Finite Group Databasebeta

← #13177334

Subgroups — #13177334

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×D3×C4⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
3C3×#64139C211yesmaximal, characteristic, complemented
4C32×C4×#67C211yesmaximal, characteristic, complemented
5#11723643C212yesmaximal, complemented
6#11723623C212yesmaximal, complemented
7#11723539C212yesmaximal, complemented
8#11723484C211yesmaximal, characteristic, complemented
9#11723439C211yesmaximal, characteristic, complemented
10#11723426C211yesmaximal, characteristic, complemented
11#11723421C212yesmaximal, complemented
12#10596021—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×#638C2212yescomplemented
22C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
23C2×C32×#67C2211yescharacteristic
24C2×C32×C4×D3C2211yescharacteristic
25C2×#6292C2212yes—
26C2×#6289C2212yescomplemented
27D3×C4⋊(C3⋊D3)—24no—
28C3×#3752C2212yescomplemented
29C3×C4⋊(C3⋊Dic3)C2211yescharacteristic, complemented
30C32×#217C2211yescharacteristic, complemented
31C32×(C2×D3)⋊C4C2211yescharacteristic
32C32×Dic3⋊C4C2211yescharacteristic, complemented
33C32×C4×Dic3C2211yescharacteristic, complemented
34#88807C2212yescomplemented
35#88804C2212yescomplemented
36#88803C2212yescomplemented
37#88803C2212yescomplemented
38(C2×D3)⋊(C3⋊Dic3)C2212yes—
39C2×#3828—31no—
40C2×#3828—34no—
41C2×#3828—64no—
42C22×C4⋊(C3⋊D3)—31no—
43C2×D3×D12—34no—
44C32×C4×D4—31no—
45#64146—32no—
46#64139D311yescharacteristic, complemented
47#64133—31no—
48#64103—32no—
49#64093—32no—
50C22⋊C4×C3⋊D3—32no—
51#64084—31no—
52C4×C4⋊(C3⋊D3)—31no—
53C3×#930—34no—
54C3×C4×#67D314yescomplemented
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⋊D3C2312yes—
65C22×C32×D3C2311yescharacteristic
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—42no—
78C2×C3×C3⋊Dic3C2312yes—
79C3×C4⋊(C3⋊D3)—22no—
80C32×#67—22no—
81C2×C32×Dic3C2311yescharacteristic
82C2×C32×Dic3C2311yescharacteristic
83C2×C32×Dic3C2311yescharacteristic
84C32×C4×D3—22no—
85#6292—24no—
86#6289—42no—
87#5612—91no—
88#5550—94no—
89C23×C3⋊D3—32no—
90C22×D32—38no—
91C2×C32×D4—31no—
92C22×C32×C4—24no—
93C22×C32×C4—31no—
94C22×C32×C4C2×D314yescomplemented
95C22×C32×C4C2×D311yescharacteristic, complemented
96C2×#638—32no—
97C2×#638—38no—
98C2×#638—68no—
99C2×#638C2×D312yescomplemented
100#3828—64no—
101#3828—616no—
102#3828—632no—
103C2×C4⋊(C3⋊D3)—31no—
104C2×C4⋊(C3⋊D3)—34no—
105C2×C4⋊(C3⋊D3)—32no—
106C2×C4⋊(C3⋊D3)—64no—
107C2×C4⋊(C3⋊D3)—64no—
108C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
109C2×C4×C3⋊D3—32no—
110C2×C4×C3⋊D3—38no—
111C2×C4×C3⋊D3—32no—
112C2×C4×C3⋊D3—68no—
113C2×#634—31no—
114C2×#634—34no—
115C2×#634—64no—
116C2×C3×#67—38no—
117C2×C3×#67C2×D314yes—
118C2×C3×D12—34no—
119C2×C3×C4×D3C2×D314yes—
120C2×#626—38no—
121C2×#624—38no—
122D3×D12—616no—
123C32×C4⋊C4—31no—
124C32×C22⋊C4—31no—
125C32×C22⋊C4—31no—
126C32×C42—31no—
127#3757—32no—
128#3752—32no—
129#3752—32no—
130#3752—32no—
131#3752C2×D312yescomplemented
132C4⋊(C3⋊Dic3)C2×D311yescharacteristic, complemented
133#3750—32no—
134C3×#217C2×D314yescomplemented
135C3×(C2×D3)⋊C4—38no—
136C3×(C2×D3)⋊C4C2×D314yes—
137C3×C4⋊Dic3—34no—
138C3×Dic3⋊C4C2×D314yescomplemented
139C3×C4×Dic3C2×D314yescomplemented
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—22no—
153C2×C3×C3⋊D3—22no—
154C2×C3×C3⋊D3—22no—
155C2×C32×D3—21no—
156C2×C32×D3—21no—
157C2×C32×D3#4112yescomplemented
158D3×C3⋊D3—48no—
159C33×C4—21no—
160C33×C4C2×D412yes—
161C3⋊(C3⋊Dic3)—22no—
162C3×C3⋊Dic3—22no—
163C32×Dic3—21no—
164C32×Dic3—21no—
165C32×Dic3—21no—
166C2×D3×D4—91no—
167C2×#235—94no—
168C2×#235—91no—
169C2×#235—184no—
170C22×D12—94no—
171C3×#121—91no—
172C3×C4×D4—34no—
173#938—92no—
174#937—98no—
175#937—92no—
176#936—91no—
177#930—34no—
178C4×#67C3⋊D311yescharacteristic, complemented
179#924—94no—
180#897—91no—
181#894—98no—
182#893—91no—
183#884—98no—
184#883—92no—
185D3×C22⋊C4—98no—
186D3×C22⋊C4—92no—
187#875—94no—
188C4×D12—94no—
189C23×C32—24no—
190C23×C32—31no—
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×D312yes—
200C22×C3×D3—38no—
201C22×C3×D3C22×D314yes—
202C2×D32—616no—
203C2×D32—616no—
204C2×D32—128no—
205C32×D4—62no—
206C2×C32×C4—216no—
207C2×C32×C4—28no—
208C2×C32×C4—22no—
209C2×C32×C4—24no—
210C2×C32×C4—24no—
211C2×C32×C4—31no—
212C2×C32×C4—31no—
213C2×C32×C4—31no—
214C2×C32×C4—62no—
215C2×C32×C4C22×D314yes—
216C2×C32×C4C22×D314yes—
217C2×C32×C4C22×D311yescharacteristic
218C2×C32×C4C22×D311yescharacteristic
219#638—42no—
220#638—64no—
221#638—616no—
222#638—632no—
223C2×C3⋊Dic3—32no—
224C2×C3⋊Dic3—38no—
225C2×C3⋊Dic3—68no—
226C2×C3⋊Dic3C22×D312yes—
227C4⋊(C3⋊D3)—22no—
228C4⋊(C3⋊D3)—34no—
229C4⋊(C3⋊D3)—316no—
230C4⋊(C3⋊D3)—616no—
231C4⋊(C3⋊D3)—64no—
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—28no—
240C3×#67—128no—
241C2×C3×Dic3—38no—
242C2×C3×Dic3C22×D314yes—
243C2×C3×Dic3C22×D314yes—
244C2×C3×Dic3C22×D314yes—
245C3×D12—68no—
246C3×C4×D3—28no—
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—42no—
256C32×D3—22no—
257C23×D3—98no—
258C23×D3—92no—
259C2×C3×D4—34no—
260C2×C3×D4—92no—
261C2×C3×D4—91no—
262C22×C3×C4—24no—
263C22×C3×C4—34no—
264C22×C3×C4D3214yescomplemented
265C22×C3×C4C2×C3⋊D311yescharacteristic, complemented
266C2×#67—38no—
267C2×#67—98no—
268C2×#67—92no—
269C2×#67—92no—
270C2×#67—188no—
271C2×#67C2×C3⋊D311yescharacteristic
272D3×D4—184no—
273#235—1816no—
274#235—184no—
275#235—1832no—
276C2×D12—34no—
277C2×D12—94no—
278C2×D12—91no—
279C2×D12—98no—
280C2×D12—184no—
281C2×D12—1816no—
282C2×C4×D3—98no—
283C2×C4×D3—92no—
284C2×C4×D3—98no—
285C2×C4×D3—92no—
286C2×C4×D3—188no—
287C2×C4×D3C2×C3⋊D311yescharacteristic
288C2×Dic6—94no—
289C2×Dic6—91no—
290C2×Dic6—184no—
291C3×C4⋊C4—34no—
292C3×C4⋊C4—91no—
293C3×C22⋊C4—34no—
294C3×C22⋊C4—34no—
295C3×C22⋊C4—92no—
296C3×C42—34no—
297#217—98no—
298#217—92no—
299#217—92no—
300#217C2×C3⋊D311yescharacteristic, complemented
301(C2×D3)⋊C4—38no—
302(C2×D3)⋊C4—98no—
303(C2×D3)⋊C4—98no—
304(C2×D3)⋊C4—98no—
305(C2×D3)⋊C4—92no—
306(C2×D3)⋊C4—92no—
307(C2×D3)⋊C4C2×C3⋊D311yescharacteristic
308C4⋊Dic3—34no—
309C4⋊Dic3—92no—
310Dic3⋊C4—98no—
311Dic3⋊C4C2×C3⋊D311yescharacteristic, complemented
312C4×Dic3C2×C3⋊D311yescharacteristic, complemented
313C22×C32—28no—
314C22×C32—24no—
315C22×C32—21no—
316C22×C32—21no—
317C22×C32—21no—
318C22×C32—24no—
319C22×C32—24no—
320C22×C32—24no—
321C22×C32—32no—
322C22×C32—44no—
323C22×C32—44no—
324C22×C32—61no—
325C22×C32—61no—
326C22×C32C23×D314yes—
327C22×C32C23×D311yescharacteristic
328C2×C3⋊D3—22no—
329C2×C3⋊D3—22no—
330C2×C3⋊D3—22no—
331C2×C3⋊D3—34no—
332C2×C3⋊D3—316no—
333C2×C3⋊D3—616no—
334C2×C3⋊D3—62no—
335C2×C3⋊D3—62no—
336C2×C3⋊D3—68no—
337C2×C3⋊D3—68no—
338C2×C3⋊D3—128no—
339C2×C3⋊D3—128no—
340C2×C3⋊D3—128no—
341C2×C3×D3—24no—
342C2×C3×D3—24no—
343C2×C3×D3—68no—
344C2×C3×D3—68no—
345C2×C3×D3—68no—
346C2×C3×D3#23518yescomplemented
347D32—1232no—
348C32×C4—28no—
349C32×C4—24no—
350C32×C4—21no—
351C32×C4—28no—
352C32×C4—61no—
353C32×C4—61no—
354C32×C4—61no—
355C32×C4D3×D418yes—
356C32×C4D3×D412yes—
357C3⋊Dic3—22no—
358C3⋊Dic3—616no—
359C3⋊Dic3—68no—
360C3⋊Dic3—62no—
361C3×Dic3—24no—
362C3×Dic3—24no—
363C3×Dic3—24no—
364C3×Dic3—68no—
365C2×#41—271no—
366C22×D4—271no—
367#124—271no—
368#123—272no—
369C22⋊Q8—271no—
370#121—91no—
371#121—272no—
372#120—272no—
373C4×D4—31no—
374C4×D4—271no—
375C2×C22⋊C4—272no—
376C33#54611yescharacteristic, complemented, Sylow
377C23×C3—24no—
378C23×C3—34no—
379C23×C3—92no—
380C23×C3C2×D3214yes—
381C23×C3C22×C3⋊D311yescharacteristic
382C22×D3—38no—
383C22×D3—98no—
384C22×D3—92no—
385C22×D3—188no—
386C22×D3—184no—
387C22×D3—1816no—
388C22×D3—1816no—
389C22×D3—184no—
390C22×D3—362no—
391C22×D3—368no—
392C22×D3C22×C3⋊D311yescharacteristic
393C3×D4—68no—
394C3×D4—182no—
395C3×D4—362no—
396C2×C3×C4—216no—
397C2×C3×C4—28no—
398C2×C3×C4—22no—
399C2×C3×C4—24no—
400C2×C3×C4—24no—
401C2×C3×C4—34no—
402C2×C3×C4—34no—
403C2×C3×C4—34no—
404C2×C3×C4—68no—
405C2×C3×C4—92no—
406C2×C3×C4C2×D3214yes—
407C2×C3×C4C2×D3214yes—
408C2×C3×C4C22×C3⋊D311yescharacteristic
409C2×C3×C4C22×C3⋊D311yescharacteristic
410#67—22no—
411#67—128no—
412#67—184no—
413#67—1816no—
414#67—184no—
415#67—1832no—
416C2×Dic3—38no—
417C2×Dic3—98no—
418C2×Dic3—92no—
419C2×Dic3—188no—
420C2×Dic3C22×C3⋊D311yescharacteristic
421C2×Dic3C22×C3⋊D311yescharacteristic
422C2×Dic3C22×C3⋊D311yescharacteristic
423D12—68no—
424D12—916no—
425D12—94no—
426D12—1816no—
427D12—1816no—
428C4×D3—22no—
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—44no—
442C2×C32—41no—
443C2×C32—62no—
444C2×C32C2×#23514yes—
445C2×C32C2×D3×D414yes—
446C2×C32C2×D3×D411yescharacteristic
447C2×C32#100411yescharacteristic
448C2×C32D3×#4111yescharacteristic
449C2×C32#100914yes—
450C3⋊D3—42no—
451C3⋊D3—64no—
452C3⋊D3—616no—
453C3⋊D3—1216no—
454C3×D3—28no—
455C3×D3—128no—
456C24—272no—
457#41—544no—
458C2×Q8—271no—
459C2×D4—31no—
460C2×D4—92no—
461C2×D4—91no—
462C2×D4—272no—
463C2×D4—272no—
464C2×D4—271no—
465C2×D4—544no—
466C22×C4—31no—
467C22×C4—272no—
468C22×C4—272no—
469C22×C4D3×C3⋊D311yescharacteristic, complemented
470C4⋊C4—31no—
471C4⋊C4—91no—
472C4⋊C4—272no—
473C22⋊C4—31no—
474C22⋊C4—31no—
475C22⋊C4—92no—
476C22⋊C4—272no—
477C22⋊C4—272no—
478C22⋊C4—272no—
479C22⋊C4—272no—
480C42—31no—
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—38no—
490C22×C3—44no—
491C22×C3—44no—
492C22×C3—64no—
493C22×C3—64no—
494C22×C3—182no—
495C22×C3—182no—
496C22×C3—182no—
497C22×C3C22×D3214yes—
498C22×C3C23×C3⋊D311yescharacteristic
499C2×D3—21no—
500C2×D3—21no—
501C2×D3—68no—
502C2×D3—68no—
503C2×D3—68no—
504C2×D3—916no—
505C2×D3—94no—
506C2×D3—1816no—
507C2×D3—188no—
508C2×D3—188no—
509C2×D3—182no—
510C2×D3—182no—
511C2×D3—368no—
512C2×D3—368no—
513C2×D3—368no—
514C2×D3—3632no—
515C2×D3#382812yescomplemented
516C3×C4—28no—
517C3×C4—24no—
518C3×C4—21no—
519C3×C4—28no—
520C3×C4—64no—
521C3×C4—64no—
522C3×C4—64no—
523C3×C4—182no—
524C3×C4#381118yes—
525C3×C4D4×C3⋊D312yes—
526Dic3—21no—
527Dic3—21no—
528Dic3—21no—
529Dic3—68no—
530Dic3—1816no—
531Dic3—188no—
532Dic3—182no—
533C32—24no—
534C32#555014yescomplemented
535C32#561211yescharacteristic, complemented
536C23—31no—
537C23—92no—
538C23—272no—
539C23—544no—
540C23—544no—
541C23—1082no—
542C23C2×D3×C3⋊D311yescharacteristic
543Q8—542no—
544D4—62no—
545D4—182no—
546D4—274no—
547D4—362no—
548D4—544no—
549D4—544no—
550C2×C4—22no—
551C2×C4—31no—
552C2×C4—31no—
553C2×C4—31no—
554C2×C4—62no—
555C2×C4—92no—
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—68no—
568C2×C3—362no—
569C2×C3D3×#23514yes—
570C2×C3#6430714yes—
571C2×C3C2×#381114yes—
572C2×C3C2×#382811yescharacteristic
573C2×C3C2×D4×C3⋊D311yescharacteristic
574C2×C3#6436611yescharacteristic
575D3—22no—
576D3—128no—
577D3—1816no—
578D3—184no—
579D3—3616no—
580C22—21no—
581C22—21no—
582C22—21no—
583C22—32no—
584C22—61no—
585C22—61no—
586C22—182no—
587C22—182no—
588C22—182no—
589C22—274no—
590C22—542no—
591C22—542no—
592C22—1088no—
593C22C22×D3×C3⋊D311yescharacteristic
594C4—21no—
595C4—61no—
596C4—61no—
597C4—61no—
598C4—182no—
599C4—542no—
600C4#8903912yes—
601C3—24no—
602C3#1059507814yescomplemented
603C3#1059602111yescharacteristic, complemented
604C2—41no—
605C2—62no—
606C2—362no—
607C2—544no—
608C2D3×#382811yescharacteristic
609C2#1172458011yescharacteristic
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×D3×C4⋊(C3⋊D3)C2semidirect
3. C3×#64139C2semidirect
4. C32×C4×#67C2semidirect
5. #11723643C2semidirect
6. #11723623C2semidirect
7. #11723539C2semidirect
8. #11723484C2semidirect
9. #11723439C2semidirect
10. #11723426C2semidirect
11. #11723421C2semidirect
15. C22×C33×C4C22semidirect
16. C2×#6312C22semidirect
19. C2×C4×C3⋊(C3⋊D3)C22semidirect
20. C2×#6308C22semidirect
21. C2×C3×#638C22semidirect
26. C2×#6289C22semidirect
28. C3×#3752C22semidirect
29. C3×C4⋊(C3⋊Dic3)C22semidirect
30. C32×#217C22semidirect
32. C32×Dic3⋊C4C22semidirect
33. C32×C4×Dic3C22semidirect
34. #88807C22semidirect
35. #88804C22semidirect
36. #88803C22semidirect
37. #88803C22semidirect
46. #64139D3semidirect
54. C3×C4×#67D3semidirect
74. C4⋊(C3⋊(C3⋊D3))D4semidirect
94. C22×C32×C4C2×D3semidirect
95. C22×C32×C4C2×D3semidirect
99. C2×#638C2×D3semidirect
131. #3752C2×D3semidirect
132. C4⋊(C3⋊Dic3)C2×D3semidirect
134. C3×#217C2×D3semidirect
138. C3×Dic3⋊C4C2×D3semidirect
139. C3×C4×Dic3C2×D3semidirect
157. C2×C32×D3#41semidirect
178. C4×#67C3⋊D3semidirect
264. C22×C3×C4D32semidirect
265. C22×C3×C4C2×C3⋊D3semidirect
300. #217C2×C3⋊D3semidirect
311. Dic3⋊C4C2×C3⋊D3semidirect
312. C4×Dic3C2×C3⋊D3semidirect
346. C2×C3×D3#235semidirect
376. C33#546semidirect
469. C22×C4D3×C3⋊D3semidirect
515. C2×D3#3828semidirect
534. C32#5550semidirect
535. C32#5612semidirect
602. C3#10595078semidirect
603. C3#10596021semidirect

Maximal subgroups

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

subgroupindex
C2×#890762
C2×D3×C4⋊(C3⋊D3)2
C3×#641392
C32×C4×#672
#117236432
#117236232
#117235392
#117234842
#117234392
#117234262
#117234212
#105960213
#105950783