Finite Group Databasebeta

← #13176556

Subgroups — #13176556

Order 1728.

2308 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89026C211yesmaximal, characteristic, complemented
2C2×#89020C211yesmaximal, characteristic, complemented
3#11723933C211yesmaximal, characteristic, complemented
4C3×#64104C211yesmaximal, characteristic, complemented
5C32×#895C211yesmaximal, characteristic, complemented
6#11723541C212yesmaximal, complemented
7#11723540C212yesmaximal, complemented
8#11723530C212yesmaximal, complemented
9#11723529C212yesmaximal, complemented
10#11723522C211yesmaximal, characteristic, complemented
11#11723515C211yesmaximal, characteristic, complemented
12#10596078—31nomaximal
13#10594634—34nomaximal
14C22×D3×C3⋊D3C2212yes—
15C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
16C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
17C2×C3×C4⋊(C3⋊D3)C2212yescomplemented
18C2×C3×C4×C3⋊D3C2211yescharacteristic, complemented
19C2×C32×D12C2211yescharacteristic
20C2×C32×D12C2212yescomplemented
21C2×C32×C4×D3C2211yescharacteristic, complemented
22C2×#6293C2211yescharacteristic, complemented
23C2×#6292C2211yescharacteristic
24C2×#6291C2211yescharacteristic
25C2×#6290C2212yes—
26C2×#6290C2211yescharacteristic, complemented
27C2×#6290C2212yescomplemented
28C2×#6290C2212yescomplemented
29#89026—24no—
30#89020—24no—
31C33×C4⋊C4C2211yescharacteristic, complemented
32#88859C2212yes—
33C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic, complemented
34C4×C3⋊(C3⋊Dic3)C2212yescomplemented
35C3×#3752C2212yescomplemented
36C32×(C2×D3)⋊C4C2212yescomplemented
37#88803C2212yescomplemented
38(C2×D3)⋊(C3⋊Dic3)C2212yescomplemented
39C2×D4×C3⋊D3—31no—
40C2×#3828—31no—
41C2×#3802—34no—
42C2×#3798—34no—
43C32×#121—31no—
44#64149—32no—
45#64147—31no—
46#64146—32no—
47#64139—31no—
48#64104D311yescharacteristic, complemented
49#64101—31no—
50#64101—34no—
51#64101—64no—
52#64094—32no—
53#64091—32no—
54C3×#895—34no—
55C3×#895D314yescomplemented
56#63922—38no—
57#63922—38no—
58#63915—38no—
59#63914—38no—
60#63908—34no—
61#63908—34no—
62C22×C3⋊(C3⋊D3)C2311yescharacteristic
63C22×C3×C3⋊D3C2312yes—
64C22×C3×C3⋊D3C2311yescharacteristic
65C22×C32×D3C2312yes—
66C22×C32×D3C2311yescharacteristic
67C2×D3×C3⋊D3—24no—
68C2×D3×C3⋊D3—42no—
69C2×D3×C3⋊D3—42no—
70C2×C33×C4C2312yes—
71C2×C33×C4C2311yescharacteristic
72C2×C3⋊(C3⋊Dic3)C2312yes—
73C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
74C4×C3⋊(C3⋊D3)D414yescomplemented
75C2×C3×C3⋊Dic3C2311yescharacteristic
76C3×C4⋊(C3⋊D3)—22no—
77C3×C4⋊(C3⋊D3)—42no—
78C3×C4×C3⋊D3—22no—
79C2×C32×Dic3C2311yescharacteristic
80C32×D12—22no—
81C32×D12—42no—
82C32×C4×D3—22no—
83#6293—22no—
84#6292—22no—
85#6291—22no—
86#6290—22no—
87#6290—24no—
88#6290—42no—
89#6290—42no—
90#5607—94no—
91#5607—91no—
92C23×C3⋊D3—32no—
93C22×D32—38no—
94C2×C32×D4—31no—
95C2×C32×D4—32no—
96C22×C32×C4—31no—
97C2×#638—32no—
98C2×#638—31no—
99C2×#638—32no—
100C2×#638—32no—
101C2×#638—31no—
102D4×C3⋊D3—64no—
103#3828—64no—
104C2×C4⋊(C3⋊D3)—31no—
105C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
106C2×C4⋊(C3⋊D3)C2×D312yescomplemented
107C2×C4×C3⋊D3—31no—
108C2×C4×C3⋊D3—34no—
109C2×C4×C3⋊D3—64no—
110C2×C4×C3⋊D3C2×D311yescharacteristic, complemented
111C2×#634—31no—
112C2×C3×D12—34no—
113C2×C3×D12—38no—
114C2×C3×D12C2×D314yes—
115C2×C3×D12C2×D318yescomplemented
116C2×C3×C4×D3—34no—
117C2×C3×C4×D3C2×D314yescomplemented
118C2×#627—34no—
119C2×#626—34no—
120C2×#626—34no—
121C2×#625—34no—
122C2×#625—38no—
123C2×#625—38no—
124C2×#625—38no—
125#3802—616no—
126#3798—616no—
127C32×C4⋊C4—24no—
128C32×C4⋊C4C2×D314yescomplemented
129C32×C4⋊C4C2×D311yescharacteristic, complemented
130C32×C22⋊C4—32no—
131#3757—32no—
132#3752—32no—
133#3752—32no—
134#3752—38no—
135#3752—68no—
136#3752C2×D312yescomplemented
137C4⋊(C3⋊Dic3)—31no—
138C4⋊(C3⋊Dic3)—34no—
139C4⋊(C3⋊Dic3)—64no—
140C4×C3⋊Dic3—32no—
141C4×C3⋊Dic3—38no—
142C4×C3⋊Dic3—68no—
143C3×(C2×D3)⋊C4—38no—
144C3×(C2×D3)⋊C4C2×D318yescomplemented
145(C2×D3)⋊Dic3—38no—
146(C2×D3)⋊Dic3—38no—
147C22×C33C2411yescharacteristic
148C2×C3⋊(C3⋊D3)—21no—
149C2×C3⋊(C3⋊D3)—21no—
150C2×C3⋊(C3⋊D3)C2×D412yes—
151C2×C3×C3⋊D3—21no—
152C2×C3×C3⋊D3—22no—
153C2×C3×C3⋊D3—21no—
154C2×C3×C3⋊D3—22no—
155C2×C3×C3⋊D3—21no—
156C2×C3×C3⋊D3—22no—
157C2×C32×D3—21no—
158C2×C32×D3—22no—
159C2×C32×D3—21no—
160C2×C32×D3—22no—
161C2×C32×D3—21no—
162C2×C32×D3—22no—
163D3×C3⋊D3—48no—
164C33×C4—22no—
165C33×C4C2×D412yes—
166C3⋊(C3⋊Dic3)—22no—
167C3⋊(C3⋊Dic3)C2×D412yes—
168C3×C3⋊Dic3—21no—
169C32×Dic3—21no—
170C2×D3×D4—94no—
171C2×D3×D4—91no—
172C2×#235—94no—
173C2×#235—91no—
174C3×#121—34no—
175C3×#121—91no—
176#940—98no—
177#940—92no—
178#938—94no—
179#938—91no—
180#937—98no—
181#937—92no—
182#930—94no—
183#930—91no—
184#895—34no—
185#895C3⋊D311yescharacteristic, complemented
186#892—94no—
187#892—91no—
188#892—184no—
189#885—98no—
190#885—92no—
191#882—98no—
192#882—92no—
193C23×C32—32no—
194C23×C32—31no—
195C22×C3⋊D3—31no—
196C22×C3⋊D3—34no—
197C22×C3⋊D3—64no—
198C22×C3⋊D3—64no—
199C22×C3⋊D3—122no—
200C22×C3⋊D3—122no—
201C22×C3⋊D3C22×D312yes—
202C22×C3⋊D3C22×D311yescharacteristic
203C22×C3×D3—38no—
204C22×C3×D3—34no—
205C22×C3×D3C22×D318yes—
206C22×C3×D3C22×D314yes—
207C2×D32—616no—
208C2×D32—128no—
209C2×D32—128no—
210C32×D4—62no—
211C32×D4—122no—
212C2×C32×C4—24no—
213C2×C32×C4—28no—
214C2×C32×C4—31no—
215C2×C32×C4—62no—
216C2×C32×C4C22×D318yes—
217C2×C32×C4C22×D314yes—
218C2×C32×C4C22×D312yes—
219C2×C32×C4C22×D311yescharacteristic
220#638—62no—
221#638—62no—
222#638—64no—
223#638—122no—
224#638—122no—
225C2×C3⋊Dic3—31no—
226C2×C3⋊Dic3—32no—
227C2×C3⋊Dic3—34no—
228C2×C3⋊Dic3—38no—
229C2×C3⋊Dic3—64no—
230C2×C3⋊Dic3—68no—
231C2×C3⋊Dic3C22×D311yescharacteristic
232C4⋊(C3⋊D3)—22no—
233C4⋊(C3⋊D3)—42no—
234C4⋊(C3⋊D3)—62no—
235C4×C3⋊D3—22no—
236C4×C3⋊D3—34no—
237C4×C3⋊D3—316no—
238C4×C3⋊D3—616no—
239#634—62no—
240C2×C3×Dic3—34no—
241C2×C3×Dic3C22×D314yes—
242C3×D12—28no—
243C3×D12—48no—
244C3×D12—68no—
245C3×D12—128no—
246C3×C4×D3—28no—
247C3×C4×D3—68no—
248#627—68no—
249#626—68no—
250#626—68no—
251#625—68no—
252#625—616no—
253#625—128no—
254#625—128no—
255#535—271noSylow
256C2×C33C22×D411yescharacteristic
257C2×C33H(5,𝔽2)11yescharacteristic
258C2×C33H(5,𝔽2)11yescharacteristic
259C3⋊(C3⋊D3)—22no—
260C3×C3⋊D3—42no—
261C3×C3⋊D3—41no—
262C32×D3—42no—
263C32×D3—41no—
264C23×D3—98no—
265C23×D3—92no—
266C2×C3×D4—34no—
267C2×C3×D4—38no—
268C2×C3×D4—91no—
269C2×C3×D4—92no—
270C22×C3×C4—34no—
271C22×C3×C4—91no—
272C2×#67—94no—
273C2×#67—98no—
274C2×#67—94no—
275C2×#67—92no—
276C2×#67—91no—
277C2×#67—98no—
278C2×#67—98no—
279C2×#67—92no—
280C2×#67—92no—
281C2×#67—91no—
282D3×D4—184no—
283D3×D4—1816no—
284#235—184no—
285#235—1816no—
286C2×D12—34no—
287C2×D12—38no—
288C2×D12—94no—
289C2×D12—91no—
290C2×D12C2×C3⋊D311yescharacteristic
291C2×D12C2×C3⋊D312yescomplemented
292C2×C4×D3—34no—
293C2×C4×D3—94no—
294C2×C4×D3—91no—
295C2×C4×D3—184no—
296C2×C4×D3C2×C3⋊D311yescharacteristic, complemented
297C2×Dic6—94no—
298C2×Dic6—91no—
299C3×C4⋊C4—24no—
300C3×C4⋊C4D3214yescomplemented
301C3×C4⋊C4C2×C3⋊D311yescharacteristic, complemented
302C3×C22⋊C4—38no—
303C3×C22⋊C4—92no—
304#217—98no—
305#217—92no—
306(C2×D3)⋊C4—38no—
307(C2×D3)⋊C4—98no—
308(C2×D3)⋊C4—92no—
309(C2×D3)⋊C4—98no—
310(C2×D3)⋊C4—92no—
311(C2×D3)⋊C4—188no—
312(C2×D3)⋊C4C2×C3⋊D312yescomplemented
313C4⋊Dic3—94no—
314C4⋊Dic3—91no—
315C4⋊Dic3—184no—
316C4×Dic3—98no—
317C4×Dic3—92no—
318C4×Dic3—188no—
319C22×C32—24no—
320C22×C32—61no—
321C22×C32—62no—
322C22×C32—61no—
323C22×C32—62no—
324C22×C32—61no—
325C22×C32—62no—
326C22×C32C23×D314yes—
327C22×C32C23×D311yescharacteristic
328C2×C3⋊D3—21no—
329C2×C3⋊D3—22no—
330C2×C3⋊D3—21no—
331C2×C3⋊D3—22no—
332C2×C3⋊D3—21no—
333C2×C3⋊D3—22no—
334C2×C3⋊D3—32no—
335C2×C3⋊D3—38no—
336C2×C3⋊D3—68no—
337C2×C3⋊D3—61no—
338C2×C3⋊D3—61no—
339C2×C3⋊D3—64no—
340C2×C3⋊D3—64no—
341C2×C3⋊D3—124no—
342C2×C3⋊D3—124no—
343C2×C3⋊D3—128no—
344C2×C3×D3—24no—
345C2×C3×D3—28no—
346C2×C3×D3—24no—
347C2×C3×D3—28no—
348C2×C3×D3—24no—
349C2×C3×D3—28no—
350C2×C3×D3—64no—
351C2×C3×D3—68no—
352C2×C3×D3—64no—
353C2×C3×D3—68no—
354C2×C3×D3—64no—
355C2×C3×D3—68no—
356D32—1232no—
357C32×C4—22no—
358C32×C4—28no—
359C32×C4—28no—
360C32×C4—48no—
361C32×C4—61no—
362C32×C4D3×D418yes—
363C32×C4D3×D412yes—
364C3⋊Dic3—21no—
365C3⋊Dic3—32no—
366C3⋊Dic3—38no—
367C3⋊Dic3—68no—
368C3⋊Dic3—62no—
369C3⋊Dic3—68no—
370C3⋊Dic3—128no—
371C3×Dic3—24no—
372C3×Dic3—64no—
373C2×#41—271no—
374C22×D4—271no—
375#127—272no—
376#124—272no—
377#121—31no—
378#121—91no—
379#121—271no—
380#121—271no—
381#120—272no—
382#120—272no—
383#117—271no—
384C33#53511yescharacteristic, complemented, Sylow
385C23×C3—38no—
386C23×C3—34no—
387C23×C3—92no—
388C23×C3—91no—
389C22×D3—38no—
390C22×D3—34no—
391C22×D3—94no—
392C22×D3—91no—
393C22×D3—184no—
394C22×D3—184no—
395C22×D3—1816no—
396C22×D3—362no—
397C22×D3—362no—
398C22×D3—368no—
399C22×D3—368no—
400C22×D3C22×C3⋊D312yes—
401C22×D3C22×C3⋊D311yescharacteristic
402C3×D4—68no—
403C3×D4—128no—
404C3×D4—182no—
405C3×D4—362no—
406C2×C3×C4—24no—
407C2×C3×C4—28no—
408C2×C3×C4—34no—
409C2×C3×C4—68no—
410C2×C3×C4—91no—
411C2×C3×C4—182no—
412C2×C3×C4C2×D3218yes—
413C2×C3×C4C2×D3214yes—
414C2×C3×C4C22×C3⋊D312yes—
415C2×C3×C4C22×C3⋊D311yescharacteristic
416#67—182no—
417#67—188no—
418#67—182no—
419#67—184no—
420#67—188no—
421#67—1816no—
422#67—362no—
423#67—362no—
424#67—368no—
425#67—368no—
426C2×Dic3—34no—
427C2×Dic3—94no—
428C2×Dic3—91no—
429C2×Dic3—98no—
430C2×Dic3—92no—
431C2×Dic3—184no—
432C2×Dic3—188no—
433C2×Dic3C22×C3⋊D311yescharacteristic
434D12—22no—
435D12—42no—
436D12—68no—
437D12—128no—
438D12—188no—
439D12—182no—
440C4×D3—22no—
441C4×D3—68no—
442C4×D3—916no—
443C4×D3—94no—
444C4×D3—1816no—
445Dic6—182no—
446Dic6—188no—
447C2×C32—24no—
448C2×C32—24no—
449C2×C32—24no—
450C2×C32—122no—
451C2×C32—121no—
452C2×C32C2×D3×D414yes—
453C2×C32C2×D3×D411yescharacteristic
454C2×C32#100414yes—
455C2×C32#100411yescharacteristic
456C2×C32#100914yes—
457C2×C32#100911yescharacteristic
458C3⋊D3—41no—
459C3⋊D3—42no—
460C3⋊D3—62no—
461C3⋊D3—68no—
462C3⋊D3—128no—
463C3×D3—48no—
464C3×D3—44no—
465C3×D3—128no—
466C3×D3—124no—
467C24—272no—
468#41—544no—
469C2×Q8—271no—
470C2×D4—31no—
471C2×D4—32no—
472C2×D4—91no—
473C2×D4—92no—
474C2×D4—271no—
475C2×D4—271no—
476C2×D4—272no—
477C2×D4—271no—
478C2×D4—272no—
479C2×D4—272no—
480C2×D4—544no—
481C22×C4—31no—
482C22×C4—91no—
483C22×C4—271no—
484C4⋊C4—271no—
485C4⋊C4D3×C3⋊D311yescharacteristic, complemented
486C22⋊C4—32no—
487C22⋊C4—92no—
488C22⋊C4—272no—
489C22⋊C4—272no—
490C22⋊C4—272no—
491C42—272no—
492C22×C3—24no—
493C22×C3—64no—
494C22×C3—68no—
495C22×C3—64no—
496C22×C3—68no—
497C22×C3—64no—
498C22×C3—68no—
499C22×C3—181no—
500C22×C3—182no—
501C22×C3—181no—
502C22×C3—182no—
503C22×C3—181no—
504C22×C3—182no—
505C22×C3C22×D3214yes—
506C22×C3C23×C3⋊D311yescharacteristic
507C2×D3—21no—
508C2×D3—22no—
509C2×D3—21no—
510C2×D3—22no—
511C2×D3—21no—
512C2×D3—22no—
513C2×D3—64no—
514C2×D3—68no—
515C2×D3—64no—
516C2×D3—68no—
517C2×D3—64no—
518C2×D3—68no—
519C2×D3—98no—
520C2×D3—92no—
521C2×D3—188no—
522C2×D3—184no—
523C2×D3—184no—
524C2×D3—181no—
525C2×D3—181no—
526C2×D3—364no—
527C2×D3—364no—
528C2×D3—368no—
529C2×D3—3632no—
530C3×C4—28no—
531C3×C4—22no—
532C3×C4—28no—
533C3×C4—48no—
534C3×C4—64no—
535C3×C4—181no—
536C3×C4#380218yes—
537C3×C4D4×C3⋊D312yes—
538Dic3—21no—
539Dic3—64no—
540Dic3—98no—
541Dic3—92no—
542Dic3—188no—
543Dic3—188no—
544Dic3—182no—
545Dic3—368no—
546C32—24no—
547C32#560714yescomplemented
548C32#560711yescharacteristic, complemented
549C23—32no—
550C23—31no—
551C23—92no—
552C23—91no—
553C23—271no—
554C23—544no—
555C23—1082no—
556C23—1082no—
557Q8—542no—
558D4—62no—
559D4—122no—
560D4—182no—
561D4—362no—
562D4—542no—
563D4—542no—
564D4—544no—
565D4—542no—
566D4—1082no—
567D4—1082no—
568C2×C4—31no—
569C2×C4—62no—
570C2×C4—91no—
571C2×C4—182no—
572C2×C4—271no—
573C2×C4—274no—
574C2×C4—272no—
575C2×C4C2×D3×C3⋊D312yes—
576C2×C4C2×D3×C3⋊D311yescharacteristic
577C2×C3—24no—
578C2×C3—24no—
579C2×C3—24no—
580C2×C3—128no—
581C2×C3—124no—
582C2×C3—362no—
583C2×C3—361no—
584C2×C3C2×#380214yes—
585C2×C3#6431314yes—
586C2×C3#6432014yes—
587C2×C3C2×D4×C3⋊D311yescharacteristic
588C2×C3#6436111yescharacteristic
589C2×C3#6436611yescharacteristic
590D3—42no—
591D3—41no—
592D3—128no—
593D3—124no—
594D3—188no—
595D3—182no—
596D3—368no—
597C22—61no—
598C22—62no—
599C22—61no—
600C22—62no—
601C22—61no—
602C22—62no—
603C22—181no—
604C22—182no—
605C22—181no—
606C22—182no—
607C22—181no—
608C22—182no—
609C22—272no—
610C22—541no—
611C22—541no—
612C22—1088no—
613C22C22×D3×C3⋊D311yescharacteristic
614C4—22no—
615C4—61no—
616C4—181no—
617C4—272no—
618C4—542no—
619C4#8902612yes—
620C3—24no—
621C3#1059463414yescomplemented
622C3#1059607811yescharacteristic, complemented
623C2—122no—
624C2—121no—
625C2—362no—
626C2—361no—
627C2—542no—
628C2C2×#8902611yescharacteristic
629C2#1172458711yescharacteristic
630C2#1172459711yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC33×C4⋊C4C22
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33#535
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#89026C2semidirect
2. C2×#89020C2semidirect
3. #11723933C2semidirect
4. C3×#64104C2semidirect
5. C32×#895C2semidirect
6. #11723541C2semidirect
7. #11723540C2semidirect
8. #11723530C2semidirect
9. #11723529C2semidirect
10. #11723522C2semidirect
11. #11723515C2semidirect
17. C2×C3×C4⋊(C3⋊D3)C22semidirect
18. C2×C3×C4×C3⋊D3C22semidirect
20. C2×C32×D12C22semidirect
21. C2×C32×C4×D3C22semidirect
22. C2×#6293C22semidirect
26. C2×#6290C22semidirect
27. C2×#6290C22semidirect
28. C2×#6290C22semidirect
31. C33×C4⋊C4C22semidirect
33. C4⋊(C3⋊(C3⋊Dic3))C22semidirect
34. C4×C3⋊(C3⋊Dic3)C22semidirect
35. C3×#3752C22semidirect
36. C32×(C2×D3)⋊C4C22semidirect
37. #88803C22semidirect
38. (C2×D3)⋊(C3⋊Dic3)C22semidirect
48. #64104D3semidirect
55. C3×#895D3semidirect
74. C4×C3⋊(C3⋊D3)D4semidirect
106. C2×C4⋊(C3⋊D3)C2×D3semidirect
110. C2×C4×C3⋊D3C2×D3semidirect
115. C2×C3×D12C2×D3semidirect
117. C2×C3×C4×D3C2×D3semidirect
128. C32×C4⋊C4C2×D3semidirect
129. C32×C4⋊C4C2×D3semidirect
136. #3752C2×D3semidirect
144. C3×(C2×D3)⋊C4C2×D3semidirect
185. #895C3⋊D3semidirect
291. C2×D12C2×C3⋊D3semidirect
296. C2×C4×D3C2×C3⋊D3semidirect
300. C3×C4⋊C4D32semidirect
301. C3×C4⋊C4C2×C3⋊D3semidirect
312. (C2×D3)⋊C4C2×C3⋊D3semidirect
384. C33#535semidirect
485. C4⋊C4D3×C3⋊D3semidirect
547. C32#5607semidirect
548. C32#5607semidirect
621. C3#10594634semidirect
622. C3#10596078semidirect

Maximal subgroups

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

subgroupindex
C2×#890262
C2×#890202
#117239332
C3×#641042
C32×#8952
#117235412
#117235402
#117235302
#117235292
#117235222
#117235152
#105960783
#105946343