Finite Group Databasebeta

← #13177218

Subgroups — #13177218

Order 1728.

1804 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723971C211yesmaximal, characteristic, complemented
2C3×#64135C211yesmaximal, characteristic, complemented
3C32×C4×#67C211yesmaximal, characteristic, complemented
4#11723628C212yesmaximal, complemented
5#11723616C212yesmaximal, complemented
6#11723516C211yesmaximal, characteristic, complemented
7#11723484C211yesmaximal, characteristic, complemented
8D3×C4⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
9#11723461C212yesmaximal, complemented
10#11723454C211yesmaximal, characteristic, complemented
11#11723446C212yesmaximal, complemented
12#10596036—31nomaximal
13#10594946—34nomaximal
14C22×C33×C4C2211yescharacteristic
15C2×#6312C2212yes—
16C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
17C2×C32×#67C2211yescharacteristic
18C2×C32×C4×D3C2211yescharacteristic
19C2×#6291C2212yes—
20C2×#6289C2212yes—
21C2×D3×C3⋊Dic3C2212yes—
22#88859C2212yes—
23C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
24C3×#3757C2212yes—
25C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
26C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
27C3×C4×C3⋊Dic3C2212yes—
28C32×#217C2211yescharacteristic
29C32×(C2×D3)⋊C4C2211yescharacteristic
30C32×Dic3⋊C4C2211yescharacteristic
31C32×C4×Dic3C2211yescharacteristic
32#88807C2212yes—
33#88806C2212yes—
34Dic3⋊(C3⋊Dic3)C2212yes—
35#88804C2212yes—
36#88804C2212yes—
37C32×C4×D4—31no—
38#64148—32no—
39#64139—31no—
40#64139—31no—
41#64139—34no—
42#64139—64no—
43#64135D311yescharacteristic, complemented
44C2×C4⋊(C3⋊Dic3)—31no—
45#64107—32no—
46#64092—32no—
47#64090—32no—
48C4×C4⋊(C3⋊D3)—31no—
49#64079—31no—
50C3×C4×#67D314yescomplemented
51C3×#926—34no—
52#63975—38no—
53#63971—38no—
54#63909—34no—
55#63894—34no—
56D3×C4⋊Dic3—34no—
57#63880—38no—
58#63869—34no—
59#63868—38no—
60C23×C33C2311yescharacteristic
61C22×C3⋊(C3⋊D3)C2312yes—
62C22×C32×D3C2311yescharacteristic
63C2×C33×C4—22no—
64C2×C33×C4C2311yescharacteristic
65C2×C33×C4C2311yescharacteristic
66#6312—42no—
67C2×C3⋊(C3⋊Dic3)C2312yes—
68C4⋊(C3⋊(C3⋊D3))—22no—
69C2×C3×C3⋊Dic3C2312yes—
70C2×C3×C3⋊Dic3C2312yes—
71C32×#67—22no—
72C2×C32×Dic3C2311yescharacteristic
73C2×C32×Dic3C2311yescharacteristic
74C2×C32×Dic3C2311yescharacteristic
75C32×C4×D3—22no—
76#6291—24no—
77#6289—42no—
78D3×C3⋊Dic3—42no—
79#5565—94no—
80#5530—91no—
81C2×C32×D4—31no—
82C22×C32×C4—24no—
83C22×C32×C4—31no—
84C22×C32×C4C2×D314yes—
85C22×C32×C4C2×D311yescharacteristic
86C2×#638—32no—
87C2×#638—38no—
88C2×#638—32no—
89C2×#638—68no—
90C22×C3⋊Dic3—32no—
91C2×C4⋊(C3⋊D3)—31no—
92C2×C4⋊(C3⋊D3)—34no—
93C2×C4⋊(C3⋊D3)—64no—
94C2×C4×C3⋊D3—32no—
95C2×C3×#67C2×D314yes—
96C2×C3×C4×D3C2×D314yes—
97C2×#626—38no—
98C2×#624—38no—
99C2×D3×Dic3—38no—
100C32×C4⋊C4—31no—
101C32×C22⋊C4—31no—
102C32×C22⋊C4—31no—
103C32×C42—31no—
104#3757C2×D312yes—
105#3752—32no—
106#3752—38no—
107#3752—32no—
108#3752—32no—
109#3752—68no—
110C4⋊(C3⋊Dic3)—31no—
111C4⋊(C3⋊Dic3)—34no—
112C4⋊(C3⋊Dic3)—32no—
113C4⋊(C3⋊Dic3)—64no—
114C4⋊(C3⋊Dic3)C2×D311yescharacteristic
115C4⋊(C3⋊Dic3)C2×D311yescharacteristic
116#3750—32no—
117#3750—32no—
118C4×C3⋊Dic3C2×D312yes—
119C3×#217—38no—
120C3×#217C2×D314yes—
121C3×(C2×D3)⋊C4C2×D314yes—
122C3×C4⋊Dic3—34no—
123C3×C4⋊Dic3—34no—
124C3×Dic3⋊C4C2×D314yes—
125C3×C4×Dic3—38no—
126C3×C4×Dic3C2×D314yes—
127#3724—38no—
128Dic3⋊Dic3—38no—
129Dic3⋊Dic3—38no—
130#3722—38no—
131#3722—38no—
132C22×C33—21no—
133C22×C33—21no—
134C22×C33—21no—
135C22×C33C2411yescharacteristic
136C2×C3⋊(C3⋊D3)—22no—
137C2×C3⋊(C3⋊D3)—22no—
138C2×C3⋊(C3⋊D3)—22no—
139C2×C32×D3—21no—
140C2×C32×D3—21no—
141C2×C32×D3#4112yes—
142C33×C4—21no—
143C33×C4#4112yes—
144C3⋊(C3⋊Dic3)—22no—
145C3×C3⋊Dic3—22no—
146C3×C3⋊Dic3—22no—
147C32×Dic3—21no—
148C32×Dic3—21no—
149C32×Dic3—21no—
150C3×C4×D4—34no—
151C3×#117—91no—
152#939—98no—
153#930—94no—
154#930—94no—
155#930—91no—
156#930—184no—
157C4×#67C3⋊D311yescharacteristic, complemented
158#926—34no—
159C2×C4⋊Dic3—94no—
160#898—98no—
161#895—91no—
162#893—91no—
163D3×C4⋊C4—91no—
164#890—91no—
165#884—92no—
166#883—98no—
167#883—92no—
168#67⋊C4—98no—
169#876—92no—
170C4×D12—94no—
171C4×D12—92no—
172#870—94no—
173C23×C32—24no—
174C23×C32—31no—
175C23×C32C22×D314yes—
176C23×C32C22×D311yescharacteristic
177C22×C3⋊D3—32no—
178C22×C3⋊D3—38no—
179C22×C3⋊D3—68no—
180C22×C3×D3C22×D314yes—
181C32×D4—62no—
182C2×C32×C4—22no—
183C2×C32×C4—24no—
184C2×C32×C4—24no—
185C2×C32×C4—28no—
186C2×C32×C4—31no—
187C2×C32×C4—31no—
188C2×C32×C4—31no—
189C2×C32×C4—48no—
190C2×C32×C4—62no—
191C2×C32×C4C22×D314yes—
192C2×C32×C4C22×D314yes—
193C2×C32×C4C22×D311yescharacteristic
194C2×C32×C4C22×D311yescharacteristic
195#638—64no—
196#638—122no—
197#638—128no—
198#638—1216no—
199C2×C3⋊Dic3—32no—
200C2×C3⋊Dic3—38no—
201C2×C3⋊Dic3—68no—
202C2×C3⋊Dic3—122no—
203C2×C3⋊Dic3C22×D312yes—
204C2×C3⋊Dic3C22×D312yes—
205C4⋊(C3⋊D3)—62no—
206C4⋊(C3⋊D3)—68no—
207C4⋊(C3⋊D3)—128no—
208C4×C3⋊D3—122no—
209C3×#67—28no—
210C2×C3×Dic3—38no—
211C2×C3×Dic3—38no—
212C2×C3×Dic3C22×D314yes—
213C2×C3×Dic3C22×D314yes—
214C2×C3×Dic3C22×D314yes—
215C3×C4×D3—28no—
216#626—616no—
217#624—128no—
218D3×Dic3—128no—
219#553—271noSylow
220C2×C33—41no—
221C2×C33C2×#4111yescharacteristic
222C2×C33C2×#4111yescharacteristic
223C2×C33H(5,𝔽2)11yescharacteristic
224C3⋊(C3⋊D3)—42no—
225C32×D3—22no—
226C2×C3×D4—34no—
227C22×C3×C4—24no—
228C22×C3×C4—34no—
229C22×C3×C4D3214yes—
230C22×C3×C4C2×C3⋊D311yescharacteristic
231C2×#67—98no—
232C2×#67—98no—
233C2×#67—92no—
234C2×#67—188no—
235C2×#67C2×C3⋊D311yescharacteristic
236C22×Dic3—98no—
237C2×D12—92no—
238C2×D12—94no—
239C2×D12—91no—
240C2×D12—184no—
241C2×C4×D3—98no—
242C2×C4×D3—92no—
243C2×C4×D3—92no—
244C2×C4×D3C2×C3⋊D311yescharacteristic
245C3×C4⋊C4—34no—
246C3×C4⋊C4—91no—
247C3×C4⋊C4—91no—
248C3×C22⋊C4—34no—
249C3×C22⋊C4—34no—
250C3×C22⋊C4—92no—
251C3×C42—34no—
252C3×C42—92no—
253#217—38no—
254#217C2×C3⋊D311yescharacteristic
255(C2×D3)⋊C4—98no—
256(C2×D3)⋊C4—98no—
257(C2×D3)⋊C4—98no—
258(C2×D3)⋊C4—92no—
259(C2×D3)⋊C4—92no—
260(C2×D3)⋊C4—92no—
261(C2×D3)⋊C4—188no—
262(C2×D3)⋊C4C2×C3⋊D311yescharacteristic
263C4⋊Dic3—34no—
264C4⋊Dic3—34no—
265C4⋊Dic3—98no—
266C4⋊Dic3—94no—
267C4⋊Dic3—91no—
268C4⋊Dic3—92no—
269C4⋊Dic3—184no—
270Dic3⋊C4—98no—
271Dic3⋊C4—98no—
272Dic3⋊C4—92no—
273Dic3⋊C4—92no—
274Dic3⋊C4C2×C3⋊D311yescharacteristic
275C4×Dic3—38no—
276C4×Dic3C2×C3⋊D311yescharacteristic
277C22×C32—28no—
278C22×C32—21no—
279C22×C32—21no—
280C22×C32—24no—
281C22×C32—21no—
282C22×C32—24no—
283C22×C32—24no—
284C22×C32—24no—
285C22×C32—32no—
286C22×C32—44no—
287C22×C32—44no—
288C22×C32—61no—
289C22×C32—61no—
290C22×C32C23×D314yes—
291C22×C32C23×D311yescharacteristic
292C2×C3⋊D3—62no—
293C2×C3⋊D3—62no—
294C2×C3⋊D3—62no—
295C2×C3⋊D3—68no—
296C2×C3⋊D3—68no—
297C2×C3⋊D3—68no—
298C2×C3⋊D3—128no—
299C2×C3⋊D3—128no—
300C2×C3⋊D3—128no—
301C2×C3×D3—24no—
302C2×C3×D3—24no—
303C2×C3×D3#23518yes—
304C32×C4—21no—
305C32×C4—28no—
306C32×C4—24no—
307C32×C4—44no—
308C32×C4—61no—
309C32×C4—61no—
310C32×C4—61no—
311C32×C4#23512yes—
312C32×C4#23718yes—
313C3⋊Dic3—22no—
314C3⋊Dic3—22no—
315C3⋊Dic3—62no—
316C3⋊Dic3—68no—
317C3⋊Dic3—128no—
318C3×Dic3—24no—
319C3×Dic3—24no—
320C3×Dic3—24no—
321C3×Dic3—68no—
322C3×Dic3—68no—
323#126—272no—
324#125—271no—
325#123—272no—
326#121—272no—
327#121—271no—
328#121—271no—
329C4×D4—31no—
330C4×D4—271no—
331C4×D4—272no—
332#117—91no—
333C2×C4⋊C4—271no—
334C33#55311yescharacteristic, complemented, Sylow
335C23×C3—24no—
336C23×C3—34no—
337C23×C3C2×D3214yes—
338C23×C3C22×C3⋊D311yescharacteristic
339C22×D3—98no—
340C22×D3—92no—
341C22×D3—188no—
342C22×D3C22×C3⋊D311yescharacteristic
343C3×D4—68no—
344C2×C3×C4—28no—
345C2×C3×C4—22no—
346C2×C3×C4—24no—
347C2×C3×C4—24no—
348C2×C3×C4—34no—
349C2×C3×C4—34no—
350C2×C3×C4—34no—
351C2×C3×C4—48no—
352C2×C3×C4—68no—
353C2×C3×C4—92no—
354C2×C3×C4—92no—
355C2×C3×C4C2×D3214yes—
356C2×C3×C4C2×D3214yes—
357C2×C3×C4C22×C3⋊D311yescharacteristic
358C2×C3×C4C22×C3⋊D311yescharacteristic
359#67—22no—
360#67—1816no—
361#67—368no—
362#67—362no—
363#67—3616no—
364C2×Dic3—38no—
365C2×Dic3—38no—
366C2×Dic3—98no—
367C2×Dic3—92no—
368C2×Dic3—188no—
369C2×Dic3—368no—
370C2×Dic3C22×C3⋊D311yescharacteristic
371C2×Dic3C22×C3⋊D311yescharacteristic
372C2×Dic3C22×C3⋊D311yescharacteristic
373D12—184no—
374D12—188no—
375D12—182no—
376D12—368no—
377C4×D3—22no—
378C4×D3—362no—
379C4×D3—368no—
380C4×D3—362no—
381C2×C32—24no—
382C2×C32—24no—
383C2×C32—24no—
384C2×C32—48no—
385C2×C32—41no—
386C2×C32—44no—
387C2×C32—62no—
388C2×C32C2×#23514yes—
389C2×C32C2×#23511yescharacteristic
390C2×C32C2×#23714yes—
391C2×C32D3×#4111yescharacteristic
392C2×C32#100914yes—
393C2×C32#100911yescharacteristic
394C3⋊D3—122no—
395C3⋊D3—128no—
396C3⋊D3—248no—
397C3×D3—28no—
398C2×D4—31no—
399C2×D4—272no—
400C2×D4—272no—
401C2×D4—271no—
402C22×C4—31no—
403C22×C4—272no—
404C22×C4—272no—
405C22×C4D3×C3⋊D311yescharacteristic
406C4⋊C4—31no—
407C4⋊C4—91no—
408C4⋊C4—91no—
409C4⋊C4—272no—
410C4⋊C4—272no—
411C4⋊C4—272no—
412C4⋊C4—271no—
413C22⋊C4—31no—
414C22⋊C4—31no—
415C22⋊C4—92no—
416C22⋊C4—272no—
417C22⋊C4—272no—
418C22⋊C4—272no—
419C42—31no—
420C42—92no—
421C22×C3—28no—
422C22×C3—24no—
423C22×C3—24no—
424C22×C3—24no—
425C22×C3—21no—
426C22×C3—21no—
427C22×C3—21no—
428C22×C3—24no—
429C22×C3—38no—
430C22×C3—44no—
431C22×C3—44no—
432C22×C3—64no—
433C22×C3—64no—
434C22×C3C22×D3214yes—
435C22×C3C23×C3⋊D311yescharacteristic
436C2×D3—21no—
437C2×D3—21no—
438C2×D3—188no—
439C2×D3—188no—
440C2×D3—188no—
441C2×D3—182no—
442C2×D3—182no—
443C2×D3—182no—
444C2×D3—368no—
445C2×D3—368no—
446C2×D3—368no—
447C2×D3#382812yes—
448C3×C4—24no—
449C3×C4—21no—
450C3×C4—28no—
451C3×C4—44no—
452C3×C4—64no—
453C3×C4—64no—
454C3×C4—64no—
455C3×C4—182no—
456C3×C4—182no—
457C3×C4#380418yes—
458C3×C4#383012yes—
459Dic3—21no—
460Dic3—21no—
461Dic3—21no—
462Dic3—68no—
463Dic3—68no—
464Dic3—188no—
465Dic3—182no—
466Dic3—368no—
467C32—24no—
468C32#553011yescharacteristic, complemented
469C32#556514yescomplemented
470C23—31no—
471C23—272no—
472C23C2×D3×C3⋊D311yescharacteristic
473D4—62no—
474D4—544no—
475D4—542no—
476D4—1082no—
477C2×C4—22no—
478C2×C4—31no—
479C2×C4—31no—
480C2×C4—31no—
481C2×C4—62no—
482C2×C4—92no—
483C2×C4—92no—
484C2×C4—272no—
485C2×C4—1082no—
486C2×C4—1082no—
487C2×C4C2×D3×C3⋊D311yescharacteristic
488C2×C4C2×D3×C3⋊D311yescharacteristic
489C2×C3—24no—
490C2×C3—24no—
491C2×C3—24no—
492C2×C3—48no—
493C2×C3—44no—
494C2×C3—41no—
495C2×C3—68no—
496C2×C3D3×#23514yes—
497C2×C3#6430814yes—
498C2×C3C2×#380414yes—
499C2×C3C2×#382811yescharacteristic
500C2×C3C2×#383011yescharacteristic
501C2×C3#6436611yescharacteristic
502D3—22no—
503D3—368no—
504D3—362no—
505D3—728no—
506C22—21no—
507C22—21no—
508C22—21no—
509C22—32no—
510C22—61no—
511C22—61no—
512C22—542no—
513C22—542no—
514C22—542no—
515C22C22×D3×C3⋊D311yescharacteristic
516C4—21no—
517C4—61no—
518C4—61no—
519C4—61no—
520C4—182no—
521C4—182no—
522C4—542no—
523C4#8902812yes—
524C3—24no—
525C3#1059494614yescomplemented
526C3#1059603611yescharacteristic, complemented
527C2—41no—
528C2—62no—
529C2—1082no—
530C2D3×#382811yescharacteristic
531C2#1172458111yescharacteristic
532C2C2×#8902811yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC22×C33×C4C22
hypercenterC22×C4D3×C3⋊D3
hypocenterC33#553
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. #11723971C2semidirect
2. C3×#64135C2semidirect
3. C32×C4×#67C2semidirect
4. #11723628C2semidirect
5. #11723616C2semidirect
6. #11723516C2semidirect
7. #11723484C2semidirect
8. D3×C4⋊(C3⋊Dic3)C2semidirect
9. #11723461C2semidirect
10. #11723454C2semidirect
11. #11723446C2semidirect
43. #64135D3semidirect
50. C3×C4×#67D3semidirect
157. C4×#67C3⋊D3semidirect
334. C33#553semidirect
468. C32#5530semidirect
469. C32#5565semidirect
525. C3#10594946semidirect
526. C3#10596036semidirect

Maximal subgroups

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

subgroupindex
#117239712
C3×#641352
C32×C4×#672
#117236282
#117236162
#117235162
#117234842
D3×C4⋊(C3⋊Dic3)2
#117234612
#117234542
#117234462
#105960363
#105949463