Finite Group Databasebeta

← #13175344

Subgroups — #13175344

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×#64079C211yesmaximal, characteristic, complemented
3C32×C4×D12C211yesmaximal, characteristic, complemented
4#11723516C211yesmaximal, characteristic, complemented
5#11723516C211yesmaximal, characteristic, complemented
6D3×C4⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
7#11723475C212yesmaximal, complemented
8#11723462C212yesmaximal, complemented
9#11723424C211yesmaximal, characteristic, complemented
10#11723420C212yesmaximal, complemented
11#11723411C212yesmaximal, complemented
12#10596036—31nomaximal
13#10593925—34nomaximal
14C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
15C2×C4×C3⋊(C3⋊D3)C2212yes—
16C2×C32×D12C2211yescharacteristic
17C2×C32×C4×D3C2211yescharacteristic
18C2×C32×C4×D3C2211yescharacteristic
19C2×#6291C2212yes—
20C2×#6291C2212yes—
21C2×D3×C3⋊Dic3C2212yes—
22C33×C42C2211yescharacteristic
23#88859C2212yes—
24C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
25C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
26C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
27C3×#3750C2212yes—
28C3×#3750C2212yes—
29C32×(C2×D3)⋊C4C2211yescharacteristic
30C32×(C2×D3)⋊C4C2211yescharacteristic
31C32×C4⋊Dic3C2211yescharacteristic
32Dic3⋊(C3⋊Dic3)C2212yes—
33#88804C2212yes—
34#88804C2212yes—
35(C2×D3)⋊(C3⋊Dic3)C2212yes—
36Dic3×C3⋊Dic3C2212yes—
37C32×C4×D4—31no—
38#64148—32no—
39#64139—31no—
40#64139—31no—
41#64135—31no—
42C2×C4⋊(C3⋊Dic3)—31no—
43#64107—32no—
44#64092—32no—
45#64090—32no—
46C4×C4⋊(C3⋊D3)—31no—
47C4×C4⋊(C3⋊D3)—34no—
48C4×C4⋊(C3⋊D3)—64no—
49#64079D311yescharacteristic, complemented
50C3×C4×D12D314yescomplemented
51C3×#870—34no—
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×C32×D3C2311yescharacteristic
62C22×C32×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⋊Dic3C2312yes—
70C2×C3×C3⋊Dic3C2312yes—
71C2×C32×Dic3C2311yescharacteristic
72C2×C32×Dic3C2311yescharacteristic
73C32×D12—22no—
74C32×C4×D3—22no—
75C32×C4×D3—22no—
76#6291—24no—
77#6291—42no—
78D3×C3⋊Dic3—42no—
79#5695—91no—
80#5565—94no—
81C2×C32×D4—31no—
82C22×C32×C4—31no—
83C22×C32×C4—31no—
84C2×#638—32no—
85C2×#638—32no—
86C22×C3⋊Dic3—32no—
87C2×C4⋊(C3⋊D3)—31no—
88C2×C4⋊(C3⋊D3)—34no—
89C2×C4⋊(C3⋊D3)—64no—
90C2×C4×C3⋊D3—32no—
91C2×C4×C3⋊D3—38no—
92C2×C4×C3⋊D3—68no—
93C2×C3×D12C2×D314yes—
94C2×C3×C4×D3C2×D314yes—
95C2×C3×C4×D3C2×D314yes—
96C2×#626—38no—
97C2×#626—38no—
98C2×D3×Dic3—38no—
99C32×C4⋊C4—31no—
100C32×C22⋊C4—31no—
101C32×C22⋊C4—31no—
102C32×C42—24no—
103C32×C42C2×D314yes—
104C32×C42C2×D311yescharacteristic
105#3757—32no—
106#3752—32no—
107#3752—38no—
108#3752—32no—
109#3752—32no—
110#3752—68no—
111C4⋊(C3⋊Dic3)—31no—
112C4⋊(C3⋊Dic3)—34no—
113C4⋊(C3⋊Dic3)—32no—
114C4⋊(C3⋊Dic3)—64no—
115C4⋊(C3⋊Dic3)C2×D311yescharacteristic
116C4⋊(C3⋊Dic3)C2×D311yescharacteristic
117#3750C2×D312yes—
118#3750C2×D312yes—
119C4×C3⋊Dic3—32no—
120C3×(C2×D3)⋊C4C2×D314yes—
121C3×(C2×D3)⋊C4C2×D314yes—
122C3×C4⋊Dic3—34no—
123C3×C4⋊Dic3—34no—
124C3×C4⋊Dic3C2×D314yes—
125C3×Dic3⋊C4—38no—
126C3×Dic3⋊C4—38no—
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×C32×D3—21no—
137C2×C32×D3—21no—
138C2×C32×D3—21no—
139C2×C32×D3—21no—
140C2×C32×D3—21no—
141C2×C32×D3#4112yes—
142C33×C4—21no—
143C33×C4—21no—
144C33×C4#4112yes—
145C3⋊(C3⋊Dic3)—22no—
146C3×C3⋊Dic3—22no—
147C3×C3⋊Dic3—22no—
148C32×Dic3—21no—
149C32×Dic3—21no—
150C3×#125—91no—
151C3×C4×D4—34no—
152#939—98no—
153#930—94no—
154#930—94no—
155#926—94no—
156C2×C4⋊Dic3—94no—
157#898—98no—
158#898—92no—
159#895—92no—
160#895—91no—
161#895—91no—
162#894—92no—
163#893—92no—
164#892—91no—
165D3×C4⋊C4—91no—
166#883—98no—
167#67⋊C4—98no—
168C4×D12—94no—
169C4×D12—91no—
170C4×D12—184no—
171C4×D12C3⋊D311yescharacteristic, complemented
172#870—34no—
173C23×C32—31no—
174C23×C32—31no—
175C22×C3⋊D3—32no—
176C22×C3⋊D3—38no—
177C22×C3⋊D3—68no—
178C22×C3×D3C22×D314yes—
179C22×C3×D3C22×D314yes—
180C32×D4—62no—
181C2×C32×C4—24no—
182C2×C32×C4—24no—
183C2×C32×C4—24no—
184C2×C32×C4—31no—
185C2×C32×C4—31no—
186C2×C32×C4—62no—
187C2×C32×C4—62no—
188C2×C32×C4C22×D314yes—
189C2×C32×C4C22×D314yes—
190C2×C32×C4C22×D314yes—
191C2×C32×C4C22×D311yescharacteristic
192C2×C32×C4C22×D311yescharacteristic
193C2×C32×C4C22×D311yescharacteristic
194#638—64no—
195#638—122no—
196C2×C3⋊Dic3—32no—
197C2×C3⋊Dic3—38no—
198C2×C3⋊Dic3—68no—
199C2×C3⋊Dic3—122no—
200C2×C3⋊Dic3C22×D312yes—
201C2×C3⋊Dic3C22×D312yes—
202C4⋊(C3⋊D3)—616no—
203C4⋊(C3⋊D3)—68no—
204C4⋊(C3⋊D3)—62no—
205C4×C3⋊D3—122no—
206C4×C3⋊D3—128no—
207C4×C3⋊D3—1216no—
208C2×C3×Dic3—38no—
209C2×C3×Dic3—38no—
210C2×C3×Dic3C22×D314yes—
211C2×C3×Dic3C22×D314yes—
212C3×D12—28no—
213C3×C4×D3—28no—
214C3×C4×D3—28no—
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—
223C32×D3—22no—
224C32×D3—41no—
225C2×C3×D4—34no—
226C22×C3×C4—34no—
227C22×C3×C4—34no—
228C2×#67—98no—
229C2×#67—98no—
230C22×Dic3—98no—
231C2×D12—92no—
232C2×D12—94no—
233C2×D12—91no—
234C2×D12—92no—
235C2×D12—184no—
236C2×D12C2×C3⋊D311yescharacteristic
237C2×C4×D3—92no—
238C2×C4×D3—98no—
239C2×C4×D3—92no—
240C2×C4×D3—188no—
241C2×C4×D3C2×C3⋊D311yescharacteristic
242C2×C4×D3C2×C3⋊D311yescharacteristic
243C3×C4⋊C4—34no—
244C3×C4⋊C4—91no—
245C3×C4⋊C4—91no—
246C3×C4⋊C4—92no—
247C3×C4⋊C4—92no—
248C3×C22⋊C4—34no—
249C3×C22⋊C4—34no—
250C3×C42—24no—
251C3×C42D3214yes—
252C3×C42C2×C3⋊D311yescharacteristic
253#217—98no—
254(C2×D3)⋊C4—98no—
255(C2×D3)⋊C4—98no—
256(C2×D3)⋊C4—92no—
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
263(C2×D3)⋊C4C2×C3⋊D311yescharacteristic
264C4⋊Dic3—34no—
265C4⋊Dic3—34no—
266C4⋊Dic3—98no—
267C4⋊Dic3—94no—
268C4⋊Dic3—91no—
269C4⋊Dic3—184no—
270C4⋊Dic3C2×C3⋊D311yescharacteristic
271Dic3⋊C4—38no—
272Dic3⋊C4—38no—
273Dic3⋊C4—92no—
274C4×Dic3—98no—
275C4×Dic3—92no—
276C22×C32—24no—
277C22×C32—32no—
278C22×C32—61no—
279C22×C32—61no—
280C22×C32—61no—
281C22×C32—61no—
282C22×C32—61no—
283C22×C32C23×D314yes—
284C22×C32C23×D311yescharacteristic
285C2×C3⋊D3—616no—
286C2×C3⋊D3—68no—
287C2×C3⋊D3—62no—
288C2×C3⋊D3—62no—
289C2×C3⋊D3—62no—
290C2×C3⋊D3—68no—
291C2×C3⋊D3—68no—
292C2×C3⋊D3—128no—
293C2×C3⋊D3—128no—
294C2×C3×D3—24no—
295C2×C3×D3—24no—
296C2×C3×D3—24no—
297C2×C3×D3—24no—
298C2×C3×D3—24no—
299C2×C3×D3#23518yes—
300C32×C4—28no—
301C32×C4—21no—
302C32×C4—24no—
303C32×C4—21no—
304C32×C4—28no—
305C32×C4—24no—
306C32×C4—44no—
307C32×C4—61no—
308C32×C4—61no—
309C32×C4#23718yes—
310C32×C4#23912yes—
311C3⋊Dic3—22no—
312C3⋊Dic3—22no—
313C3⋊Dic3—62no—
314C3⋊Dic3—68no—
315C3⋊Dic3—128no—
316C3×Dic3—24no—
317C3×Dic3—24no—
318C3×Dic3—68no—
319C3×Dic3—68no—
320#126—272no—
321#125—91no—
322#123—272no—
323#121—271no—
324#121—271no—
325#121—272no—
326C4×D4—31no—
327C4×D4—272no—
328C4×D4—271no—
329#117—271no—
330C2×C4⋊C4—271no—
331C33#55311yescharacteristic, complemented, Sylow
332C23×C3—34no—
333C23×C3—34no—
334C22×D3—98no—
335C22×D3—92no—
336C22×D3—188no—
337C22×D3C22×C3⋊D311yescharacteristic
338C22×D3C22×C3⋊D311yescharacteristic
339C3×D4—68no—
340C2×C3×C4—24no—
341C2×C3×C4—24no—
342C2×C3×C4—24no—
343C2×C3×C4—34no—
344C2×C3×C4—34no—
345C2×C3×C4—68no—
346C2×C3×C4—68no—
347C2×C3×C4—92no—
348C2×C3×C4—92no—
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—1816no—
356#67—368no—
357C2×Dic3—38no—
358C2×Dic3—38no—
359C2×Dic3—98no—
360C2×Dic3—92no—
361C2×Dic3—188no—
362C2×Dic3—368no—
363C2×Dic3C22×C3⋊D311yescharacteristic
364C2×Dic3C22×C3⋊D311yescharacteristic
365D12—22no—
366D12—1816no—
367D12—184no—
368D12—188no—
369D12—182no—
370D12—362no—
371C4×D3—22no—
372C4×D3—22no—
373C4×D3—368no—
374C4×D3—362no—
375C4×D3—362no—
376C4×D3—3616no—
377C2×C32—24no—
378C2×C32—24no—
379C2×C32—24no—
380C2×C32—62no—
381C2×C32—121no—
382C2×C32C2×#23514yes—
383C2×C32C2×#23714yes—
384C2×C32C2×#23911yescharacteristic
385C2×C32D3×#4111yescharacteristic
386C2×C32#100914yes—
387C2×C32#100911yescharacteristic
388C3⋊D3—1216no—
389C3⋊D3—122no—
390C3⋊D3—128no—
391C3×D3—28no—
392C3×D3—44no—
393C2×D4—31no—
394C2×D4—272no—
395C2×D4—272no—
396C2×D4—271no—
397C22×C4—31no—
398C22×C4—31no—
399C22×C4—272no—
400C22×C4—272no—
401C4⋊C4—31no—
402C4⋊C4—91no—
403C4⋊C4—91no—
404C4⋊C4—92no—
405C4⋊C4—92no—
406C4⋊C4—272no—
407C4⋊C4—271no—
408C22⋊C4—31no—
409C22⋊C4—31no—
410C22⋊C4—272no—
411C22⋊C4—272no—
412C22⋊C4—272no—
413C22⋊C4—272no—
414C42—272no—
415C42D3×C3⋊D311yescharacteristic
416C22×C3—24no—
417C22×C3—38no—
418C22×C3—64no—
419C22×C3—64no—
420C22×C3—64no—
421C22×C3—64no—
422C22×C3—64no—
423C22×C3C22×D3214yes—
424C22×C3C23×C3⋊D311yescharacteristic
425C2×D3—21no—
426C2×D3—21no—
427C2×D3—21no—
428C2×D3—21no—
429C2×D3—21no—
430C2×D3—1816no—
431C2×D3—188no—
432C2×D3—188no—
433C2×D3—188no—
434C2×D3—182no—
435C2×D3—182no—
436C2×D3—182no—
437C2×D3—368no—
438C2×D3—368no—
439C2×D3#382812yes—
440C3×C4—28no—
441C3×C4—24no—
442C3×C4—24no—
443C3×C4—21no—
444C3×C4—21no—
445C3×C4—28no—
446C3×C4—44no—
447C3×C4—64no—
448C3×C4—64no—
449C3×C4—182no—
450C3×C4—182no—
451C3×C4#379618yes—
452C3×C4#383012yes—
453Dic3—21no—
454Dic3—21no—
455Dic3—68no—
456Dic3—68no—
457Dic3—188no—
458Dic3—182no—
459Dic3—368no—
460C32—24no—
461C32#556514yescomplemented
462C32#569511yescharacteristic, complemented
463C23—31no—
464C23—31no—
465C23—272no—
466D4—62no—
467D4—544no—
468D4—542no—
469D4—1082no—
470C2×C4—31no—
471C2×C4—31no—
472C2×C4—62no—
473C2×C4—62no—
474C2×C4—92no—
475C2×C4—92no—
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—68no—
486C2×C3—124no—
487C2×C3C2×#379614yes—
488C2×C3D3×#23514yes—
489C2×C3#6430814yes—
490C2×C3C2×#382811yescharacteristic
491C2×C3C2×#383011yescharacteristic
492C2×C3#6436611yescharacteristic
493D3—22no—
494D3—41no—
495D3—3616no—
496D3—368no—
497D3—362no—
498C22—32no—
499C22—61no—
500C22—61no—
501C22—61no—
502C22—61no—
503C22—61no—
504C22—542no—
505C22—542no—
506C22—542no—
507C22C22×D3×C3⋊D311yescharacteristic
508C4—21no—
509C4—21no—
510C4—61no—
511C4—61no—
512C4—182no—
513C4—182no—
514C4—542no—
515C4#8901512yes—
516C3—24no—
517C3#1059392514yescomplemented
518C3#1059603611yescharacteristic, complemented
519C2—62no—
520C2—121no—
521C2—1082no—
522C2C2×#8901511yescharacteristic
523C2D3×#382811yescharacteristic
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×#64079C2semidirect
3. C32×C4×D12C2semidirect
4. #11723516C2semidirect
5. #11723516C2semidirect
6. D3×C4⋊(C3⋊Dic3)C2semidirect
7. #11723475C2semidirect
8. #11723462C2semidirect
9. #11723424C2semidirect
10. #11723420C2semidirect
11. #11723411C2semidirect
49. #64079D3semidirect
50. C3×C4×D12D3semidirect
171. C4×D12C3⋊D3semidirect
331. C33#553semidirect
461. C32#5565semidirect
462. C32#5695semidirect
517. C3#10593925semidirect
518. C3#10596036semidirect

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×#640792
C32×C4×D122
#117235162
D3×C4⋊(C3⋊Dic3)2
#117234752
#117234622
#117234242
#117234202
#117234112
#105960363
#105939253