Finite Group Databasebeta

← #13175363

Subgroups — #13175363

Order 1728.

1816 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×C4⋊(C3⋊(C3⋊D3))C211yesmaximal, characteristic, complemented
2C3×#64081C211yesmaximal, characteristic, complemented
3C32×C4×D12C211yesmaximal, characteristic, complemented
4#11723527C212yesmaximal, complemented
5#11723526C211yesmaximal, characteristic, complemented
6#11723524C211yesmaximal, characteristic, complemented
7#11723522C211yesmaximal, characteristic, complemented
8#11723514C212yesmaximal, complemented
9#11723447C211yesmaximal, characteristic, complemented
10#11723420C212yesmaximal, complemented
11#11723414C212yesmaximal, complemented
12#10596033—31nomaximal
13#10593986—34nomaximal
14C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
15C2×C4×C3⋊(C3⋊D3)C2212yes—
16C2×C3×C4×C3⋊D3C2211yescharacteristic
17C2×C32×D12C2211yescharacteristic
18C2×C32×C4×D3C2212yes—
19C2×#6292C2212yes—
20C2×#6291C2212yes—
21C2×#6291C2212yes—
22C2×#6290C2212yes—
23C33×C42C2211yescharacteristic
24#88859C2212yes—
25C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
26C3×#3752C2211yescharacteristic
27C3×#3752C2211yescharacteristic
28C3×#3750C2211yescharacteristic
29C3×#3750C2211yescharacteristic
30C3×C4×C3⋊Dic3C2211yescharacteristic
31C32×(C2×D3)⋊C4C2212yes—
32C32×C4⋊Dic3C2211yescharacteristic
33#88806C2212yes—
34Dic3⋊(C3⋊Dic3)C2212yes—
35#88804C2212yes—
36(C2×D3)⋊(C3⋊Dic3)C2212yes—
37C32×C4×D4—31no—
38#64149—31no—
39#64147—31no—
40#64139—32no—
41#64138—32no—
42#64104—31no—
43#64099—31no—
44#64093—32no—
45#64092—32no—
46C4×C4⋊(C3⋊D3)—31no—
47C4×C4⋊(C3⋊D3)—34no—
48C4×C4⋊(C3⋊D3)—64no—
49#64081D311yescharacteristic, complemented
50C3×C4×D12D314yescomplemented
51C3×#872—34no—
52#63912—38no—
53#63911—34no—
54#63909—34no—
55#63908—34no—
56#63906—38no—
57#63869—34no—
58#63853—38no—
59#63850—38no—
60C22×C3⋊(C3⋊D3)C2312yes—
61C22×C3×C3⋊D3C2311yescharacteristic
62C22×C32×D3C2312yes—
63C2×C33×C4C2311yescharacteristic
64C2×C33×C4C2311yescharacteristic
65C2×C33×C4C2311yescharacteristic
66C2×C3⋊(C3⋊Dic3)C2312yes—
67C4⋊(C3⋊(C3⋊D3))—22no—
68C4×C3⋊(C3⋊D3)—42no—
69C2×C3×C3⋊Dic3C2311yescharacteristic
70C2×C3×C3⋊Dic3C2311yescharacteristic
71C2×C3×C3⋊Dic3C2311yescharacteristic
72C3×C4×C3⋊D3—22no—
73C2×C32×Dic3C2312yes—
74C32×D12—22no—
75C32×C4×D3—42no—
76#6292—42no—
77#6291—42no—
78#6291—42no—
79#6290—42no—
80#5562—94no—
81#5535—91no—
82C2×C32×D4—31no—
83C22×C32×C4—32no—
84C2×#638—32no—
85C2×#638—32no—
86C2×#638—32no—
87C2×C4⋊(C3⋊D3)—31no—
88C2×C4⋊(C3⋊D3)—34no—
89C2×C4⋊(C3⋊D3)—32no—
90C2×C4⋊(C3⋊D3)—64no—
91C2×C4×C3⋊D3—32no—
92C2×C4×C3⋊D3—38no—
93C2×C4×C3⋊D3—68no—
94C2×C4×C3⋊D3C2×D311yescharacteristic
95C2×C3×D12C2×D314yes—
96C2×C3×C4×D3—34no—
97C2×C3×C4×D3C2×D318yes—
98C2×#626—38no—
99C2×#626—38no—
100C2×#626—38no—
101C2×#625—38no—
102C32×C4⋊C4—31no—
103C32×C22⋊C4—32no—
104C32×C42—24no—
105C32×C42C2×D314yes—
106C32×C42C2×D311yescharacteristic
107#3757—32no—
108#3752—32no—
109#3752—38no—
110#3752—32no—
111#3752—68no—
112#3752C2×D311yescharacteristic
113#3752C2×D311yescharacteristic
114C4⋊(C3⋊Dic3)—31no—
115C4⋊(C3⋊Dic3)—34no—
116C4⋊(C3⋊Dic3)—32no—
117C4⋊(C3⋊Dic3)—64no—
118#3750—32no—
119#3750C2×D311yescharacteristic
120#3750C2×D311yescharacteristic
121C4×C3⋊Dic3C2×D311yescharacteristic
122C3×(C2×D3)⋊C4—34no—
123C3×(C2×D3)⋊C4—34no—
124C3×(C2×D3)⋊C4C2×D318yes—
125C3×C4⋊Dic3C2×D314yes—
126C3×Dic3⋊C4—34no—
127C3×Dic3⋊C4—34no—
128C3×C4×Dic3—34no—
129#3724—38no—
130Dic3⋊Dic3—38no—
131#3722—38no—
132(C2×D3)⋊Dic3—38no—
133C22×C33C2411yescharacteristic
134C2×C3⋊(C3⋊D3)—22no—
135C2×C3⋊(C3⋊D3)—22no—
136C2×C3⋊(C3⋊D3)—22no—
137C2×C3×C3⋊D3—21no—
138C2×C3×C3⋊D3—21no—
139C2×C3×C3⋊D3—21no—
140C2×C32×D3—22no—
141C2×C32×D3—22no—
142C2×C32×D3—22no—
143C33×C4—21no—
144C33×C4—21no—
145C33×C4#4112yes—
146C3⋊(C3⋊Dic3)—22no—
147C3×C3⋊Dic3—21no—
148C3×C3⋊Dic3—21no—
149C3×C3⋊Dic3—21no—
150C32×Dic3—22no—
151C3×C4×D4—34no—
152C3×#117—91no—
153#940—94no—
154#938—94no—
155#930—98no—
156#930—91no—
157#930—91no—
158#929—98no—
159#895—94no—
160#894—92no—
161#894—92no—
162#890—94no—
163#884—98no—
164#884—92no—
165#884—92no—
166#883—98no—
167#874—91no—
168C4×D12—94no—
169C4×D12—91no—
170C4×D12—184no—
171C4×D12C3⋊D311yescharacteristic, complemented
172#872—34no—
173#870—91no—
174C23×C32—32no—
175C22×C3⋊D3—32no—
176C22×C3⋊D3—38no—
177C22×C3⋊D3—68no—
178C22×C3⋊D3C22×D311yescharacteristic
179C22×C3×D3—34no—
180C22×C3×D3C22×D318yes—
181C32×D4—62no—
182C2×C32×C4—24no—
183C2×C32×C4—24no—
184C2×C32×C4—24no—
185C2×C32×C4—32no—
186C2×C32×C4—122no—
187C2×C32×C4C22×D314yes—
188C2×C32×C4C22×D314yes—
189C2×C32×C4C22×D314yes—
190C2×C32×C4C22×D311yescharacteristic
191C2×C32×C4C22×D311yescharacteristic
192C2×C32×C4C22×D311yescharacteristic
193#638—122no—
194#638—122no—
195#638—122no—
196C2×C3⋊Dic3—32no—
197C2×C3⋊Dic3—38no—
198C2×C3⋊Dic3—68no—
199C2×C3⋊Dic3C22×D311yescharacteristic
200C2×C3⋊Dic3C22×D311yescharacteristic
201C2×C3⋊Dic3C22×D311yescharacteristic
202C4⋊(C3⋊D3)—616no—
203C4⋊(C3⋊D3)—68no—
204C4⋊(C3⋊D3)—62no—
205C4⋊(C3⋊D3)—122no—
206C4×C3⋊D3—22no—
207C4×C3⋊D3—122no—
208C4×C3⋊D3—128no—
209C4×C3⋊D3—1216no—
210C2×C3×Dic3—34no—
211C2×C3×Dic3—34no—
212C2×C3×Dic3—34no—
213C2×C3×Dic3C22×D318yes—
214C3×D12—28no—
215C3×C4×D3—48no—
216C3×C4×D3—68no—
217#626—128no—
218#626—128no—
219#626—128no—
220#625—128no—
221#540—271noSylow
222C2×C33C2×#4111yescharacteristic
223C2×C33H(5,𝔽2)11yescharacteristic
224C2×C33H(5,𝔽2)11yescharacteristic
225C3⋊(C3⋊D3)—42no—
226C3×C3⋊D3—41no—
227C32×D3—42no—
228C2×C3×D4—34no—
229C22×C3×C4—38no—
230C22×C3×C4—91no—
231C2×#67—98no—
232C2×#67—98no—
233C2×#67—98no—
234C2×#67—92no—
235C2×#67—92no—
236C2×D12—94no—
237C2×D12—91no—
238C2×D12—98no—
239C2×D12—92no—
240C2×D12—92no—
241C2×D12—184no—
242C2×D12C2×C3⋊D311yescharacteristic
243C2×C4×D3—34no—
244C2×C4×D3—98no—
245C2×C4×D3—92no—
246C2×C4×D3—188no—
247C2×C4×D3C2×C3⋊D312yes—
248C3×C4⋊C4—34no—
249C3×C4⋊C4—91no—
250C3×C4⋊C4—91no—
251C3×C22⋊C4—38no—
252C3×C22⋊C4—91no—
253C3×C22⋊C4—91no—
254C3×C42—24no—
255C3×C42—91no—
256C3×C42D3214yes—
257C3×C42C2×C3⋊D311yescharacteristic
258#217—98no—
259(C2×D3)⋊C4—34no—
260(C2×D3)⋊C4—34no—
261(C2×D3)⋊C4—98no—
262(C2×D3)⋊C4—92no—
263(C2×D3)⋊C4—98no—
264(C2×D3)⋊C4—92no—
265(C2×D3)⋊C4—92no—
266(C2×D3)⋊C4—188no—
267(C2×D3)⋊C4C2×C3⋊D312yes—
268C4⋊Dic3—98no—
269C4⋊Dic3—94no—
270C4⋊Dic3—91no—
271C4⋊Dic3—184no—
272C4⋊Dic3C2×C3⋊D311yescharacteristic
273Dic3⋊C4—34no—
274Dic3⋊C4—34no—
275Dic3⋊C4—98no—
276Dic3⋊C4—92no—
277Dic3⋊C4—92no—
278C4×Dic3—34no—
279C22×C32—24no—
280C22×C32—62no—
281C22×C32—62no—
282C22×C32—62no—
283C22×C32C23×D314yes—
284C22×C32C23×D311yescharacteristic
285C2×C3⋊D3—21no—
286C2×C3⋊D3—21no—
287C2×C3⋊D3—21no—
288C2×C3⋊D3—616no—
289C2×C3⋊D3—68no—
290C2×C3⋊D3—62no—
291C2×C3⋊D3—62no—
292C2×C3⋊D3—62no—
293C2×C3⋊D3—68no—
294C2×C3⋊D3—68no—
295C2×C3⋊D3—128no—
296C2×C3⋊D3—128no—
297C2×C3×D3—28no—
298C2×C3×D3—28no—
299C2×C3×D3—28no—
300C2×C3×D3—64no—
301C2×C3×D3—64no—
302C2×C3×D3—64no—
303C32×C4—28no—
304C32×C4—24no—
305C32×C4—21no—
306C32×C4—21no—
307C32×C4—28no—
308C32×C4—24no—
309C32×C4—44no—
310C32×C4—62no—
311C32×C4#23518yes—
312C32×C4#23512yes—
313C3⋊Dic3—21no—
314C3⋊Dic3—21no—
315C3⋊Dic3—21no—
316C3⋊Dic3—62no—
317C3⋊Dic3—68no—
318C3⋊Dic3—128no—
319C3×Dic3—28no—
320C3×Dic3—64no—
321C3×Dic3—64no—
322C3×Dic3—64no—
323#127—271no—
324#125—271no—
325#123—272no—
326#123—272no—
327#121—272no—
328#121—272no—
329#121—271no—
330#121—271no—
331C4×D4—31no—
332C4×D4—271no—
333#117—91no—
334C33#54011yescharacteristic, complemented, Sylow
335C23×C3—38no—
336C23×C3—91no—
337C22×D3—34no—
338C22×D3—98no—
339C22×D3—92no—
340C22×D3—188no—
341C22×D3C22×C3⋊D312yes—
342C3×D4—68no—
343C2×C3×C4—24no—
344C2×C3×C4—24no—
345C2×C3×C4—24no—
346C2×C3×C4—38no—
347C2×C3×C4—91no—
348C2×C3×C4—91no—
349C2×C3×C4—91no—
350C2×C3×C4—128no—
351C2×C3×C4—182no—
352C2×C3×C4C2×D3214yes—
353C2×C3×C4C2×D3214yes—
354C2×C3×C4C2×D3214yes—
355C2×C3×C4C22×C3⋊D311yescharacteristic
356C2×C3×C4C22×C3⋊D311yescharacteristic
357C2×C3×C4C22×C3⋊D311yescharacteristic
358#67—362no—
359#67—368no—
360#67—368no—
361#67—362no—
362#67—368no—
363C2×Dic3—34no—
364C2×Dic3—34no—
365C2×Dic3—34no—
366C2×Dic3—98no—
367C2×Dic3—92no—
368C2×Dic3—188no—
369C2×Dic3C22×C3⋊D312yes—
370D12—22no—
371D12—1816no—
372D12—188no—
373D12—182no—
374D12—368no—
375D12—362no—
376D12—362no—
377C4×D3—42no—
378C4×D3—68no—
379C4×D3—368no—
380C4×D3—362no—
381C4×D3—3616no—
382C2×C32—24no—
383C2×C32—24no—
384C2×C32—24no—
385C2×C32—122no—
386C2×C32C2×#23514yes—
387C2×C32C2×#23511yescharacteristic
388C2×C32#100414yes—
389C2×C32#100914yes—
390C2×C32#100911yescharacteristic
391C2×C32#100911yescharacteristic
392C3⋊D3—41no—
393C3⋊D3—1216no—
394C3⋊D3—122no—
395C3⋊D3—128no—
396C3×D3—48no—
397C3×D3—124no—
398C2×D4—31no—
399C2×D4—272no—
400C2×D4—272no—
401C2×D4—272no—
402C2×D4—272no—
403C2×D4—271no—
404C22×C4—32no—
405C22×C4—91no—
406C22×C4—272no—
407C4⋊C4—31no—
408C4⋊C4—91no—
409C4⋊C4—91no—
410C4⋊C4—272no—
411C4⋊C4—272no—
412C4⋊C4—271no—
413C22⋊C4—32no—
414C22⋊C4—91no—
415C22⋊C4—91no—
416C22⋊C4—272no—
417C22⋊C4—272no—
418C22⋊C4—272no—
419C42—91no—
420C42D3×C3⋊D311yescharacteristic
421C22×C3—24no—
422C22×C3—68no—
423C22×C3—68no—
424C22×C3—68no—
425C22×C3—181no—
426C22×C3—181no—
427C22×C3—181no—
428C22×C3C22×D3214yes—
429C22×C3C23×C3⋊D311yescharacteristic
430C2×D3—22no—
431C2×D3—22no—
432C2×D3—22no—
433C2×D3—64no—
434C2×D3—64no—
435C2×D3—64no—
436C2×D3—1816no—
437C2×D3—188no—
438C2×D3—188no—
439C2×D3—188no—
440C2×D3—182no—
441C2×D3—182no—
442C2×D3—182no—
443C2×D3—368no—
444C2×D3—368no—
445C3×C4—28no—
446C3×C4—24no—
447C3×C4—24no—
448C3×C4—21no—
449C3×C4—21no—
450C3×C4—28no—
451C3×C4—44no—
452C3×C4—68no—
453C3×C4—181no—
454C3×C4—181no—
455C3×C4—181no—
456C3×C4#379818yes—
457C3×C4#382812yes—
458Dic3—22no—
459Dic3—64no—
460Dic3—64no—
461Dic3—64no—
462Dic3—188no—
463Dic3—182no—
464Dic3—368no—
465C32—24no—
466C32#553511yescharacteristic, complemented
467C32#556214yescomplemented
468C23—32no—
469C23—91no—
470C23—272no—
471D4—62no—
472D4—542no—
473D4—1082no—
474D4—1082no—
475D4—1082no—
476D4—1082no—
477C2×C4—32no—
478C2×C4—91no—
479C2×C4—91no—
480C2×C4—91no—
481C2×C4—122no—
482C2×C4—182no—
483C2×C4—272no—
484C2×C4—1082no—
485C2×C4C2×D3×C3⋊D311yescharacteristic
486C2×C4C2×D3×C3⋊D311yescharacteristic
487C2×C4C2×D3×C3⋊D311yescharacteristic
488C2×C3—24no—
489C2×C3—24no—
490C2×C3—24no—
491C2×C3—128no—
492C2×C3—361no—
493C2×C3C2×#379814yes—
494C2×C3#6430714yes—
495C2×C3#6430814yes—
496C2×C3C2×#382811yescharacteristic
497C2×C3#6436111yescharacteristic
498C2×C3#6436611yescharacteristic
499D3—42no—
500D3—124no—
501D3—3616no—
502D3—362no—
503D3—368no—
504C22—62no—
505C22—62no—
506C22—62no—
507C22—181no—
508C22—181no—
509C22—181no—
510C22—542no—
511C22—542no—
512C22—542no—
513C22C22×D3×C3⋊D311yescharacteristic
514C4—21no—
515C4—21no—
516C4—62no—
517C4—181no—
518C4—181no—
519C4—181no—
520C4—542no—
521C4#8902012yes—
522C3—24no—
523C3#1059398614yescomplemented
524C3#1059603311yescharacteristic, complemented
525C2—122no—
526C2—361no—
527C2—1082no—
528C2C2×#8902011yescharacteristic
529C2#1172457911yescharacteristic
530C2#1172458111yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC33×C42C22
hypercenterC42D3×C3⋊D3
hypocenterC33#540
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. C4×C4⋊(C3⋊(C3⋊D3))C2semidirect
2. C3×#64081C2semidirect
3. C32×C4×D12C2semidirect
4. #11723527C2semidirect
5. #11723526C2semidirect
6. #11723524C2semidirect
7. #11723522C2semidirect
8. #11723514C2semidirect
9. #11723447C2semidirect
10. #11723420C2semidirect
11. #11723414C2semidirect
49. #64081D3semidirect
50. C3×C4×D12D3semidirect
171. C4×D12C3⋊D3semidirect
334. C33#540semidirect
466. C32#5535semidirect
467. C32#5562semidirect
523. C3#10593986semidirect
524. C3#10596033semidirect

Maximal subgroups

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

subgroupindex
C4×C4⋊(C3⋊(C3⋊D3))2
C3×#640812
C32×C4×D122
#117235272
#117235262
#117235242
#117235222
#117235142
#117234472
#117234202
#117234142
#105960333
#105939863