Finite Group Databasebeta

← #13177415

Subgroups — #13177415

Order 1728.

2380 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×D3×D4C211yesmaximal, characteristic, complemented
2C2×#89015C211yesmaximal, characteristic, complemented
3#11723979C211yesmaximal, characteristic, complemented
4C3×#64145C211yesmaximal, characteristic, complemented
5#11723630C212yesmaximal, complemented
6#11723616C212yesmaximal, complemented
7#11723614C212yesmaximal, complemented
8D3×#3757C212yesmaximal, complemented
9C4×#6291C211yesmaximal, characteristic, complemented
10D12×C3⋊Dic3C211yesmaximal, characteristic, complemented
11#11723425C211yesmaximal, characteristic, complemented
12#10596277—31nomaximal
13#10595229—34nomaximal
14C23×C32×D3C2212yes—
15C2×C33×D4C2211yescharacteristic
16C2×#6312C2212yescomplemented
17C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic, complemented
18C2×C3×#634C2211yescharacteristic, complemented
19C2×C32×#67C2212yes—
20C32×D3×D4—24no—
21C2×C32×D12C2211yescharacteristic
22C2×C32×C4×D3C2211yescharacteristic
23C2×#6291C2211yescharacteristic
24C2×#6291C2212yescomplemented
25C2×D3×C3⋊Dic3C2212yescomplemented
26C2×D3×C3⋊Dic3C2212yescomplemented
27#89015—24no—
28#88864C2212yescomplemented
29C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic, complemented
30C3×#3757C2212yescomplemented
31C3×#3757C2212yescomplemented
32C3×C4×C3⋊Dic3C2211yescharacteristic, complemented
33#88807C2211yescharacteristic, complemented
34#88806C2212yescomplemented
35#88804C2211yescharacteristic, complemented
36(C2×D3)⋊(C3⋊Dic3)C2211yescharacteristic
37(C2×D3)⋊(C3⋊Dic3)C2212yes—
38(C2×D3)⋊(C3⋊Dic3)C2212yescomplemented
39C22×C32×D4—31no—
40C2×#3830—31no—
41C2×C3×D3×D4D314yescomplemented
42C2×#3796—34no—
43#64162—32no—
44C2×#3757—32no—
45#64148—32no—
46#64147—31no—
47#64147—34no—
48#64147—64no—
49#64145D311yescharacteristic, complemented
50#64144—32no—
51D4×C3⋊Dic3—31no—
52C4×#638—31no—
53#64133—31no—
54C3×#936—34no—
55#63977—38no—
56#63971—38no—
57#63969—38no—
58D3×#217—38no—
59C4×#626—34no—
60Dic3×D12—34no—
61#63856—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×D4—22no—
70C2×C33×C4C2311yescharacteristic
71#6312—42no—
72C2×C3⋊(C3⋊Dic3)C2312yes—
73C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
74C4×C3⋊(C3⋊D3)—22no—
75C2×C3×C3⋊Dic3C2311yescharacteristic
76C2×C3×C3⋊Dic3C2311yescharacteristic
77C2×C3×C3⋊Dic3C2312yes—
78C3×#634—22no—
79C32×#67—24no—
80C2×C32×Dic3C2311yescharacteristic
81C32×D12D414yescomplemented
82C32×C4×D3—22no—
83#6291—24no—
84#6291—22no—
85D3×C3⋊Dic3—24no—
86D3×C3⋊Dic3—42no—
87#5806—94no—
88#5676—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×D314yes—
96C2×C32×D4C2×D311yescharacteristic
97C22×C32×C4—31no—
98C2×#638—32no—
99C2×#638—38no—
100C2×#638—32no—
101C2×#638—31no—
102C2×#638—68no—
103C22×C3⋊Dic3—32no—
104C22×C3⋊Dic3—32no—
105#3830—64no—
106C2×C4×C3⋊D3—31no—
107C2×C4×C3⋊D3—34no—
108C2×C4×C3⋊D3—64no—
109C2×#634C2×D311yescharacteristic, complemented
110C2×C3×#67C2×D318yes—
111C3×D3×D4—216no—
112C2×C3×D12C2×D314yes—
113C2×C3×C4×D3C2×D314yes—
114C2×C3×Dic6—34no—
115C2×#626—38no—
116C2×#626—34no—
117C2×D3×Dic3—38no—
118C2×D3×Dic3—38no—
119#3796—616no—
120#3757—32no—
121#3757—38no—
122#3757—32no—
123#3757—31no—
124#3757—32no—
125#3757—68no—
126#3757C2×D312yescomplemented
127#3757C2×D312yescomplemented
128#3752—31no—
129C4⋊(C3⋊Dic3)—31no—
130C4⋊(C3⋊Dic3)—34no—
131C4⋊(C3⋊Dic3)—64no—
132#3750—32no—
133#3750—31no—
134C4×C3⋊Dic3C2×D311yescharacteristic, complemented
135C3×#217—38no—
136C3×#217—38no—
137C3×C4×Dic3—34no—
138#3724—38no—
139Dic3⋊Dic3—34no—
140#3722—34no—
141(C2×D3)⋊Dic3—38no—
142(C2×D3)⋊Dic3—38no—
143(C2×D3)⋊Dic3—34no—
144C22×C33—22no—
145C22×C33—22no—
146C22×C33—22no—
147C22×C33C2411yescharacteristic
148C2×C3⋊(C3⋊D3)—21no—
149C2×C3⋊(C3⋊D3)—21no—
150C2×C3⋊(C3⋊D3)—21no—
151C2×C32×D3—21no—
152C2×C32×D3—22no—
153C2×C32×D3—22no—
154C2×C32×D3—21no—
155C2×C32×D3—48no—
156C2×C32×D3C2×D414yes—
157C2×C32×D3#4112yescomplemented
158C33×C4C2×D412yes—
159C3⋊(C3⋊Dic3)—21no—
160C3⋊(C3⋊Dic3)—22no—
161C3×C3⋊Dic3—22no—
162C3×C3⋊Dic3—21no—
163C3×C3⋊Dic3—21no—
164C32×Dic3—21no—
165C22×C3×D4—34no—
166C2×#239—91no—
167C2×#237—94no—
168C2×D3×D4C3⋊D311yescharacteristic, complemented
169C3×#124—91no—
170#953—98no—
171C2×#217—98no—
172#946—91no—
173#939—98no—
174#938—94no—
175#938—91no—
176#938—184no—
177#936—34no—
178#935—98no—
179D4×Dic3—94no—
180C4×#67—94no—
181#924—94no—
182#884—92no—
183#883—92no—
184#882—92no—
185D3×C22⋊C4—92no—
186C4×D12—91no—
187C4×D12—91no—
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⋊D3—31no—
197C22×C3⋊D3—34no—
198C22×C3⋊D3—64no—
199C22×C3×D3—216no—
200C22×C3×D3—216no—
201C22×C3×D3—48no—
202C22×C3×D3C22×D314yes—
203C22×C3×D3C22×D318yes—
204C32×D4—22no—
205C32×D4—28no—
206C32×D4—34no—
207C32×D4—48no—
208C32×D4—64no—
209C2×C32×C4—24no—
210C2×C32×C4—31no—
211C2×C32×C4—62no—
212C2×C32×C4C22×D314yes—
213C2×C32×C4C22×D311yescharacteristic
214#638—64no—
215#638—62no—
216#638—122no—
217#638—128no—
218#638—1216no—
219C2×C3⋊Dic3—32no—
220C2×C3⋊Dic3—31no—
221C2×C3⋊Dic3—38no—
222C2×C3⋊Dic3—34no—
223C2×C3⋊Dic3—68no—
224C2×C3⋊Dic3—64no—
225C2×C3⋊Dic3—64no—
226C2×C3⋊Dic3—122no—
227C2×C3⋊Dic3C22×D311yescharacteristic
228C2×C3⋊Dic3C22×D311yescharacteristic
229C2×C3⋊Dic3C22×D312yes—
230C4×C3⋊D3—62no—
231C4×C3⋊D3—68no—
232C4×C3⋊D3—128no—
233#634—22no—
234C3×#67—216no—
235C2×C3×Dic3—34no—
236C2×C3×Dic3—34no—
237C2×C3×Dic3—38no—
238C2×C3×Dic3C22×D314yes—
239C3×D12#67116yescomplemented
240C3×C4×D3—28no—
241C3×Dic6—68no—
242#626—68no—
243#626—616no—
244D3×Dic3—616no—
245D3×Dic3—128no—
246#546—271noSylow
247C2×C33—42no—
248C2×C33C22×D411yescharacteristic
249C2×C33C2×#4111yescharacteristic
250C2×C33H(5,𝔽2)11yescharacteristic
251C3⋊(C3⋊D3)—41no—
252C32×D3—22no—
253C32×D3—24no—
254C24×C3—38no—
255C23×D3C2×C3⋊D312yes—
256C2×C3×Q8—91no—
257C2×C3×D4—24no—
258C2×C3×D4—38no—
259C2×C3×D4—34no—
260C2×C3×D4—616no—
261C2×C3×D4D3214yes—
262C2×C3×D4C2×C3⋊D311yescharacteristic
263C22×C3×C4—34no—
264C2×#67—94no—
265C2×#67—98no—
266C2×#67—98no—
267C2×#67—92no—
268C2×#67—188no—
269C2×#67C2×C3⋊D312yes—
270C22×Dic3—98no—
271C22×Dic3—98no—
272#239—184no—
273#237—1816no—
274D3×D4—24no—
275C2×D12—91no—
276C2×D12—92no—
277C2×D12C2×C3⋊D311yescharacteristic
278C2×C4×D3—92no—
279C2×C4×D3—94no—
280C2×C4×D3—91no—
281C2×C4×D3—92no—
282C2×C4×D3—184no—
283C2×C4×D3C2×C3⋊D311yescharacteristic
284C2×Dic6—34no—
285C3×C22⋊C4—92no—
286C3×C22⋊C4—92no—
287C3×C42—91no—
288#217—38no—
289#217—38no—
290#217—98no—
291#217—94no—
292#217—98no—
293#217—98no—
294#217—92no—
295#217—188no—
296(C2×D3)⋊C4—94no—
297(C2×D3)⋊C4—91no—
298(C2×D3)⋊C4—92no—
299(C2×D3)⋊C4—92no—
300(C2×D3)⋊C4—91no—
301C4⋊Dic3—94no—
302C4⋊Dic3—91no—
303C4⋊Dic3—91no—
304C4⋊Dic3—184no—
305Dic3⋊C4—94no—
306Dic3⋊C4—98no—
307Dic3⋊C4—92no—
308C4×Dic3—34no—
309C22×C32—22no—
310C22×C32—22no—
311C22×C32—22no—
312C22×C32—24no—
313C22×C32—28no—
314C22×C32—28no—
315C22×C32—28no—
316C22×C32—34no—
317C22×C32—32no—
318C22×C32—48no—
319C22×C32—48no—
320C22×C32—48no—
321C22×C32—61no—
322C22×C32—62no—
323C22×C32—62no—
324C22×C32—61no—
325C22×C32—128no—
326C22×C32C23×D314yes—
327C22×C32C23×D311yescharacteristic
328C2×C3⋊D3—68no—
329C2×C3⋊D3—61no—
330C2×C3⋊D3—64no—
331C2×C3⋊D3—61no—
332C2×C3⋊D3—61no—
333C2×C3⋊D3—64no—
334C2×C3⋊D3—64no—
335C2×C3⋊D3—124no—
336C2×C3⋊D3—124no—
337C2×C3×D3—28no—
338C2×C3×D3—28no—
339C2×C3×D3—24no—
340C2×C3×D3—24no—
341C2×C3×D3—432no—
342C2×C3×D3#23718yescomplemented
343C2×C3×D3C2×#67116yes—
344C32×C4—28no—
345C32×C4—61no—
346C32×C4D3×D412yes—
347C32×C4C2×#6718yes—
348C3⋊Dic3—22no—
349C3⋊Dic3—21no—
350C3⋊Dic3—21no—
351C3⋊Dic3—61no—
352C3⋊Dic3—62no—
353C3⋊Dic3—68no—
354C3⋊Dic3—64no—
355C3⋊Dic3—124no—
356C3⋊Dic3—128no—
357C3×Dic3—24no—
358C3×Dic3—64no—
359C3×Dic3—64no—
360C3×Dic3—68no—
361C2×#41—271no—
362C22×D4—31no—
363#124—91no—
364#123—272no—
365C22⋊Q8—271no—
366#121—272no—
367#121—271no—
368#120—272no—
369C4×D4—271no—
370C4×D4—271no—
371C2×C22⋊C4—272no—
372C33#54611yescharacteristic, complemented, Sylow
373C23×C3—28no—
374C23×C3—34no—
375C23×C3—38no—
376C23×C3—616no—
377C23×C3—616no—
378C23×C3—128no—
379C23×C3C2×D3218yes—
380C23×C3C22×C3⋊D312yes—
381C22×D3—24no—
382C22×D3—24no—
383C22×D3—42no—
384C22×D3—94no—
385C22×D3—91no—
386C22×D3—184no—
387C22×D3C22×C3⋊D311yescharacteristic
388C22×D3C22×C3⋊D312yes—
389C3×Q8—182no—
390C3×D4—28no—
391C3×D4—22no—
392C3×D4—316no—
393C3×D4—48no—
394C3×D4—616no—
395C2×C3×C4—24no—
396C2×C3×C4—34no—
397C2×C3×C4—68no—
398C2×C3×C4—92no—
399C2×C3×C4—91no—
400C2×C3×C4—91no—
401C2×C3×C4C2×D3214yes—
402C2×C3×C4C22×C3⋊D311yescharacteristic
403#67—24no—
404#67—1816no—
405#67—188no—
406#67—368no—
407#67—362no—
408#67—3616no—
409C2×Dic3—34no—
410C2×Dic3—34no—
411C2×Dic3—38no—
412C2×Dic3—98no—
413C2×Dic3—94no—
414C2×Dic3—92no—
415C2×Dic3—91no—
416C2×Dic3—188no—
417C2×Dic3—184no—
418C2×Dic3—1816no—
419C2×Dic3—368no—
420C2×Dic3C22×C3⋊D311yescharacteristic
421D12—182no—
422D12—184no—
423D12#63814yescomplemented
424C4×D3—22no—
425C4×D3—184no—
426C4×D3—188no—
427C4×D3—182no—
428C4×D3—368no—
429C4×D3—362no—
430Dic6—68no—
431C2×C32—24no—
432C2×C32—24no—
433C2×C32—24no—
434C2×C32—42no—
435C2×C32—48no—
436C2×C32—64no—
437C2×C32—62no—
438C2×C32—88no—
439C2×C32C2×D3×D411yescharacteristic
440C2×C32C2×#23714yes—
441C2×C32#100414yes—
442C2×C32D3×#4111yescharacteristic
443C2×C32#100911yescharacteristic
444C2×C32C22×#6714yes—
445C3⋊D3—128no—
446C3⋊D3—121no—
447C3⋊D3—124no—
448C3×D3—28no—
449C3×D3—216no—
450C24—32no—
451#41—544no—
452C2×Q8—91no—
453C2×D4—32no—
454C2×D4—31no—
455C2×D4—64no—
456C2×D4—271no—
457C2×D4—272no—
458C2×D4—272no—
459C2×D4D3×C3⋊D311yescharacteristic
460C22×C4—31no—
461C22×C4—272no—
462C22×C4—272no—
463C22×C4—271no—
464C4⋊C4—271no—
465C4⋊C4—272no—
466C4⋊C4—271no—
467C22⋊C4—92no—
468C22⋊C4—92no—
469C22⋊C4—271no—
470C22⋊C4—271no—
471C22⋊C4—272no—
472C22⋊C4—272no—
473C22⋊C4—272no—
474C42—91no—
475C22×C3—28no—
476C22×C3—28no—
477C22×C3—28no—
478C22×C3—22no—
479C22×C3—22no—
480C22×C3—22no—
481C22×C3—24no—
482C22×C3—38no—
483C22×C3—316no—
484C22×C3—48no—
485C22×C3—48no—
486C22×C3—48no—
487C22×C3—68no—
488C22×C3—68no—
489C22×C3—64no—
490C22×C3—64no—
491C22×C3—1232no—
492C22×C3C22×D3214yes—
493C22×C3C23×C3⋊D311yescharacteristic
494C2×D3—22no—
495C2×D3—22no—
496C2×D3—21no—
497C2×D3—21no—
498C2×D3—48no—
499C2×D3—188no—
500C2×D3—184no—
501C2×D3—184no—
502C2×D3—184no—
503C2×D3—181no—
504C2×D3—181no—
505C2×D3—181no—
506C2×D3—364no—
507C2×D3—364no—
508C2×D3#383012yescomplemented
509C2×D3C2×#63814yes—
510C3×C4—28no—
511C3×C4—64no—
512C3×C4—181no—
513C3×C4—181no—
514C3×C4—182no—
515C3×C4D3×#6718yes—
516C3×C4C2×#63812yes—
517Dic3—21no—
518Dic3—64no—
519Dic3—64no—
520Dic3—68no—
521Dic3—188no—
522Dic3—184no—
523Dic3—182no—
524Dic3—181no—
525Dic3—364no—
526Dic3—368no—
527C32—24no—
528C32#567611yescharacteristic, complemented
529C32#580614yescomplemented
530C23—31no—
531C23—32no—
532C23—64no—
533C23—64no—
534C23—122no—
535C23—271no—
536C23C2×D3×C3⋊D312yes—
537Q8—182no—
538D4—22no—
539D4—34no—
540D4—64no—
541D4—544no—
542D4—542no—
543D4—1082no—
544C2×C4—31no—
545C2×C4—62no—
546C2×C4—91no—
547C2×C4—91no—
548C2×C4—92no—
549C2×C4—272no—
550C2×C4—271no—
551C2×C4—544no—
552C2×C4—542no—
553C2×C4—1082no—
554C2×C4C2×D3×C3⋊D311yescharacteristic
555C2×C3—24no—
556C2×C3—24no—
557C2×C3—24no—
558C2×C3—48no—
559C2×C3—42no—
560C2×C3—68no—
561C2×C3—616no—
562C2×C3—88no—
563C2×C3D3×#23714yes—
564C2×C3#6431414yes—
565C2×C3C2×D3×#6714yes—
566C2×C3C2×#383011yescharacteristic
567C2×C3#6436111yescharacteristic
568C2×C3C22×#63811yescharacteristic
569D3—22no—
570D3—24no—
571D3—368no—
572D3—364no—
573D3—361no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—32no—
578C22—34no—
579C22—62no—
580C22—62no—
581C22—61no—
582C22—61no—
583C22—128no—
584C22—541no—
585C22—541no—
586C22—541no—
587C22C22×D3×C3⋊D311yescharacteristic
588C4—61no—
589C4—181no—
590C4—181no—
591C4—182no—
592C4—542no—
593C4—541no—
594C4D3×#63812yes—
595C3—24no—
596C3#1059522914yescomplemented
597C3#1059627711yescharacteristic, complemented
598C2—42no—
599C2—62no—
600C2—64no—
601C2—1081no—
602C2D3×#383011yescharacteristic
603C2#1172458811yescharacteristic
604C2C2×D3×#63811yescharacteristic
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×C32×D3×D4C2semidirect
2. C2×#89015C2semidirect
3. #11723979C2semidirect
4. C3×#64145C2semidirect
5. #11723630C2semidirect
6. #11723616C2semidirect
7. #11723614C2semidirect
8. D3×#3757C2semidirect
9. C4×#6291C2semidirect
10. D12×C3⋊Dic3C2semidirect
11. #11723425C2semidirect
16. C2×#6312C22semidirect
17. C2×C4×C3⋊(C3⋊D3)C22semidirect
18. C2×C3×#634C22semidirect
24. C2×#6291C22semidirect
25. C2×D3×C3⋊Dic3C22semidirect
26. C2×D3×C3⋊Dic3C22semidirect
28. #88864C22semidirect
29. C4⋊(C3⋊(C3⋊Dic3))C22semidirect
30. C3×#3757C22semidirect
31. C3×#3757C22semidirect
32. C3×C4×C3⋊Dic3C22semidirect
33. #88807C22semidirect
34. #88806C22semidirect
35. #88804C22semidirect
38. (C2×D3)⋊(C3⋊Dic3)C22semidirect
41. C2×C3×D3×D4D3semidirect
49. #64145D3semidirect
81. C32×D12D4semidirect
109. C2×#634C2×D3semidirect
126. #3757C2×D3semidirect
127. #3757C2×D3semidirect
134. C4×C3⋊Dic3C2×D3semidirect
157. C2×C32×D3#41semidirect
168. C2×D3×D4C3⋊D3semidirect
239. C3×D12#67semidirect
342. C2×C3×D3#237semidirect
372. C33#546semidirect
423. D12#638semidirect
508. C2×D3#3830semidirect
528. C32#5676semidirect
529. C32#5806semidirect
596. C3#10595229semidirect
597. C3#10596277semidirect

Maximal subgroups

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

subgroupindex
C2×C32×D3×D42
C2×#890152
#117239792
C3×#641452
#117236302
#117236162
#117236142
D3×#37572
C4×#62912
D12×C3⋊Dic32
#117234252
#105962773
#105952293