Finite Group Databasebeta

← #13177627

Subgroups — #13177627

Order 1728.

2080 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723978C211yesmaximal, characteristic, complemented
2C3×#64146C211yesmaximal, characteristic, complemented
3C32×#936C211yesmaximal, characteristic, complemented
4#11723639C212yesmaximal, complemented
5#11723636C212yesmaximal, complemented
6#11723592C212yesmaximal, complemented
7#11723591C212yesmaximal, complemented
8#11723416C212yesmaximal, complemented
9#11723392C212yesmaximal, complemented
10#10596154—31nomaximal
11#10595262—34nomaximal
12C23×C3⋊(C3⋊D3)C2211yescharacteristic, complemented
13C23×C3×C3⋊D3C2211yescharacteristic, complemented
14C2×C33×D4C2211yescharacteristic
15C2×#6312C2212yes—
16C2×C3×#638C2212yes—
17C2×C32×Dic6C2211yescharacteristic, complemented
18C2×#6293C2212yescomplemented
19C2×#6292C2212yes—
20C2×#6292C2212yes—
21#88864C2211yescharacteristic, complemented
22#88859C2212yescomplemented
23C3×#3757C2211yescharacteristic, complemented
24C3×#3752C2212yescomplemented
25C32×#217C2212yescomplemented
26C32×#217C2212yescomplemented
27C32×C4×Dic3C2211yescharacteristic, complemented
28#88804C2212yescomplemented
29#88804C2212yescomplemented
30#88803C2212yescomplemented
31#88803C2212yescomplemented
32Dic3×C3⋊Dic3C2212yescomplemented
33C32×#124—31no—
34#64156—32no—
35#64146—31no—
36#64146—34no—
37#64146—64no—
38#64146D311yescharacteristic, complemented
39#64094—32no—
40#64094—32no—
41#64091—32no—
42#64091—32no—
43#64084—32no—
44C3×#937—34no—
45C3×#936D314yescomplemented
46#63981—38no—
47#63977—38no—
48#63959—38no—
49#63953—38no—
50#63907—38no—
51#63852—38no—
52C23×C33C2311yescharacteristic
53C23×C33C2311yescharacteristic
54C22×C3⋊(C3⋊D3)—42no—
55C22×C3⋊(C3⋊D3)—41no—
56C22×C3⋊(C3⋊D3)C2312yes—
57C22×C3×C3⋊D3—41no—
58C22×C3×C3⋊D3—42no—
59C22×C3×C3⋊D3C2312yes—
60C33×D4—41no—
61C2×C33×C4C2311yescharacteristic
62#6312—42no—
63C2×C3⋊(C3⋊Dic3)C2312yes—
64C3×#638—42no—
65C2×C3×C3⋊Dic3C2312yes—
66C2×C32×Dic3C2312yes—
67C2×C32×Dic3C2312yes—
68C32×Dic6—41no—
69#6293—42no—
70#6292—42no—
71#6292—42no—
72#5683—94no—
73#5594—91no—
74C23×C3⋊D3—31no—
75C23×C3⋊D3—34no—
76C23×C3⋊D3—64no—
77C23×C3⋊D3C2×D311yescharacteristic, complemented
78C23×C3×D3—34no—
79C2×C32×Q8—31no—
80C2×C32×D4—24no—
81C2×C32×D4C2×D314yes—
82C2×C32×D4C2×D311yescharacteristic
83C2×#638—32no—
84C2×#638—38no—
85C2×#638—68no—
86C2×#638C2×D312yes—
87C2×C4⋊(C3⋊D3)—32no—
88C2×C4⋊(C3⋊D3)—32no—
89C2×#634—32no—
90C2×C3×#67—38no—
91C2×C3×Dic6C2×D314yescomplemented
92C2×#627—38no—
93C2×#626—38no—
94C2×#626—38no—
95C32×C22⋊C4—32no—
96C32×C22⋊C4—32no—
97C32×C42—31no—
98#3757—31no—
99#3757—34no—
100#3757—64no—
101#3757C2×D311yescharacteristic, complemented
102#3752—32no—
103#3752—32no—
104#3752—32no—
105#3752—38no—
106#3752—32no—
107#3752—32no—
108#3752—68no—
109#3752C2×D312yescomplemented
110C4×C3⋊Dic3—32no—
111C3×#217—34no—
112C3×#217C2×D318yescomplemented
113C3×#217C2×D318yescomplemented
114C3×(C2×D3)⋊C4—38no—
115C3×C4×Dic3C2×D314yescomplemented
116#3722—38no—
117#3722—38no—
118(C2×D3)⋊Dic3—38no—
119(C2×D3)⋊Dic3—38no—
120Dic32—38no—
121C22×C33—21no—
122C22×C33—22no—
123C22×C33—22no—
124C22×C33—21no—
125C22×C33C2411yescharacteristic
126C2×C3⋊(C3⋊D3)—22no—
127C2×C3⋊(C3⋊D3)—22no—
128C2×C3⋊(C3⋊D3)—22no—
129C2×C3⋊(C3⋊D3)—44no—
130C2×C3×C3⋊D3—22no—
131C2×C3×C3⋊D3—22no—
132C2×C3×C3⋊D3—22no—
133C2×C3×C3⋊D3—44no—
134C33×C4—21no—
135C3⋊(C3⋊Dic3)—22no—
136C3×C3⋊Dic3—22no—
137C32×Dic3—22no—
138C32×Dic3—22no—
139C3×#124—34no—
140C3×#120—91no—
141#953—92no—
142#947—98no—
143#937—34no—
144#937—94no—
145#937—91no—
146#937—92no—
147#937—184no—
148#936—92no—
149#936C3⋊D311yescharacteristic, complemented
150#885—98no—
151#885—98no—
152#885—92no—
153#885—92no—
154#885—92no—
155#882—98no—
156#882—98no—
157#875—98no—
158C23×C32—24no—
159C23×C32—24no—
160C23×C32C22×D314yes—
161C23×C32C22×D314yes—
162C23×C32C22×D311yescharacteristic
163C23×C32C22×D311yescharacteristic
164C22×C3⋊D3—32no—
165C22×C3⋊D3—38no—
166C22×C3⋊D3—41no—
167C22×C3⋊D3—42no—
168C22×C3⋊D3—68no—
169C22×C3⋊D3—121no—
170C22×C3⋊D3—122no—
171C22×C3⋊D3—128no—
172C22×C3⋊D3—124no—
173C22×C3⋊D3—1216no—
174C22×C3⋊D3—128no—
175C22×C3⋊D3C22×D312yes—
176C22×C3×D3—38no—
177C22×C3×D3—124no—
178C22×C3×D3—128no—
179C32×Q8—121no—
180C32×D4—41no—
181C32×D4—44no—
182C32×D4—48no—
183C2×C32×C4—24no—
184C2×C32×C4—32no—
185C2×C32×C4—32no—
186C2×C32×C4C22×D314yes—
187C2×C32×C4C22×D311yescharacteristic
188#638—42no—
189#638—122no—
190#638—128no—
191#638—1216no—
192C2×C3⋊Dic3—32no—
193C2×C3⋊Dic3—38no—
194C2×C3⋊Dic3—68no—
195C2×C3⋊Dic3C22×D312yes—
196C4⋊(C3⋊D3)—122no—
197C4⋊(C3⋊D3)—122no—
198#634—122no—
199C3×#67—128no—
200C2×C3×Dic3—38no—
201C2×C3×Dic3C22×D318yes—
202C2×C3×Dic3C22×D318yes—
203C3×Dic6—44no—
204#627—128no—
205#626—128no—
206#626—128no—
207H(3,𝔽4)—271noSylow
208C2×C33—41no—
209C2×C33—41no—
210C2×C33H(5,𝔽2)11yescharacteristic
211C2×C33H(5,𝔽2)12yes—
212C3⋊(C3⋊D3)—42no—
213C3×C3⋊D3—42no—
214C24×C3—91no—
215C23×D3—34no—
216C23×D3—94no—
217C23×D3—91no—
218C23×D3—184no—
219C2×C3×Q8—34no—
220C2×C3×D4—24no—
221C2×C3×D4—92no—
222C2×C3×D4D3214yes—
223C2×C3×D4C2×C3⋊D311yescharacteristic
224C2×#67—38no—
225C2×#67—98no—
226C2×#67—92no—
227C2×#67—92no—
228C2×#67—92no—
229C2×#67—188no—
230C2×D12—98no—
231C2×D12—98no—
232C2×Dic6—98no—
233C2×Dic6—92no—
234C2×Dic6C2×C3⋊D311yescharacteristic, complemented
235C3×C22⋊C4—38no—
236C3×C22⋊C4—38no—
237C3×C22⋊C4—91no—
238C3×C22⋊C4—92no—
239C3×C42—34no—
240#217—34no—
241#217—94no—
242#217—91no—
243#217—92no—
244#217—92no—
245#217—184no—
246#217C2×C3⋊D312yescomplemented
247#217C2×C3⋊D312yescomplemented
248(C2×D3)⋊C4—38no—
249(C2×D3)⋊C4—98no—
250(C2×D3)⋊C4—98no—
251(C2×D3)⋊C4—98no—
252(C2×D3)⋊C4—98no—
253(C2×D3)⋊C4—98no—
254(C2×D3)⋊C4—92no—
255(C2×D3)⋊C4—92no—
256(C2×D3)⋊C4—92no—
257(C2×D3)⋊C4—188no—
258C4×Dic3—98no—
259C4×Dic3—92no—
260C4×Dic3C2×C3⋊D311yescharacteristic, complemented
261C22×C32—28no—
262C22×C32—22no—
263C22×C32—22no—
264C22×C32—21no—
265C22×C32—24no—
266C22×C32—21no—
267C22×C32—24no—
268C22×C32—24no—
269C22×C32—28no—
270C22×C32—28no—
271C22×C32—48no—
272C22×C32—48no—
273C22×C32—44no—
274C22×C32C23×D314yes—
275C22×C32C23×D311yescharacteristic
276C2×C3⋊D3—22no—
277C2×C3⋊D3—22no—
278C2×C3⋊D3—22no—
279C2×C3⋊D3—44no—
280C2×C3⋊D3—616no—
281C2×C3⋊D3—62no—
282C2×C3⋊D3—62no—
283C2×C3⋊D3—62no—
284C2×C3⋊D3—68no—
285C2×C3⋊D3—68no—
286C2×C3⋊D3—68no—
287C2×C3⋊D3—128no—
288C2×C3⋊D3—128no—
289C2×C3⋊D3—1232no—
290C2×C3⋊D3—1216no—
291C2×C3⋊D3—124no—
292C2×C3×D3—68no—
293C2×C3×D3—68no—
294C2×C3×D3—68no—
295C2×C3×D3—1216no—
296C32×C4—21no—
297C32×C4—24no—
298C32×C4—44no—
299C32×C4—62no—
300C32×C4—62no—
301C3⋊Dic3—22no—
302C3⋊Dic3—62no—
303C3⋊Dic3—68no—
304C3⋊Dic3—128no—
305C3×Dic3—28no—
306C3×Dic3—28no—
307C3×Dic3—68no—
308#124—31no—
309#124—272no—
310#124—272no—
311#124—272no—
312#124—272no—
313#120—91no—
314#120—272no—
315#120—272no—
316#120—271no—
317C33H(3,𝔽4)11yescharacteristic, complemented, Sylow
318C23×C3—24no—
319C23×C3—24no—
320C23×C3—92no—
321C23×C3—361no—
322C23×C3—362no—
323C23×C3C2×D3214yes—
324C23×C3C2×D3214yes—
325C23×C3C22×C3⋊D311yescharacteristic
326C23×C3C22×C3⋊D311yescharacteristic
327C22×D3—38no—
328C22×D3—98no—
329C22×D3—92no—
330C22×D3—128no—
331C22×D3—124no—
332C22×D3—188no—
333C22×D3—368no—
334C22×D3—364no—
335C22×D3—361no—
336C22×D3—362no—
337C22×D3—3616no—
338C22×D3—368no—
339C3×Q8—124no—
340C3×D4—44no—
341C3×D4—41no—
342C3×D4—48no—
343C3×D4—362no—
344C2×C3×C4—24no—
345C2×C3×C4—38no—
346C2×C3×C4—38no—
347C2×C3×C4—92no—
348C2×C3×C4C2×D3214yes—
349C2×C3×C4C22×C3⋊D311yescharacteristic
350#67—128no—
351#67—362no—
352#67—362no—
353#67—368no—
354#67—362no—
355#67—3616no—
356C2×Dic3—38no—
357C2×Dic3—98no—
358C2×Dic3—92no—
359C2×Dic3—188no—
360C2×Dic3C22×C3⋊D312yes—
361C2×Dic3C22×C3⋊D312yes—
362D12—368no—
363D12—368no—
364Dic6—41no—
365Dic6—362no—
366Dic6—368no—
367C2×C32—28no—
368C2×C32—24no—
369C2×C32—48no—
370C2×C32—44no—
371C2×C32—41no—
372C2×C32—41no—
373C2×C32—44no—
374C2×C32—84no—
375C2×C32#100414yes—
376C2×C32#100412yes—
377C2×C32#100411yescharacteristic
378C2×C32#100918yes—
379C3⋊D3—42no—
380C3⋊D3—1216no—
381C3⋊D3—128no—
382C3⋊D3—122no—
383C3×D3—128no—
384C24—91no—
385C24—271no—
386C2×Q8—31no—
387C2×Q8—272no—
388C2×D4—92no—
389C2×D4—272no—
390C2×D4—272no—
391C2×D4—272no—
392C2×D4D3×C3⋊D311yescharacteristic
393C22⋊C4—32no—
394C22⋊C4—32no—
395C22⋊C4—91no—
396C22⋊C4—92no—
397C22⋊C4—272no—
398C22⋊C4—272no—
399C22⋊C4—272no—
400C22⋊C4—272no—
401C22⋊C4—271no—
402C22⋊C4—272no—
403C42—31no—
404C42—272no—
405C22×C3—28no—
406C22×C3—28no—
407C22×C3—28no—
408C22×C3—24no—
409C22×C3—24no—
410C22×C3—21no—
411C22×C3—21no—
412C22×C3—22no—
413C22×C3—22no—
414C22×C3—24no—
415C22×C3—48no—
416C22×C3—48no—
417C22×C3—44no—
418C22×C3—182no—
419C22×C3—182no—
420C22×C3—182no—
421C22×C3—364no—
422C22×C3C22×D3214yes—
423C22×C3C23×C3⋊D311yescharacteristic
424C2×D3—68no—
425C2×D3—68no—
426C2×D3—68no—
427C2×D3—1216no—
428C2×D3—1816no—
429C2×D3—188no—
430C2×D3—188no—
431C2×D3—188no—
432C2×D3—182no—
433C2×D3—182no—
434C2×D3—182no—
435C2×D3—368no—
436C2×D3—368no—
437C2×D3—3632no—
438C2×D3—364no—
439C2×D3—3616no—
440C3×C4—24no—
441C3×C4—21no—
442C3×C4—44no—
443C3×C4—68no—
444C3×C4—68no—
445C3×C4—182no—
446Dic3—22no—
447Dic3—22no—
448Dic3—68no—
449Dic3—188no—
450Dic3—182no—
451Dic3—368no—
452C32—24no—
453C32#559411yescharacteristic, complemented
454C32#568314yescomplemented
455C23—92no—
456C23—272no—
457C23—362no—
458C23—361no—
459C23—1082no—
460C23—1081no—
461C23C2×D3×C3⋊D311yescharacteristic
462C23C2×D3×C3⋊D311yescharacteristic
463Q8—121no—
464Q8—1082no—
465D4—41no—
466D4—362no—
467D4—1082no—
468D4—1082no—
469D4—1082no—
470C2×C4—32no—
471C2×C4—32no—
472C2×C4—92no—
473C2×C4—272no—
474C2×C4C2×D3×C3⋊D311yescharacteristic
475C2×C3—28no—
476C2×C3—24no—
477C2×C3—48no—
478C2×C3—41no—
479C2×C3—41no—
480C2×C3—44no—
481C2×C3—44no—
482C2×C3—84no—
483C2×C3—362no—
484C2×C3#6431418yes—
485C2×C3#6433014yes—
486C2×C3#6436111yescharacteristic
487C2×C3#6436612yes—
488D3—128no—
489D3—3616no—
490D3—362no—
491D3—368no—
492C22—22no—
493C22—22no—
494C22—21no—
495C22—21no—
496C22—182no—
497C22—182no—
498C22—182no—
499C22—364no—
500C22—542no—
501C22—542no—
502C22—542no—
503C22—1084no—
504C22C22×D3×C3⋊D311yescharacteristic
505C4—21no—
506C4—62no—
507C4—62no—
508C4—182no—
509C4—542no—
510C3—24no—
511C3#1059526214yescomplemented
512C3#1059615411yescharacteristic, complemented
513C2—41no—
514C2—41no—
515C2—362no—
516C2—1082no—
517C2#1172458912yes—
518C2#1172461111yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC2×C33×D4C22
hypercenterC2×D4D3×C3⋊D3
hypocenterC33H(3,𝔽4)
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. #11723978C2semidirect
2. C3×#64146C2semidirect
3. C32×#936C2semidirect
4. #11723639C2semidirect
5. #11723636C2semidirect
6. #11723592C2semidirect
7. #11723591C2semidirect
8. #11723416C2semidirect
9. #11723392C2semidirect
12. C23×C3⋊(C3⋊D3)C22semidirect
13. C23×C3×C3⋊D3C22semidirect
17. C2×C32×Dic6C22semidirect
18. C2×#6293C22semidirect
21. #88864C22semidirect
22. #88859C22semidirect
23. C3×#3757C22semidirect
24. C3×#3752C22semidirect
25. C32×#217C22semidirect
26. C32×#217C22semidirect
27. C32×C4×Dic3C22semidirect
28. #88804C22semidirect
29. #88804C22semidirect
30. #88803C22semidirect
31. #88803C22semidirect
32. Dic3×C3⋊Dic3C22semidirect
38. #64146D3semidirect
45. C3×#936D3semidirect
77. C23×C3⋊D3C2×D3semidirect
91. C2×C3×Dic6C2×D3semidirect
101. #3757C2×D3semidirect
109. #3752C2×D3semidirect
112. C3×#217C2×D3semidirect
113. C3×#217C2×D3semidirect
115. C3×C4×Dic3C2×D3semidirect
149. #936C3⋊D3semidirect
234. C2×Dic6C2×C3⋊D3semidirect
246. #217C2×C3⋊D3semidirect
247. #217C2×C3⋊D3semidirect
260. C4×Dic3C2×C3⋊D3semidirect
317. C33H(3,𝔽4)semidirect
453. C32#5594semidirect
454. C32#5683semidirect
511. C3#10595262semidirect
512. C3#10596154semidirect

Maximal subgroups

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

subgroupindex
#117239782
C3×#641462
C32×#9362
#117236392
#117236362
#117235922
#117235912
#117234162
#117233922
#105961543
#105952623