Finite Group Databasebeta

← #13175506

Subgroups — #13175506

Order 1728.

2308 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6290C211yesmaximal, characteristic, complemented
2C2×#89029C211yesmaximal, characteristic, complemented
3#11723918C211yesmaximal, characteristic, complemented
4C3×#64091C211yesmaximal, characteristic, complemented
5C32×#882C211yesmaximal, characteristic, complemented
6#11723634C211yesmaximal, characteristic, complemented
7#11723629C211yesmaximal, characteristic, complemented
8#11723530C212yesmaximal, complemented
9#11723529C212yesmaximal, complemented
10#11723519C212yesmaximal, complemented
11#11723474C212yesmaximal, complemented
12#10596063—31nomaximal
13#10594192—34nomaximal
14C23×C3×C3⋊D3C2211yescharacteristic
15C23×C32×D3C2211yescharacteristic
16C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
17C2×C3×#638C2211yescharacteristic
18C2×C3×C4⋊(C3⋊D3)C2212yescomplemented
19C2×C32×#67C2211yescharacteristic
20C2×C32×D12C2212yescomplemented
21C2×#6293C2211yescharacteristic, complemented
22C2×#6290—24no—
23C2×#6290C2212yes—
24C2×#6290C2212yes—
25C2×#6290C2211yescharacteristic, complemented
26C2×#6290C2212yescomplemented
27C2×#6290C2212yescomplemented
28C2×Dic3×C3⋊D3C2211yescharacteristic, complemented
29#89029—24no—
30C2×D3×C3⋊Dic3C2211yescharacteristic, complemented
31C33×C22⋊C4C2211yescharacteristic
32#88864C2211yescharacteristic
33#88857C2212yescomplemented
34C4×C3⋊(C3⋊Dic3)C2212yescomplemented
35C3×#3752C2212yescomplemented
36C32×(C2×D3)⋊C4C2212yescomplemented
37#88803C2212yescomplemented
38(C2×D3)⋊(C3⋊Dic3)C2212yescomplemented
39C22×#638—31no—
40C2×#3830—31no—
41C22×#625—34no—
42C2×#3805—34no—
43C32×#120—31no—
44#64162—31no—
45#64149—32no—
46#64148—32no—
47#64146—31no—
48#64094—32no—
49#64093—32no—
50#64091D311yescharacteristic, complemented
51#64086—31no—
52#64086—34no—
53#64086—64no—
54C3×#882—34no—
55C3×#882D314yescomplemented
56#63976—34no—
57#63976—34no—
58#63915—38no—
59#63914—38no—
60#63885—38no—
61#63885—38no—
62C23×C33C2311yescharacteristic
63C22×C3×C3⋊D3—22no—
64C22×C3×C3⋊D3—42no—
65C22×C3×C3⋊D3C2312yes—
66C22×C3×C3⋊D3C2311yescharacteristic
67C22×C32×D3—22no—
68C22×C32×D3—42no—
69C22×C32×D3C2312yes—
70C22×C32×D3C2311yescharacteristic
71C2×C33×C4C2312yes—
72C2×C3⋊(C3⋊Dic3)D414yescomplemented
73C2×C3⋊(C3⋊Dic3)C2312yes—
74C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
75C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
76C3×#638—22no—
77C2×C3×C3⋊Dic3C2311yescharacteristic
78C3×C4⋊(C3⋊D3)—42no—
79C32×#67—22no—
80C2×C32×Dic3C2311yescharacteristic
81C32×D12—42no—
82#6293—22no—
83#6290—22no—
84#6290—24no—
85#6290—24no—
86#6290—42no—
87#6290—42no—
88Dic3×C3⋊D3—22no—
89D3×C3⋊Dic3—22no—
90#5592—94no—
91#5592—91no—
92C24×C32—31no—
93C23×C3⋊D3C2×D311yescharacteristic
94C23×C3×D3—34no—
95C23×C3×D3C2×D314yes—
96C2×C32×D4—31no—
97C2×C32×D4—32no—
98C2×#638—32no—
99C2×#638—32no—
100C2×#638—31no—
101C2×#638—32no—
102C2×#638—32no—
103C2×#638—64no—
104C2×#638C2×D311yescharacteristic
105C22×C3⋊Dic3—31no—
106C22×C3⋊Dic3—34no—
107C22×C3⋊Dic3—31no—
108C22×C3⋊Dic3—64no—
109#3830—64no—
110C2×C4⋊(C3⋊D3)C2×D312yescomplemented
111C2×C4×C3⋊D3—31no—
112C2×#634—31no—
113C2×C3×#67—34no—
114C2×C3×#67C2×D314yes—
115C2×C3×D12—38no—
116C2×C3×D12C2×D318yescomplemented
117C2×#627—34no—
118C2×#625—34no—
119C2×#625—38no—
120C2×#625—38no—
121C2×#625—38no—
122C2×#625—38no—
123C2×#625—616no—
124#3805—616no—
125C2×D3×Dic3—34no—
126C2×D3×Dic3—34no—
127C32×C22⋊C4—24no—
128C32×C22⋊C4—32no—
129C32×C22⋊C4C2×D314yes—
130C32×C22⋊C4C2×D311yescharacteristic
131#3757—31no—
132#3757—34no—
133#3757—32no—
134#3757—64no—
135#3752—32no—
136#3752C2×D312yescomplemented
137#3750—32no—
138#3750—38no—
139#3750—68no—
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×C33—22no—
148C22×C33C2×D412yes—
149C22×C33C2411yescharacteristic
150C2×C3×C3⋊D3—21no—
151C2×C3×C3⋊D3—22no—
152C2×C3×C3⋊D3—22no—
153C2×C3×C3⋊D3—22no—
154C2×C3×C3⋊D3—22no—
155C2×C3×C3⋊D3—44no—
156C2×C32×D3—21no—
157C2×C32×D3—22no—
158C2×C32×D3—22no—
159C2×C32×D3—22no—
160C2×C32×D3—22no—
161C2×C32×D3—44no—
162C33×C4—22no—
163C3⋊(C3⋊Dic3)—22no—
164C3⋊(C3⋊Dic3)C2×D412yes—
165C3⋊(C3⋊Dic3)C2×D412yes—
166C3×C3⋊Dic3—21no—
167C32×Dic3—21no—
168C22×#67—94no—
169C22×#67—91no—
170C2×#237—94no—
171C2×#237—91no—
172C3×#120—34no—
173C3×#120—91no—
174#953—94no—
175#953—91no—
176#940—98no—
177#940—92no—
178#939—98no—
179#939—92no—
180#937—94no—
181#937—91no—
182#885—98no—
183#885—92no—
184#884—98no—
185#884—92no—
186#882—34no—
187#882C3⋊D311yescharacteristic, complemented
188#877—94no—
189#877—91no—
190#877—184no—
191C23×C32—24no—
192C23×C32—32no—
193C23×C32—31no—
194C23×C32—62no—
195C23×C32—122no—
196C23×C32C22×D314yes—
197C23×C32C22×D311yescharacteristic
198C22×C3⋊D3—22no—
199C22×C3⋊D3—42no—
200C22×C3⋊D3C22×D312yes—
201C22×C3⋊D3C22×D311yescharacteristic
202C22×C3×D3—28no—
203C22×C3×D3—38no—
204C22×C3×D3—34no—
205C22×C3×D3—48no—
206C22×C3×D3—68no—
207C22×C3×D3—128no—
208C22×C3×D3C22×D318yes—
209C22×C3×D3C22×D314yes—
210C32×D4—62no—
211C32×D4—122no—
212C2×C32×C4—28no—
213C2×C32×C4—31no—
214C2×C32×C4C22×D318yes—
215C2×C32×C4C22×D312yes—
216#638—22no—
217#638—62no—
218#638—64no—
219#638—64no—
220#638—122no—
221#638—122no—
222C2×C3⋊Dic3—34no—
223C2×C3⋊Dic3—31no—
224C2×C3⋊Dic3—31no—
225C2×C3⋊Dic3—32no—
226C2×C3⋊Dic3—316no—
227C2×C3⋊Dic3—34no—
228C2×C3⋊Dic3—34no—
229C2×C3⋊Dic3—38no—
230C2×C3⋊Dic3—64no—
231C2×C3⋊Dic3—64no—
232C2×C3⋊Dic3—616no—
233C2×C3⋊Dic3—68no—
234C2×C3⋊Dic3—62no—
235C2×C3⋊Dic3C22×D311yescharacteristic
236C4⋊(C3⋊D3)—42no—
237C4×C3⋊D3—62no—
238#634—62no—
239C3×#67—28no—
240C3×#67—68no—
241C2×C3×Dic3—34no—
242C2×C3×Dic3C22×D314yes—
243C3×D12—48no—
244C3×D12—128no—
245#627—68no—
246#625—68no—
247#625—616no—
248#625—616no—
249#625—128no—
250#625—128no—
251D3×Dic3—68no—
252D3×Dic3—68no—
253#535—271noSylow
254C2×C33—22no—
255C2×C33C22×D411yescharacteristic
256C2×C33H(5,𝔽2)12yes—
257C3×C3⋊D3—41no—
258C3×C3⋊D3—42no—
259C32×D3—41no—
260C32×D3—42no—
261C24×C3—34no—
262C24×C3—91no—
263C23×D3—34no—
264C23×D3C2×C3⋊D311yescharacteristic
265C2×C3×D4—34no—
266C2×C3×D4—38no—
267C2×C3×D4—91no—
268C2×C3×D4—92no—
269C2×#67—34no—
270C2×#67—98no—
271C2×#67—98no—
272C2×#67—94no—
273C2×#67—92no—
274C2×#67—92no—
275C2×#67—91no—
276C2×#67—98no—
277C2×#67—98no—
278C2×#67—92no—
279C2×#67—92no—
280C2×#67—184no—
281C2×#67—1816no—
282C2×#67C2×C3⋊D311yescharacteristic
283C22×Dic3—94no—
284C22×Dic3—94no—
285C22×Dic3—91no—
286C22×Dic3—91no—
287C22×Dic3—184no—
288#237—184no—
289#237—1816no—
290C2×D12—38no—
291C2×D12C2×C3⋊D312yescomplemented
292C2×C4×D3—94no—
293C2×C4×D3—91no—
294C2×Dic6—94no—
295C2×Dic6—91no—
296C3×C22⋊C4—24no—
297C3×C22⋊C4—38no—
298C3×C22⋊C4—92no—
299C3×C22⋊C4D3214yes—
300C3×C22⋊C4C2×C3⋊D311yescharacteristic
301#217—98no—
302#217—94no—
303#217—91no—
304#217—92no—
305#217—184no—
306(C2×D3)⋊C4—38no—
307(C2×D3)⋊C4—98no—
308(C2×D3)⋊C4—92no—
309(C2×D3)⋊C4C2×C3⋊D312yescomplemented
310Dic3⋊C4—98no—
311Dic3⋊C4—92no—
312Dic3⋊C4—188no—
313C4×Dic3—98no—
314C4×Dic3—92no—
315C4×Dic3—188no—
316C22×C32—22no—
317C22×C32—24no—
318C22×C32—28no—
319C22×C32—28no—
320C22×C32—48no—
321C22×C32—61no—
322C22×C32—62no—
323C22×C32—62no—
324C22×C32—62no—
325C22×C32—62no—
326C22×C32—124no—
327C22×C32D3×D418yes—
328C22×C32D3×D412yes—
329C22×C32C23×D314yes—
330C22×C32C23×D311yescharacteristic
331C2×C3⋊D3—21no—
332C2×C3⋊D3—22no—
333C2×C3⋊D3—22no—
334C2×C3⋊D3—22no—
335C2×C3⋊D3—22no—
336C2×C3⋊D3—44no—
337C2×C3×D3—24no—
338C2×C3×D3—28no—
339C2×C3×D3—28no—
340C2×C3×D3—28no—
341C2×C3×D3—28no—
342C2×C3×D3—416no—
343C2×C3×D3—64no—
344C2×C3×D3—68no—
345C2×C3×D3—68no—
346C2×C3×D3—68no—
347C2×C3×D3—68no—
348C2×C3×D3—1216no—
349C32×C4—22no—
350C32×C4—28no—
351C32×C4—48no—
352C32×C4—61no—
353C3⋊Dic3—21no—
354C3⋊Dic3—32no—
355C3⋊Dic3—32no—
356C3⋊Dic3—38no—
357C3⋊Dic3—38no—
358C3⋊Dic3—68no—
359C3⋊Dic3—68no—
360C3⋊Dic3—62no—
361C3⋊Dic3—68no—
362C3⋊Dic3—128no—
363C3×Dic3—24no—
364C3×Dic3—64no—
365C2×#41—271no—
366C22×D4—271no—
367#127—272no—
368#124—272no—
369#121—272no—
370#121—272no—
371#120—31no—
372#120—91no—
373#120—271no—
374#120—271no—
375#117—271no—
376C33#53511yescharacteristic, complemented, Sylow
377C23×C3—24no—
378C23×C3—38no—
379C23×C3—34no—
380C23×C3—68no—
381C23×C3—92no—
382C23×C3—91no—
383C23×C3—128no—
384C23×C3—182no—
385C23×C3—362no—
386C23×C3C2×D3214yes—
387C23×C3C22×C3⋊D311yescharacteristic
388C22×D3—22no—
389C22×D3—38no—
390C22×D3—34no—
391C22×D3—42no—
392C22×D3—68no—
393C22×D3—128no—
394C22×D3C22×C3⋊D312yes—
395C22×D3C22×C3⋊D311yescharacteristic
396C3×D4—68no—
397C3×D4—128no—
398C3×D4—182no—
399C3×D4—362no—
400C2×C3×C4—28no—
401C2×C3×C4—34no—
402C2×C3×C4—91no—
403C2×C3×C4C2×D3218yes—
404C2×C3×C4C22×C3⋊D312yes—
405#67—22no—
406#67—68no—
407#67—182no—
408#67—184no—
409#67—184no—
410#67—188no—
411#67—1816no—
412#67—1816no—
413#67—362no—
414#67—362no—
415#67—368no—
416#67—368no—
417C2×Dic3—34no—
418C2×Dic3—94no—
419C2×Dic3—94no—
420C2×Dic3—91no—
421C2×Dic3—91no—
422C2×Dic3—916no—
423C2×Dic3—94no—
424C2×Dic3—98no—
425C2×Dic3—92no—
426C2×Dic3—184no—
427C2×Dic3—184no—
428C2×Dic3—1816no—
429C2×Dic3—188no—
430C2×Dic3—182no—
431C2×Dic3—188no—
432C2×Dic3C22×C3⋊D311yescharacteristic
433D12—42no—
434D12—128no—
435C4×D3—188no—
436C4×D3—182no—
437Dic6—182no—
438Dic6—188no—
439C2×C32—22no—
440C2×C32—24no—
441C2×C32—28no—
442C2×C32—28no—
443C2×C32—48no—
444C2×C32—121no—
445C2×C32—122no—
446C2×C32C2×D3×D414yes—
447C2×C32C2×D3×D411yescharacteristic
448C2×C32#100418yes—
449C2×C32#100412yes—
450C3⋊D3—41no—
451C3⋊D3—42no—
452C3×D3—44no—
453C3×D3—48no—
454C3×D3—124no—
455C3×D3—128no—
456C24—31no—
457C24—91no—
458#41—544no—
459C2×Q8—271no—
460C2×D4—31no—
461C2×D4—32no—
462C2×D4—91no—
463C2×D4—92no—
464C2×D4—272no—
465C2×D4—272no—
466C2×D4—271no—
467C2×D4—272no—
468C2×D4—272no—
469C2×D4—544no—
470C22×C4—271no—
471C22×C4—271no—
472C22×C4—271no—
473C4⋊C4—272no—
474C22⋊C4—32no—
475C22⋊C4—92no—
476C22⋊C4—272no—
477C22⋊C4—272no—
478C22⋊C4—271no—
479C22⋊C4D3×C3⋊D311yescharacteristic
480C42—272no—
481C22×C3—28no—
482C22×C3—22no—
483C22×C3—24no—
484C22×C3—28no—
485C22×C3—48no—
486C22×C3—64no—
487C22×C3—68no—
488C22×C3—68no—
489C22×C3—68no—
490C22×C3—68no—
491C22×C3—1216no—
492C22×C3—181no—
493C22×C3—182no—
494C22×C3—182no—
495C22×C3—182no—
496C22×C3—182no—
497C22×C3—364no—
498C22×C3#380218yes—
499C22×C3D4×C3⋊D312yes—
500C22×C3C22×D3214yes—
501C22×C3C23×C3⋊D311yescharacteristic
502C2×D3—21no—
503C2×D3—22no—
504C2×D3—22no—
505C2×D3—22no—
506C2×D3—22no—
507C2×D3—44no—
508C2×D3—64no—
509C2×D3—68no—
510C2×D3—68no—
511C2×D3—68no—
512C2×D3—68no—
513C2×D3—1216no—
514C3×C4—28no—
515C3×C4—22no—
516C3×C4—48no—
517C3×C4—64no—
518C3×C4—181no—
519Dic3—21no—
520Dic3—64no—
521Dic3—98no—
522Dic3—98no—
523Dic3—92no—
524Dic3—92no—
525Dic3—188no—
526Dic3—188no—
527Dic3—188no—
528Dic3—182no—
529Dic3—368no—
530C32—24no—
531C32#559214yescomplemented
532C32#559211yescharacteristic, complemented
533C23—32no—
534C23—31no—
535C23—62no—
536C23—92no—
537C23—91no—
538C23—122no—
539C23—182no—
540C23—362no—
541C23C2×D3×C3⋊D311yescharacteristic
542Q8—542no—
543D4—62no—
544D4—122no—
545D4—182no—
546D4—362no—
547D4—544no—
548D4—544no—
549D4—542no—
550D4—1082no—
551D4—1082no—
552C2×C4—31no—
553C2×C4—91no—
554C2×C4—271no—
555C2×C4—271no—
556C2×C4—274no—
557C2×C4—272no—
558C2×C4—542no—
559C2×C4—542no—
560C2×C4C2×D3×C3⋊D312yes—
561C2×C3—28no—
562C2×C3—22no—
563C2×C3—24no—
564C2×C3—28no—
565C2×C3—48no—
566C2×C3—128no—
567C2×C3—124no—
568C2×C3—362no—
569C2×C3—361no—
570C2×C3C2×#380214yes—
571C2×C3#6431318yes—
572C2×C3C2×D4×C3⋊D311yescharacteristic
573C2×C3#6436112yes—
574D3—42no—
575D3—41no—
576D3—128no—
577D3—124no—
578C22—22no—
579C22—61no—
580C22—62no—
581C22—62no—
582C22—62no—
583C22—62no—
584C22—124no—
585C22—181no—
586C22—182no—
587C22—182no—
588C22—182no—
589C22—182no—
590C22—364no—
591C22#8902612yes—
592C22C22×D3×C3⋊D311yescharacteristic
593C4—22no—
594C4—61no—
595C4—181no—
596C4—272no—
597C4—272no—
598C4—542no—
599C3—24no—
600C3#1059419214yescomplemented
601C3#1059606311yescharacteristic, complemented
602C2—22no—
603C2—122no—
604C2—121no—
605C2—362no—
606C2—361no—
607C2C2×#8902611yescharacteristic
608C2#1172458712yes—
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC33×C22⋊C4C22
hypercenterC22⋊C4D3×C3⋊D3
hypocenterC33#535
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C22×#6290C2semidirect
2. C2×#89029C2semidirect
3. #11723918C2semidirect
4. C3×#64091C2semidirect
5. C32×#882C2semidirect
6. #11723634C2semidirect
7. #11723629C2semidirect
8. #11723530C2semidirect
9. #11723529C2semidirect
10. #11723519C2semidirect
11. #11723474C2semidirect
18. C2×C3×C4⋊(C3⋊D3)C22semidirect
20. C2×C32×D12C22semidirect
21. C2×#6293C22semidirect
25. C2×#6290C22semidirect
26. C2×#6290C22semidirect
27. C2×#6290C22semidirect
28. C2×Dic3×C3⋊D3C22semidirect
30. C2×D3×C3⋊Dic3C22semidirect
33. #88857C22semidirect
34. C4×C3⋊(C3⋊Dic3)C22semidirect
35. C3×#3752C22semidirect
36. C32×(C2×D3)⋊C4C22semidirect
37. #88803C22semidirect
38. (C2×D3)⋊(C3⋊Dic3)C22semidirect
50. #64091D3semidirect
55. C3×#882D3semidirect
72. C2×C3⋊(C3⋊Dic3)D4semidirect
110. C2×C4⋊(C3⋊D3)C2×D3semidirect
116. C2×C3×D12C2×D3semidirect
136. #3752C2×D3semidirect
144. C3×(C2×D3)⋊C4C2×D3semidirect
187. #882C3⋊D3semidirect
291. C2×D12C2×C3⋊D3semidirect
309. (C2×D3)⋊C4C2×C3⋊D3semidirect
376. C33#535semidirect
531. C32#5592semidirect
532. C32#5592semidirect
600. C3#10594192semidirect
601. C3#10596063semidirect

Maximal subgroups

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

subgroupindex
C22×#62902
C2×#890292
#117239182
C3×#640912
C32×#8822
#117236342
#117236292
#117235302
#117235292
#117235192
#117234742
#105960633
#105941923