Finite Group Databasebeta

← #13177397

Subgroups — #13177397

Order 1728.

2248 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3830C211yesmaximal, characteristic, complemented
2C2×#89015C211yesmaximal, characteristic, complemented
3#11723979C211yesmaximal, characteristic, complemented
4C32×#940C211yesmaximal, characteristic, complemented
5#11723635C212yesmaximal, complemented
6#11723633C212yesmaximal, complemented
7#67×C3⋊Dic3C212yesmaximal, complemented
8#11723615C212yesmaximal, complemented
9#11723522C211yesmaximal, characteristic, complemented
10C4×#6292C211yesmaximal, characteristic, complemented
11Dic3×#634C211yesmaximal, characteristic, complemented
12#10596189—31nomaximal
13#10595158—34nomaximal
14C2×C33×D4C2211yescharacteristic
15C2×#6312C2212yes—
16C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
17C2×C3×#638C2212yes—
18C22×C3×C3⋊Dic3C2212yes—
19C3×#3830—24no—
20C2×C3×C4×C3⋊D3C2211yescharacteristic
21C2×C3×#634C2211yescharacteristic
22C2×C32×#67C2212yes—
23C2×C32×#67C2212yes—
24C2×C32×D12C2211yescharacteristic
25C2×#6292C2211yescharacteristic
26C2×#6291C2212yes—
27C2×#6290C2212yes—
28C2×D3×C3⋊Dic3C2212yes—
29#89015—24no—
30#88864C2212yes—
31C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
32C32×C4×Dic3C2211yescharacteristic
33#88807C2212yes—
34Dic3⋊(C3⋊Dic3)C2211yescharacteristic
35#88804C2211yescharacteristic
36#88803C2211yescharacteristic
37(C2×D3)⋊(C3⋊Dic3)C2212yes—
38Dic3×C3⋊Dic3C2212yes—
39C2×#3830—31no—
40C2×#3830D311yescharacteristic, complemented
41C2×C3×#237—34no—
42C2×#3796—34no—
43C32×#127—31no—
44#64149—32no—
45#64148—32no—
46#64148—32no—
47#64147—31no—
48#64147—31no—
49#64147—34no—
50#64147—64no—
51D4×C3⋊Dic3—32no—
52C4×C4⋊(C3⋊D3)—31no—
53C4×#634—31no—
54C3×#940D314yescomplemented
55#63979—38no—
56#63978—38no—
57Dic3×#67—38no—
58#63970—38no—
59#63908—34no—
60C4×#626—34no—
61Dic3×Dic6—34no—
62C23×C33C2312yes—
63C22×C3⋊(C3⋊D3)C2311yescharacteristic
64C22×C3×C3⋊D3C2311yescharacteristic
65C22×C32×D3C2312yes—
66C33×D4—22no—
67C2×C33×C4C2311yescharacteristic
68#6312—42no—
69C2×C3⋊(C3⋊Dic3)C2312yes—
70C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
71C4×C3⋊(C3⋊D3)—22no—
72C3×#638—24no—
73C2×C3×C3⋊Dic3—24no—
74C2×C3×C3⋊Dic3C2311yescharacteristic
75C2×C3×C3⋊Dic3C2312yes—
76C3×C4×C3⋊D3—22no—
77C3×#634D414yescomplemented
78C32×#67—24no—
79C32×#67—42no—
80C2×C32×Dic3C2311yescharacteristic
81C2×C32×Dic3C2311yescharacteristic
82C32×D12—22no—
83#6292—22no—
84#6291—24no—
85#6290—42no—
86D3×C3⋊Dic3—24no—
87#5834—91no—
88#5718—94no—
89C2×C32×D4—24no—
90C2×C32×D4—32no—
91C2×C32×D4—32no—
92C2×C32×D4—31no—
93C2×C32×D4C2×D314yes—
94C2×C32×D4C2×D311yescharacteristic
95C2×#638—32no—
96C2×#638—38no—
97C2×#638—32no—
98C2×#638—32no—
99C2×#638—68no—
100C2×#638C2×D312yes—
101C22×C3⋊Dic3—32no—
102C22×C3⋊Dic3C2×D312yes—
103#3830—24no—
104#3830—64no—
105C2×C4⋊(C3⋊D3)—31no—
106C2×C4×C3⋊D3—31no—
107C2×C4×C3⋊D3—34no—
108C2×C4×C3⋊D3—64no—
109C2×C4×C3⋊D3C2×D311yescharacteristic
110C2×#634C2×D311yescharacteristic
111C2×C3×#67—38no—
112C2×C3×#67C2×D318yes—
113C2×C3×#67C2×D318yes—
114C22×C3×Dic3—38no—
115C3×#237—616no—
116C2×C3×D12C2×D314yes—
117C2×C3×C4×D3—34no—
118C2×C3×Dic6—34no—
119C2×#626—38no—
120C2×#626—34no—
121C2×#625—38no—
122C2×D3×Dic3—38no—
123#3796—616no—
124C32×C42—31no—
125#3757—32no—
126#3757—38no—
127#3757—32no—
128#3757—68no—
129#3752—31no—
130#3752—31no—
131C4⋊(C3⋊Dic3)—31no—
132C4⋊(C3⋊Dic3)—34no—
133C4⋊(C3⋊Dic3)—31no—
134C4⋊(C3⋊Dic3)—64no—
135#3750—32no—
136C4×C3⋊Dic3—32no—
137C3×C4×Dic3C2×D314yes—
138Dic3⋊Dic3—38no—
139Dic3⋊Dic3—34no—
140#3722—34no—
141(C2×D3)⋊Dic3—38no—
142(C2×D3)⋊Dic3—34no—
143Dic32—38no—
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×C3×C3⋊D3—21no—
152C2×C3×C3⋊D3—21no—
153C2×C3×C3⋊D3—21no—
154C2×C32×D3—22no—
155C2×C32×D3—22no—
156C2×C32×D3—22no—
157C33×C4C2×D412yes—
158C3⋊(C3⋊Dic3)—21no—
159C3⋊(C3⋊Dic3)—22no—
160C3×C3⋊Dic3—21no—
161C3×C3⋊Dic3C2×D414yes—
162C32×Dic3—21no—
163C32×Dic3#4112yescomplemented
164C2×C3×#41—91no—
165C2×#239—91no—
166C2×#237—34no—
167C2×#237—94no—
168C3×#127—34no—
169Q8×Dic3—91no—
170#940—98no—
171#940—92no—
172#940C3⋊D311yescharacteristic, complemented
173#939—98no—
174#939—98no—
175#938—94no—
176#938—94no—
177#938—91no—
178#938—92no—
179#938—184no—
180D4×Dic3—98no—
181#930—92no—
182#930—91no—
183C4×#67—92no—
184C4×#67—91no—
185C4×D12—94no—
186C4×Dic6—94no—
187C23×C32—28no—
188C23×C32—32no—
189C23×C32C22×D318yes—
190C23×C32C22×D312yes—
191C22×C3⋊D3—31no—
192C22×C3⋊D3—34no—
193C22×C3⋊D3—64no—
194C22×C3⋊D3C22×D311yescharacteristic
195C22×C3×D3—34no—
196C22×C3×D3C22×D318yes—
197C32×D4—22no—
198C32×D4—28no—
199C32×D4—48no—
200C32×D4—64no—
201C32×D4—62no—
202C32×D4—122no—
203C2×C32×C4—24no—
204C2×C32×C4—31no—
205C2×C32×C4—31no—
206C2×C32×C4C22×D314yes—
207C2×C32×C4C22×D311yescharacteristic
208#638—24no—
209#638—64no—
210#638—122no—
211#638—122no—
212#638—128no—
213#638—1216no—
214C2×C3⋊Dic3—24no—
215C2×C3⋊Dic3—32no—
216C2×C3⋊Dic3—31no—
217C2×C3⋊Dic3—38no—
218C2×C3⋊Dic3—34no—
219C2×C3⋊Dic3—68no—
220C2×C3⋊Dic3—64no—
221C2×C3⋊Dic3—64no—
222C2×C3⋊Dic3C22×D311yescharacteristic
223C2×C3⋊Dic3C22×D312yes—
224C4⋊(C3⋊D3)—62no—
225C4×C3⋊D3—22no—
226C4×C3⋊D3—62no—
227C4×C3⋊D3—68no—
228C4×C3⋊D3—128no—
229#634#6714yescomplemented
230C3×#67—216no—
231C3×#67—48no—
232C3×#67—616no—
233C2×C3×Dic3—38no—
234C2×C3×Dic3—34no—
235C2×C3×Dic3—616no—
236C2×C3×Dic3C22×D314yes—
237C2×C3×Dic3C22×D314yes—
238C3×D12—28no—
239C3×C4×D3—68no—
240C3×Dic6—316no—
241#626—616no—
242#626—68no—
243#625—128no—
244D3×Dic3—616no—
245#550—271noSylow
246C2×C33—42no—
247C2×C33C22×D411yescharacteristic
248C2×C33C2×#4111yescharacteristic
249C2×C33H(5,𝔽2)11yescharacteristic
250C3⋊(C3⋊D3)—41no—
251C3×C3⋊D3—41no—
252C32×D3—42no—
253C3×#41—184no—
254C2×C3×Q8—91no—
255C2×C3×D4—24no—
256C2×C3×D4—38no—
257C2×C3×D4—38no—
258C2×C3×D4—34no—
259C2×C3×D4—92no—
260C2×C3×D4D3214yes—
261C2×C3×D4C2×C3⋊D311yescharacteristic
262C22×C3×C4—92no—
263C22×C3×C4—91no—
264C2×#67—38no—
265C2×#67—98no—
266C2×#67—98no—
267C2×#67—98no—
268C2×#67—92no—
269C2×#67—92no—
270C2×#67—91no—
271C2×#67—188no—
272C2×#67C2×C3⋊D312yes—
273C2×#67C2×C3⋊D312yes—
274C22×Dic3—38no—
275C22×Dic3—98no—
276#239—184no—
277#237—616no—
278#237—1816no—
279C2×D12—94no—
280C2×D12—92no—
281C2×D12C2×C3⋊D311yescharacteristic
282C2×C4×D3—34no—
283C2×C4×D3—92no—
284C2×C4×D3—94no—
285C2×C4×D3—91no—
286C2×C4×D3—184no—
287C2×Dic6—34no—
288C3×C42—34no—
289#217—98no—
290#217—98no—
291#217—92no—
292#217—91no—
293#217—188no—
294(C2×D3)⋊C4—94no—
295(C2×D3)⋊C4—94no—
296(C2×D3)⋊C4—92no—
297(C2×D3)⋊C4—91no—
298C4⋊Dic3—94no—
299C4⋊Dic3—94no—
300C4⋊Dic3—91no—
301C4⋊Dic3—92no—
302C4⋊Dic3—184no—
303Dic3⋊C4—98no—
304Dic3⋊C4—91no—
305C4×Dic3—98no—
306C4×Dic3—92no—
307C4×Dic3C2×C3⋊D311yescharacteristic
308C22×C32—22no—
309C22×C32—22no—
310C22×C32—22no—
311C22×C32—24no—
312C22×C32—28no—
313C22×C32—28no—
314C22×C32—28no—
315C22×C32—48no—
316C22×C32—48no—
317C22×C32—48no—
318C22×C32—62no—
319C22×C32—62no—
320C22×C32—62no—
321C22×C32C23×D314yes—
322C22×C32C23×D311yescharacteristic
323C2×C3⋊D3—21no—
324C2×C3⋊D3—21no—
325C2×C3⋊D3—21no—
326C2×C3⋊D3—68no—
327C2×C3⋊D3—64no—
328C2×C3⋊D3—61no—
329C2×C3⋊D3—61no—
330C2×C3⋊D3—61no—
331C2×C3⋊D3—64no—
332C2×C3⋊D3—64no—
333C2×C3⋊D3—124no—
334C2×C3⋊D3—124no—
335C2×C3×D3—28no—
336C2×C3×D3—28no—
337C2×C3×D3—28no—
338C2×C3×D3—64no—
339C2×C3×D3—64no—
340C2×C3×D3—64no—
341C32×C4—28no—
342C32×C4—32no—
343C32×C4—61no—
344C32×C4D3×D418yes—
345C32×C4C2×#6712yes—
346C3⋊Dic3—21no—
347C3⋊Dic3—62no—
348C3⋊Dic3—61no—
349C3⋊Dic3—64no—
350C3⋊Dic3—68no—
351C3⋊Dic3—124no—
352C3⋊Dic3—128no—
353C3⋊Dic3C2×#6714yes—
354C3×Dic3—24no—
355C3×Dic3—316no—
356C3×Dic3—64no—
357C3×Dic3#23718yescomplemented
358C2×#41—91no—
359C2×#41—271no—
360#127—31no—
361#127—272no—
362#121—271no—
363#121—272no—
364#121—272no—
365#121—271no—
366C4×Q8—271no—
367C4×D4—271no—
368C4×D4—272no—
369C33#55011yescharacteristic, complemented, Sylow
370C23×C3—28no—
371C23×C3—38no—
372C23×C3—91no—
373C23×C3C2×D3218yes—
374C23×C3C22×C3⋊D312yes—
375C22×D3—34no—
376C22×D3—94no—
377C22×D3—91no—
378C22×D3—184no—
379C22×D3C22×C3⋊D312yes—
380C3×Q8—94no—
381C3×D4—28no—
382C3×D4—22no—
383C3×D4—48no—
384C3×D4—616no—
385C3×D4—68no—
386C3×D4—128no—
387C3×D4—184no—
388C2×C3×C4—24no—
389C2×C3×C4—34no—
390C2×C3×C4—34no—
391C2×C3×C4—92no—
392C2×C3×C4—91no—
393C2×C3×C4—184no—
394C2×C3×C4—182no—
395C2×C3×C4C2×D3214yes—
396C2×C3×C4C22×C3⋊D311yescharacteristic
397#67—24no—
398#67—42no—
399#67—616no—
400#67—1816no—
401#67—182no—
402#67—362no—
403#67—368no—
404#67—368no—
405#67—362no—
406#67—3616no—
407C2×Dic3—38no—
408C2×Dic3—34no—
409C2×Dic3—616no—
410C2×Dic3—98no—
411C2×Dic3—94no—
412C2×Dic3—92no—
413C2×Dic3—91no—
414C2×Dic3—188no—
415C2×Dic3—184no—
416C2×Dic3—1816no—
417C2×Dic3C22×C3⋊D311yescharacteristic
418C2×Dic3C22×C3⋊D311yescharacteristic
419D12—22no—
420D12—188no—
421D12—184no—
422C4×D3—68no—
423C4×D3—184no—
424C4×D3—188no—
425C4×D3—182no—
426C4×D3—368no—
427Dic6—316no—
428C2×C32—24no—
429C2×C32—24no—
430C2×C32—24no—
431C2×C32—42no—
432C2×C32—48no—
433C2×C32—88no—
434C2×C32—122no—
435C2×C32C2×D3×D414yes—
436C2×C32C2×#23714yes—
437C2×C32#100414yes—
438C2×C32D3×#4111yescharacteristic
439C2×C32#100911yescharacteristic
440C2×C32C22×#6711yescharacteristic
441C3⋊D3—41no—
442C3⋊D3—128no—
443C3⋊D3—124no—
444C3⋊D3—121no—
445C3×D3—48no—
446C3×D3—124no—
447#41—184no—
448#41—544no—
449C2×Q8—91no—
450C2×D4—32no—
451C2×D4—32no—
452C2×D4—31no—
453C2×D4—92no—
454C2×D4—271no—
455C2×D4—272no—
456C2×D4—272no—
457C2×D4—272no—
458C2×D4D3×C3⋊D311yescharacteristic
459C22×C4—92no—
460C22×C4—91no—
461C22×C4—272no—
462C22×C4—271no—
463C4⋊C4—271no—
464C4⋊C4—272no—
465C4⋊C4—271no—
466C22⋊C4—271no—
467C22⋊C4—271no—
468C22⋊C4—272no—
469C22⋊C4—272no—
470C42—31no—
471C42—272no—
472C22×C3—28no—
473C22×C3—28no—
474C22×C3—28no—
475C22×C3—22no—
476C22×C3—22no—
477C22×C3—22no—
478C22×C3—24no—
479C22×C3—48no—
480C22×C3—48no—
481C22×C3—48no—
482C22×C3—68no—
483C22×C3—68no—
484C22×C3—68no—
485C22×C3—181no—
486C22×C3—181no—
487C22×C3—181no—
488C22×C3C22×D3214yes—
489C22×C3C23×C3⋊D311yescharacteristic
490C2×D3—22no—
491C2×D3—22no—
492C2×D3—22no—
493C2×D3—64no—
494C2×D3—64no—
495C2×D3—64no—
496C2×D3—188no—
497C2×D3—184no—
498C2×D3—184no—
499C2×D3—184no—
500C2×D3—181no—
501C2×D3—181no—
502C2×D3—181no—
503C2×D3—364no—
504C2×D3—364no—
505C3×C4—28no—
506C3×C4—38no—
507C3×C4—64no—
508C3×C4—94no—
509C3×C4—181no—
510C3×C4D3×#6718yes—
511C3×C4D4×C3⋊D312yes—
512Dic3—21no—
513Dic3—316no—
514Dic3—64no—
515Dic3—188no—
516Dic3—184no—
517Dic3—181no—
518Dic3—182no—
519Dic3—364no—
520Dic3—368no—
521Dic3#383012yescomplemented
522C32—24no—
523C32#571814yescomplemented
524C32#583411yescharacteristic, complemented
525C23—32no—
526C23—91no—
527C23—271no—
528C23C2×D3×C3⋊D312yes—
529Q8—94no—
530D4—22no—
531D4—64no—
532D4—62no—
533D4—122no—
534D4—184no—
535D4—542no—
536D4—544no—
537D4—1082no—
538D4—1082no—
539C2×C4—31no—
540C2×C4—31no—
541C2×C4—92no—
542C2×C4—91no—
543C2×C4—184no—
544C2×C4—182no—
545C2×C4—272no—
546C2×C4—271no—
547C2×C4—544no—
548C2×C4—542no—
549C2×C4C2×D3×C3⋊D311yescharacteristic
550C2×C3—24no—
551C2×C3—24no—
552C2×C3—24no—
553C2×C3—48no—
554C2×C3—42no—
555C2×C3—88no—
556C2×C3—128no—
557C2×C3—361no—
558C2×C3D3×#23714yes—
559C2×C3#6431414yes—
560C2×C3C2×D3×#6714yes—
561C2×C3C2×D4×C3⋊D311yescharacteristic
562C2×C3C2×#383011yescharacteristic
563C2×C3#6436111yescharacteristic
564D3—42no—
565D3—124no—
566D3—368no—
567D3—364no—
568D3—361no—
569C22—22no—
570C22—22no—
571C22—22no—
572C22—62no—
573C22—62no—
574C22—62no—
575C22—181no—
576C22—181no—
577C22—181no—
578C22—541no—
579C22—541no—
580C22—541no—
581C22C22×D3×C3⋊D311yescharacteristic
582C4—32no—
583C4—61no—
584C4—94no—
585C4—181no—
586C4—542no—
587C4—541no—
588C4C3⋊D3×#6712yes—
589C3—24no—
590C3#1059515814yescomplemented
591C3#1059618911yescharacteristic, complemented
592C2—42no—
593C2—122no—
594C2—361no—
595C2—1081no—
596C2D3×#383011yescharacteristic
597C2#1172458811yescharacteristic
598C2C2×C3⋊D3×#6711yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC2×C33×D4C22
hypercenterC2×D4D3×C3⋊D3
hypocenterC33#550
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×#89015C2semidirect
3. #11723979C2semidirect
4. C32×#940C2semidirect
5. #11723635C2semidirect
6. #11723633C2semidirect
7. #67×C3⋊Dic3C2semidirect
8. #11723615C2semidirect
9. #11723522C2semidirect
10. C4×#6292C2semidirect
11. Dic3×#634C2semidirect
40. C2×#3830D3semidirect
54. C3×#940D3semidirect
77. C3×#634D4semidirect
163. C32×Dic3#41semidirect
172. #940C3⋊D3semidirect
229. #634#67semidirect
357. C3×Dic3#237semidirect
369. C33#550semidirect
521. Dic3#3830semidirect
523. C32#5718semidirect
524. C32#5834semidirect
590. C3#10595158semidirect
591. C3#10596189semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#38302
C2×#890152
#117239792
C32×#9402
#117236352
#117236332
#67×C3⋊Dic32
#117236152
#117235222
C4×#62922
Dic3×#6342
#105961893
#105951583