Finite Group Databasebeta

← #13177632

Subgroups — #13177632

Order 1728.

2032 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723978C211yesmaximal, characteristic, complemented
2C3×#64149C211yesmaximal, characteristic, complemented
3C32×#935C211yesmaximal, characteristic, complemented
4#11723639C212yesmaximal, complemented
5#11723632C212yesmaximal, complemented
6#11723624C212yesmaximal, complemented
7#11723621C212yesmaximal, complemented
8#11723482C212yesmaximal, complemented
9#11723446C212yesmaximal, complemented
10#10596137—31nomaximal
11#10595266—34nomaximal
12C23×C3⋊(C3⋊D3)C2211yescharacteristic
13C2×C33×D4C2211yescharacteristic
14C2×#6312C2212yes—
15C2×C3×#638C2212yes—
16C2×C3×#638C2212yescomplemented
17C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic, complemented
18C22×C32×Dic3C2211yescharacteristic, complemented
19C2×#6292C2212yes—
20C2×#6292C2212yes—
21C2×Dic3×C3⋊D3C2212yescomplemented
22#88864C2211yescharacteristic
23#88859C2212yes—
24C3×C4×C3⋊Dic3C2211yescharacteristic, complemented
25C32×#217C2212yes—
26C32×#217C2211yescharacteristic, complemented
27C32×Dic3⋊C4C2212yescomplemented
28#88806C2212yescomplemented
29Dic3⋊(C3⋊Dic3)C2212yescomplemented
30#88804C2212yes—
31#88804C2212yes—
32#88803C2212yescomplemented
33C32×#123—31no—
34#64149D311yescharacteristic, complemented
35#64146—31no—
36#64146—34no—
37#64146—64no—
38#64139—32no—
39#64107—32no—
40#64104—32no—
41#64093—32no—
42#64092—32no—
43#64091—32no—
44C3×#940—34no—
45C3×#935D314yescomplemented
46#63981—38no—
47#63978—38no—
48#63971—38no—
49#63969—38no—
50#63886—38no—
51#63868—38no—
52C23×C33C2311yescharacteristic
53C23×C33C2311yescharacteristic
54C22×C3⋊(C3⋊D3)—42no—
55C22×C3⋊(C3⋊D3)—41no—
56C22×C3⋊(C3⋊D3)C2312yes—
57C22×C3×C3⋊D3C2312yes—
58C33×D4—41no—
59C2×C33×C4C2311yescharacteristic
60#6312—42no—
61C2×C3⋊(C3⋊Dic3)C2312yes—
62C3×#638—42no—
63C3×#638—42no—
64C2×C3×C3⋊Dic3C2312yes—
65C3×C4⋊(C3⋊D3)—41no—
66C2×C32×Dic3—41no—
67C2×C32×Dic3C2312yes—
68C2×C32×Dic3C2312yes—
69#6292—42no—
70#6292—42no—
71Dic3×C3⋊D3—42no—
72#5720—91no—
73#5666—94no—
74C23×C3⋊D3—31no—
75C23×C3⋊D3—34no—
76C23×C3⋊D3—64no—
77C2×C32×D4—24no—
78C2×C32×D4C2×D314yes—
79C2×C32×D4C2×D311yescharacteristic
80C22×C32×C4—31no—
81C2×#638—32no—
82C2×#638—38no—
83C2×#638—68no—
84C2×#638C2×D312yes—
85C2×#638C2×D312yescomplemented
86C2×C4⋊(C3⋊D3)—32no—
87C2×C4⋊(C3⋊D3)—32no—
88C2×C4⋊(C3⋊D3)C2×D311yescharacteristic, complemented
89C2×C4×C3⋊D3—32no—
90C2×C3×#67—38no—
91C2×C3×#67—38no—
92C22×C3×Dic3C2×D314yescomplemented
93C2×C3×D12—34no—
94C2×#626—38no—
95C2×#626—38no—
96C2×D3×Dic3—38no—
97C32×C4⋊C4—32no—
98C32×C22⋊C4—32no—
99C32×C22⋊C4—31no—
100#3757—31no—
101#3757—34no—
102#3757—64no—
103#3752—32no—
104#3752—32no—
105#3752—38no—
106#3752—32no—
107#3752—32no—
108#3752—68no—
109C4⋊(C3⋊Dic3)—32no—
110#3750—32no—
111C4×C3⋊Dic3C2×D311yescharacteristic, complemented
112C3×#217C2×D318yes—
113C3×#217C2×D314yescomplemented
114C3×Dic3⋊C4C2×D318yescomplemented
115C3×C4×Dic3—34no—
116#3724—38no—
117Dic3⋊Dic3—38no—
118#3722—38no—
119#3722—38no—
120(C2×D3)⋊Dic3—38no—
121C22×C33—21no—
122C22×C33—21no—
123C22×C33—22no—
124C22×C33—22no—
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—
133C33×C4—21no—
134C3⋊(C3⋊Dic3)—22no—
135C3×C3⋊Dic3—22no—
136C32×Dic3—22no—
137C32×Dic3—22no—
138C3×#127—91no—
139C3×#123—34no—
140#940—34no—
141#939—92no—
142#939—92no—
143#939—92no—
144#937—94no—
145#937—91no—
146#937—92no—
147#937—184no—
148#935—92no—
149#935C3⋊D311yescharacteristic, complemented
150#930—98no—
151#898—98no—
152#895—98no—
153#884—98no—
154#883—98no—
155#882—98no—
156#876—92no—
157C23×C32—24no—
158C23×C32—24no—
159C23×C32C22×D314yes—
160C23×C32C22×D314yes—
161C23×C32C22×D311yescharacteristic
162C23×C32C22×D311yescharacteristic
163C22×C3⋊D3—32no—
164C22×C3⋊D3—38no—
165C22×C3⋊D3—68no—
166C22×C3⋊D3—121no—
167C22×C3⋊D3—122no—
168C22×C3⋊D3—128no—
169C22×C3⋊D3—124no—
170C22×C3⋊D3—1216no—
171C22×C3⋊D3—128no—
172C22×C3⋊D3C22×D312yes—
173C22×C3×D3—38no—
174C32×D4—41no—
175C32×D4—44no—
176C32×D4—48no—
177C2×C32×C4—24no—
178C2×C32×C4—32no—
179C2×C32×C4—32no—
180C2×C32×C4—121no—
181C2×C32×C4C22×D314yes—
182C2×C32×C4C22×D311yescharacteristic
183#638—42no—
184#638—42no—
185#638—122no—
186#638—128no—
187#638—1216no—
188C2×C3⋊Dic3—32no—
189C2×C3⋊Dic3—38no—
190C2×C3⋊Dic3—68no—
191C2×C3⋊Dic3C22×D312yes—
192C4⋊(C3⋊D3)—41no—
193C4⋊(C3⋊D3)—122no—
194C4⋊(C3⋊D3)—122no—
195C4×C3⋊D3—122no—
196C3×#67—128no—
197C3×#67—128no—
198C2×C3×Dic3—38no—
199C2×C3×Dic3—44no—
200C2×C3×Dic3C22×D318yes—
201C2×C3×Dic3C22×D318yes—
202C3×D12—124no—
203#626—128no—
204#626—128no—
205D3×Dic3—128no—
206#560—271noSylow
207C2×C33—41no—
208C2×C33—41no—
209C2×C33H(5,𝔽2)11yescharacteristic
210C2×C33H(5,𝔽2)12yes—
211C3⋊(C3⋊D3)—42no—
212C3×C3⋊D3—42no—
213C23×D3—94no—
214C23×D3—91no—
215C23×D3—184no—
216C2×C3×D4—24no—
217C2×C3×D4—92no—
218C2×C3×D4—92no—
219C2×C3×D4—91no—
220C2×C3×D4D3214yes—
221C2×C3×D4C2×C3⋊D311yescharacteristic
222C22×C3×C4—34no—
223C2×#67—38no—
224C2×#67—38no—
225C2×#67—98no—
226C2×#67—92no—
227C2×#67—92no—
228C2×#67—92no—
229C2×#67—188no—
230C22×Dic3—92no—
231C22×Dic3C2×C3⋊D311yescharacteristic, complemented
232C2×D12—34no—
233C2×D12—98no—
234C2×D12—98no—
235C2×C4×D3—98no—
236C3×C4⋊C4—38no—
237C3×C22⋊C4—38no—
238C3×C22⋊C4—34no—
239C3×C42—91no—
240#217—94no—
241#217—91no—
242#217—92no—
243#217—184no—
244#217C2×C3⋊D312yes—
245#217C2×C3⋊D311yescharacteristic, complemented
246(C2×D3)⋊C4—98no—
247(C2×D3)⋊C4—98no—
248(C2×D3)⋊C4—98no—
249(C2×D3)⋊C4—98no—
250(C2×D3)⋊C4—92no—
251(C2×D3)⋊C4—92no—
252(C2×D3)⋊C4—92no—
253(C2×D3)⋊C4—188no—
254C4⋊Dic3—98no—
255Dic3⋊C4—98no—
256Dic3⋊C4—92no—
257Dic3⋊C4—92no—
258Dic3⋊C4C2×C3⋊D312yescomplemented
259C4×Dic3—34no—
260C22×C32—28no—
261C22×C32—22no—
262C22×C32—22no—
263C22×C32—21no—
264C22×C32—24no—
265C22×C32—21no—
266C22×C32—24no—
267C22×C32—24no—
268C22×C32—28no—
269C22×C32—28no—
270C22×C32—48no—
271C22×C32—48no—
272C22×C32—44no—
273C22×C32C23×D314yes—
274C22×C32C23×D311yescharacteristic
275C2×C3⋊D3—22no—
276C2×C3⋊D3—22no—
277C2×C3⋊D3—22no—
278C2×C3⋊D3—616no—
279C2×C3⋊D3—62no—
280C2×C3⋊D3—62no—
281C2×C3⋊D3—62no—
282C2×C3⋊D3—68no—
283C2×C3⋊D3—68no—
284C2×C3⋊D3—68no—
285C2×C3⋊D3—128no—
286C2×C3⋊D3—128no—
287C2×C3⋊D3—1232no—
288C2×C3⋊D3—1216no—
289C2×C3⋊D3—124no—
290C2×C3×D3—68no—
291C2×C3×D3—68no—
292C2×C3×D3—68no—
293C32×C4—21no—
294C32×C4—24no—
295C32×C4—44no—
296C32×C4—62no—
297C32×C4—62no—
298C3⋊Dic3—22no—
299C3⋊Dic3—62no—
300C3⋊Dic3—68no—
301C3⋊Dic3—128no—
302C3×Dic3—28no—
303C3×Dic3—28no—
304C3×Dic3—68no—
305#127—91no—
306#126—272no—
307#123—31no—
308#123—272no—
309#121—272no—
310#121—272no—
311#121—272no—
312#120—272no—
313#120—271no—
314C33#56011yescharacteristic, complemented, Sylow
315C23×C3—24no—
316C23×C3—24no—
317C23×C3—92no—
318C23×C3C2×D3214yes—
319C23×C3C2×D3214yes—
320C23×C3C22×C3⋊D311yescharacteristic
321C23×C3C22×C3⋊D311yescharacteristic
322C22×D3—38no—
323C22×D3—98no—
324C22×D3—92no—
325C22×D3—188no—
326C22×D3—368no—
327C22×D3—364no—
328C22×D3—361no—
329C22×D3—362no—
330C22×D3—3616no—
331C22×D3—368no—
332C3×D4—44no—
333C3×D4—41no—
334C3×D4—48no—
335C3×D4—361no—
336C3×D4—362no—
337C3×D4—362no—
338C2×C3×C4—24no—
339C2×C3×C4—38no—
340C2×C3×C4—38no—
341C2×C3×C4—92no—
342C2×C3×C4—124no—
343C2×C3×C4C2×D3214yes—
344C2×C3×C4C22×C3⋊D311yescharacteristic
345#67—128no—
346#67—128no—
347#67—362no—
348#67—362no—
349#67—368no—
350#67—362no—
351#67—3616no—
352C2×Dic3—38no—
353C2×Dic3—41no—
354C2×Dic3—98no—
355C2×Dic3—92no—
356C2×Dic3—188no—
357C2×Dic3—362no—
358C2×Dic3C22×C3⋊D312yes—
359C2×Dic3C22×C3⋊D312yes—
360D12—124no—
361D12—368no—
362D12—368no—
363C4×D3—368no—
364C2×C32—28no—
365C2×C32—24no—
366C2×C32—48no—
367C2×C32—44no—
368C2×C32—41no—
369C2×C32—41no—
370C2×C32—44no—
371C2×C32—84no—
372C2×C32#100414yes—
373C2×C32#100412yes—
374C2×C32#100411yescharacteristic
375C2×C32#100918yes—
376C3⋊D3—42no—
377C3⋊D3—1216no—
378C3⋊D3—128no—
379C3⋊D3—122no—
380C3×D3—128no—
381C24—271no—
382C2×D4—92no—
383C2×D4—92no—
384C2×D4—91no—
385C2×D4—272no—
386C2×D4—272no—
387C2×D4—272no—
388C2×D4D3×C3⋊D311yescharacteristic
389C22×C4—31no—
390C22×C4—272no—
391C4⋊C4—32no—
392C4⋊C4—272no—
393C4⋊C4—272no—
394C22⋊C4—32no—
395C22⋊C4—31no—
396C22⋊C4—272no—
397C22⋊C4—272no—
398C22⋊C4—272no—
399C22⋊C4—271no—
400C22⋊C4—272no—
401C42—91no—
402C22×C3—28no—
403C22×C3—28no—
404C22×C3—28no—
405C22×C3—24no—
406C22×C3—24no—
407C22×C3—21no—
408C22×C3—21no—
409C22×C3—22no—
410C22×C3—22no—
411C22×C3—24no—
412C22×C3—48no—
413C22×C3—48no—
414C22×C3—44no—
415C22×C3—182no—
416C22×C3—182no—
417C22×C3—182no—
418C22×C3C22×D3214yes—
419C22×C3C23×C3⋊D311yescharacteristic
420C2×D3—68no—
421C2×D3—68no—
422C2×D3—68no—
423C2×D3—1816no—
424C2×D3—188no—
425C2×D3—188no—
426C2×D3—188no—
427C2×D3—182no—
428C2×D3—182no—
429C2×D3—182no—
430C2×D3—368no—
431C2×D3—368no—
432C2×D3—3632no—
433C2×D3—364no—
434C2×D3—3616no—
435C3×C4—24no—
436C3×C4—21no—
437C3×C4—44no—
438C3×C4—68no—
439C3×C4—68no—
440C3×C4—182no—
441Dic3—22no—
442Dic3—22no—
443Dic3—68no—
444Dic3—188no—
445Dic3—182no—
446Dic3—368no—
447C32—24no—
448C32#566614yescomplemented
449C32#572011yescharacteristic, complemented
450C23—92no—
451C23—272no—
452C23—1082no—
453C23—1081no—
454C23C2×D3×C3⋊D311yescharacteristic
455C23C2×D3×C3⋊D311yescharacteristic
456D4—41no—
457D4—361no—
458D4—362no—
459D4—362no—
460D4—1082no—
461D4—1082no—
462D4—1082no—
463C2×C4—32no—
464C2×C4—32no—
465C2×C4—92no—
466C2×C4—121no—
467C2×C4—272no—
468C2×C4—1082no—
469C2×C4C2×D3×C3⋊D311yescharacteristic
470C2×C3—28no—
471C2×C3—24no—
472C2×C3—48no—
473C2×C3—41no—
474C2×C3—41no—
475C2×C3—44no—
476C2×C3—44no—
477C2×C3—84no—
478C2×C3—362no—
479C2×C3#6431418yes—
480C2×C3#6433014yes—
481C2×C3#6436111yescharacteristic
482C2×C3#6436612yes—
483D3—128no—
484D3—3616no—
485D3—362no—
486D3—368no—
487C22—22no—
488C22—22no—
489C22—21no—
490C22—21no—
491C22—182no—
492C22—182no—
493C22—182no—
494C22—542no—
495C22—542no—
496C22—542no—
497C22—1084no—
498C22C22×D3×C3⋊D311yescharacteristic
499C4—21no—
500C4—62no—
501C4—62no—
502C4—182no—
503C4—542no—
504C3—24no—
505C3#1059526614yescomplemented
506C3#1059613711yescharacteristic, complemented
507C2—41no—
508C2—41no—
509C2—362no—
510C2—1082no—
511C2#1172458912yes—
512C2#1172461111yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC2×C33×D4C22
hypercenterC2×D4D3×C3⋊D3
hypocenterC33#560
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×#64149C2semidirect
3. C32×#935C2semidirect
4. #11723639C2semidirect
5. #11723632C2semidirect
6. #11723624C2semidirect
7. #11723621C2semidirect
8. #11723482C2semidirect
9. #11723446C2semidirect
16. C2×C3×#638C22semidirect
17. C2×C3×C4⋊(C3⋊D3)C22semidirect
18. C22×C32×Dic3C22semidirect
21. C2×Dic3×C3⋊D3C22semidirect
24. C3×C4×C3⋊Dic3C22semidirect
26. C32×#217C22semidirect
27. C32×Dic3⋊C4C22semidirect
28. #88806C22semidirect
29. Dic3⋊(C3⋊Dic3)C22semidirect
32. #88803C22semidirect
34. #64149D3semidirect
45. C3×#935D3semidirect
85. C2×#638C2×D3semidirect
88. C2×C4⋊(C3⋊D3)C2×D3semidirect
92. C22×C3×Dic3C2×D3semidirect
111. C4×C3⋊Dic3C2×D3semidirect
113. C3×#217C2×D3semidirect
114. C3×Dic3⋊C4C2×D3semidirect
149. #935C3⋊D3semidirect
231. C22×Dic3C2×C3⋊D3semidirect
245. #217C2×C3⋊D3semidirect
258. Dic3⋊C4C2×C3⋊D3semidirect
314. C33#560semidirect
448. C32#5666semidirect
449. C32#5720semidirect
505. C3#10595266semidirect
506. C3#10596137semidirect

Maximal subgroups

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

subgroupindex
#117239782
C3×#641492
C32×#9352
#117236392
#117236322
#117236242
#117236212
#117234822
#117234462
#105961373
#105952663