Finite Group Databasebeta

← #13177332

Subgroups — #13177332

Order 1728.

2660 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×#89076C211yesmaximal, characteristic, complemented
2C2×C3⋊D3×D12C211yesmaximal, characteristic, complemented
3C3×C4×#638C211yesmaximal, characteristic, complemented
4C32×#930C211yesmaximal, characteristic, complemented
5#11723643C212yesmaximal, complemented
6#11723599C212yesmaximal, complemented
7#11723539C212yesmaximal, complemented
8#11723494C211yesmaximal, characteristic, complemented
9#11723431C211yesmaximal, characteristic, complemented
10#11723429C211yesmaximal, characteristic, complemented
11#11723420C212yesmaximal, complemented
12#10596083—31nomaximal
13#10595078—34nomaximal
14C22×D3×C3⋊D3C2212yes—
15C22×C33×C4C2211yescharacteristic, complemented
16C2×#6312C2212yescomplemented
17#89076—24no—
18C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
19C2×C4×C3⋊(C3⋊D3)C2212yescomplemented
20C2×#6308C2211yescharacteristic, complemented
21C2×C3×#638C2211yescharacteristic
22C2×C3×C4×C3⋊D3C2211yescharacteristic
23C2×C32×#67C2212yescomplemented
24C2×C32×D12C2211yescharacteristic
25C2×#6291C2212yes—
26C2×#6289C2212yescomplemented
27C3⋊D3×D12—24no—
28C3×#3757C2211yescharacteristic, complemented
29C3×#3752C2211yescharacteristic
30C3×#3750C2211yescharacteristic, complemented
31C3×C4×C3⋊Dic3C2211yescharacteristic, complemented
32C32×(C2×D3)⋊C4C2212yescomplemented
33C32×C4⋊Dic3C2211yescharacteristic, complemented
34Dic3⋊(C3⋊Dic3)C2212yescomplemented
35#88804C2212yescomplemented
36#88803C2212yes—
37(C2×D3)⋊(C3⋊Dic3)C2212yescomplemented
38(C2×D3)⋊(C3⋊Dic3)C2212yescomplemented
39C2×D4×C3⋊D3—31no—
40C2×#3828—31no—
41C2×#3828—34no—
42C2×#3828—64no—
43C2×D3×D12—34no—
44C32×#121—31no—
45#64147—32no—
46#64146—32no—
47#64145—31no—
48C4×#638D311yescharacteristic, complemented
49#64106—31no—
50#64102—31no—
51#64092—32no—
52C22⋊C4×C3⋊D3—32no—
53C3×#930D314yescomplemented
54C3×C4×#67—34no—
55#63983—38no—
56#63958—38no—
57#63921—38no—
58#63894—34no—
59#63860—34no—
60#63857—34no—
61#63853—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—24no—
73C2×C3⋊(C3⋊Dic3)C2312yes—
74C4⋊(C3⋊(C3⋊D3))D414yescomplemented
75C4×C3⋊(C3⋊D3)—24no—
76#6308—22no—
77C3×#638—22no—
78C2×C3×C3⋊Dic3C2311yescharacteristic
79C2×C3×C3⋊Dic3C2311yescharacteristic
80C2×C3×C3⋊Dic3C2311yescharacteristic
81C3×C4×C3⋊D3—22no—
82C32×#67—42no—
83C2×C32×Dic3C2312yes—
84C32×D12—22no—
85#6291—24no—
86#6289—42no—
87#5612—94no—
88#5550—91no—
89C23×C3⋊D3—32no—
90C22×D32—38no—
91C2×C32×D4—32no—
92C2×C32×D4—31no—
93C22×C32×C4—24no—
94C22×C32×C4C2×D314yescomplemented
95C22×C32×C4C2×D311yescharacteristic, complemented
96C2×#638—32no—
97C2×#638—38no—
98C2×#638—32no—
99C2×#638—68no—
100C2×#638C2×D311yescharacteristic
101D4×C3⋊D3—64no—
102#3828—64no—
103#3828—616no—
104#3828—632no—
105C2×C4⋊(C3⋊D3)—31no—
106C2×C4⋊(C3⋊D3)—34no—
107C2×C4⋊(C3⋊D3)—64no—
108C2×C4×C3⋊D3—32no—
109C2×C4×C3⋊D3—38no—
110C2×C4×C3⋊D3—32no—
111C2×C4×C3⋊D3—68no—
112C2×C4×C3⋊D3C2×D311yescharacteristic
113C2×#634—31no—
114C2×#634—34no—
115C2×#634—64no—
116C2×C3×#67—34no—
117C2×C3×#67C2×D318yescomplemented
118C2×C3×D12C2×D314yes—
119C2×C3×C4×D3—34no—
120C2×#626—38no—
121C2×#624—38no—
122D3×D12—616no—
123C32×C4⋊C4—31no—
124C32×C22⋊C4—32no—
125#3757—32no—
126#3757—32no—
127#3757C2×D311yescharacteristic, complemented
128#3752—32no—
129#3752—32no—
130#3752C2×D311yescharacteristic
131C4⋊(C3⋊Dic3)—32no—
132#3750C2×D311yescharacteristic, complemented
133C4×C3⋊Dic3C2×D311yescharacteristic, complemented
134C3×#217—34no—
135C3×(C2×D3)⋊C4—34no—
136C3×(C2×D3)⋊C4C2×D318yescomplemented
137C3×C4⋊Dic3C2×D314yescomplemented
138C3×Dic3⋊C4—34no—
139C3×C4×Dic3—34no—
140Dic3⋊Dic3—38no—
141#3722—38no—
142(C2×D3)⋊Dic3—38no—
143(C2×D3)⋊Dic3—38no—
144(C2×D3)⋊Dic3—38no—
145C22×C33—21no—
146C22×C33—21no—
147C22×C33—21no—
148C22×C33C2411yescharacteristic
149C2×C3⋊(C3⋊D3)—22no—
150C2×C3⋊(C3⋊D3)—22no—
151C2×C3⋊(C3⋊D3)C2×D414yes—
152C2×C3×C3⋊D3—21no—
153C2×C3×C3⋊D3—21no—
154C2×C3×C3⋊D3#4112yescomplemented
155C2×C32×D3—22no—
156C2×C32×D3—22no—
157C2×C32×D3—22no—
158D3×C3⋊D3—48no—
159C33×C4—21no—
160C33×C4C2×D412yes—
161C3⋊(C3⋊Dic3)—22no—
162C3×C3⋊Dic3—21no—
163C3×C3⋊Dic3—21no—
164C3×C3⋊Dic3—21no—
165C32×Dic3—22no—
166C2×D3×D4—94no—
167C2×#235—94no—
168C2×#235—91no—
169C2×#235—184no—
170C22×D12—91no—
171C3×#121—34no—
172C3×C4×D4—91no—
173#938—98no—
174#937—98no—
175#937—92no—
176#936—94no—
177#930C3⋊D311yescharacteristic, complemented
178C4×#67—34no—
179#924—91no—
180#897—94no—
181#894—92no—
182#893—94no—
183#884—92no—
184#883—98no—
185D3×C22⋊C4—98no—
186D3×C22⋊C4—92no—
187#875—91no—
188C4×D12—91no—
189C23×C32—24no—
190C23×C32—32no—
191C23×C32C22×D314yes—
192C23×C32C22×D311yescharacteristic
193C22×C3⋊D3—32no—
194C22×C3⋊D3—38no—
195C22×C3⋊D3—68no—
196C22×C3⋊D3—64no—
197C22×C3⋊D3—64no—
198C22×C3⋊D3—122no—
199C22×C3⋊D3C22×D311yescharacteristic
200C22×C3×D3—34no—
201C22×C3×D3C22×D318yes—
202C2×D32—616no—
203C2×D32—616no—
204C2×D32—128no—
205C32×D4—62no—
206C32×D4—122no—
207C2×C32×C4—216no—
208C2×C32×C4—28no—
209C2×C32×C4—22no—
210C2×C32×C4—24no—
211C2×C32×C4—24no—
212C2×C32×C4—32no—
213C2×C32×C4C22×D314yes—
214C2×C32×C4C22×D314yes—
215C2×C32×C4C22×D311yescharacteristic
216C2×C32×C4C22×D311yescharacteristic
217#638—22no—
218#638—64no—
219#638—64no—
220#638—616no—
221#638—632no—
222C2×C3⋊Dic3—32no—
223C2×C3⋊Dic3—38no—
224C2×C3⋊Dic3—68no—
225C2×C3⋊Dic3C22×D311yescharacteristic
226C2×C3⋊Dic3C22×D311yescharacteristic
227C2×C3⋊Dic3C22×D311yescharacteristic
228C4⋊(C3⋊D3)—34no—
229C4⋊(C3⋊D3)—316no—
230C4⋊(C3⋊D3)—616no—
231C4×C3⋊D3—22no—
232C4×C3⋊D3—64no—
233C4×C3⋊D3—616no—
234C4×C3⋊D3—632no—
235C4×C3⋊D3—122no—
236#634—616no—
237#634—68no—
238#634—62no—
239C3×#67—48no—
240C3×#67—68no—
241C2×C3×Dic3—34no—
242C2×C3×Dic3—34no—
243C2×C3×Dic3—34no—
244C2×C3×Dic3C22×D318yes—
245C3×D12—28no—
246C3×C4×D3—68no—
247#626—616no—
248#624—128no—
249#546—271noSylow
250C2×C33—41no—
251C2×C33C22×D411yescharacteristic
252C2×C33C2×#4111yescharacteristic
253C2×C33H(5,𝔽2)11yescharacteristic
254C3⋊(C3⋊D3)—24no—
255C3×C3⋊D3—22no—
256C32×D3—42no—
257C23×D3—98no—
258C23×D3—92no—
259C2×C3×D4—38no—
260C2×C3×D4—34no—
261C2×C3×D4—91no—
262C22×C3×C4—24no—
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
272D3×D4—1816no—
273#235—1816no—
274#235—184no—
275#235—1832no—
276C2×D12—94no—
277C2×D12—91no—
278C2×D12—92no—
279C2×D12—184no—
280C2×D12—184no—
281C2×D12C2×C3⋊D311yescharacteristic
282C2×C4×D3—34no—
283C2×C4×D3—98no—
284C2×C4×D3—92no—
285C2×C4×D3—98no—
286C2×C4×D3—92no—
287C2×C4×D3—188no—
288C2×Dic6—94no—
289C2×Dic6—91no—
290C2×Dic6—184no—
291C3×C4⋊C4—34no—
292C3×C4⋊C4—91no—
293C3×C22⋊C4—38no—
294C3×C22⋊C4—91no—
295C3×C22⋊C4—91no—
296C3×C42—91no—
297#217—34no—
298#217—98no—
299#217—98no—
300#217—92no—
301(C2×D3)⋊C4—34no—
302(C2×D3)⋊C4—98no—
303(C2×D3)⋊C4—98no—
304(C2×D3)⋊C4—92no—
305(C2×D3)⋊C4—92no—
306(C2×D3)⋊C4—92no—
307(C2×D3)⋊C4C2×C3⋊D312yescomplemented
308C4⋊Dic3—98no—
309C4⋊Dic3C2×C3⋊D311yescharacteristic, complemented
310Dic3⋊C4—34no—
311Dic3⋊C4—92no—
312C4×Dic3—34no—
313C22×C32—28no—
314C22×C32—21no—
315C22×C32—24no—
316C22×C32—21no—
317C22×C32—21no—
318C22×C32—24no—
319C22×C32—24no—
320C22×C32—24no—
321C22×C32—44no—
322C22×C32—44no—
323C22×C32—62no—
324C22×C32—62no—
325C22×C32—62no—
326C22×C32C23×D314yes—
327C22×C32C23×D311yescharacteristic
328C2×C3⋊D3—21no—
329C2×C3⋊D3—21no—
330C2×C3⋊D3—34no—
331C2×C3⋊D3—316no—
332C2×C3⋊D3—616no—
333C2×C3⋊D3—62no—
334C2×C3⋊D3—62no—
335C2×C3⋊D3—68no—
336C2×C3⋊D3—68no—
337C2×C3⋊D3—128no—
338C2×C3⋊D3—128no—
339C2×C3⋊D3—128no—
340C2×C3⋊D3#23512yescomplemented
341C2×C3×D3—28no—
342C2×C3×D3—28no—
343C2×C3×D3—28no—
344C2×C3×D3—38no—
345C2×C3×D3—64no—
346C2×C3×D3—64no—
347D32—1232no—
348C32×C4—28no—
349C32×C4—24no—
350C32×C4—21no—
351C32×C4—28no—
352C32×C4—62no—
353C32×C4D3×D418yes—
354C32×C4D3×D412yes—
355C3⋊Dic3—21no—
356C3⋊Dic3—21no—
357C3⋊Dic3—21no—
358C3⋊Dic3—616no—
359C3⋊Dic3—68no—
360C3⋊Dic3—62no—
361C3×Dic3—28no—
362C3×Dic3—64no—
363C3×Dic3—64no—
364C3×Dic3—64no—
365C2×#41—271no—
366C22×D4—271no—
367#124—271no—
368#123—272no—
369C22⋊Q8—271no—
370#121—31no—
371#121—272no—
372#120—272no—
373C4×D4—91no—
374C4×D4—271no—
375C2×C22⋊C4—272no—
376C33#54611yescharacteristic, complemented, Sylow
377C23×C3—24no—
378C23×C3—38no—
379C23×C3—91no—
380C23×C3C2×D3214yes—
381C23×C3C22×C3⋊D311yescharacteristic
382C22×D3—34no—
383C22×D3—98no—
384C22×D3—92no—
385C22×D3—188no—
386C22×D3—184no—
387C22×D3—184no—
388C22×D3—1816no—
389C22×D3—1816no—
390C22×D3—362no—
391C22×D3—368no—
392C22×D3C22×C3⋊D312yes—
393C3×D4—68no—
394C3×D4—128no—
395C3×D4—182no—
396C2×C3×C4—216no—
397C2×C3×C4—28no—
398C2×C3×C4—22no—
399C2×C3×C4—24no—
400C2×C3×C4—24no—
401C2×C3×C4—38no—
402C2×C3×C4—91no—
403C2×C3×C4—91no—
404C2×C3×C4—91no—
405C2×C3×C4—182no—
406C2×C3×C4C2×D3214yes—
407C2×C3×C4C2×D3214yes—
408C2×C3×C4C22×C3⋊D311yescharacteristic
409C2×C3×C4C22×C3⋊D311yescharacteristic
410#67—42no—
411#67—68no—
412#67—1816no—
413#67—1816no—
414#67—184no—
415#67—1832no—
416C2×Dic3—34no—
417C2×Dic3—34no—
418C2×Dic3—34no—
419C2×Dic3—98no—
420C2×Dic3—92no—
421C2×Dic3—188no—
422C2×Dic3C22×C3⋊D312yes—
423D12—22no—
424D12—916no—
425D12—94no—
426D12—1816no—
427D12—184no—
428C4×D3—68no—
429C4×D3—1816no—
430C4×D3—184no—
431C4×D3—1832no—
432C4×D3—362no—
433C4×D3—368no—
434Dic6—1816no—
435Dic6—188no—
436Dic6—182no—
437C2×C32—24no—
438C2×C32—24no—
439C2×C32—24no—
440C2×C32—48no—
441C2×C32—41no—
442C2×C32—44no—
443C2×C32—122no—
444C2×C32C2×#23511yescharacteristic
445C2×C32C2×D3×D414yes—
446C2×C32C2×D3×D411yescharacteristic
447C2×C32#100414yes—
448C2×C32D3×#4114yes—
449C2×C32#100911yescharacteristic
450C3⋊D3—22no—
451C3⋊D3—64no—
452C3⋊D3—616no—
453C3⋊D3—1216no—
454C3×D3—48no—
455C3×D3—68no—
456C24—272no—
457#41—544no—
458C2×Q8—271no—
459C2×D4—32no—
460C2×D4—31no—
461C2×D4—91no—
462C2×D4—272no—
463C2×D4—272no—
464C2×D4—271no—
465C2×D4—544no—
466C22×C4—91no—
467C22×C4—272no—
468C22×C4—272no—
469C22×C4D3×C3⋊D311yescharacteristic, complemented
470C4⋊C4—31no—
471C4⋊C4—91no—
472C4⋊C4—272no—
473C22⋊C4—32no—
474C22⋊C4—91no—
475C22⋊C4—91no—
476C22⋊C4—272no—
477C22⋊C4—272no—
478C22⋊C4—272no—
479C22⋊C4—272no—
480C42—91no—
481C22×C3—28no—
482C22×C3—24no—
483C22×C3—24no—
484C22×C3—24no—
485C22×C3—21no—
486C22×C3—21no—
487C22×C3—21no—
488C22×C3—24no—
489C22×C3—44no—
490C22×C3—44no—
491C22×C3—68no—
492C22×C3—68no—
493C22×C3—68no—
494C22×C3—92no—
495C22×C3—181no—
496C22×C3—181no—
497C22×C3C22×D3214yes—
498C22×C3C23×C3⋊D311yescharacteristic
499C2×D3—22no—
500C2×D3—22no—
501C2×D3—22no—
502C2×D3—38no—
503C2×D3—64no—
504C2×D3—64no—
505C2×D3—916no—
506C2×D3—94no—
507C2×D3—1816no—
508C2×D3—188no—
509C2×D3—188no—
510C2×D3—182no—
511C2×D3—182no—
512C2×D3—368no—
513C2×D3—368no—
514C2×D3—368no—
515C2×D3—3632no—
516C3×C4—28no—
517C3×C4—24no—
518C3×C4—21no—
519C3×C4—28no—
520C3×C4—68no—
521C3×C4—181no—
522C3×C4—181no—
523C3×C4—181no—
524C3×C4#381118yes—
525C3×C4D4×C3⋊D312yes—
526Dic3—22no—
527Dic3—64no—
528Dic3—64no—
529Dic3—64no—
530Dic3—1816no—
531Dic3—188no—
532Dic3—182no—
533C32—24no—
534C32#555011yescharacteristic, complemented
535C32#561214yescomplemented
536C23—32no—
537C23—91no—
538C23—272no—
539C23—544no—
540C23—544no—
541C23—1082no—
542C23C2×D3×C3⋊D311yescharacteristic
543Q8—542no—
544D4—62no—
545D4—122no—
546D4—182no—
547D4—274no—
548D4—544no—
549D4—544no—
550C2×C4—22no—
551C2×C4—32no—
552C2×C4—91no—
553C2×C4—91no—
554C2×C4—91no—
555C2×C4—182no—
556C2×C4—272no—
557C2×C4—544no—
558C2×C4—1082no—
559C2×C4C2×D3×C3⋊D311yescharacteristic
560C2×C4C2×D3×C3⋊D311yescharacteristic
561C2×C3—24no—
562C2×C3—24no—
563C2×C3—24no—
564C2×C3—48no—
565C2×C3—44no—
566C2×C3—41no—
567C2×C3—128no—
568C2×C3—182no—
569C2×C3D3×#23514yes—
570C2×C3#6430714yes—
571C2×C3C2×#381114yes—
572C2×C3C2×D4×C3⋊D311yescharacteristic
573C2×C3#6436111yescharacteristic
574C2×C3#41×C3⋊D311yescharacteristic
575D3—42no—
576D3—68no—
577D3—1816no—
578D3—184no—
579D3—3616no—
580C22—21no—
581C22—21no—
582C22—21no—
583C22—62no—
584C22—62no—
585C22—62no—
586C22—92no—
587C22—181no—
588C22—181no—
589C22—274no—
590C22—542no—
591C22—542no—
592C22—1088no—
593C22C22×D3×C3⋊D311yescharacteristic
594C4—21no—
595C4—62no—
596C4—181no—
597C4—181no—
598C4—181no—
599C4—542no—
600C4#8903912yes—
601C3—24no—
602C3#1059507814yescomplemented
603C3#1059608311yescharacteristic, complemented
604C2—41no—
605C2—122no—
606C2—182no—
607C2—544no—
608C2C3⋊D3×#23511yescharacteristic
609C2#1172457911yescharacteristic
610C2C2×#8903911yescharacteristic
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×#89076C2semidirect
2. C2×C3⋊D3×D12C2semidirect
3. C3×C4×#638C2semidirect
4. C32×#930C2semidirect
5. #11723643C2semidirect
6. #11723599C2semidirect
7. #11723539C2semidirect
8. #11723494C2semidirect
9. #11723431C2semidirect
10. #11723429C2semidirect
11. #11723420C2semidirect
15. C22×C33×C4C22semidirect
16. C2×#6312C22semidirect
19. C2×C4×C3⋊(C3⋊D3)C22semidirect
20. C2×#6308C22semidirect
23. C2×C32×#67C22semidirect
26. C2×#6289C22semidirect
28. C3×#3757C22semidirect
30. C3×#3750C22semidirect
31. C3×C4×C3⋊Dic3C22semidirect
32. C32×(C2×D3)⋊C4C22semidirect
33. C32×C4⋊Dic3C22semidirect
34. Dic3⋊(C3⋊Dic3)C22semidirect
35. #88804C22semidirect
37. (C2×D3)⋊(C3⋊Dic3)C22semidirect
38. (C2×D3)⋊(C3⋊Dic3)C22semidirect
48. C4×#638D3semidirect
53. C3×#930D3semidirect
74. C4⋊(C3⋊(C3⋊D3))D4semidirect
94. C22×C32×C4C2×D3semidirect
95. C22×C32×C4C2×D3semidirect
117. C2×C3×#67C2×D3semidirect
127. #3757C2×D3semidirect
132. #3750C2×D3semidirect
133. C4×C3⋊Dic3C2×D3semidirect
136. C3×(C2×D3)⋊C4C2×D3semidirect
137. C3×C4⋊Dic3C2×D3semidirect
154. C2×C3×C3⋊D3#41semidirect
177. #930C3⋊D3semidirect
264. C22×C3×C4D32semidirect
265. C22×C3×C4C2×C3⋊D3semidirect
271. C2×#67C2×C3⋊D3semidirect
307. (C2×D3)⋊C4C2×C3⋊D3semidirect
309. C4⋊Dic3C2×C3⋊D3semidirect
340. C2×C3⋊D3#235semidirect
376. C33#546semidirect
469. C22×C4D3×C3⋊D3semidirect
534. C32#5550semidirect
535. C32#5612semidirect
602. C3#10595078semidirect
603. C3#10596083semidirect

Maximal subgroups

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

subgroupindex
C2×#890762
C2×C3⋊D3×D122
C3×C4×#6382
C32×#9302
#117236432
#117235992
#117235392
#117234942
#117234312
#117234292
#117234202
#105960833
#105950783