Finite Group Databasebeta

← #13177599

Subgroups — #13177599

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×#89078C211yesmaximal, characteristic, complemented
2C2×C32×D3×D4C211yesmaximal, characteristic, complemented
3C3×D4×C3⋊Dic3C211yesmaximal, characteristic, complemented
4#11723635C212yesmaximal, complemented
5#11723630C212yesmaximal, complemented
6#11723603C212yesmaximal, complemented
7C2×(C2×D3)⋊(C3⋊Dic3)C212yesmaximal, complemented
8#11723531C211yesmaximal, characteristic, complemented
9C4×#6291C211yesmaximal, characteristic, complemented
10#11723429C211yesmaximal, characteristic, complemented
11#11723426C211yesmaximal, characteristic, complemented
12#10596277—31nomaximal
13#10595246—34nomaximal
14C23×C32×D3C2212yes—
15C2×C33×D4C2211yescharacteristic
16C2×#6312C2212yescomplemented
17C22×C3⋊(C3⋊Dic3)C2212yescomplemented
18#89078—24no—
19C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic, complemented
20C2×#6308C2211yescharacteristic, complemented
21C22×C3×C3⋊Dic3C2212yescomplemented
22C2×C32×#67C2212yes—
23C32×D3×D4—24no—
24C2×C32×D12C2211yescharacteristic
25C2×C32×C4×D3C2211yescharacteristic
26C2×#6291C2211yescharacteristic
27C2×#6291C2212yescomplemented
28C3×#3757C2212yescomplemented
29C3×C4⋊(C3⋊Dic3)C2211yescharacteristic, complemented
30C3×C4×C3⋊Dic3C2211yescharacteristic, complemented
31#88807C2211yescharacteristic, complemented
32#88807C2212yescomplemented
33#88804C2211yescharacteristic, complemented
34(C2×D3)⋊(C3⋊Dic3)C2211yescharacteristic
35(C2×D3)⋊(C3⋊Dic3)C2212yes—
36(C2×D3)⋊(C3⋊Dic3)C2212yescomplemented
37(C2×D3)⋊(C3⋊Dic3)C2212yescomplemented
38(C2×D3)⋊(C3⋊Dic3)C2212yescomplemented
39C22×C32×D4—31no—
40C2×#3830—31no—
41C2×#3830—34no—
42C2×#3830—64no—
43C2×C3×D3×D4D314yescomplemented
44#64162—32no—
45C2×#3757—32no—
46#64148—32no—
47#64147—31no—
48#64145—31no—
49#64144—32no—
50D4×C3⋊Dic3D311yescharacteristic, complemented
51C4×#638—31no—
52#64133—31no—
53C3×D4×Dic3—34no—
54#63978—38no—
55#63977—38no—
56#63962—38no—
57C2×(C2×D3)⋊Dic3—38no—
58#63916—34no—
59C4×#626—34no—
60#63860—34no—
61#63857—34no—
62C23×C33C2312yes—
63C22×C3⋊(C3⋊D3)C2311yescharacteristic
64C22×C32×D3—24no—
65C22×C32×D3—24no—
66C22×C32×D3—42no—
67C22×C32×D3C2312yes—
68C22×C32×D3C2311yescharacteristic
69C33×D4D414yescomplemented
70C2×C33×C4C2311yescharacteristic
71#6312—24no—
72C2×C3⋊(C3⋊Dic3)—24no—
73C2×C3⋊(C3⋊Dic3)C2312yes—
74C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
75C4×C3⋊(C3⋊D3)—22no—
76#6308—22no—
77C2×C3×C3⋊Dic3—42no—
78C2×C3×C3⋊Dic3C2311yescharacteristic
79C2×C3×C3⋊Dic3C2311yescharacteristic
80C2×C3×C3⋊Dic3C2312yes—
81C32×#67—24no—
82C2×C32×Dic3C2311yescharacteristic
83C32×D12—22no—
84C32×C4×D3—22no—
85#6291—22no—
86#6291—42no—
87#5806—94no—
88#5554—91no—
89C24×C32—32no—
90C23×C3×D3C2×D318yes—
91C2×C32×D4—24no—
92C2×C32×D4—32no—
93C2×C32×D4—31no—
94C2×C32×D4—64no—
95C2×C32×D4C2×D311yescharacteristic
96C2×C32×D4C2×D314yes—
97C22×C32×C4—31no—
98C2×#638—32no—
99C2×#638—38no—
100C2×#638—31no—
101C2×#638—32no—
102C2×#638—68no—
103C22×C3⋊Dic3—32no—
104C22×C3⋊Dic3—38no—
105C22×C3⋊Dic3—68no—
106C22×C3⋊Dic3C2×D312yescomplemented
107#3830—632no—
108#3830—64no—
109#3830—616no—
110C2×C4×C3⋊D3—31no—
111C2×C4×C3⋊D3—34no—
112C2×C4×C3⋊D3—64no—
113C2×#634—31no—
114C2×#634—34no—
115C2×#634—64no—
116C2×C3×#67C2×D318yes—
117C22×C3×Dic3—38no—
118C3×D3×D4—216no—
119C2×C3×D12C2×D314yes—
120C2×C3×C4×D3C2×D314yes—
121C2×#626—34no—
122C2×#626—38no—
123#3757—32no—
124#3757—31no—
125#3757—32no—
126#3757—32no—
127#3757—32no—
128#3757C2×D312yescomplemented
129#3752—31no—
130C4⋊(C3⋊Dic3)C2×D311yescharacteristic, complemented
131#3750—32no—
132#3750—31no—
133C4×C3⋊Dic3C2×D311yescharacteristic, complemented
134C3×#217—38no—
135C3×C4⋊Dic3—34no—
136C3×C4×Dic3—34no—
137Dic3⋊Dic3—38no—
138Dic3⋊Dic3—34no—
139#3722—34no—
140(C2×D3)⋊Dic3—38no—
141(C2×D3)⋊Dic3—38no—
142(C2×D3)⋊Dic3—38no—
143(C2×D3)⋊Dic3—34no—
144(C2×D3)⋊Dic3—38no—
145C22×C33—22no—
146C22×C33—22no—
147C22×C33C2×D414yes—
148C22×C33C2411yescharacteristic
149C2×C3⋊(C3⋊D3)—21no—
150C2×C3⋊(C3⋊D3)—21no—
151C2×C3⋊(C3⋊D3)—21no—
152C2×C32×D3—21no—
153C2×C32×D3—21no—
154C2×C32×D3—22no—
155C2×C32×D3—22no—
156C2×C32×D3—22no—
157C2×C32×D3—48no—
158C2×C32×D3#4112yescomplemented
159C33×C4C2×D412yes—
160C3⋊(C3⋊Dic3)—22no—
161C3⋊(C3⋊Dic3)—21no—
162C3×C3⋊Dic3—21no—
163C3×C3⋊Dic3—22no—
164C3×C3⋊Dic3—21no—
165C32×Dic3—21no—
166C22×C3×D4—34no—
167C2×#237—94no—
168C2×#237—91no—
169C2×#237—184no—
170C2×D3×D4C3⋊D311yescharacteristic, complemented
171C3×C4×D4—91no—
172#953—98no—
173C2×#217—98no—
174#939—98no—
175#938—94no—
176#936—94no—
177#935—98no—
178D4×Dic3—34no—
179#930—92no—
180C4×#67—94no—
181C2×(C2×D3)⋊C4—92no—
182#924—94no—
183#897—91no—
184#895—91no—
185#886—92no—
186#882—92no—
187#875—91no—
188C4×D12—91no—
189C23×C32—28no—
190C23×C32—31no—
191C23×C32—32no—
192C23×C32—64no—
193C23×C32—64no—
194C23×C32—122no—
195C23×C32C22×D318yes—
196C23×C32C22×D312yes—
197C22×C3⋊D3—31no—
198C22×C3⋊D3—34no—
199C22×C3⋊D3—64no—
200C22×C3×D3—216no—
201C22×C3×D3—216no—
202C22×C3×D3—48no—
203C22×C3×D3C22×D318yes—
204C22×C3×D3C22×D314yes—
205C32×D4—216no—
206C32×D4—64no—
207C32×D4—62no—
208C32×D4D1214yescomplemented
209C32×D4#67116yescomplemented
210C2×C32×C4—24no—
211C2×C32×C4—31no—
212C2×C32×C4—62no—
213C2×C32×C4C22×D314yes—
214C2×C32×C4C22×D311yescharacteristic
215#638—62no—
216#638—632no—
217#638—64no—
218#638—616no—
219#638—122no—
220C2×C3⋊Dic3—32no—
221C2×C3⋊Dic3—31no—
222C2×C3⋊Dic3—38no—
223C2×C3⋊Dic3—34no—
224C2×C3⋊Dic3—42no—
225C2×C3⋊Dic3—68no—
226C2×C3⋊Dic3—64no—
227C2×C3⋊Dic3—632no—
228C2×C3⋊Dic3—64no—
229C2×C3⋊Dic3—616no—
230C2×C3⋊Dic3C22×D311yescharacteristic
231C2×C3⋊Dic3C22×D311yescharacteristic
232C2×C3⋊Dic3C22×D312yes—
233C4×C3⋊D3—616no—
234C4×C3⋊D3—62no—
235C4×C3⋊D3—68no—
236#634—616no—
237#634—62no—
238#634—68no—
239C3×#67—216no—
240C2×C3×Dic3—34no—
241C2×C3×Dic3—34no—
242C2×C3×Dic3—38no—
243C2×C3×Dic3—128no—
244C2×C3×Dic3C22×D314yes—
245C3×D12—28no—
246C3×C4×D3—28no—
247#626—68no—
248#626—128no—
249#546—271noSylow
250C2×C33—24no—
251C2×C33C22×D411yescharacteristic
252C2×C33C2×#4111yescharacteristic
253C2×C33H(5,𝔽2)11yescharacteristic
254C3⋊(C3⋊D3)—41no—
255C32×D3—22no—
256C32×D3—42no—
257C24×C3—38no—
258C23×D3C2×C3⋊D312yes—
259C2×C3×D4—24no—
260C2×C3×D4—38no—
261C2×C3×D4—34no—
262C2×C3×D4—616no—
263C2×C3×D4D3214yes—
264C2×C3×D4C2×C3⋊D311yescharacteristic
265C22×C3×C4—34no—
266C22×C3×C4—92no—
267C2×#67—94no—
268C2×#67—98no—
269C2×#67—98no—
270C2×#67—92no—
271C2×#67—188no—
272C2×#67C2×C3⋊D312yes—
273C22×Dic3—38no—
274C22×Dic3—98no—
275C22×Dic3—92no—
276C22×Dic3—188no—
277#237—1832no—
278#237—1816no—
279#237—184no—
280D3×D4—24no—
281C2×D12—91no—
282C2×D12—92no—
283C2×D12C2×C3⋊D311yescharacteristic
284C2×C4×D3—94no—
285C2×C4×D3—91no—
286C2×C4×D3—184no—
287C2×C4×D3C2×C3⋊D311yescharacteristic
288C2×Dic6—94no—
289C2×Dic6—91no—
290C2×Dic6—184no—
291C3×C4⋊C4—91no—
292C3×C22⋊C4—92no—
293C3×C42—91no—
294#217—38no—
295#217—98no—
296#217—94no—
297#217—98no—
298#217—98no—
299#217—98no—
300(C2×D3)⋊C4—94no—
301(C2×D3)⋊C4—91no—
302(C2×D3)⋊C4—92no—
303(C2×D3)⋊C4—92no—
304(C2×D3)⋊C4—92no—
305(C2×D3)⋊C4—92no—
306(C2×D3)⋊C4—91no—
307C4⋊Dic3—34no—
308C4⋊Dic3—91no—
309C4⋊Dic3—92no—
310Dic3⋊C4—98no—
311Dic3⋊C4—94no—
312C4×Dic3—34no—
313C22×C32—22no—
314C22×C32—22no—
315C22×C32—24no—
316C22×C32—216no—
317C22×C32—28no—
318C22×C32—28no—
319C22×C32—32no—
320C22×C32—48no—
321C22×C32—48no—
322C22×C32—61no—
323C22×C32—61no—
324C22×C32—62no—
325C22×C32—62no—
326C22×C32—62no—
327C22×C32—128no—
328C22×C32C2×D1214yes—
329C22×C32C2×#67116yes—
330C22×C32C23×D314yes—
331C22×C32C23×D311yescharacteristic
332C2×C3⋊D3—68no—
333C2×C3⋊D3—61no—
334C2×C3⋊D3—61no—
335C2×C3⋊D3—61no—
336C2×C3⋊D3—64no—
337C2×C3⋊D3—64no—
338C2×C3⋊D3—64no—
339C2×C3⋊D3—124no—
340C2×C3⋊D3—124no—
341C2×C3×D3—24no—
342C2×C3×D3—24no—
343C2×C3×D3—28no—
344C2×C3×D3—28no—
345C2×C3×D3—28no—
346C2×C3×D3—432no—
347C2×C3×D3#23718yescomplemented
348C32×C4—28no—
349C32×C4—61no—
350C32×C4C2×D1212yes—
351C32×C4C2×#6718yes—
352C3⋊Dic3—21no—
353C3⋊Dic3—22no—
354C3⋊Dic3—21no—
355C3⋊Dic3—616no—
356C3⋊Dic3—68no—
357C3⋊Dic3—62no—
358C3⋊Dic3—61no—
359C3⋊Dic3—68no—
360C3⋊Dic3—64no—
361C3×Dic3—24no—
362C3×Dic3—64no—
363C3×Dic3—68no—
364C3×Dic3—64no—
365C2×#41—271no—
366C22×D4—31no—
367#124—271no—
368#123—272no—
369C22⋊Q8—271no—
370#121—271no—
371#121—272no—
372#120—272no—
373C4×D4—91no—
374C4×D4—271no—
375C2×C22⋊C4—272no—
376C33#54611yescharacteristic, complemented, Sylow
377C23×C3—28no—
378C23×C3—34no—
379C23×C3—38no—
380C23×C3—616no—
381C23×C3—616no—
382C23×C3—128no—
383C23×C3C2×D3218yes—
384C23×C3C22×C3⋊D312yes—
385C22×D3—24no—
386C22×D3—24no—
387C22×D3—42no—
388C22×D3—94no—
389C22×D3—91no—
390C22×D3—184no—
391C22×D3C22×C3⋊D312yes—
392C22×D3C22×C3⋊D311yescharacteristic
393C3×D4—216no—
394C3×D4—616no—
395C3×D4—68no—
396C3×D4#626116yescomplemented
397C3×D4#63814yescomplemented
398C2×C3×C4—24no—
399C2×C3×C4—34no—
400C2×C3×C4—68no—
401C2×C3×C4—91no—
402C2×C3×C4—92no—
403C2×C3×C4—91no—
404C2×C3×C4—362no—
405C2×C3×C4C2×D3214yes—
406C2×C3×C4C22×C3⋊D311yescharacteristic
407#67—24no—
408#67—188no—
409#67—1832no—
410#67—1816no—
411#67—184no—
412#67—368no—
413C2×Dic3—34no—
414C2×Dic3—34no—
415C2×Dic3—38no—
416C2×Dic3—98no—
417C2×Dic3—94no—
418C2×Dic3—92no—
419C2×Dic3—91no—
420C2×Dic3—128no—
421C2×Dic3—188no—
422C2×Dic3—184no—
423C2×Dic3—1832no—
424C2×Dic3—1816no—
425C2×Dic3—184no—
426C2×Dic3C22×C3⋊D311yescharacteristic
427D12—22no—
428D12—182no—
429D12—362no—
430C4×D3—22no—
431C4×D3—1816no—
432C4×D3—188no—
433C4×D3—182no—
434Dic6—1816no—
435Dic6—188no—
436Dic6—182no—
437C2×C32—24no—
438C2×C32—24no—
439C2×C32—24no—
440C2×C32—24no—
441C2×C32—216no—
442C2×C32—416no—
443C2×C32—62no—
444C2×C32—122no—
445C2×C32C22×D1211yescharacteristic
446C2×C32C2×#23714yes—
447C2×C32#100414yes—
448C2×C32#100411yescharacteristic
449C2×C32D3×#4111yescharacteristic
450C2×C32C22×#6714yes—
451C3⋊D3—128no—
452C3⋊D3—124no—
453C3⋊D3—121no—
454C3×D3—28no—
455C3×D3—48no—
456C24—32no—
457#41—544no—
458C2×Q8—271no—
459C2×D4—32no—
460C2×D4—31no—
461C2×D4—64no—
462C2×D4—271no—
463C2×D4—272no—
464C2×D4—272no—
465C2×D4D3×C3⋊D311yescharacteristic
466C22×C4—31no—
467C22×C4—92no—
468C22×C4—272no—
469C22×C4—271no—
470C4⋊C4—91no—
471C4⋊C4—271no—
472C4⋊C4—272no—
473C22⋊C4—92no—
474C22⋊C4—271no—
475C22⋊C4—271no—
476C22⋊C4—272no—
477C22⋊C4—272no—
478C22⋊C4—272no—
479C22⋊C4—272no—
480C42—91no—
481C22×C3—28no—
482C22×C3—28no—
483C22×C3—22no—
484C22×C3—22no—
485C22×C3—24no—
486C22×C3—216no—
487C22×C3—38no—
488C22×C3—48no—
489C22×C3—48no—
490C22×C3—64no—
491C22×C3—64no—
492C22×C3—68no—
493C22×C3—68no—
494C22×C3—68no—
495C22×C3—1232no—
496C22×C3C2×#626116yes—
497C22×C3C2×#63814yes—
498C22×C3C22×D3214yes—
499C22×C3C23×C3⋊D311yescharacteristic
500C2×D3—21no—
501C2×D3—21no—
502C2×D3—22no—
503C2×D3—22no—
504C2×D3—22no—
505C2×D3—48no—
506C2×D3—188no—
507C2×D3—184no—
508C2×D3—184no—
509C2×D3—184no—
510C2×D3—181no—
511C2×D3—181no—
512C2×D3—181no—
513C2×D3—364no—
514C2×D3—364no—
515C2×D3#383012yescomplemented
516C3×C4—28no—
517C3×C4—64no—
518C3×C4—181no—
519C3×C4—182no—
520C3×C4—181no—
521C3×C4C2×#62618yes—
522C3×C4C2×#63812yes—
523Dic3—21no—
524Dic3—64no—
525Dic3—64no—
526Dic3—68no—
527Dic3—1816no—
528Dic3—188no—
529Dic3—188no—
530Dic3—184no—
531Dic3—182no—
532Dic3—181no—
533C32—24no—
534C32#555411yescharacteristic, complemented
535C32#580614yescomplemented
536C23—31no—
537C23—32no—
538C23—64no—
539C23—64no—
540C23—122no—
541C23—271no—
542C23C2×D3×C3⋊D312yes—
543Q8—542no—
544D4—64no—
545D4—62no—
546D4—542no—
547D4—544no—
548D4—1082no—
549D4#629114yescomplemented
550C2×C4—31no—
551C2×C4—62no—
552C2×C4—91no—
553C2×C4—91no—
554C2×C4—92no—
555C2×C4—271no—
556C2×C4—272no—
557C2×C4—362no—
558C2×C4—544no—
559C2×C4—542no—
560C2×C4C2×D3×C3⋊D311yescharacteristic
561C2×C3—216no—
562C2×C3—24no—
563C2×C3—24no—
564C2×C3—24no—
565C2×C3—24no—
566C2×C3—416no—
567C2×C3—68no—
568C2×C3—128no—
569C2×C3D3×#23714yes—
570C2×C3#6431314yes—
571C2×C3C22×#62614yes—
572C2×C3C2×#383011yescharacteristic
573C2×C3#6436111yescharacteristic
574C2×C3C22×#63811yescharacteristic
575D3—22no—
576D3—42no—
577D3—368no—
578D3—364no—
579D3—361no—
580C22—22no—
581C22—22no—
582C22—32no—
583C22—61no—
584C22—61no—
585C22—62no—
586C22—62no—
587C22—62no—
588C22—128no—
589C22—541no—
590C22—541no—
591C22—541no—
592C22C2×#629114yes—
593C22C22×D3×C3⋊D311yescharacteristic
594C4—61no—
595C4—181no—
596C4—181no—
597C4—182no—
598C4—541no—
599C4—542no—
600C4C2×#629112yes—
601C3—24no—
602C3#1059524614yescomplemented
603C3#1059627711yescharacteristic, complemented
604C2—24no—
605C2—62no—
606C2—122no—
607C2—1081no—
608C2D3×#383011yescharacteristic
609C2#1172458711yescharacteristic
610C2C22×#629111yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC2×C33×D4C22
hypercenterC2×D4D3×C3⋊D3
hypocenterC33#546
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×#89078C2semidirect
2. C2×C32×D3×D4C2semidirect
3. C3×D4×C3⋊Dic3C2semidirect
4. #11723635C2semidirect
5. #11723630C2semidirect
6. #11723603C2semidirect
7. C2×(C2×D3)⋊(C3⋊Dic3)C2semidirect
8. #11723531C2semidirect
9. C4×#6291C2semidirect
10. #11723429C2semidirect
11. #11723426C2semidirect
16. C2×#6312C22semidirect
17. C22×C3⋊(C3⋊Dic3)C22semidirect
19. C2×C4×C3⋊(C3⋊D3)C22semidirect
20. C2×#6308C22semidirect
21. C22×C3×C3⋊Dic3C22semidirect
27. C2×#6291C22semidirect
28. C3×#3757C22semidirect
29. C3×C4⋊(C3⋊Dic3)C22semidirect
30. C3×C4×C3⋊Dic3C22semidirect
31. #88807C22semidirect
32. #88807C22semidirect
33. #88804C22semidirect
36. (C2×D3)⋊(C3⋊Dic3)C22semidirect
37. (C2×D3)⋊(C3⋊Dic3)C22semidirect
38. (C2×D3)⋊(C3⋊Dic3)C22semidirect
43. C2×C3×D3×D4D3semidirect
50. D4×C3⋊Dic3D3semidirect
69. C33×D4D4semidirect
106. C22×C3⋊Dic3C2×D3semidirect
128. #3757C2×D3semidirect
130. C4⋊(C3⋊Dic3)C2×D3semidirect
133. C4×C3⋊Dic3C2×D3semidirect
158. C2×C32×D3#41semidirect
170. C2×D3×D4C3⋊D3semidirect
208. C32×D4D12semidirect
209. C32×D4#67semidirect
347. C2×C3×D3#237semidirect
376. C33#546semidirect
396. C3×D4#626semidirect
397. C3×D4#638semidirect
515. C2×D3#3830semidirect
534. C32#5554semidirect
535. C32#5806semidirect
549. D4#6291semidirect
602. C3#10595246semidirect
603. C3#10596277semidirect

Maximal subgroups

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

subgroupindex
C2×#890782
C2×C32×D3×D42
C3×D4×C3⋊Dic32
#117236352
#117236302
#117236032
C2×(C2×D3)⋊(C3⋊Dic3)2
#117235312
C4×#62912
#117234292
#117234262
#105962773
#105952463