Finite Group Databasebeta

← #13178046

Subgroups — #13178046

Order 1728.

2500 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C3×C3⋊Dic3C211yesmaximal, characteristic, complemented
2C2×#89028C211yesmaximal, characteristic, complemented
3#11723994C211yesmaximal, characteristic, complemented
4C32×#953C211yesmaximal, characteristic, complemented
5#11723641C212yesmaximal, complemented
6#11723635C212yesmaximal, complemented
7#11723622C212yesmaximal, complemented
8#67×C3⋊Dic3C212yesmaximal, complemented
9#11723608C211yesmaximal, characteristic, complemented
10#11723602C211yesmaximal, characteristic, complemented
11#11723595C211yesmaximal, characteristic, complemented
12#10596057—31nomaximal
13#10595498—34nomaximal
14C24×C33C2211yescharacteristic
15C2×#6312C2211yescharacteristic
16C2×#6312C2212yes—
17C22×C3×C3⋊Dic3—24no—
18C22×C3×C3⋊Dic3C2211yescharacteristic
19C22×C3×C3⋊Dic3C2212yes—
20C22×C3×C3⋊Dic3C2212yes—
21C22×C3×C3⋊Dic3C2211yescharacteristic
22C2×C32×#67C2211yescharacteristic
23C2×C32×#67C2212yes—
24C2×#6293C2211yescharacteristic, complemented
25C2×#6291C2211yescharacteristic, complemented
26C2×#6289C2211yescharacteristic, complemented
27#89028—24no—
28C2×D3×C3⋊Dic3C2211yescharacteristic, complemented
29#88864C2211yescharacteristic
30#88864C2212yes—
31C32×#217C2211yescharacteristic
32C32×#217C2212yes—
33#88807C2212yescomplemented
34#88806C2212yescomplemented
35Dic3⋊(C3⋊Dic3)C2212yescomplemented
36#88804C2212yescomplemented
37(C2×D3)⋊(C3⋊Dic3)C2212yescomplemented
38Dic3×C3⋊Dic3C2212yescomplemented
39C23×C3⋊Dic3D311yescharacteristic, complemented
40C2×#3830—31no—
41C23×C3×Dic3—34no—
42C2×#3804—34no—
43C32×#120—31no—
44#64162—31no—
45#64162—34no—
46#64162—64no—
47#64148—32no—
48#64144—31no—
49D4×C3⋊Dic3—32no—
50#64095—31no—
51#64090—32no—
52#64087—32no—
53#64086—31no—
54C3×#953D314yescomplemented
55#63978—38no—
56Dic3×#67—38no—
57#63966—34no—
58#63964—38no—
59#63961—34no—
60#63955—34no—
61#63952—38no—
62C23×C33—22no—
63C23×C33—24no—
64C23×C33C2311yescharacteristic
65C23×C33C2312yes—
66C22×C3⋊(C3⋊D3)C2311yescharacteristic
67C22×C32×D3C2311yescharacteristic
68#6312—22no—
69#6312—24no—
70C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
71C2×C3⋊(C3⋊Dic3)C2312yes—
72C2×C3×C3⋊Dic3—22no—
73C2×C3×C3⋊Dic3—24no—
74C2×C3×C3⋊Dic3—24no—
75C2×C3×C3⋊Dic3D414yescomplemented
76C2×C3×C3⋊Dic3C2312yes—
77C2×C3×C3⋊Dic3C2311yescharacteristic
78C2×C3×C3⋊Dic3C2311yescharacteristic
79C32×#67—22no—
80C32×#67—24no—
81C2×C32×Dic3C2311yescharacteristic
82C2×C32×Dic3C2312yes—
83#6293—22no—
84#6291—22no—
85#6289—22no—
86D3×C3⋊Dic3—22no—
87#5791—91no—
88#5586—94no—
89C24×C32—24no—
90C24×C32C2×D314yes—
91C24×C32C2×D311yescharacteristic
92C2×C32×D4—31no—
93C2×C32×D4—32no—
94C2×#638—32no—
95C2×#638—31no—
96C2×#638—34no—
97C2×#638—38no—
98C2×#638—31no—
99C2×#638—68no—
100C2×#638—64no—
101C22×C3⋊Dic3—24no—
102C22×C3⋊Dic3—31no—
103C22×C3⋊Dic3C2×D312yes—
104C22×C3⋊Dic3C2×D312yes—
105C22×C3⋊Dic3C2×D311yescharacteristic
106C22×C3⋊Dic3C2×D311yescharacteristic
107#3830—64no—
108C2×C4×C3⋊D3—31no—
109C2×#634—31no—
110C2×C3×#67C2×D314yes—
111C2×C3×#67C2×D318yes—
112C22×C3×Dic3—34no—
113C22×C3×Dic3—38no—
114C22×C3×Dic3—34no—
115C22×C3×Dic3—38no—
116C22×C3×Dic3—616no—
117C2×#627—34no—
118C2×#626—34no—
119C2×#624—34no—
120#3804—616no—
121C2×D3×Dic3—34no—
122C32×C22⋊C4—31no—
123C32×C22⋊C4—32no—
124#3757—31no—
125#3757—32no—
126#3757—34no—
127#3757—38no—
128#3757—32no—
129#3757—64no—
130#3757—68no—
131#3752—32no—
132C4⋊(C3⋊Dic3)—32no—
133#3750—32no—
134#3750—32no—
135C4×C3⋊Dic3—32no—
136C3×#217C2×D318yes—
137C3×#217C2×D314yes—
138#3724—38no—
139Dic3⋊Dic3—38no—
140Dic3⋊Dic3—38no—
141#3722—38no—
142(C2×D3)⋊Dic3—38no—
143Dic32—38no—
144C22×C33—28no—
145C22×C33—22no—
146C22×C33—22no—
147C22×C33—22no—
148C22×C33C2×D412yes—
149C22×C33#4114yescomplemented
150C22×C33C2411yescharacteristic
151C2×C3⋊(C3⋊D3)—21no—
152C2×C3⋊(C3⋊D3)—22no—
153C2×C32×D3—21no—
154C2×C32×D3—22no—
155C3⋊(C3⋊Dic3)—21no—
156C3⋊(C3⋊Dic3)—22no—
157C3×C3⋊Dic3—22no—
158C3×C3⋊Dic3C2×D412yes—
159C3×C3⋊Dic3C2×D412yes—
160C32×Dic3—21no—
161C32×Dic3—22no—
162C23×C3×C4—91no—
163C23×Dic3—34no—
164C2×#237—94no—
165C2×#235—91no—
166C3×#120—34no—
167#953—94no—
168#953—91no—
169#953—184no—
170#953C3⋊D311yescharacteristic, complemented
171#939—98no—
172#935—94no—
173D4×Dic3—98no—
174#930—92no—
175#929—91no—
176#929—91no—
177C4×#67—92no—
178C4×#67—92no—
179#926—91no—
180#924—92no—
181#886—94no—
182#67⋊C4—98no—
183#878—98no—
184#877—94no—
185C23×C32—24no—
186C23×C32—22no—
187C23×C32—28no—
188C23×C32—24no—
189C23×C32—28no—
190C23×C32—216no—
191C23×C32—31no—
192C23×C32—416no—
193C23×C32—48no—
194C23×C32C22×D314yes—
195C23×C32C22×D318yes—
196C23×C32C22×D312yes—
197C23×C32C22×D311yescharacteristic
198C22×C3⋊D3—31no—
199C22×C3⋊D3—34no—
200C22×C3⋊D3—64no—
201C22×C3×D3C22×D314yes—
202C32×D4—64no—
203C32×D4—62no—
204C2×C32×C4—32no—
205C2×C32×C4—31no—
206#638—64no—
207#638—62no—
208#638—62no—
209#638—68no—
210#638—616no—
211#638—1216no—
212#638—128no—
213C2×C3⋊Dic3—22no—
214C2×C3⋊Dic3—24no—
215C2×C3⋊Dic3—24no—
216C2×C3⋊Dic3—32no—
217C2×C3⋊Dic3—31no—
218C2×C3⋊Dic3—38no—
219C2×C3⋊Dic3—34no—
220C2×C3⋊Dic3—68no—
221C2×C3⋊Dic3—64no—
222C2×C3⋊Dic3—62no—
223C2×C3⋊Dic3#6714yescomplemented
224C2×C3⋊Dic3C22×D312yes—
225C2×C3⋊Dic3C22×D311yescharacteristic
226C2×C3⋊Dic3C22×D311yescharacteristic
227C4×C3⋊D3—62no—
228#634—62no—
229C3×#67—28no—
230C3×#67—216no—
231C2×C3×Dic3—316no—
232C2×C3×Dic3—34no—
233C2×C3×Dic3—34no—
234C2×C3×Dic3—38no—
235C2×C3×Dic3—616no—
236C2×C3×Dic3—616no—
237C2×C3×Dic3—68no—
238C2×C3×Dic3C22×D318yes—
239C2×C3×Dic3C22×D314yes—
240#627—68no—
241#626—68no—
242#624—68no—
243D3×Dic3—68no—
244#525—271noSylow
245C2×C33—22no—
246C2×C33—24no—
247C2×C33C22×D411yescharacteristic
248C2×C33C2×#4112yes—
249C3⋊(C3⋊D3)—41no—
250C32×D3—41no—
251C24×C3—24no—
252C24×C3D3214yes—
253C24×C3C2×C3⋊D311yescharacteristic
254C2×C3×D4—38no—
255C2×C3×D4—34no—
256C22×C3×C4—91no—
257C22×C3×C4—92no—
258C22×C3×C4—92no—
259C22×C3×C4—91no—
260C22×C3×C4—184no—
261C2×#67—94no—
262C2×#67—98no—
263C2×#67—94no—
264C2×#67—92no—
265C2×#67—91no—
266C2×#67—188no—
267C2×#67—184no—
268C2×#67C2×C3⋊D312yes—
269C2×#67C2×C3⋊D311yescharacteristic
270C22×Dic3—34no—
271C22×Dic3—38no—
272C22×Dic3—38no—
273C22×Dic3—34no—
274C22×Dic3—616no—
275C22×Dic3—94no—
276#237—1816no—
277#235—184no—
278C2×D12—91no—
279C2×C4×D3—94no—
280C2×C4×D3—91no—
281C2×C4×D3—91no—
282C2×Dic6—94no—
283C2×Dic6—91no—
284C3×C22⋊C4—34no—
285C3×C22⋊C4—38no—
286#217—98no—
287#217—94no—
288#217—98no—
289#217—92no—
290#217—91no—
291#217—184no—
292#217—188no—
293#217C2×C3⋊D312yes—
294#217C2×C3⋊D311yescharacteristic
295(C2×D3)⋊C4—98no—
296(C2×D3)⋊C4—92no—
297(C2×D3)⋊C4—92no—
298C4⋊Dic3—98no—
299C4⋊Dic3—92no—
300Dic3⋊C4—98no—
301Dic3⋊C4—98no—
302Dic3⋊C4—92no—
303Dic3⋊C4—92no—
304C4×Dic3—98no—
305C4×Dic3—92no—
306C22×C32—28no—
307C22×C32—22no—
308C22×C32—22no—
309C22×C32—22no—
310C22×C32—24no—
311C22×C32—216no—
312C22×C32—28no—
313C22×C32—232no—
314C22×C32—28no—
315C22×C32—28no—
316C22×C32—28no—
317C22×C32—48no—
318C22×C32—48no—
319C22×C32—48no—
320C22×C32—432no—
321C22×C32—62no—
322C22×C32—61no—
323C22×C32#23514yescomplemented
324C22×C32D3×D418yes—
325C22×C32#237116yescomplemented
326C22×C32C2×#6712yes—
327C22×C32C23×D314yes—
328C22×C32C23×D311yescharacteristic
329C2×C3⋊D3—62no—
330C2×C3⋊D3—61no—
331C2×C3⋊D3—64no—
332C2×C3⋊D3—68no—
333C2×C3⋊D3—128no—
334C2×C3⋊D3—124no—
335C2×C3×D3—24no—
336C2×C3×D3—28no—
337C32×C4—62no—
338C32×C4—61no—
339C3⋊Dic3—22no—
340C3⋊Dic3—62no—
341C3⋊Dic3—61no—
342C3⋊Dic3—64no—
343C3⋊Dic3—68no—
344C3⋊Dic3—128no—
345C3⋊Dic3—124no—
346C3⋊Dic3C2×#6712yes—
347C3⋊Dic3C2×#6712yes—
348C3×Dic3—24no—
349C3×Dic3—28no—
350C3×Dic3—38no—
351C3×Dic3—38no—
352C3×Dic3—68no—
353C2×#41—271no—
354C23×C4—91no—
355#123—271no—
356#123—271no—
357C22⋊Q8—272no—
358#121—272no—
359#120—31no—
360#120—271no—
361C4×D4—272no—
362C4×D4—272no—
363#117—271no—
364C33#52511yescharacteristic, complemented, Sylow
365C23×C3—216no—
366C23×C3—28no—
367C23×C3—22no—
368C23×C3—24no—
369C23×C3—28no—
370C23×C3—24no—
371C23×C3—34no—
372C23×C3—416no—
373C23×C3—48no—
374C23×C3C2×D3218yes—
375C23×C3C2×D3214yes—
376C23×C3C22×C3⋊D311yescharacteristic
377C23×C3C22×C3⋊D312yes—
378C22×D3—94no—
379C22×D3—91no—
380C22×D3—184no—
381C22×D3C22×C3⋊D311yescharacteristic
382C3×D4—616no—
383C3×D4—68no—
384C2×C3×C4—38no—
385C2×C3×C4—34no—
386C2×C3×C4—91no—
387C2×C3×C4—91no—
388C2×C3×C4—94no—
389C2×C3×C4—92no—
390C2×C3×C4—184no—
391C2×C3×C4—184no—
392C2×C3×C4—182no—
393#67—22no—
394#67—24no—
395#67—188no—
396#67—1816no—
397#67—188no—
398#67—182no—
399#67—184no—
400#67—3616no—
401#67—368no—
402C2×Dic3—34no—
403C2×Dic3—34no—
404C2×Dic3—316no—
405C2×Dic3—38no—
406C2×Dic3—616no—
407C2×Dic3—616no—
408C2×Dic3—68no—
409C2×Dic3—98no—
410C2×Dic3—94no—
411C2×Dic3—92no—
412C2×Dic3—91no—
413C2×Dic3—188no—
414C2×Dic3—184no—
415C2×Dic3—188no—
416C2×Dic3C22×C3⋊D312yes—
417C2×Dic3C22×C3⋊D311yescharacteristic
418D12—182no—
419C4×D3—182no—
420C4×D3—188no—
421C4×D3—182no—
422Dic6—182no—
423Dic6—188no—
424C2×C32—24no—
425C2×C32—22no—
426C2×C32—28no—
427C2×C32—24no—
428C2×C32—28no—
429C2×C32—216no—
430C2×C32—416no—
431C2×C32—48no—
432C2×C32—121no—
433C2×C32C2×#23512yes—
434C2×C32C2×D3×D414yes—
435C2×C32C2×#23718yes—
436C2×C32C22×#6711yescharacteristic
437C3⋊D3—121no—
438C3⋊D3—124no—
439C3⋊D3—244no—
440C3×D3—44no—
441C24D3×C3⋊D311yescharacteristic
442#41—544no—
443C2×Q8—271no—
444C2×D4—32no—
445C2×D4—31no—
446C2×D4—271no—
447C2×D4—272no—
448C2×D4—271no—
449C22×C4—91no—
450C22×C4—92no—
451C22×C4—92no—
452C22×C4—91no—
453C22×C4—184no—
454C22×C4—271no—
455C22×C4—271no—
456C4⋊C4—272no—
457C4⋊C4—272no—
458C4⋊C4—272no—
459C22⋊C4—31no—
460C22⋊C4—32no—
461C22⋊C4—272no—
462C22⋊C4—272no—
463C22⋊C4—271no—
464C22⋊C4—272no—
465C42—272no—
466C22×C3—28no—
467C22×C3—232no—
468C22×C3—28no—
469C22×C3—28no—
470C22×C3—28no—
471C22×C3—22no—
472C22×C3—22no—
473C22×C3—22no—
474C22×C3—24no—
475C22×C3—216no—
476C22×C3—28no—
477C22×C3—48no—
478C22×C3—48no—
479C22×C3—48no—
480C22×C3—432no—
481C22×C3—68no—
482C22×C3—64no—
483C22×C3#3804116yescomplemented
484C22×C3D3×#6718yes—
485C22×C3D4×C3⋊D312yes—
486C22×C3#383014yescomplemented
487C22×C3C22×D3214yes—
488C22×C3C23×C3⋊D311yescharacteristic
489C2×D3—21no—
490C2×D3—22no—
491C2×D3—188no—
492C2×D3—184no—
493C2×D3—181no—
494C2×D3—182no—
495C2×D3—368no—
496C2×D3—364no—
497C3×C4—68no—
498C3×C4—64no—
499C3×C4—92no—
500C3×C4—92no—
501C3×C4—182no—
502Dic3—21no—
503Dic3—22no—
504Dic3—38no—
505Dic3—38no—
506Dic3—68no—
507Dic3—188no—
508Dic3—184no—
509Dic3—181no—
510Dic3—182no—
511Dic3—368no—
512Dic3—364no—
513C32—24no—
514C32#558614yescomplemented
515C32#579111yescharacteristic, complemented
516C23—24no—
517C23—22no—
518C23—31no—
519C23—271no—
520C23C2×D3×C3⋊D312yes—
521C23C2×D3×C3⋊D311yescharacteristic
522Q8—542no—
523D4—64no—
524D4—62no—
525D4—542no—
526D4—544no—
527D4—542no—
528C2×C4—32no—
529C2×C4—31no—
530C2×C4—91no—
531C2×C4—91no—
532C2×C4—94no—
533C2×C4—92no—
534C2×C4—184no—
535C2×C4—184no—
536C2×C4—182no—
537C2×C4—272no—
538C2×C4—271no—
539C2×C4—542no—
540C2×C4—542no—
541C2×C3—216no—
542C2×C3—28no—
543C2×C3—22no—
544C2×C3—24no—
545C2×C3—28no—
546C2×C3—24no—
547C2×C3—416no—
548C2×C3—48no—
549C2×C3—124no—
550C2×C3C2×#380418yes—
551C2×C3C2×D3×#6714yes—
552C2×C3C2×D4×C3⋊D311yescharacteristic
553C2×C3C2×#383012yes—
554D3—41no—
555D3—364no—
556D3—361no—
557D3—724no—
558C22—28no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—62no—
563C22—61no—
564C22—542no—
565C22—541no—
566C22#8902814yescomplemented
567C22C3⋊D3×#6712yes—
568C22C22×D3×C3⋊D311yescharacteristic
569C4—62no—
570C4—61no—
571C4—92no—
572C4—92no—
573C4—182no—
574C4—542no—
575C4—541no—
576C3—24no—
577C3#1059549814yescomplemented
578C3#1059605711yescharacteristic, complemented
579C2—24no—
580C2—22no—
581C2—121no—
582C2—1081no—
583C2C2×#8902812yes—
584C2C2×C3⋊D3×#6711yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC24×C33C22
hypercenterC24D3×C3⋊D3
hypocenterC33#525
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C24

Complements

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

NH ≅ G/Ntype
1. C23×C3×C3⋊Dic3C2semidirect
2. C2×#89028C2semidirect
3. #11723994C2semidirect
4. C32×#953C2semidirect
5. #11723641C2semidirect
6. #11723635C2semidirect
7. #11723622C2semidirect
8. #67×C3⋊Dic3C2semidirect
9. #11723608C2semidirect
10. #11723602C2semidirect
11. #11723595C2semidirect
24. C2×#6293C22semidirect
25. C2×#6291C22semidirect
26. C2×#6289C22semidirect
28. C2×D3×C3⋊Dic3C22semidirect
33. #88807C22semidirect
34. #88806C22semidirect
35. Dic3⋊(C3⋊Dic3)C22semidirect
36. #88804C22semidirect
37. (C2×D3)⋊(C3⋊Dic3)C22semidirect
38. Dic3×C3⋊Dic3C22semidirect
39. C23×C3⋊Dic3D3semidirect
54. C3×#953D3semidirect
75. C2×C3×C3⋊Dic3D4semidirect
149. C22×C33#41semidirect
170. #953C3⋊D3semidirect
223. C2×C3⋊Dic3#67semidirect
323. C22×C32#235semidirect
325. C22×C32#237semidirect
364. C33#525semidirect
483. C22×C3#3804semidirect
486. C22×C3#3830semidirect
514. C32#5586semidirect
515. C32#5791semidirect
566. C22#89028semidirect
577. C3#10595498semidirect
578. C3#10596057semidirect

Maximal subgroups

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

subgroupindex
C23×C3×C3⋊Dic32
C2×#890282
#117239942
C32×#9532
#117236412
#117236352
#117236222
#67×C3⋊Dic32
#117236082
#117236022
#117235952
#105960573
#105954983