Finite Group Databasebeta

← #13175325

Subgroups — #13175325

Order 1728.

1804 conjugacy classes of subgroups.

The subgroup lattice has 526 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×C4×C4⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
3C32×#870C211yesmaximal, characteristic, complemented
4#11723526C211yesmaximal, characteristic, complemented
5#11723526C211yesmaximal, characteristic, complemented
6C3⋊D3×C4⋊Dic3C211yesmaximal, characteristic, complemented
7#11723482C212yesmaximal, complemented
8#11723472C212yesmaximal, complemented
9#11723421C212yesmaximal, complemented
10#11723404C211yesmaximal, characteristic, complemented
11#11723403C212yesmaximal, complemented
12#10596166—31nomaximal
13#10593925—34nomaximal
14C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
15C2×C4×C3⋊(C3⋊D3)C2212yes—
16C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
17C2×C3×C4×C3⋊D3C2211yescharacteristic
18C2×C3×C4×C3⋊D3C2211yescharacteristic
19C2×#6292C2212yes—
20C2×#6292C2212yes—
21C2×Dic3×C3⋊D3C2212yes—
22C33×C42C2211yescharacteristic
23#88859C2212yes—
24C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
25C3×#3752C2211yescharacteristic
26C3×#3752C2211yescharacteristic
27C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
28C32×C4⋊Dic3C2211yescharacteristic
29C32×C4⋊Dic3C2211yescharacteristic
30C32×Dic3⋊C4C2212yes—
31C32×Dic3⋊C4C2212yes—
32#88807C2212yes—
33#88804C2212yes—
34#88804C2212yes—
35#88803C2212yes—
36Dic3×C3⋊Dic3C2212yes—
37C32×#125—31no—
38#64107—32no—
39#64104—32no—
40#64104—31no—
41#64104—31no—
42#64103—32no—
43#64102—32no—
44#64101—31no—
45C4⋊C4×C3⋊D3—31no—
46C4×C4⋊(C3⋊D3)—31no—
47C4×C4⋊(C3⋊D3)—34no—
48C4×C4⋊(C3⋊D3)—64no—
49C4×C4⋊(C3⋊D3)D311yescharacteristic, complemented
50C3×C4×D12—34no—
51C3×#870D314yescomplemented
52#63909—34no—
53#63909—34no—
54D3×C4⋊Dic3—34no—
55#63886—38no—
56#63881—38no—
57#63853—38no—
58#63848—38no—
59#63841—34no—
60C22×C3⋊(C3⋊D3)C2312yes—
61C22×C3×C3⋊D3C2311yescharacteristic
62C22×C3×C3⋊D3C2311yescharacteristic
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
71C3×C4⋊(C3⋊D3)—22no—
72C3×C4×C3⋊D3—22no—
73C3×C4×C3⋊D3—22no—
74C2×C32×Dic3C2312yes—
75C2×C32×Dic3C2312yes—
76#6292—24no—
77#6292—42no—
78Dic3×C3⋊D3—42no—
79#5695—94no—
80#5565—91no—
81C2×C4⋊(C3⋊D3)—32no—
82C2×C4⋊(C3⋊D3)—31no—
83C2×C4⋊(C3⋊D3)—34no—
84C2×C4⋊(C3⋊D3)—32no—
85C2×C4⋊(C3⋊D3)—64no—
86C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
87C2×C4×C3⋊D3—32no—
88C2×C4×C3⋊D3—32no—
89C2×C4×C3⋊D3—38no—
90C2×C4×C3⋊D3—68no—
91C2×C4×C3⋊D3C2×D311yescharacteristic
92C2×C4×C3⋊D3C2×D311yescharacteristic
93C2×C3×D12—34no—
94C2×C3×C4×D3—34no—
95C2×C3×C4×D3—34no—
96C2×#626—38no—
97C2×#626—38no—
98C2×D3×Dic3—38no—
99C32×C4⋊C4—31no—
100C32×C4⋊C4—31no—
101C32×C4⋊C4—32no—
102C32×C4⋊C4—32no—
103C32×C42—24no—
104C32×C42C2×D314yes—
105C32×C42C2×D311yescharacteristic
106#3752—32no—
107#3752—32no—
108#3752—38no—
109#3752—32no—
110#3752—32no—
111#3752—68no—
112#3752C2×D311yescharacteristic
113#3752C2×D311yescharacteristic
114C4⋊(C3⋊Dic3)—31no—
115C4⋊(C3⋊Dic3)—34no—
116C4⋊(C3⋊Dic3)—64no—
117C4⋊(C3⋊Dic3)C2×D311yescharacteristic
118#3750—32no—
119C4×C3⋊Dic3—32no—
120C3×(C2×D3)⋊C4—34no—
121C3×(C2×D3)⋊C4—34no—
122C3×C4⋊Dic3—34no—
123C3×C4⋊Dic3C2×D314yes—
124C3×C4⋊Dic3C2×D314yes—
125C3×Dic3⋊C4C2×D318yes—
126C3×Dic3⋊C4C2×D318yes—
127Dic3⋊Dic3—38no—
128#3722—38no—
129#3722—38no—
130(C2×D3)⋊Dic3—38no—
131Dic32—38no—
132C22×C33C2411yescharacteristic
133C2×C3⋊(C3⋊D3)—22no—
134C2×C3⋊(C3⋊D3)—22no—
135C2×C3⋊(C3⋊D3)—22no—
136C2×C3×C3⋊D3—21no—
137C2×C3×C3⋊D3—21no—
138C2×C3×C3⋊D3—21no—
139C2×C3×C3⋊D3—21no—
140C2×C3×C3⋊D3—21no—
141C2×C3×C3⋊D3#4112yes—
142C33×C4—21no—
143C33×C4—21no—
144C33×C4#4112yes—
145C3⋊(C3⋊Dic3)—22no—
146C3×C3⋊Dic3—21no—
147C3×C3⋊Dic3—21no—
148C32×Dic3—22no—
149C32×Dic3—22no—
150C3×#125—34no—
151C3×C4×D4—91no—
152#939—92no—
153#930—91no—
154#930—91no—
155#926—91no—
156C2×C4⋊Dic3—91no—
157#898—98no—
158#898—92no—
159#895—94no—
160#895—94no—
161#895—98no—
162#894—98no—
163#893—98no—
164#892—94no—
165D3×C4⋊C4—94no—
166#883—92no—
167#67⋊C4—92no—
168C4×D12—34no—
169C4×D12—94no—
170C4×D12—91no—
171C4×D12—184no—
172#870C3⋊D311yescharacteristic, complemented
173C22×C3⋊D3—32no—
174C22×C3⋊D3—38no—
175C22×C3⋊D3—68no—
176C22×C3⋊D3C22×D311yescharacteristic
177C22×C3⋊D3C22×D311yescharacteristic
178C22×C3×D3—34no—
179C22×C3×D3—34no—
180C2×C32×C4—24no—
181C2×C32×C4—24no—
182C2×C32×C4—24no—
183C2×C32×C4—32no—
184C2×C32×C4—32no—
185C2×C32×C4C22×D314yes—
186C2×C32×C4C22×D314yes—
187C2×C32×C4C22×D314yes—
188C2×C32×C4C22×D311yescharacteristic
189C2×C32×C4C22×D311yescharacteristic
190C2×C32×C4C22×D311yescharacteristic
191C2×C3⋊Dic3—32no—
192C2×C3⋊Dic3—38no—
193C2×C3⋊Dic3—68no—
194C2×C3⋊Dic3C22×D311yescharacteristic
195C2×C3⋊Dic3C22×D311yescharacteristic
196C4⋊(C3⋊D3)—22no—
197C4⋊(C3⋊D3)—616no—
198C4⋊(C3⋊D3)—64no—
199C4⋊(C3⋊D3)—62no—
200C4⋊(C3⋊D3)—68no—
201C4⋊(C3⋊D3)—122no—
202C4×C3⋊D3—22no—
203C4×C3⋊D3—22no—
204C4×C3⋊D3—122no—
205C4×C3⋊D3—128no—
206C4×C3⋊D3—122no—
207C4×C3⋊D3—1216no—
208C2×C3×Dic3—34no—
209C2×C3×Dic3—34no—
210C2×C3×Dic3C22×D318yes—
211C2×C3×Dic3C22×D318yes—
212C3×D12—68no—
213C3×C4×D3—68no—
214C3×C4×D3—68no—
215#626—616no—
216#626—128no—
217D3×Dic3—128no—
218#553—271noSylow
219C2×C33C2×#4111yescharacteristic
220C2×C33C2×#4111yescharacteristic
221C2×C33H(5,𝔽2)11yescharacteristic
222C3⋊(C3⋊D3)—42no—
223C3×C3⋊D3—22no—
224C3×C3⋊D3—41no—
225C2×C3×D4—91no—
226C22×C3×C4—91no—
227C22×C3×C4—91no—
228C2×#67—92no—
229C2×#67—92no—
230C22×Dic3—92no—
231C2×D12—34no—
232C2×D12—98no—
233C2×D12—94no—
234C2×D12—91no—
235C2×D12—98no—
236C2×D12—184no—
237C2×C4×D3—34no—
238C2×C4×D3—34no—
239C2×C4×D3—98no—
240C2×C4×D3—98no—
241C2×C4×D3—92no—
242C2×C4×D3—188no—
243C3×C4⋊C4—34no—
244C3×C4⋊C4—34no—
245C3×C4⋊C4—38no—
246C3×C4⋊C4—38no—
247C3×C4⋊C4—91no—
248C3×C22⋊C4—91no—
249C3×C22⋊C4—91no—
250C3×C42—24no—
251C3×C42D3214yes—
252C3×C42C2×C3⋊D311yescharacteristic
253#217—92no—
254(C2×D3)⋊C4—34no—
255(C2×D3)⋊C4—34no—
256(C2×D3)⋊C4—98no—
257(C2×D3)⋊C4—98no—
258(C2×D3)⋊C4—98no—
259(C2×D3)⋊C4—98no—
260(C2×D3)⋊C4—92no—
261(C2×D3)⋊C4—92no—
262(C2×D3)⋊C4—92no—
263(C2×D3)⋊C4—188no—
264C4⋊Dic3—34no—
265C4⋊Dic3—94no—
266C4⋊Dic3—91no—
267C4⋊Dic3—92no—
268C4⋊Dic3—184no—
269C4⋊Dic3C2×C3⋊D311yescharacteristic
270C4⋊Dic3C2×C3⋊D311yescharacteristic
271Dic3⋊C4—98no—
272Dic3⋊C4C2×C3⋊D312yes—
273Dic3⋊C4C2×C3⋊D312yes—
274C4×Dic3—98no—
275C4×Dic3—92no—
276C22×C32—24no—
277C22×C32C23×D314yes—
278C22×C32C23×D311yescharacteristic
279C2×C3⋊D3—21no—
280C2×C3⋊D3—21no—
281C2×C3⋊D3—21no—
282C2×C3⋊D3—21no—
283C2×C3⋊D3—21no—
284C2×C3⋊D3—616no—
285C2×C3⋊D3—62no—
286C2×C3⋊D3—62no—
287C2×C3⋊D3—68no—
288C2×C3⋊D3—62no—
289C2×C3⋊D3—68no—
290C2×C3⋊D3—68no—
291C2×C3⋊D3—128no—
292C2×C3⋊D3—128no—
293C2×C3⋊D3#23512yes—
294C2×C3×D3—38no—
295C2×C3×D3—64no—
296C2×C3×D3—64no—
297C2×C3×D3—64no—
298C2×C3×D3—64no—
299C2×C3×D3—64no—
300C32×C4—28no—
301C32×C4—21no—
302C32×C4—24no—
303C32×C4—21no—
304C32×C4—28no—
305C32×C4—24no—
306C32×C4—44no—
307C32×C4—62no—
308C32×C4—62no—
309C32×C4#23712yes—
310C32×C4#23918yes—
311C3⋊Dic3—21no—
312C3⋊Dic3—21no—
313C3⋊Dic3—62no—
314C3⋊Dic3—68no—
315C3⋊Dic3—128no—
316C3×Dic3—28no—
317C3×Dic3—28no—
318C3×Dic3—64no—
319C3×Dic3—64no—
320#126—272no—
321#125—31no—
322#123—272no—
323#121—271no—
324#121—271no—
325#121—272no—
326C4×D4—91no—
327C4×D4—272no—
328C4×D4—271no—
329#117—271no—
330C2×C4⋊C4—271no—
331C33#55311yescharacteristic, complemented, Sylow
332C23×C3—91no—
333C23×C3—91no—
334C22×D3—34no—
335C22×D3—34no—
336C22×D3—98no—
337C22×D3—92no—
338C22×D3—188no—
339C3×D4—182no—
340C2×C3×C4—24no—
341C2×C3×C4—24no—
342C2×C3×C4—24no—
343C2×C3×C4—38no—
344C2×C3×C4—38no—
345C2×C3×C4—91no—
346C2×C3×C4—91no—
347C2×C3×C4—182no—
348C2×C3×C4—182no—
349C2×C3×C4C2×D3214yes—
350C2×C3×C4C2×D3214yes—
351C2×C3×C4C2×D3214yes—
352C2×C3×C4C22×C3⋊D311yescharacteristic
353C2×C3×C4C22×C3⋊D311yescharacteristic
354C2×C3×C4C22×C3⋊D311yescharacteristic
355#67—184no—
356#67—362no—
357C2×Dic3—34no—
358C2×Dic3—34no—
359C2×Dic3—98no—
360C2×Dic3—92no—
361C2×Dic3—188no—
362C2×Dic3—362no—
363C2×Dic3C22×C3⋊D312yes—
364C2×Dic3C22×C3⋊D312yes—
365D12—68no—
366D12—1816no—
367D12—1816no—
368D12—188no—
369D12—182no—
370D12—368no—
371C4×D3—68no—
372C4×D3—68no—
373C4×D3—368no—
374C4×D3—362no—
375C4×D3—368no—
376C4×D3—3616no—
377C2×C32—24no—
378C2×C32—24no—
379C2×C32—24no—
380C2×C32C2×#23511yescharacteristic
381C2×C32C2×#23711yescharacteristic
382C2×C32C2×#23914yes—
383C2×C32D3×#4114yes—
384C2×C32#100914yes—
385C2×C32#100911yescharacteristic
386C3⋊D3—22no—
387C3⋊D3—41no—
388C3⋊D3—1216no—
389C3⋊D3—128no—
390C3⋊D3—122no—
391C3×D3—68no—
392C3×D3—124no—
393C2×D4—91no—
394C2×D4—272no—
395C2×D4—272no—
396C2×D4—271no—
397C22×C4—91no—
398C22×C4—91no—
399C22×C4—272no—
400C22×C4—272no—
401C4⋊C4—31no—
402C4⋊C4—31no—
403C4⋊C4—32no—
404C4⋊C4—32no—
405C4⋊C4—91no—
406C4⋊C4—272no—
407C4⋊C4—271no—
408C22⋊C4—91no—
409C22⋊C4—91no—
410C22⋊C4—272no—
411C22⋊C4—272no—
412C22⋊C4—272no—
413C22⋊C4—272no—
414C42—272no—
415C42D3×C3⋊D311yescharacteristic
416C22×C3—24no—
417C22×C3—92no—
418C22×C3—181no—
419C22×C3—181no—
420C22×C3—181no—
421C22×C3—181no—
422C22×C3—181no—
423C22×C3C22×D3214yes—
424C22×C3C23×C3⋊D311yescharacteristic
425C2×D3—38no—
426C2×D3—64no—
427C2×D3—64no—
428C2×D3—64no—
429C2×D3—64no—
430C2×D3—64no—
431C2×D3—1816no—
432C2×D3—188no—
433C2×D3—188no—
434C2×D3—188no—
435C2×D3—182no—
436C2×D3—182no—
437C2×D3—182no—
438C2×D3—368no—
439C2×D3—368no—
440C3×C4—28no—
441C3×C4—24no—
442C3×C4—24no—
443C3×C4—21no—
444C3×C4—21no—
445C3×C4—28no—
446C3×C4—44no—
447C3×C4—68no—
448C3×C4—68no—
449C3×C4—181no—
450C3×C4—181no—
451C3×C4#379618yes—
452C3×C4#383212yes—
453Dic3—22no—
454Dic3—22no—
455Dic3—64no—
456Dic3—64no—
457Dic3—188no—
458Dic3—182no—
459Dic3—368no—
460C32—24no—
461C32#556511yescharacteristic, complemented
462C32#569514yescomplemented
463C23—91no—
464C23—91no—
465C23—272no—
466D4—182no—
467D4—544no—
468D4—542no—
469D4—1082no—
470C2×C4—32no—
471C2×C4—32no—
472C2×C4—91no—
473C2×C4—91no—
474C2×C4—182no—
475C2×C4—182no—
476C2×C4—272no—
477C2×C4—1082no—
478C2×C4—1082no—
479C2×C4C2×D3×C3⋊D311yescharacteristic
480C2×C4C2×D3×C3⋊D311yescharacteristic
481C2×C4C2×D3×C3⋊D311yescharacteristic
482C2×C3—24no—
483C2×C3—24no—
484C2×C3—24no—
485C2×C3—182no—
486C2×C3—361no—
487C2×C3C2×#379614yes—
488C2×C3D3×#23514yes—
489C2×C3#6430814yes—
490C2×C3C2×#383211yescharacteristic
491C2×C3#41×C3⋊D311yescharacteristic
492C2×C3#6436611yescharacteristic
493D3—68no—
494D3—124no—
495D3—3616no—
496D3—362no—
497D3—368no—
498C22—92no—
499C22—181no—
500C22—181no—
501C22—181no—
502C22—181no—
503C22—181no—
504C22—542no—
505C22—542no—
506C22—542no—
507C22C22×D3×C3⋊D311yescharacteristic
508C4—21no—
509C4—21no—
510C4—62no—
511C4—62no—
512C4—181no—
513C4—181no—
514C4—542no—
515C4#8901712yes—
516C3—24no—
517C3#1059392514yescomplemented
518C3#1059616611yescharacteristic, complemented
519C2—182no—
520C2—361no—
521C2—1082no—
522C2C2×#8901711yescharacteristic
523C2C3⋊D3×#23511yescharacteristic
524C2#1172458111yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC33×C42C22
hypercenterC42D3×C3⋊D3
hypocenterC33#553
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×C4×C4⋊(C3⋊D3)C2semidirect
3. C32×#870C2semidirect
4. #11723526C2semidirect
5. #11723526C2semidirect
6. C3⋊D3×C4⋊Dic3C2semidirect
7. #11723482C2semidirect
8. #11723472C2semidirect
9. #11723421C2semidirect
10. #11723404C2semidirect
11. #11723403C2semidirect
49. C4×C4⋊(C3⋊D3)D3semidirect
51. C3×#870D3semidirect
172. #870C3⋊D3semidirect
331. C33#553semidirect
461. C32#5565semidirect
462. C32#5695semidirect
517. C3#10593925semidirect
518. C3#10596166semidirect

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×C4×C4⋊(C3⋊D3)2
C32×#8702
#117235262
C3⋊D3×C4⋊Dic32
#117234822
#117234722
#117234212
#117234042
#117234032
#105961663
#105939253