Finite Group Databasebeta

← #13177590

Subgroups — #13177590

Order 1728.

2380 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3830C211yesmaximal, characteristic, complemented
2C2×C32×D3×D4C211yesmaximal, characteristic, complemented
3D4×C3⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
4#11723637C212yesmaximal, complemented
5#11723629C212yesmaximal, complemented
6#11723602C212yesmaximal, complemented
7C2×(C2×D3)⋊(C3⋊Dic3)C212yesmaximal, complemented
8#11723522C211yesmaximal, characteristic, complemented
9C4×#6290C211yesmaximal, characteristic, complemented
10#11723434C211yesmaximal, characteristic, complemented
11#11723425C211yesmaximal, characteristic, complemented
12#10596277—31nomaximal
13#10595244—34nomaximal
14C23×C32×D3C2212yes—
15C2×C33×D4C2211yescharacteristic
16C22×C3⋊(C3⋊Dic3)C2212yescomplemented
17C2×C3×#638C2212yescomplemented
18C22×C3×C3⋊Dic3C2212yescomplemented
19C3×#3830—24no—
20C2×C3×C4×C3⋊D3C2211yescharacteristic, complemented
21C2×C3×#634C2211yescharacteristic, complemented
22C2×C32×#67C2212yes—
23C32×D3×D4—24no—
24C2×C32×D12C2211yescharacteristic
25C2×C32×C4×D3C2211yescharacteristic
26C2×#6290C2211yescharacteristic
27C2×#6290C2212yescomplemented
28#88864C2212yescomplemented
29C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic, complemented
30C4×C3⋊(C3⋊Dic3)C2211yescharacteristic, complemented
31#88806C2211yescharacteristic, complemented
32#88806C2212yescomplemented
33#88803C2211yescharacteristic, 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×#3830D311yescharacteristic, complemented
41C2×C3×#237—34no—
42C2×C3×D3×D4D314yescomplemented
43#64162—32no—
44C2×#3757—32no—
45#64148—32no—
46#64147—31no—
47#64145—31no—
48#64144—32no—
49D4×C3⋊Dic3—31no—
50D4×C3⋊Dic3—34no—
51D4×C3⋊Dic3—64no—
52C4×#638—31no—
53#64133—31no—
54#63980—38no—
55#63976—38no—
56#63961—38no—
57C2×(C2×D3)⋊Dic3—38no—
58#63908—34no—
59C4×#625—34no—
60#63863—34no—
61#63856—34no—
62C23×C33C2312yes—
63C22×C3×C3⋊D3C2311yescharacteristic
64C22×C32×D3—24no—
65C22×C32×D3—24no—
66C22×C32×D3—42no—
67C22×C32×D3C2312yes—
68C22×C32×D3C2311yescharacteristic
69C33×D4D414yescomplemented
70C2×C33×C4C2311yescharacteristic
71C2×C3⋊(C3⋊Dic3)—42no—
72C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
73C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
74C2×C3⋊(C3⋊Dic3)C2312yes—
75C3×#638—24no—
76C2×C3×C3⋊Dic3—24no—
77C2×C3×C3⋊Dic3C2312yes—
78C2×C3×C3⋊Dic3C2311yescharacteristic
79C3×C4×C3⋊D3—22no—
80C3×#634—22no—
81C32×#67—24no—
82C2×C32×Dic3C2311yescharacteristic
83C32×D12—22no—
84C32×C4×D3—22no—
85#6290—22no—
86#6290—42no—
87#5829—91no—
88#5806—94no—
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×D314yes—
96C2×C32×D4C2×D311yescharacteristic
97C22×C32×C4—31no—
98C2×#638—31no—
99C2×#638—32no—
100C2×#638C2×D312yescomplemented
101C22×C3⋊Dic3—32no—
102C22×C3⋊Dic3—38no—
103C22×C3⋊Dic3—68no—
104C22×C3⋊Dic3C2×D312yescomplemented
105#3830—24no—
106C2×C4×C3⋊D3C2×D311yescharacteristic, complemented
107C2×#634C2×D311yescharacteristic, complemented
108C2×C3×#67—38no—
109C2×C3×#67C2×D318yes—
110C22×C3×Dic3—38no—
111C3×#237—616no—
112C3×D3×D4—216no—
113C2×C3×D12C2×D314yes—
114C2×C3×C4×D3—34no—
115C2×C3×C4×D3C2×D314yes—
116C2×C3×Dic6—34no—
117C2×#625—34no—
118C2×#625—38no—
119#3757—32no—
120#3757—38no—
121#3757—32no—
122#3757—31no—
123#3757—32no—
124#3757—32no—
125#3757—32no—
126#3757—68no—
127#3752—31no—
128C4⋊(C3⋊Dic3)—31no—
129C4⋊(C3⋊Dic3)—34no—
130C4⋊(C3⋊Dic3)—64no—
131#3750—32no—
132#3750—31no—
133C4×C3⋊Dic3—31no—
134C4×C3⋊Dic3—34no—
135C4×C3⋊Dic3—64no—
136#3724—38no—
137#3724—34no—
138(C2×D3)⋊Dic3—38no—
139(C2×D3)⋊Dic3—38no—
140(C2×D3)⋊Dic3—38no—
141(C2×D3)⋊Dic3—34no—
142(C2×D3)⋊Dic3—38no—
143(C2×D3)⋊Dic3—34no—
144C22×C33—22no—
145C22×C33—22no—
146C22×C33C2×D414yes—
147C22×C33C2411yescharacteristic
148C2×C3×C3⋊D3—21no—
149C2×C3×C3⋊D3—21no—
150C2×C3×C3⋊D3—21no—
151C2×C32×D3—21no—
152C2×C32×D3—21no—
153C2×C32×D3—22no—
154C2×C32×D3—22no—
155C2×C32×D3—22no—
156C2×C32×D3—48no—
157C2×C32×D3#4112yescomplemented
158C33×C4C2×D412yes—
159C3⋊(C3⋊Dic3)—21no—
160C3⋊(C3⋊Dic3)—21no—
161C3⋊(C3⋊Dic3)—22no—
162C3×C3⋊Dic3—22no—
163C3×C3⋊Dic3—21no—
164C32×Dic3—21no—
165C2×C3×#41—91no—
166C22×C3×D4—34no—
167C2×#237—34no—
168C2×D3×D4C3⋊D311yescharacteristic, complemented
169#953—98no—
170C2×#217—98no—
171#947—91no—
172#946—91no—
173#939—98no—
174#939—92no—
175#938—94no—
176#937—92no—
177#936—94no—
178#935—98no—
179D4×Dic3—94no—
180D4×Dic3—91no—
181D4×Dic3—184no—
182#930—91no—
183#929—92no—
184C4×#67—94no—
185C4×#67—91no—
186C2×(C2×D3)⋊C4—92no—
187#924—94no—
188C23×C32—28no—
189C23×C32—31no—
190C23×C32—32no—
191C23×C32—64no—
192C23×C32—64no—
193C23×C32—122no—
194C23×C32C22×D318yes—
195C23×C32C22×D312yes—
196C22×C3⋊D3C22×D311yescharacteristic
197C22×C3×D3—216no—
198C22×C3×D3—216no—
199C22×C3×D3—34no—
200C22×C3×D3—48no—
201C22×C3×D3C22×D318yes—
202C22×C3×D3C22×D314yes—
203C32×D4—216no—
204C32×D4—64no—
205C32×D4—62no—
206C32×D4#67116yescomplemented
207C32×D4#6714yescomplemented
208C2×C32×C4—24no—
209C2×C32×C4—31no—
210C2×C32×C4—62no—
211C2×C32×C4C22×D314yes—
212C2×C32×C4C22×D311yescharacteristic
213#638—24no—
214#638—62no—
215#638—122no—
216C2×C3⋊Dic3—24no—
217C2×C3⋊Dic3—31no—
218C2×C3⋊Dic3—31no—
219C2×C3⋊Dic3—32no—
220C2×C3⋊Dic3—34no—
221C2×C3⋊Dic3—34no—
222C2×C3⋊Dic3—38no—
223C2×C3⋊Dic3—64no—
224C2×C3⋊Dic3—64no—
225C2×C3⋊Dic3—68no—
226C2×C3⋊Dic3—122no—
227C2×C3⋊Dic3—128no—
228C2×C3⋊Dic3—1216no—
229C2×C3⋊Dic3C22×D312yes—
230C2×C3⋊Dic3C22×D311yescharacteristic
231C4×C3⋊D3—22no—
232#634—22no—
233C3×#67—216no—
234C3×#67—616no—
235C2×C3×Dic3—38no—
236C2×C3×Dic3—34no—
237C2×C3×Dic3—616no—
238C2×C3×Dic3C22×D314yes—
239C3×D12—28no—
240C3×C4×D3—28no—
241C3×C4×D3—68no—
242C3×Dic6—68no—
243#625—68no—
244#625—128no—
245#546—271noSylow
246C2×C33—24no—
247C2×C33C22×D411yescharacteristic
248C2×C33C2×#4111yescharacteristic
249C2×C33H(5,𝔽2)11yescharacteristic
250C3×C3⋊D3—41no—
251C32×D3—22no—
252C32×D3—42no—
253C24×C3—38no—
254C23×D3C2×C3⋊D312yes—
255C3×#41—184no—
256C2×C3×Q8—91no—
257C2×C3×D4—24no—
258C2×C3×D4—38no—
259C2×C3×D4—34no—
260C2×C3×D4—616no—
261C2×C3×D4—92no—
262C2×C3×D4D3214yes—
263C2×C3×D4C2×C3⋊D311yescharacteristic
264C22×C3×C4—34no—
265C22×C3×C4—92no—
266C22×C3×C4—91no—
267C2×#67—38no—
268C2×#67—94no—
269C2×#67—91no—
270C2×#67—98no—
271C2×#67—92no—
272C2×#67C2×C3⋊D312yes—
273C22×Dic3—38no—
274C22×Dic3—98no—
275C22×Dic3—92no—
276C22×Dic3—188no—
277#237—616no—
278D3×D4—24no—
279C2×D12C2×C3⋊D311yescharacteristic
280C2×C4×D3—34no—
281C2×C4×D3C2×C3⋊D311yescharacteristic
282C2×Dic6—34no—
283#217—98no—
284#217—94no—
285#217—98no—
286#217—98no—
287#217—98no—
288#217—98no—
289#217—92no—
290#217—91no—
291#217—188no—
292(C2×D3)⋊C4—94no—
293(C2×D3)⋊C4—91no—
294(C2×D3)⋊C4—92no—
295(C2×D3)⋊C4—92no—
296(C2×D3)⋊C4—92no—
297(C2×D3)⋊C4—92no—
298C4⋊Dic3—94no—
299C4⋊Dic3—91no—
300C4⋊Dic3—184no—
301Dic3⋊C4—98no—
302Dic3⋊C4—94no—
303Dic3⋊C4—91no—
304Dic3⋊C4—92no—
305C4×Dic3—94no—
306C4×Dic3—91no—
307C4×Dic3—184no—
308C22×C32—22no—
309C22×C32—22no—
310C22×C32—24no—
311C22×C32—216no—
312C22×C32—28no—
313C22×C32—28no—
314C22×C32—32no—
315C22×C32—48no—
316C22×C32—48no—
317C22×C32—61no—
318C22×C32—61no—
319C22×C32—62no—
320C22×C32—62no—
321C22×C32—62no—
322C22×C32—128no—
323C22×C32C2×#67116yes—
324C22×C32C2×#6714yes—
325C22×C32C23×D314yes—
326C22×C32C23×D311yescharacteristic
327C2×C3⋊D3—21no—
328C2×C3⋊D3—21no—
329C2×C3⋊D3—21no—
330C2×C3×D3—24no—
331C2×C3×D3—24no—
332C2×C3×D3—28no—
333C2×C3×D3—28no—
334C2×C3×D3—28no—
335C2×C3×D3—432no—
336C2×C3×D3—64no—
337C2×C3×D3—64no—
338C2×C3×D3—64no—
339C2×C3×D3#23718yescomplemented
340C32×C4—28no—
341C32×C4—61no—
342C32×C4C2×#6718yes—
343C32×C4C2×#6712yes—
344C3⋊Dic3—22no—
345C3⋊Dic3—21no—
346C3⋊Dic3—68no—
347C3⋊Dic3—64no—
348C3⋊Dic3—61no—
349C3⋊Dic3—61no—
350C3⋊Dic3—62no—
351C3⋊Dic3—64no—
352C3⋊Dic3—68no—
353C3⋊Dic3—124no—
354C3⋊Dic3—128no—
355C3×Dic3—24no—
356C3×Dic3—68no—
357C3×Dic3—64no—
358C2×#41—91no—
359C22×D4—31no—
360#124—271no—
361#123—272no—
362C22⋊Q8—271no—
363#121—271no—
364#121—272no—
365#120—272no—
366C4×D4—271no—
367C4×D4—271no—
368C2×C22⋊C4—272no—
369C33#54611yescharacteristic, complemented, Sylow
370C23×C3—28no—
371C23×C3—34no—
372C23×C3—38no—
373C23×C3—616no—
374C23×C3—616no—
375C23×C3—91no—
376C23×C3—128no—
377C23×C3C2×D3218yes—
378C23×C3C22×C3⋊D312yes—
379C22×D3—24no—
380C22×D3—24no—
381C22×D3—34no—
382C22×D3—42no—
383C22×D3C22×C3⋊D312yes—
384C22×D3C22×C3⋊D311yescharacteristic
385C3×Q8—182no—
386C3×D4—216no—
387C3×D4—616no—
388C3×D4—68no—
389C3×D4—184no—
390C3×D4#625116yescomplemented
391C3×D4#63814yescomplemented
392C2×C3×C4—24no—
393C2×C3×C4—34no—
394C2×C3×C4—68no—
395C2×C3×C4—91no—
396C2×C3×C4—92no—
397C2×C3×C4—184no—
398C2×C3×C4—182no—
399C2×C3×C4C2×D3214yes—
400C2×C3×C4C22×C3⋊D311yescharacteristic
401#67—24no—
402#67—616no—
403#67—188no—
404#67—182no—
405#67—362no—
406#67—368no—
407C2×Dic3—38no—
408C2×Dic3—34no—
409C2×Dic3—616no—
410C2×Dic3—94no—
411C2×Dic3—94no—
412C2×Dic3—98no—
413C2×Dic3—91no—
414C2×Dic3—91no—
415C2×Dic3—92no—
416C2×Dic3—184no—
417C2×Dic3—184no—
418C2×Dic3—188no—
419C2×Dic3—368no—
420C2×Dic3—362no—
421C2×Dic3—3616no—
422C2×Dic3C22×C3⋊D311yescharacteristic
423D12—22no—
424C4×D3—22no—
425C4×D3—68no—
426Dic6—68no—
427C2×C32—24no—
428C2×C32—24no—
429C2×C32—24no—
430C2×C32—24no—
431C2×C32—216no—
432C2×C32—416no—
433C2×C32—62no—
434C2×C32—122no—
435C2×C32C2×#23714yes—
436C2×C32#100414yes—
437C2×C32D3×#4111yescharacteristic
438C2×C32#100911yescharacteristic
439C2×C32C22×#6714yes—
440C2×C32C22×#6711yescharacteristic
441C3⋊D3—41no—
442C3×D3—28no—
443C3×D3—48no—
444C3×D3—124no—
445C24—32no—
446#41—184no—
447C2×Q8—91no—
448C2×D4—32no—
449C2×D4—31no—
450C2×D4—64no—
451C2×D4—92no—
452C2×D4—271no—
453C2×D4—272no—
454C2×D4D3×C3⋊D311yescharacteristic
455C22×C4—31no—
456C22×C4—92no—
457C22×C4—91no—
458C22×C4—272no—
459C4⋊C4—271no—
460C4⋊C4—272no—
461C4⋊C4—271no—
462C22⋊C4—271no—
463C22⋊C4—271no—
464C22⋊C4—272no—
465C22⋊C4—272no—
466C22⋊C4—272no—
467C22⋊C4—272no—
468C22⋊C4—272no—
469C42—271no—
470C22×C3—28no—
471C22×C3—28no—
472C22×C3—22no—
473C22×C3—22no—
474C22×C3—24no—
475C22×C3—216no—
476C22×C3—38no—
477C22×C3—48no—
478C22×C3—48no—
479C22×C3—64no—
480C22×C3—64no—
481C22×C3—68no—
482C22×C3—68no—
483C22×C3—68no—
484C22×C3—1232no—
485C22×C3—181no—
486C22×C3—181no—
487C22×C3—181no—
488C22×C3C2×#625116yes—
489C22×C3C2×#63814yes—
490C22×C3C22×D3214yes—
491C22×C3C23×C3⋊D311yescharacteristic
492C2×D3—21no—
493C2×D3—21no—
494C2×D3—22no—
495C2×D3—22no—
496C2×D3—22no—
497C2×D3—48no—
498C2×D3—64no—
499C2×D3—64no—
500C2×D3—64no—
501C2×D3#383012yescomplemented
502C3×C4—28no—
503C3×C4—64no—
504C3×C4—181no—
505C3×C4—182no—
506C3×C4C2×#62518yes—
507C3×C4C2×#63812yes—
508Dic3—21no—
509Dic3—68no—
510Dic3—64no—
511Dic3—188no—
512Dic3—184no—
513Dic3—184no—
514Dic3—188no—
515Dic3—181no—
516Dic3—181no—
517Dic3—182no—
518Dic3—368no—
519Dic3—364no—
520C32—24no—
521C32#580614yescomplemented
522C32#582911yescharacteristic, complemented
523C23—31no—
524C23—32no—
525C23—64no—
526C23—64no—
527C23—91no—
528C23—122no—
529C23C2×D3×C3⋊D312yes—
530Q8—182no—
531D4—64no—
532D4—62no—
533D4—184no—
534D4—542no—
535D4—1082no—
536D4#629014yescomplemented
537C2×C4—31no—
538C2×C4—62no—
539C2×C4—91no—
540C2×C4—92no—
541C2×C4—184no—
542C2×C4—182no—
543C2×C4—271no—
544C2×C4—271no—
545C2×C4—272no—
546C2×C4—1082no—
547C2×C4C2×D3×C3⋊D311yescharacteristic
548C2×C3—216no—
549C2×C3—24no—
550C2×C3—24no—
551C2×C3—24no—
552C2×C3—24no—
553C2×C3—416no—
554C2×C3—68no—
555C2×C3—128no—
556C2×C3—361no—
557C2×C3D3×#23714yes—
558C2×C3#6431414yes—
559C2×C3C22×#62514yes—
560C2×C3C2×#383011yescharacteristic
561C2×C3#6436111yescharacteristic
562C2×C3C22×#63811yescharacteristic
563D3—22no—
564D3—42no—
565D3—124no—
566C22—22no—
567C22—22no—
568C22—32no—
569C22—61no—
570C22—61no—
571C22—62no—
572C22—62no—
573C22—62no—
574C22—128no—
575C22—181no—
576C22—181no—
577C22—181no—
578C22C2×#629014yes—
579C22C22×D3×C3⋊D311yescharacteristic
580C4—61no—
581C4—181no—
582C4—182no—
583C4—541no—
584C4—541no—
585C4—542no—
586C4C2×#629012yes—
587C3—24no—
588C3#1059524414yescomplemented
589C3#1059627711yescharacteristic, complemented
590C2—24no—
591C2—62no—
592C2—122no—
593C2—361no—
594C2D3×#383011yescharacteristic
595C2#1172458811yescharacteristic
596C2C22×#629011yescharacteristic
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×C3×#3830C2semidirect
2. C2×C32×D3×D4C2semidirect
3. D4×C3⋊(C3⋊Dic3)C2semidirect
4. #11723637C2semidirect
5. #11723629C2semidirect
6. #11723602C2semidirect
7. C2×(C2×D3)⋊(C3⋊Dic3)C2semidirect
8. #11723522C2semidirect
9. C4×#6290C2semidirect
10. #11723434C2semidirect
11. #11723425C2semidirect
16. C22×C3⋊(C3⋊Dic3)C22semidirect
17. C2×C3×#638C22semidirect
18. C22×C3×C3⋊Dic3C22semidirect
20. C2×C3×C4×C3⋊D3C22semidirect
21. C2×C3×#634C22semidirect
27. C2×#6290C22semidirect
28. #88864C22semidirect
29. C4⋊(C3⋊(C3⋊Dic3))C22semidirect
30. C4×C3⋊(C3⋊Dic3)C22semidirect
31. #88806C22semidirect
32. #88806C22semidirect
33. #88803C22semidirect
36. (C2×D3)⋊(C3⋊Dic3)C22semidirect
37. (C2×D3)⋊(C3⋊Dic3)C22semidirect
38. (C2×D3)⋊(C3⋊Dic3)C22semidirect
40. C2×#3830D3semidirect
42. C2×C3×D3×D4D3semidirect
69. C33×D4D4semidirect
100. C2×#638C2×D3semidirect
104. C22×C3⋊Dic3C2×D3semidirect
106. C2×C4×C3⋊D3C2×D3semidirect
107. C2×#634C2×D3semidirect
157. C2×C32×D3#41semidirect
168. C2×D3×D4C3⋊D3semidirect
206. C32×D4#67semidirect
207. C32×D4#67semidirect
339. C2×C3×D3#237semidirect
369. C33#546semidirect
390. C3×D4#625semidirect
391. C3×D4#638semidirect
501. C2×D3#3830semidirect
521. C32#5806semidirect
522. C32#5829semidirect
536. D4#6290semidirect
588. C3#10595244semidirect
589. C3#10596277semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#38302
C2×C32×D3×D42
D4×C3⋊(C3⋊Dic3)2
#117236372
#117236292
#117236022
C2×(C2×D3)⋊(C3⋊Dic3)2
#117235222
C4×#62902
#117234342
#117234252
#105962773
#105952443