3⋊Dic3—62no—
355C3⋊Dic3—68no—
356C3⋊Dic3—128no—
357C3×Dic3—28no—
358C3×Dic3—64no—
359C3×Dic3—64no—
360C3×Dic3—64no—
361C2×#41—31no—
362C22×D4—271no—
363#124—271no—
364#123—272no—
365C22⋊Q8—271no—
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—24no—
374C23×C3—38no—
375C23×C3—91no—
376C23×C3C2×D3214yes—
377C23×C3C22×C3⋊D311yescharacteristic
378C22×D3—34no—
379C22×D3—98no—
380C22×D3—92no—
381C22×D3—188no—
382C22×D3—184no—
383C22×D3—1816no—
384C22×D3—1816no—
385C22×D3—184no—
386C22×D3—362no—
387C22×D3—368no—
388C22×D3C22×C3⋊D312yes—
389C3×Q8—68no—
390C3×D4—316no—
391C3×D4—616no—
392C3×D4—182no—
393C2×C3×C4—28no—
394C2×C3×C4—22no—
395C2×C3×C4—24no—
396C2×C3×C4—24no—
397C2×C3×C4—38no—
398C2×C3×C4—48no—
399C2×C3×C4—616no—
400C2×C3×C4—91no—
401C2×C3×C4—91no—
402C2×C3×C4—91no—
403C2×C3×C4—182no—
404C2×C3×C4C2×D3214yes—
405C2×C3×C4C2×D3214yes—
406C2×C3×C4C22×C3⋊D311yescharacteristic
407C2×C3×C4C22×C3⋊D311yescharacteristic
408#67—24no—
409#67—68no—
410#67—1816no—
411#67—368no—
412#67—362no—
413#67—3616no—
414C2×Dic3—34no—
415C2×Dic3—34no—
416C2×Dic3—34no—
417C2×Dic3—98no—
418C2×Dic3—92no—
419C2×Dic3—188no—
420C2×Dic3—368no—
421C2×Dic3C22×C3⋊D312yes—
422D12—184no—
423D12—188no—
424D12—182no—
425D12—368no—
426D12#63814yescomplemented
427C4×D3—24no—
428C4×D3—68no—
429C4×D3—362no—
430Dic6—22no—
431C2×C32—24no—
432C2×C32—24no—
433C2×C32—24no—
434C2×C32—48no—
435C2×C32—41no—
436C2×C32—44no—
437C2×C32—64no—
438C2×C32C2×#23511yescharacteristic
439C2×C32C2×D3×D411yescharacteristic
440C2×C32D3×#4114yes—
441C2×C32#100914yes—
442C2×C32#100911yescharacteristic
443C2×C32C22×#6714yes—
444C3⋊D3—22no—
445C3⋊D3—122no—
446C3⋊D3—128no—
447C3⋊D3—248no—
448C3×D3—216no—
449C3×D3—68no—
450C24—272no—
451#41—64no—
452C2×Q8—31no—
453C2×D4—32no—
454C2×D4—31no—
455C2×D4—91no—
456C2×D4—272no—
457C2×D4—272no—
458C2×D4—271no—
459C2×D4—544no—
460C22×C4—32no—
461C22×C4—91no—
462C22×C4—272no—
463C22×C4D3×C3⋊D311yescharacteristic, complemented
464C4⋊C4—91no—
465C4⋊C4—272no—
466C4⋊C4—271no—
467C22⋊C4—91no—
468C22⋊C4—91no—
469C22⋊C4—272no—
470C22⋊C4—272no—
471C22⋊C4—272no—
472C22⋊C4—272no—
473C22⋊C4—272no—
474C42—91no—
475C22×C3—28no—
476C22×C3—24no—
477C22×C3—24no—
478C22×C3—24no—
479C22×C3—21no—
480C22×C3—21no—
481C22×C3—21no—
482C22×C3—24no—
483C22×C3—316no—
484C22×C3—44no—
485C22×C3—44no—
486C22×C3—68no—
487C22×C3—68no—
488C22×C3—92no—
489C22×C3—181no—
490C22×C3—181no—
491C22×C3C22×D3214yes—
492C22×C3C23×C3⋊D311yescharacteristic
493C2×D3—22no—
494C2×D3—22no—
495C2×D3—38no—
496C2×D3—64no—
497C2×D3—64no—
498C2×D3—188no—
499C2×D3—188no—
500C2×D3—188no—
501C2×D3—182no—
502C2×D3—182no—
503C2×D3—182no—
504C2×D3—368no—
505C2×D3—368no—
506C2×D3—368no—
507C2×D3—368no—
508C2×D3—3632no—
509C2×D3C2×#63814yes—
510C3×C4—24no—
511C3×C4—21no—
512C3×C4—28no—
513C3×C4—44no—
514C3×C4—68no—
515C3×C4—181no—
516C3×C4—181no—
517C3×C4—181no—
518C3×C4D3×#6718yes—
519C3×C4C2×#63812yes—
520Dic3—22no—
521Dic3—64no—
522Dic3—64no—
523Dic3—64no—
524Dic3—188no—
525Dic3—182no—
526Dic3—368no—
527C32—24no—
528C32#555011yescharacteristic, complemented
529C32#582914yescomplemented
530C23—32no—
531C23—91no—
532C23—272no—
533C23—544no—
534C23—544no—
535C23—1082no—
536C23C2×D3×C3⋊D311yescharacteristic
537Q8—62no—
538D4—34no—
539D4—64no—
540D4—182no—
541D4—544no—
542D4—542no—
543D4—1082no—
544C2×C4—22no—
545C2×C4—32no—
546C2×C4—64no—
547C2×C4—91no—
548C2×C4—91no—
549C2×C4—91no—
550C2×C4—182no—
551C2×C4—272no—
552C2×C4—1082no—
553C2×C4C2×D3×C3⋊D311yescharacteristic
554C2×C4C2×D3×C3⋊D311yescharacteristic
555C2×C3—24no—
556C2×C3—24no—
557C2×C3—24no—
558C2×C3—48no—
559C2×C3—41no—
560C2×C3—44no—
561C2×C3—616no—
562C2×C3—182no—
563C2×C3D3×#23514yes—
564C2×C3#6430814yes—
565C2×C3C2×D3×#6714yes—
566C2×C3#41×C3⋊D311yescharacteristic
567C2×C3#6436611yescharacteristic
568C2×C3C22×#63811yescharacteristic
569D3—24no—
570D3—68no—
571D3—362no—
572D3—368no—
573D3—728no—
574C22—21no—
575C22—21no—
576C22—21no—
577C22—34no—
578C22—62no—
579C22—62no—
580C22—92no—
581C22—181no—
582C22—181no—
583C22—542no—
584C22—542no—
585C22—542no—
586C22—1088no—
587C22C22×D3×C3⋊D311yescharacteristic
588C4—21no—
589C4—62no—
590C4—181no—
591C4—181no—
592C4—181no—
593C4—542no—
594C4D3×#63812yes—
595C3—24no—
596C3#1059506714yescomplemented
597C3#1059630011yescharacteristic, complemented
598C2—41no—
599C2—64no—
600C2—182no—
601C2—1082no—
602C2C3⋊D3×#23511yescharacteristic
603C2#1172458111yescharacteristic
604C2C2×D3×#63811yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC22×C33×C4C22
hypercenterC22×C4D3×C3⋊D3
hypocenterC33#546
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C2×C32×#235C2semidirect
2. C2×C3⋊D3×D12C2semidirect
3. #11723971C2semidirect
4. C3×C4×#638C2semidirect
5. #11723643C2semidirect
6. #11723616C2semidirect
7. D3×#3752C2semidirect
8. D12×C3⋊Dic3C2semidirect
9. #11723440C2semidirect
10. #11723435C2semidirect
11. #11723414C2semidirect
15. C22×C33×C4C22semidirect
16. C2×#6312C22semidirect
20. C2×C32×#67C22semidirect
23. C2×C32×C4×D3C22semidirect
24. C2×C32×Dic6C22semidirect
26. C2×D3×C3⋊Dic3C22semidirect
28. #88859C22semidirect
29. C4⋊(C3⋊(C3⋊Dic3))C22semidirect
30. C3×#3757C22semidirect
32. C3×#3750C22semidirect
33. C3×C4×C3⋊Dic3C22semidirect
34. #88806C22semidirect
35. #88804C22semidirect
36. #88804C22semidirect
38. (C2×D3)⋊(C3⋊Dic3)C22semidirect
41. C2×C3×#235D3semidirect
52. C4×#638D3semidirect
82. C32×D12D4semidirect
97. C22×C32×C4C2×D3semidirect
98. C22×C32×C4C2×D3semidirect
111. C2×C3×#67C2×D3semidirect
115. C2×C3×C4×D3C2×D3semidirect
116. C2×C3×Dic6C2×D3semidirect
121. #3757C2×D3semidirect
133. #3750C2×D3semidirect
134. C4×C3⋊Dic3C2×D3semidirect
153. C2×C3×C3⋊D3#41semidirect
167. C2×#235C3⋊D3semidirect
240. C3×D12#67semidirect
264. C22×C3×C4D32semidirect
265. C22×C3×C4C2×C3⋊D3semidirect
271. C2×#67C2×C3⋊D3semidirect
283. C2×C4×D3C2×C3⋊D3semidirect
284. C2×Dic6C2×C3⋊D3semidirect
336. C2×C3⋊D3#235semidirect
372. C33#546semidirect
426. D12#638semidirect
463. C22×C4D3×C3⋊D3semidirect
528. C32#5550semidirect
529. C32#5829semidirect
596. C3#10595067semidirect
597. C3#10596300semidirect

Maximal subgroups

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

subgroupindex
C2×C32×#2352
C2×C3⋊D3×D122
#117239712
C3×C4×#6382
#117236432
#117236162
D3×#37522
D12×C3⋊Dic32
#117234402
#117234352
#117234142
#105963003
#105950673