Finite Group Databasebeta

← #13177399

Subgroups — #13177399

Order 1728.

2304 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3830C211yesmaximal, characteristic, complemented
2C2×#89022C211yesmaximal, characteristic, complemented
3#11723981C211yesmaximal, characteristic, complemented
4C32×#938C211yesmaximal, characteristic, complemented
5#11723635C212yesmaximal, complemented
6#11723633C212yesmaximal, complemented
7#11723623C212yesmaximal, complemented
8#11723622C212yesmaximal, complemented
9#11723526C211yesmaximal, characteristic, complemented
10C4×#6290C211yesmaximal, characteristic, complemented
11#11723410C211yesmaximal, characteristic, complemented
12#10596070—31nomaximal
13#10595236—34nomaximal
14C2×C33×D4C2211yescharacteristic
15C2×#6312C2212yes—
16C2×#6312C2212yes—
17C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
18C2×C3×#638C2212yes—
19C22×C3×C3⋊Dic3C2212yes—
20C3×#3830—24no—
21C2×C3×C4×C3⋊D3C2211yescharacteristic
22C2×C3×#634C2211yescharacteristic
23C2×C32×#67C2212yes—
24C2×C32×C4×D3C2211yescharacteristic
25C2×#6292C2212yes—
26C2×#6291C2212yes—
27C2×#6290C2211yescharacteristic
28C2×#6289C2212yes—
29#89022—24no—
30C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
31C32×#217C2212yes—
32C32×C4⋊Dic3C2211yescharacteristic
33#88807C2212yes—
34#88806C2211yescharacteristic
35#88804C2212yes—
36#88803C2211yescharacteristic
37(C2×D3)⋊(C3⋊Dic3)C2211yescharacteristic
38Dic3×C3⋊Dic3C2212yes—
39C2×#3830D311yescharacteristic, complemented
40C2×#3828—31no—
41C2×C3×#237—34no—
42C2×#3799—34no—
43C32×#121—31no—
44#64149—31no—
45#64149—32no—
46#64149—34no—
47#64149—64no—
48#64148—32no—
49C4×#638—31no—
50#64104—31no—
51#64096—31no—
52#64093—32no—
53#64090—32no—
54C3×#938D314yescomplemented
55#63979—38no—
56#63978—38no—
57#63958—38no—
58#63952—38no—
59#63909—34no—
60C4×#625—34no—
61#63843—34no—
62C23×C33C2312yes—
63C22×C3⋊(C3⋊D3)C2312yes—
64C22×C3×C3⋊D3C2311yescharacteristic
65C22×C32×D3C2311yescharacteristic
66C33×D4—22no—
67C2×C33×C4C2311yescharacteristic
68#6312—24no—
69#6312—42no—
70C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
71C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
72C4⋊(C3⋊(C3⋊D3))—22no—
73C3×#638—24no—
74C2×C3×C3⋊Dic3—24no—
75C2×C3×C3⋊Dic3C2311yescharacteristic
76C2×C3×C3⋊Dic3C2312yes—
77C3×C4×C3⋊D3—22no—
78C3×#634D414yescomplemented
79C32×#67—42no—
80C2×C32×Dic3C2311yescharacteristic
81C2×C32×Dic3C2312yes—
82C32×C4×D3—22no—
83#6292—42no—
84#6291—24no—
85#6290—22no—
86#6289—24no—
87#5834—91no—
88#5599—94no—
89C2×C32×D4—24no—
90C2×C32×D4—32no—
91C2×C32×D4C2×D314yes—
92C2×C32×D4C2×D311yescharacteristic
93C22×C32×C4—31no—
94C2×#638—32no—
95C2×#638—38no—
96C2×#638—31no—
97C2×#638—32no—
98C2×#638—38no—
99C2×#638—32no—
100C2×#638—68no—
101C2×#638—68no—
102C2×#638C2×D312yes—
103C22×C3⋊Dic3C2×D312yes—
104#3830—24no—
105#3828—64no—
106C2×C4⋊(C3⋊D3)—31no—
107C2×C4⋊(C3⋊D3)—34no—
108C2×C4⋊(C3⋊D3)—32no—
109C2×C4⋊(C3⋊D3)—64no—
110C2×C4×C3⋊D3—32no—
111C2×C4×C3⋊D3C2×D311yescharacteristic
112C2×#634C2×D311yescharacteristic
113C2×C3×#67—38no—
114C2×C3×#67C2×D318yes—
115C22×C3×Dic3—38no—
116C3×#237—616no—
117C2×C3×C4×D3—34no—
118C2×C3×C4×D3C2×D314yes—
119C2×C3×Dic6—34no—
120C2×#626—38no—
121C2×#626—38no—
122C2×#625—34no—
123C2×#624—38no—
124#3799—616no—
125C32×C4⋊C4—31no—
126C32×C22⋊C4—32no—
127#3757—31no—
128#3752—31no—
129#3752—32no—
130#3750—31no—
131#3750—32no—
132C4×C3⋊Dic3—31no—
133C4×C3⋊Dic3—34no—
134C4×C3⋊Dic3—32no—
135C4×C3⋊Dic3—64no—
136C3×#217C2×D318yes—
137C3×C4⋊Dic3C2×D314yes—
138#3724—34no—
139Dic3⋊Dic3—38no—
140#3722—38no—
141(C2×D3)⋊Dic3—34no—
142(C2×D3)⋊Dic3—34no—
143Dic32—38no—
144C22×C33—22no—
145C22×C33—22no—
146C22×C33—22no—
147C22×C33C2411yescharacteristic
148C2×C3⋊(C3⋊D3)—22no—
149C2×C3⋊(C3⋊D3)—22no—
150C2×C3⋊(C3⋊D3)—22no—
151C2×C3×C3⋊D3—21no—
152C2×C3×C3⋊D3—21no—
153C2×C3×C3⋊D3—21no—
154C2×C32×D3—21no—
155C2×C32×D3—21no—
156C2×C32×D3—21no—
157C33×C4C2×D412yes—
158C3⋊(C3⋊Dic3)—21no—
159C3⋊(C3⋊Dic3)#4112yescomplemented
160C3×C3⋊Dic3—21no—
161C3×C3⋊Dic3C2×D414yes—
162C32×Dic3—21no—
163C32×Dic3—22no—
164C2×C3×#41—91no—
165C2×#239—91no—
166C2×#237—34no—
167C2×#235—94no—
168C3×#121—34no—
169Q8×Dic3—91no—
170#940—98no—
171#940—94no—
172#940—91no—
173#940—92no—
174#940—184no—
175#939—98no—
176#938—92no—
177#938C3⋊D311yescharacteristic, complemented
178#930—92no—
179#930—91no—
180C4×#67—94no—
181C4×#67—91no—
182C4×#67—92no—
183#895—94no—
184#887—94no—
185#884—98no—
186#67⋊C4—98no—
187C23×C32—28no—
188C23×C32—31no—
189C23×C32C22×D318yes—
190C23×C32C22×D312yes—
191C22×C3⋊D3—32no—
192C22×C3⋊D3—38no—
193C22×C3⋊D3—68no—
194C22×C3⋊D3C22×D311yescharacteristic
195C22×C3×D3—34no—
196C22×C3×D3C22×D314yes—
197C32×D4—22no—
198C32×D4—28no—
199C32×D4—48no—
200C32×D4—122no—
201C2×C32×C4—24no—
202C2×C32×C4—32no—
203C2×C32×C4—31no—
204C2×C32×C4—62no—
205C2×C32×C4C22×D314yes—
206C2×C32×C4C22×D311yescharacteristic
207#638—24no—
208#638—62no—
209#638—64no—
210#638—64no—
211#638—616no—
212#638—1216no—
213#638—122no—
214#638—128no—
215#638—1216no—
216C2×C3⋊Dic3—24no—
217C2×C3⋊Dic3—31no—
218C2×C3⋊Dic3—31no—
219C2×C3⋊Dic3—34no—
220C2×C3⋊Dic3—34no—
221C2×C3⋊Dic3—64no—
222C2×C3⋊Dic3—64no—
223C2×C3⋊Dic3C22×D311yescharacteristic
224C2×C3⋊Dic3C22×D312yes—
225C4⋊(C3⋊D3)—62no—
226C4⋊(C3⋊D3)—68no—
227C4⋊(C3⋊D3)—128no—
228C4⋊(C3⋊D3)—122no—
229C4×C3⋊D3—22no—
230C4×C3⋊D3—64no—
231#634#6714yescomplemented
232C3×#67—48no—
233C3×#67—616no—
234C2×C3×Dic3—38no—
235C2×C3×Dic3—34no—
236C2×C3×Dic3—616no—
237C2×C3×Dic3C22×D318yes—
238C2×C3×Dic3C22×D314yes—
239C3×C4×D3—28no—
240C3×C4×D3—68no—
241C3×Dic6—316no—
242#626—616no—
243#626—128no—
244#625—68no—
245#624—616no—
246#550—271noSylow
247C2×C33—42no—
248C2×C33C22×D411yescharacteristic
249C2×C33C2×#4111yescharacteristic
250C2×C33H(5,𝔽2)11yescharacteristic
251C3⋊(C3⋊D3)—42no—
252C3×C3⋊D3—41no—
253C32×D3—41no—
254C3×#41—184no—
255C2×C3×Q8—91no—
256C2×C3×D4—24no—
257C2×C3×D4—38no—
258C2×C3×D4—92no—
259C2×C3×D4D3214yes—
260C2×C3×D4C2×C3⋊D311yescharacteristic
261C22×C3×C4—34no—
262C22×C3×C4—92no—
263C22×C3×C4—91no—
264C2×#67—38no—
265C2×#67—98no—
266C2×#67—94no—
267C2×#67—98no—
268C2×#67—92no—
269C2×#67—91no—
270C2×#67—98no—
271C2×#67—92no—
272C2×#67—92no—
273C2×#67—188no—
274C2×#67—188no—
275C2×#67C2×C3⋊D312yes—
276C22×Dic3—38no—
277#239—184no—
278#237—616no—
279#235—1816no—
280C2×D12—92no—
281C2×D12—94no—
282C2×D12—91no—
283C2×D12—98no—
284C2×D12—184no—
285C2×C4×D3—34no—
286C2×C4×D3—98no—
287C2×C4×D3—92no—
288C2×C4×D3C2×C3⋊D311yescharacteristic
289C2×Dic6—34no—
290C3×C4⋊C4—34no—
291C3×C22⋊C4—38no—
292#217—94no—
293#217—91no—
294#217C2×C3⋊D312yes—
295(C2×D3)⋊C4—98no—
296(C2×D3)⋊C4—94no—
297(C2×D3)⋊C4—91no—
298(C2×D3)⋊C4—92no—
299C4⋊Dic3—92no—
300C4⋊Dic3C2×C3⋊D311yescharacteristic
301Dic3⋊C4—94no—
302Dic3⋊C4—98no—
303Dic3⋊C4—91no—
304C4×Dic3—98no—
305C4×Dic3—94no—
306C4×Dic3—91no—
307C4×Dic3—92no—
308C4×Dic3—184no—
309C22×C32—22no—
310C22×C32—22no—
311C22×C32—22no—
312C22×C32—24no—
313C22×C32—28no—
314C22×C32—28no—
315C22×C32—28no—
316C22×C32—48no—
317C22×C32—48no—
318C22×C32—48no—
319C22×C32—61no—
320C22×C32—61no—
321C22×C32—61no—
322C22×C32C23×D314yes—
323C22×C32C23×D311yescharacteristic
324C2×C3⋊D3—21no—
325C2×C3⋊D3—21no—
326C2×C3⋊D3—21no—
327C2×C3⋊D3—62no—
328C2×C3⋊D3—62no—
329C2×C3⋊D3—62no—
330C2×C3⋊D3—68no—
331C2×C3⋊D3—68no—
332C2×C3⋊D3—68no—
333C2×C3⋊D3—128no—
334C2×C3⋊D3—128no—
335C2×C3⋊D3—128no—
336C2×C3×D3—24no—
337C2×C3×D3—24no—
338C2×C3×D3—24no—
339C2×C3×D3—64no—
340C2×C3×D3—64no—
341C2×C3×D3—64no—
342C32×C4—28no—
343C32×C4—62no—
344C32×C4—61no—
345C32×C4D3×D418yes—
346C32×C4C2×#6712yes—
347C3⋊Dic3—21no—
348C3⋊Dic3—32no—
349C3⋊Dic3—38no—
350C3⋊Dic3—68no—
351C3⋊Dic3—68no—
352C3⋊Dic3—64no—
353C3⋊Dic3—61no—
354C3⋊Dic3C2×#6714yes—
355C3×Dic3—24no—
356C3×Dic3—28no—
357C3×Dic3—316no—
358C3×Dic3—64no—
359C2×#41—91no—
360C2×#41—271no—
361#127—272no—
362#127—271no—
363#121—31no—
364#121—272no—
365#121—271no—
366#121—272no—
367C4×Q8—271no—
368C4×D4—272no—
369C4×D4—271no—
370C33#55011yescharacteristic, complemented, Sylow
371C23×C3—28no—
372C23×C3—34no—
373C23×C3—91no—
374C23×C3C2×D3218yes—
375C23×C3C22×C3⋊D312yes—
376C22×D3—34no—
377C22×D3—98no—
378C22×D3—92no—
379C22×D3—188no—
380C22×D3C22×C3⋊D311yescharacteristic
381C3×Q8—94no—
382C3×D4—28no—
383C3×D4—22no—
384C3×D4—48no—
385C3×D4—128no—
386C3×D4—184no—
387C2×C3×C4—24no—
388C2×C3×C4—38no—
389C2×C3×C4—34no—
390C2×C3×C4—68no—
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—42no—
398#67—616no—
399#67—1816no—
400#67—182no—
401#67—188no—
402#67—184no—
403#67—1816no—
404#67—3616no—
405#67—362no—
406#67—368no—
407#67—362no—
408#67—3616no—
409C2×Dic3—38no—
410C2×Dic3—34no—
411C2×Dic3—616no—
412C2×Dic3—94no—
413C2×Dic3—91no—
414C2×Dic3—94no—
415C2×Dic3—91no—
416C2×Dic3—184no—
417C2×Dic3—184no—
418C2×Dic3C22×C3⋊D312yes—
419C2×Dic3C22×C3⋊D311yescharacteristic
420D12—184no—
421D12—188no—
422D12—182no—
423D12—368no—
424D12—368no—
425C4×D3—22no—
426C4×D3—68no—
427C4×D3—184no—
428C4×D3—1816no—
429Dic6—316no—
430C2×C32—24no—
431C2×C32—24no—
432C2×C32—24no—
433C2×C32—42no—
434C2×C32—48no—
435C2×C32—88no—
436C2×C32—121no—
437C2×C32C2×D3×D414yes—
438C2×C32#100414yes—
439C2×C32D3×#4114yes—
440C2×C32D3×#4111yescharacteristic
441C2×C32#100911yescharacteristic
442C2×C32C22×#6711yescharacteristic
443C3⋊D3—41no—
444C3⋊D3—122no—
445C3⋊D3—128no—
446C3⋊D3—248no—
447C3×D3—44no—
448C3×D3—124no—
449#41—184no—
450#41—544no—
451C2×Q8—91no—
452C2×D4—32no—
453C2×D4—92no—
454C2×D4—272no—
455C2×D4—272no—
456C2×D4—271no—
457C2×D4—272no—
458C2×D4—272no—
459C2×D4—271no—
460C2×D4D3×C3⋊D311yescharacteristic
461C22×C4—31no—
462C22×C4—92no—
463C22×C4—91no—
464C22×C4—272no—
465C4⋊C4—31no—
466C4⋊C4—271no—
467C4⋊C4—272no—
468C22⋊C4—32no—
469C22⋊C4—272no—
470C22⋊C4—271no—
471C22⋊C4—271no—
472C42—272no—
473C42—271no—
474C22×C3—28no—
475C22×C3—28no—
476C22×C3—28no—
477C22×C3—22no—
478C22×C3—22no—
479C22×C3—22no—
480C22×C3—24no—
481C22×C3—48no—
482C22×C3—48no—
483C22×C3—48no—
484C22×C3—64no—
485C22×C3—64no—
486C22×C3—64no—
487C22×C3—181no—
488C22×C3—181no—
489C22×C3—181no—
490C22×C3C22×D3214yes—
491C22×C3C23×C3⋊D311yescharacteristic
492C2×D3—21no—
493C2×D3—21no—
494C2×D3—21no—
495C2×D3—64no—
496C2×D3—64no—
497C2×D3—64no—
498C2×D3—188no—
499C2×D3—188no—
500C2×D3—188no—
501C2×D3—182no—
502C2×D3—182no—
503C2×D3—182no—
504C2×D3—368no—
505C2×D3—368no—
506C2×D3—368no—
507C3×C4—28no—
508C3×C4—68no—
509C3×C4—64no—
510C3×C4—94no—
511C3×C4—181no—
512C3×C4D3×#6718yes—
513C3×C4D4×C3⋊D312yes—
514Dic3—21no—
515Dic3—22no—
516Dic3—316no—
517Dic3—64no—
518Dic3—98no—
519Dic3—92no—
520Dic3—188no—
521Dic3—188no—
522Dic3—184no—
523Dic3—181no—
524C32—24no—
525C32#559914yescomplemented
526C32#583411yescharacteristic, complemented
527C23—31no—
528C23—91no—
529C23—272no—
530C23C2×D3×C3⋊D312yes—
531Q8—94no—
532D4—22no—
533D4—122no—
534D4—184no—
535D4—544no—
536D4—544no—
537D4—542no—
538D4—542no—
539D4—1082no—
540D4—1082no—
541C2×C4—32no—
542C2×C4—31no—
543C2×C4—62no—
544C2×C4—92no—
545C2×C4—91no—
546C2×C4—184no—
547C2×C4—182no—
548C2×C4—271no—
549C2×C4—271no—
550C2×C4—544no—
551C2×C4C2×D3×C3⋊D311yescharacteristic
552C2×C3—24no—
553C2×C3—24no—
554C2×C3—24no—
555C2×C3—48no—
556C2×C3—42no—
557C2×C3—88no—
558C2×C3—124no—
559C2×C3—361no—
560C2×C3#6431214yes—
561C2×C3#6431414yes—
562C2×C3C2×D3×#6714yes—
563C2×C3C2×D4×C3⋊D311yescharacteristic
564C2×C3#6436111yescharacteristic
565C2×C3#41×C3⋊D311yescharacteristic
566D3—41no—
567D3—124no—
568D3—368no—
569D3—362no—
570D3—728no—
571C22—22no—
572C22—22no—
573C22—22no—
574C22—61no—
575C22—61no—
576C22—61no—
577C22—181no—
578C22—181no—
579C22—181no—
580C22—542no—
581C22—542no—
582C22—542no—
583C22C22×D3×C3⋊D311yescharacteristic
584C4—62no—
585C4—61no—
586C4—94no—
587C4—181no—
588C4—272no—
589C4—541no—
590C4C3⋊D3×#6712yes—
591C3—24no—
592C3#1059523614yescomplemented
593C3#1059607011yescharacteristic, complemented
594C2—42no—
595C2—121no—
596C2—361no—
597C2—1082no—
598C2#1172458611yescharacteristic
599C2#1172458811yescharacteristic
600C2C2×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×#89022C2semidirect
3. #11723981C2semidirect
4. C32×#938C2semidirect
5. #11723635C2semidirect
6. #11723633C2semidirect
7. #11723623C2semidirect
8. #11723622C2semidirect
9. #11723526C2semidirect
10. C4×#6290C2semidirect
11. #11723410C2semidirect
39. C2×#3830D3semidirect
54. C3×#938D3semidirect
78. C3×#634D4semidirect
159. C3⋊(C3⋊Dic3)#41semidirect
177. #938C3⋊D3semidirect
231. #634#67semidirect
370. C33#550semidirect
525. C32#5599semidirect
526. C32#5834semidirect
592. C3#10595236semidirect
593. C3#10596070semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#38302
C2×#890222
#117239812
C32×#9382
#117236352
#117236332
#117236232
#117236222
#117235262
C4×#62902
#117234102
#105960703
#105952363