Finite Group Databasebeta

← #13177541

Subgroups — #13177541

Order 1728.

2380 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×D4×C3⋊D3C211yesmaximal, characteristic, complemented
2C2×#89020C211yesmaximal, characteristic, complemented
3#11723979C211yesmaximal, characteristic, complemented
4C32×#938C211yesmaximal, characteristic, complemented
5#11723636C212yesmaximal, complemented
6#11723634C212yesmaximal, complemented
7#11723633C212yesmaximal, complemented
8#11723591C212yesmaximal, complemented
9#11723532C211yesmaximal, characteristic, complemented
10#11723515C211yesmaximal, characteristic, complemented
11#11723404C211yesmaximal, characteristic, complemented
12#10596078—31nomaximal
13#10595231—34nomaximal
14C23×C3×C3⋊D3C2212yes—
15C2×C33×D4C2211yescharacteristic
16C2×#6312C2212yescomplemented
17C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic, complemented
18C2×C3×#638C2212yes—
19C3×D4×C3⋊D3—24no—
20C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
21C2×C3×C4×C3⋊D3C2211yescharacteristic
22C2×C32×#67C2212yescomplemented
23C2×C32×C4×D3C2211yescharacteristic, complemented
24C2×#6293C2211yescharacteristic, complemented
25C2×#6292C2211yescharacteristic
26C2×#6292C2212yescomplemented
27C2×#6291C2211yescharacteristic, complemented
28C2×#6290C2211yescharacteristic
29C2×#6290C2212yescomplemented
30#89020—24no—
31#88864C2212yescomplemented
32C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic, complemented
33C32×#217C2212yescomplemented
34C32×C4⋊Dic3C2211yescharacteristic, complemented
35#88803C2212yes—
36#88803C2212yescomplemented
37#88803C2212yescomplemented
38Dic3×C3⋊Dic3C2212yescomplemented
39C2×D4×C3⋊D3D311yescharacteristic, complemented
40C2×#3828—31no—
41C2×C3×D3×D4—34no—
42C2×#3798—34no—
43C32×#121—31no—
44#64149—32no—
45#64147—31no—
46#64147—34no—
47#64147—64no—
48#64146—32no—
49#64139—31no—
50#64104—31no—
51#64101—31no—
52#64094—32no—
53#64091—32no—
54C3×#938D314yescomplemented
55#63979—38no—
56#63977—38no—
57#63976—38no—
58#63959—38no—
59#63916—34no—
60#63908—34no—
61#63841—34no—
62C23×C33C2312yes—
63C22×C3⋊(C3⋊D3)C2311yescharacteristic
64C22×C3×C3⋊D3—24no—
65C22×C3×C3⋊D3—42no—
66C22×C3×C3⋊D3—42no—
67C22×C3×C3⋊D3C2312yes—
68C22×C3×C3⋊D3C2311yescharacteristic
69C22×C32×D3C2311yescharacteristic
70C33×D4—22no—
71C2×C33×C4C2311yescharacteristic
72#6312—42no—
73C2×C3⋊(C3⋊Dic3)C2312yes—
74C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
75C4×C3⋊(C3⋊D3)—22no—
76C3×#638—24no—
77C2×C3×C3⋊Dic3C2311yescharacteristic
78C3×C4⋊(C3⋊D3)—22no—
79C3×C4×C3⋊D3D414yescomplemented
80C32×#67—42no—
81C2×C32×Dic3C2311yescharacteristic
82C2×C32×Dic3C2312yes—
83C32×C4×D3—22no—
84#6293—22no—
85#6292—22no—
86#6292—42no—
87#6291—22no—
88#6290—22no—
89#6290—42no—
90#5805—91no—
91#5607—94no—
92C23×C3⋊D3C2×D312yes—
93C23×C3×D3—38no—
94C2×C32×D4—24no—
95C2×C32×D4—32no—
96C2×C32×D4C2×D314yes—
97C2×C32×D4C2×D311yescharacteristic
98C22×C32×C4—31no—
99C2×#638—31no—
100C2×#638—32no—
101C2×#638—38no—
102C2×#638—32no—
103C2×#638—31no—
104C2×#638—68no—
105C2×#638C2×D312yes—
106D4×C3⋊D3—24no—
107#3828—64no—
108C2×C4⋊(C3⋊D3)—31no—
109C2×C4⋊(C3⋊D3)—32no—
110C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
111C2×C4×C3⋊D3—31no—
112C2×C4×C3⋊D3—34no—
113C2×C4×C3⋊D3—64no—
114C2×C4×C3⋊D3C2×D311yescharacteristic
115C2×#634—31no—
116C2×C3×#67—38no—
117C2×C3×#67C2×D318yescomplemented
118C3×D3×D4—616no—
119C2×C3×D12—34no—
120C2×C3×C4×D3—34no—
121C2×C3×C4×D3C2×D314yescomplemented
122C2×#627—34no—
123C2×#626—34no—
124C2×#626—34no—
125C2×#626—38no—
126C2×#625—34no—
127C2×#625—38no—
128#3798—616no—
129C32×C4⋊C4—31no—
130C32×C22⋊C4—32no—
131#3757—32no—
132#3757—38no—
133#3757—68no—
134#3752—32no—
135#3752—32no—
136#3752—32no—
137C4⋊(C3⋊Dic3)—31no—
138C4⋊(C3⋊Dic3)—34no—
139C4⋊(C3⋊Dic3)—64no—
140C4×C3⋊Dic3—32no—
141C3×#217C2×D318yescomplemented
142C3×C4⋊Dic3C2×D314yescomplemented
143(C2×D3)⋊Dic3—38no—
144(C2×D3)⋊Dic3—38no—
145(C2×D3)⋊Dic3—38no—
146Dic32—38no—
147C22×C33—22no—
148C22×C33—22no—
149C22×C33—22no—
150C22×C33C2411yescharacteristic
151C2×C3⋊(C3⋊D3)—21no—
152C2×C3⋊(C3⋊D3)—21no—
153C2×C3⋊(C3⋊D3)—21no—
154C2×C3×C3⋊D3—21no—
155C2×C3×C3⋊D3—21no—
156C2×C3×C3⋊D3—22no—
157C2×C3×C3⋊D3—22no—
158C2×C3×C3⋊D3—22no—
159C2×C3×C3⋊D3—48no—
160C2×C3×C3⋊D3C2×D412yes—
161C2×C32×D3—21no—
162C2×C32×D3—21no—
163C2×C32×D3—21no—
164C33×C4C2×D412yes—
165C3⋊(C3⋊Dic3)—21no—
166C3⋊(C3⋊Dic3)—22no—
167C3×C3⋊Dic3C2×D412yes—
168C32×Dic3—21no—
169C32×Dic3—22no—
170C22×C3×D4—91no—
171C2×D3×D4—34no—
172C2×#235—94no—
173C2×#235—91no—
174C3×#121—34no—
175#953—92no—
176#953—92no—
177#940—98no—
178#940—92no—
179#938—94no—
180#938—91no—
181#938—91no—
182#938—91no—
183#938—184no—
184#938C3⋊D311yescharacteristic, complemented
185#937—98no—
186#936—92no—
187#930—94no—
188#926—91no—
189#895—94no—
190#892—94no—
191#885—98no—
192#882—98no—
193C23×C32—28no—
194C23×C32—31no—
195C23×C32C22×D318yes—
196C23×C32C22×D312yes—
197C22×C3⋊D3—24no—
198C22×C3⋊D3—31no—
199C22×C3⋊D3—34no—
200C22×C3⋊D3—42no—
201C22×C3⋊D3—42no—
202C22×C3⋊D3—64no—
203C22×C3⋊D3C22×D312yes—
204C22×C3⋊D3C22×D311yescharacteristic
205C22×C3×D3—34no—
206C22×C3×D3—38no—
207C22×C3×D3—616no—
208C22×C3×D3—128no—
209C22×C3×D3—128no—
210C22×C3×D3C22×D314yes—
211C32×D4—22no—
212C32×D4—28no—
213C32×D4—48no—
214C32×D4—122no—
215C2×C32×C4—24no—
216C2×C32×C4—32no—
217C2×C32×C4—31no—
218C2×C32×C4—62no—
219C2×C32×C4C22×D314yes—
220C2×C32×C4C22×D311yescharacteristic
221#638—24no—
222#638—62no—
223#638—62no—
224#638—122no—
225#638—122no—
226#638—128no—
227#638—1216no—
228C2×C3⋊Dic3—32no—
229C2×C3⋊Dic3—31no—
230C2×C3⋊Dic3—38no—
231C2×C3⋊Dic3—34no—
232C2×C3⋊Dic3—68no—
233C2×C3⋊Dic3—64no—
234C2×C3⋊Dic3C22×D311yescharacteristic
235C4⋊(C3⋊D3)—22no—
236C4⋊(C3⋊D3)—62no—
237C4⋊(C3⋊D3)—122no—
238C4×C3⋊D3—62no—
239C4×C3⋊D3—68no—
240C4×C3⋊D3—128no—
241C4×C3⋊D3#6714yescomplemented
242#634—62no—
243C3×#67—48no—
244C3×#67—616no—
245C2×C3×Dic3—34no—
246C2×C3×Dic3C22×D318yes—
247C2×C3×Dic3C22×D314yes—
248C3×D12—68no—
249C3×C4×D3—28no—
250C3×C4×D3—316no—
251#627—68no—
252#626—68no—
253#626—68no—
254#626—128no—
255#625—68no—
256#625—128no—
257#535—271noSylow
258C2×C33—42no—
259C2×C33C22×D411yescharacteristic
260C2×C33H(5,𝔽2)11yescharacteristic
261C2×C33H(5,𝔽2)11yescharacteristic
262C3⋊(C3⋊D3)—41no—
263C3×C3⋊D3—22no—
264C3×C3⋊D3—42no—
265C32×D3—41no—
266C24×C3—92no—
267C23×D3—38no—
268C2×C3×D4—24no—
269C2×C3×D4—38no—
270C2×C3×D4—92no—
271C2×C3×D4—91no—
272C2×C3×D4—184no—
273C2×C3×D4D3214yes—
274C2×C3×D4C2×C3⋊D311yescharacteristic
275C22×C3×C4—34no—
276C22×C3×C4—91no—
277C2×#67—38no—
278C2×#67—94no—
279C2×#67—94no—
280C2×#67—91no—
281C2×#67—98no—
282C2×#67—98no—
283C2×#67—92no—
284C2×#67—92no—
285C2×#67—91no—
286C2×#67—92no—
287C2×#67—188no—
288C2×#67C2×C3⋊D312yescomplemented
289D3×D4—616no—
290#235—184no—
291#235—1816no—
292C2×D12—34no—
293C2×D12—94no—
294C2×D12—91no—
295C2×D12—98no—
296C2×C4×D3—34no—
297C2×C4×D3—94no—
298C2×C4×D3—91no—
299C2×C4×D3—184no—
300C2×C4×D3C2×C3⋊D311yescharacteristic, complemented
301C2×Dic6—94no—
302C2×Dic6—91no—
303C3×C4⋊C4—34no—
304C3×C22⋊C4—38no—
305#217—98no—
306#217—92no—
307#217—92no—
308#217—92no—
309#217—92no—
310#217—188no—
311#217C2×C3⋊D312yescomplemented
312(C2×D3)⋊C4—98no—
313(C2×D3)⋊C4—98no—
314(C2×D3)⋊C4—98no—
315C4⋊Dic3—94no—
316C4⋊Dic3—91no—
317C4⋊Dic3—184no—
318C4⋊Dic3C2×C3⋊D311yescharacteristic, complemented
319C4×Dic3—98no—
320C4×Dic3—92no—
321C22×C32—22no—
322C22×C32—22no—
323C22×C32—22no—
324C22×C32—24no—
325C22×C32—28no—
326C22×C32—28no—
327C22×C32—28no—
328C22×C32—48no—
329C22×C32—48no—
330C22×C32—48no—
331C22×C32—61no—
332C22×C32—61no—
333C22×C32—61no—
334C22×C32C23×D314yes—
335C22×C32C23×D311yescharacteristic
336C2×C3⋊D3—21no—
337C2×C3⋊D3—21no—
338C2×C3⋊D3—22no—
339C2×C3⋊D3—22no—
340C2×C3⋊D3—22no—
341C2×C3⋊D3—48no—
342C2×C3⋊D3—68no—
343C2×C3⋊D3—61no—
344C2×C3⋊D3—61no—
345C2×C3⋊D3—64no—
346C2×C3⋊D3—61no—
347C2×C3⋊D3—64no—
348C2×C3⋊D3—64no—
349C2×C3⋊D3—124no—
350C2×C3⋊D3—124no—
351C2×C3⋊D3C2×#6712yes—
352C2×C3×D3—24no—
353C2×C3×D3—24no—
354C2×C3×D3—24no—
355C2×C3×D3—38no—
356C2×C3×D3—64no—
357C2×C3×D3—64no—
358C2×C3×D3—68no—
359C2×C3×D3—68no—
360C2×C3×D3—68no—
361C2×C3×D3—1232no—
362C32×C4—28no—
363C32×C4—62no—
364C32×C4—61no—
365C32×C4D3×D418yes—
366C32×C4C2×#6712yes—
367C3⋊Dic3—62no—
368C3⋊Dic3—61no—
369C3⋊Dic3—64no—
370C3⋊Dic3—68no—
371C3⋊Dic3—124no—
372C3⋊Dic3—128no—
373C3⋊Dic3C2×#6712yes—
374C3×Dic3—24no—
375C3×Dic3—28no—
376C3×Dic3—38no—
377C2×#41—271no—
378C22×D4—91no—
379#127—272no—
380#124—272no—
381#121—31no—
382#121—271no—
383#121—271no—
384#121—271no—
385#120—272no—
386#120—272no—
387#117—271no—
388C33#53511yescharacteristic, complemented, Sylow
389C23×C3—28no—
390C23×C3—34no—
391C23×C3—91no—
392C23×C3—92no—
393C23×C3—184no—
394C23×C3—362no—
395C23×C3—362no—
396C23×C3C2×D3218yes—
397C23×C3C22×C3⋊D312yes—
398C22×D3—34no—
399C22×D3—38no—
400C22×D3—616no—
401C22×D3—94no—
402C22×D3—91no—
403C22×D3—128no—
404C22×D3—128no—
405C22×D3—184no—
406C22×D3C22×C3⋊D311yescharacteristic
407C3×D4—28no—
408C3×D4—22no—
409C3×D4—48no—
410C3×D4—128no—
411C3×D4—184no—
412C3×D4—182no—
413C2×C3×C4—24no—
414C2×C3×C4—38no—
415C2×C3×C4—34no—
416C2×C3×C4—68no—
417C2×C3×C4—91no—
418C2×C3×C4—94no—
419C2×C3×C4C2×D3214yes—
420C2×C3×C4C22×C3⋊D311yescharacteristic
421#67—42no—
422#67—616no—
423#67—182no—
424#67—188no—
425#67—182no—
426#67—188no—
427#67—362no—
428#67—362no—
429#67—368no—
430#67—368no—
431#67—362no—
432#67—3616no—
433C2×Dic3—34no—
434C2×Dic3—98no—
435C2×Dic3—94no—
436C2×Dic3—92no—
437C2×Dic3—91no—
438C2×Dic3—188no—
439C2×Dic3—184no—
440C2×Dic3C22×C3⋊D312yes—
441C2×Dic3C22×C3⋊D311yescharacteristic
442D12—68no—
443D12—188no—
444D12—182no—
445D12—368no—
446C4×D3—22no—
447C4×D3—316no—
448C4×D3—188no—
449C4×D3—182no—
450C4×D3—368no—
451Dic6—182no—
452Dic6—188no—
453C2×C32—24no—
454C2×C32—24no—
455C2×C32—24no—
456C2×C32—42no—
457C2×C32—48no—
458C2×C32—88no—
459C2×C32—121no—
460C2×C32C2×D3×D414yes—
461C2×C32#100414yes—
462C2×C32#100411yescharacteristic
463C2×C32#100411yescharacteristic
464C2×C32#100914yes—
465C2×C32C22×#6711yescharacteristic
466C3⋊D3—22no—
467C3⋊D3—42no—
468C3⋊D3—128no—
469C3⋊D3—121no—
470C3⋊D3—124no—
471C3×D3—44no—
472C3×D3—68no—
473C3×D3—128no—
474C24—92no—
475#41—544no—
476C2×Q8—271no—
477C2×D4—32no—
478C2×D4—92no—
479C2×D4—91no—
480C2×D4—184no—
481C2×D4—271no—
482C2×D4—272no—
483C2×D4—271no—
484C2×D4—271no—
485C2×D4—272no—
486C2×D4—272no—
487C2×D4D3×C3⋊D311yescharacteristic
488C22×C4—31no—
489C22×C4—91no—
490C22×C4—271no—
491C4⋊C4—31no—
492C4⋊C4—271no—
493C22⋊C4—32no—
494C22⋊C4—272no—
495C22⋊C4—272no—
496C22⋊C4—272no—
497C22⋊C4—272no—
498C42—272no—
499C22×C3—28no—
500C22×C3—28no—
501C22×C3—28no—
502C22×C3—22no—
503C22×C3—22no—
504C22×C3—22no—
505C22×C3—24no—
506C22×C3—48no—
507C22×C3—48no—
508C22×C3—48no—
509C22×C3—64no—
510C22×C3—64no—
511C22×C3—64no—
512C22×C3—92no—
513C22×C3—181no—
514C22×C3—181no—
515C22×C3—182no—
516C22×C3—182no—
517C22×C3—182no—
518C22×C3—368no—
519C22×C3C22×D3214yes—
520C22×C3C23×C3⋊D311yescharacteristic
521C2×D3—21no—
522C2×D3—21no—
523C2×D3—21no—
524C2×D3—38no—
525C2×D3—64no—
526C2×D3—64no—
527C2×D3—68no—
528C2×D3—68no—
529C2×D3—68no—
530C2×D3—1232no—
531C2×D3—188no—
532C2×D3—184no—
533C2×D3—184no—
534C2×D3—184no—
535C2×D3—181no—
536C2×D3—181no—
537C2×D3—181no—
538C2×D3—364no—
539C2×D3—364no—
540C3×C4—28no—
541C3×C4—68no—
542C3×C4—64no—
543C3×C4—92no—
544C3×C4D3×#6718yes—
545C3×C4D4×C3⋊D312yes—
546Dic3—21no—
547Dic3—22no—
548Dic3—38no—
549Dic3—188no—
550Dic3—184no—
551Dic3—181no—
552Dic3—182no—
553Dic3—364no—
554Dic3—368no—
555C32—24no—
556C32#560714yescomplemented
557C32#580511yescharacteristic, complemented
558C23—31no—
559C23—91no—
560C23—92no—
561C23—184no—
562C23—271no—
563C23—362no—
564C23—362no—
565C23C2×D3×C3⋊D312yes—
566Q8—542no—
567D4—22no—
568D4—122no—
569D4—184no—
570D4—182no—
571D4—542no—
572D4—542no—
573D4—542no—
574D4—1082no—
575D4—1082no—
576D4—1082no—
577C2×C4—32no—
578C2×C4—31no—
579C2×C4—62no—
580C2×C4—91no—
581C2×C4—94no—
582C2×C4—272no—
583C2×C4—271no—
584C2×C4—542no—
585C2×C4C2×D3×C3⋊D311yescharacteristic
586C2×C3—24no—
587C2×C3—24no—
588C2×C3—24no—
589C2×C3—48no—
590C2×C3—42no—
591C2×C3—88no—
592C2×C3—124no—
593C2×C3—182no—
594C2×C3—362no—
595C2×C3#6431314yes—
596C2×C3#6431414yes—
597C2×C3C2×D3×#6714yes—
598C2×C3C2×D4×C3⋊D311yescharacteristic
599C2×C3#6436111yescharacteristic
600C2×C3#6436611yescharacteristic
601D3—41no—
602D3—68no—
603D3—128no—
604D3—368no—
605D3—364no—
606D3—361no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—61no—
611C22—61no—
612C22—61no—
613C22—92no—
614C22—181no—
615C22—181no—
616C22—182no—
617C22—182no—
618C22—182no—
619C22—368no—
620C22—541no—
621C22—541no—
622C22—541no—
623C22C22×D3×C3⋊D311yescharacteristic
624C4—62no—
625C4—61no—
626C4—92no—
627C4—542no—
628C4—541no—
629C4C3⋊D3×#6712yes—
630C3—24no—
631C3#1059523114yescomplemented
632C3#1059607811yescharacteristic, complemented
633C2—42no—
634C2—121no—
635C2—182no—
636C2—362no—
637C2—1081no—
638C2#1172458711yescharacteristic
639C2#1172458911yescharacteristic
640C2C2×C3⋊D3×#6711yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC2×C33×D4C22
hypercenterC2×D4D3×C3⋊D3
hypocenterC33#535
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×C3×D4×C3⋊D3C2semidirect
2. C2×#89020C2semidirect
3. #11723979C2semidirect
4. C32×#938C2semidirect
5. #11723636C2semidirect
6. #11723634C2semidirect
7. #11723633C2semidirect
8. #11723591C2semidirect
9. #11723532C2semidirect
10. #11723515C2semidirect
11. #11723404C2semidirect
16. C2×#6312C22semidirect
17. C2×C4×C3⋊(C3⋊D3)C22semidirect
22. C2×C32×#67C22semidirect
23. C2×C32×C4×D3C22semidirect
24. C2×#6293C22semidirect
26. C2×#6292C22semidirect
27. C2×#6291C22semidirect
29. C2×#6290C22semidirect
31. #88864C22semidirect
32. C4⋊(C3⋊(C3⋊Dic3))C22semidirect
33. C32×#217C22semidirect
34. C32×C4⋊Dic3C22semidirect
36. #88803C22semidirect
37. #88803C22semidirect
38. Dic3×C3⋊Dic3C22semidirect
39. C2×D4×C3⋊D3D3semidirect
54. C3×#938D3semidirect
79. C3×C4×C3⋊D3D4semidirect
117. C2×C3×#67C2×D3semidirect
121. C2×C3×C4×D3C2×D3semidirect
141. C3×#217C2×D3semidirect
142. C3×C4⋊Dic3C2×D3semidirect
184. #938C3⋊D3semidirect
241. C4×C3⋊D3#67semidirect
288. C2×#67C2×C3⋊D3semidirect
300. C2×C4×D3C2×C3⋊D3semidirect
311. #217C2×C3⋊D3semidirect
318. C4⋊Dic3C2×C3⋊D3semidirect
388. C33#535semidirect
556. C32#5607semidirect
557. C32#5805semidirect
631. C3#10595231semidirect
632. C3#10596078semidirect

Maximal subgroups

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

subgroupindex
C2×C3×D4×C3⋊D32
C2×#890202
#117239792
C32×#9382
#117236362
#117236342
#117236332
#117235912
#117235322
#117235152
#117234042
#105960783
#105952313