Finite Group Databasebeta

← #13175343

Subgroups — #13175343

Order 1728.

1780 conjugacy classes of subgroups.

The subgroup lattice has 502 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×#634C211yesmaximal, characteristic, complemented
3C32×C42×D3C211yesmaximal, characteristic, complemented
4#11723516C211yesmaximal, characteristic
5C4×#6291C212yesmaximal, complemented
6#11723454C211yesmaximal, characteristic, complemented
7#11723453C212yesmaximal, complemented
8#11723452C212yesmaximal, complemented
9#11723441C211yesmaximal, characteristic, complemented
10#11723425C211yesmaximal, characteristic, complemented
11#11723414C212yesmaximal, complemented
12#10595950—31nomaximal
13#10593870—34nomaximal
14C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
15C2×C4×C3⋊(C3⋊D3)C2212yes—
16C2×C3×#634C2211yescharacteristic
17C2×C32×C4×D3C2211yescharacteristic
18C2×C32×C4×D3C2211yescharacteristic
19C2×C32×C4×D3C2211yescharacteristic
20C2×#6291C2212yes—
21C33×C42C2211yescharacteristic
22#88859C2212yes—
23C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
24C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
25C3×#3750C2212yes—
26C3×C4×C3⋊Dic3C2212yes—
27C32×C4×Dic3C2211yescharacteristic
28C32×C4×Dic3C2211yescharacteristic
29C32×C4×Dic3C2211yescharacteristic
30#88807C2212yes—
31#88806C2212yes—
32#88806C2212yes—
33#88804C2212yes—
34#88804C2212yes—
35#88804C2212yes—
36(C2×D3)⋊(C3⋊Dic3)C2212yes—
37C2×C32×C42—31no—
38#64139—31no—
39#64138—32no—
40C4×#638—32no—
41#64133—31no—
42#64085—32no—
43#64084—31no—
44C4×C4⋊(C3⋊D3)—31no—
45C4×C4⋊(C3⋊D3)—34no—
46C4×C4⋊(C3⋊D3)—64no—
47#64081—32no—
48#64079—31no—
49C4×#634D311yescharacteristic, complemented
50C3×C42×D3D314yescomplemented
51C3×C4×Dic6—34no—
52#63909—34no—
53C4×#626—38no—
54#63874—38no—
55#63869—34no—
56#63868—38no—
57#63864—34no—
58#63856—34no—
59#63850—38no—
60C22×C3⋊(C3⋊D3)C2312yes—
61C22×C32×D3C2311yescharacteristic
62C2×C33×C4C2311yescharacteristic
63C2×C33×C4C2311yescharacteristic
64C2×C33×C4C2311yescharacteristic
65C2×C3⋊(C3⋊Dic3)C2312yes—
66C4⋊(C3⋊(C3⋊D3))—22no—
67C4×C3⋊(C3⋊D3)—24no—
68C2×C3×C3⋊Dic3C2312yes—
69C2×C3×C3⋊Dic3C2312yes—
70C3×#634—22no—
71C2×C32×Dic3C2311yescharacteristic
72C2×C32×Dic3C2311yescharacteristic
73C2×C32×Dic3C2311yescharacteristic
74C32×C4×D3—22no—
75C32×C4×D3—22no—
76C32×C4×D3—22no—
77#6291—24no—
78#5580—91no—
79#5479—94no—
80C22×C32×C4—31no—
81C22×C32×C4—31no—
82C22×C32×C4—31no—
83C2×#638—32no—
84C2×C4⋊(C3⋊D3)—31no—
85C2×C4⋊(C3⋊D3)—34no—
86C2×C4⋊(C3⋊D3)—64no—
87C2×C4×C3⋊D3—32no—
88C2×C4×C3⋊D3—38no—
89C2×C4×C3⋊D3—68no—
90C2×#634C2×D311yescharacteristic
91C2×C3×C4×D3C2×D314yes—
92C2×C3×C4×D3C2×D314yes—
93C2×C3×C4×D3C2×D314yes—
94C2×C3×Dic6—34no—
95C2×#626—38no—
96C32×C42—24no—
97C32×C42—31no—
98C32×C42—31no—
99C32×C42—31no—
100C32×C42C2×D314yes—
101C32×C42C2×D311yescharacteristic
102#3757—32no—
103#3752—32no—
104#3752—38no—
105#3752—32no—
106#3752—32no—
107#3752—32no—
108#3752—68no—
109C4⋊(C3⋊Dic3)—31no—
110C4⋊(C3⋊Dic3)—34no—
111C4⋊(C3⋊Dic3)—64no—
112C4⋊(C3⋊Dic3)C2×D311yescharacteristic
113#3750—32no—
114#3750—32no—
115#3750—32no—
116#3750C2×D312yes—
117C4×C3⋊Dic3C2×D312yes—
118C3×C4⋊Dic3—34no—
119C3×Dic3⋊C4—38no—
120C3×C4×Dic3—38no—
121C3×C4×Dic3C2×D314yes—
122C3×C4×Dic3C2×D314yes—
123C3×C4×Dic3C2×D314yes—
124#3724—38no—
125#3724—38no—
126Dic3⋊Dic3—38no—
127#3722—38no—
128#3722—38no—
129#3722—38no—
130(C2×D3)⋊Dic3—38no—
131C22×C33C2411yescharacteristic
132C2×C3⋊(C3⋊D3)—22no—
133C2×C3⋊(C3⋊D3)—22no—
134C2×C3⋊(C3⋊D3)—22no—
135C2×C32×D3—21no—
136C2×C32×D3—21no—
137C2×C32×D3#4112yes—
138C33×C4—21no—
139C33×C4#4112yes—
140C33×C4#4112yes—
141C3⋊(C3⋊Dic3)—22no—
142C3×C3⋊Dic3—22no—
143C3×C3⋊Dic3—22no—
144C32×Dic3—21no—
145C32×Dic3—21no—
146C32×Dic3—21no—
147C3×C4×Q8—91no—
148C2×C3×C42—34no—
149#947—91no—
150#946—91no—
151#930—94no—
152#929—98no—
153C4×#67—98no—
154#924—94no—
155#898—92no—
156#895—91no—
157#894—92no—
158#890—91no—
159#876—98no—
160#875—94no—
161C4×D12—92no—
162C4×D12—94no—
163C4×D12—91no—
164C4×D12—184no—
165#872—98no—
166#872—92no—
167C42×D3C3⋊D311yescharacteristic, complemented
168#870—94no—
169C4×Dic6—34no—
170C23×C32—31no—
171C22×C3⋊D3—32no—
172C22×C3⋊D3—38no—
173C22×C3⋊D3—68no—
174C22×C3×D3C22×D314yes—
175C2×C32×C4—24no—
176C2×C32×C4—24no—
177C2×C32×C4—24no—
178C2×C32×C4—31no—
179C2×C32×C4—31no—
180C2×C32×C4—31no—
181C2×C32×C4—62no—
182C2×C32×C4—62no—
183C2×C32×C4—62no—
184C2×C32×C4C22×D314yes—
185C2×C32×C4C22×D314yes—
186C2×C32×C4C22×D314yes—
187C2×C32×C4C22×D311yescharacteristic
188C2×C32×C4C22×D311yescharacteristic
189C2×C32×C4C22×D311yescharacteristic
190#638—64no—
191C2×C3⋊Dic3—32no—
192C2×C3⋊Dic3—38no—
193C2×C3⋊Dic3—68no—
194C2×C3⋊Dic3C22×D312yes—
195C2×C3⋊Dic3C22×D312yes—
196C4⋊(C3⋊D3)—616no—
197C4⋊(C3⋊D3)—68no—
198C4⋊(C3⋊D3)—62no—
199C4×C3⋊D3—64no—
200C4×C3⋊D3—616no—
201C4×C3⋊D3—1216no—
202#634—22no—
203C2×C3×Dic3—38no—
204C2×C3×Dic3—38no—
205C2×C3×Dic3C22×D314yes—
206C2×C3×Dic3C22×D314yes—
207C2×C3×Dic3C22×D314yes—
208C3×C4×D3—28no—
209C3×C4×D3—28no—
210C3×C4×D3—28no—
211C3×Dic6—68no—
212#626—616no—
213#529—271noSylow
214C2×C33C2×#4111yescharacteristic
215C2×C33C2×#4111yescharacteristic
216C2×C33C2×#4111yescharacteristic
217C3⋊(C3⋊D3)—42no—
218C32×D3—22no—
219C2×C3×Q8—91no—
220C22×C3×C4—34no—
221C22×C3×C4—34no—
222C22×C3×C4—34no—
223C2×#67—98no—
224C2×D12—92no—
225C2×D12—94no—
226C2×D12—91no—
227C2×D12—184no—
228C2×C4×D3—98no—
229C2×C4×D3—92no—
230C2×C4×D3—188no—
231C2×C4×D3C2×C3⋊D311yescharacteristic
232C2×C4×D3C2×C3⋊D311yescharacteristic
233C2×C4×D3C2×C3⋊D311yescharacteristic
234C2×Dic6—34no—
235C3×C4⋊C4—91no—
236C3×C4⋊C4—92no—
237C3×C42—24no—
238C3×C42—34no—
239C3×C42—34no—
240C3×C42—34no—
241C3×C42—92no—
242C3×C42D3214yes—
243C3×C42C2×C3⋊D311yescharacteristic
244#217—98no—
245(C2×D3)⋊C4—98no—
246(C2×D3)⋊C4—98no—
247(C2×D3)⋊C4—98no—
248(C2×D3)⋊C4—92no—
249(C2×D3)⋊C4—98no—
250(C2×D3)⋊C4—92no—
251(C2×D3)⋊C4—92no—
252(C2×D3)⋊C4—92no—
253(C2×D3)⋊C4—92no—
254(C2×D3)⋊C4—188no—
255C4⋊Dic3—34no—
256C4⋊Dic3—94no—
257C4⋊Dic3—91no—
258C4⋊Dic3—92no—
259C4⋊Dic3—184no—
260Dic3⋊C4—38no—
261Dic3⋊C4—98no—
262Dic3⋊C4—98no—
263Dic3⋊C4—98no—
264Dic3⋊C4—92no—
265Dic3⋊C4—92no—
266C4×Dic3—38no—
267C4×Dic3C2×C3⋊D311yescharacteristic
268C4×Dic3C2×C3⋊D311yescharacteristic
269C4×Dic3C2×C3⋊D311yescharacteristic
270C22×C32—24no—
271C22×C32—32no—
272C22×C32—61no—
273C22×C32—61no—
274C22×C32C23×D314yes—
275C22×C32C23×D311yescharacteristic
276C2×C3⋊D3—616no—
277C2×C3⋊D3—68no—
278C2×C3⋊D3—62no—
279C2×C3⋊D3—62no—
280C2×C3⋊D3—62no—
281C2×C3⋊D3—68no—
282C2×C3⋊D3—68no—
283C2×C3⋊D3—128no—
284C2×C3⋊D3—128no—
285C2×C3×D3—24no—
286C2×C3×D3—24no—
287C2×C3×D3#23518yes—
288C32×C4—21no—
289C32×C4—28no—
290C32×C4—28no—
291C32×C4—24no—
292C32×C4—44no—
293C32×C4—61no—
294C32×C4—61no—
295C32×C4—61no—
296C32×C4#23518yes—
297C32×C4#23518yes—
298C32×C4#23512yes—
299C32×C4#23912yes—
300C3⋊Dic3—22no—
301C3⋊Dic3—22no—
302C3⋊Dic3—62no—
303C3⋊Dic3—68no—
304C3⋊Dic3—128no—
305C3×Dic3—24no—
306C3×Dic3—24no—
307C3×Dic3—24no—
308C3×Dic3—68no—
309C3×Dic3—68no—
310#126—272no—
311#125—271no—
312#124—271no—
313#123—272no—
314C22⋊Q8—271no—
315#121—271no—
316C4×Q8—91no—
317C4×D4—272no—
318C4×D4—271no—
319#117—272no—
320C2×C42—31no—
321C33#52911yescharacteristic, complemented, Sylow
322C23×C3—34no—
323C22×D3—98no—
324C22×D3—92no—
325C22×D3—188no—
326C22×D3C22×C3⋊D311yescharacteristic
327C3×Q8—182no—
328C2×C3×C4—24no—
329C2×C3×C4—24no—
330C2×C3×C4—24no—
331C2×C3×C4—34no—
332C2×C3×C4—34no—
333C2×C3×C4—34no—
334C2×C3×C4—68no—
335C2×C3×C4—68no—
336C2×C3×C4—68no—
337C2×C3×C4—92no—
338C2×C3×C4—92no—
339C2×C3×C4C2×D3214yes—
340C2×C3×C4C2×D3214yes—
341C2×C3×C4C2×D3214yes—
342C2×C3×C4C22×C3⋊D311yescharacteristic
343C2×C3×C4C22×C3⋊D311yescharacteristic
344C2×C3×C4C22×C3⋊D311yescharacteristic
345#67—1816no—
346C2×Dic3—38no—
347C2×Dic3—38no—
348C2×Dic3—98no—
349C2×Dic3—92no—
350C2×Dic3—188no—
351C2×Dic3C22×C3⋊D311yescharacteristic
352C2×Dic3C22×C3⋊D311yescharacteristic
353C2×Dic3C22×C3⋊D311yescharacteristic
354D12—1816no—
355D12—184no—
356D12—188no—
357D12—182no—
358C4×D3—22no—
359C4×D3—22no—
360C4×D3—22no—
361C4×D3—1816no—
362C4×D3—184no—
363C4×D3—3616no—
364Dic6—68no—
365C2×C32—24no—
366C2×C32—24no—
367C2×C32—24no—
368C2×C32—62no—
369C2×C32C2×#23514yes—
370C2×C32C2×#23514yes—
371C2×C32C2×#23514yes—
372C2×C32C2×#23511yescharacteristic
373C2×C32C2×#23911yescharacteristic
374C2×C32D3×#4111yescharacteristic
375C3⋊D3—1216no—
376C3⋊D3—128no—
377C3⋊D3—122no—
378C3×D3—28no—
379C2×Q8—91no—
380C2×D4—272no—
381C2×D4—271no—
382C22×C4—31no—
383C22×C4—31no—
384C22×C4—31no—
385C22×C4—272no—
386C4⋊C4—91no—
387C4⋊C4—92no—
388C4⋊C4—272no—
389C4⋊C4—272no—
390C4⋊C4—272no—
391C4⋊C4—271no—
392C22⋊C4—272no—
393C22⋊C4—272no—
394C22⋊C4—272no—
395C22⋊C4—272no—
396C22⋊C4—272no—
397C42—31no—
398C42—31no—
399C42—31no—
400C42—92no—
401C42D3×C3⋊D311yescharacteristic
402C22×C3—24no—
403C22×C3—38no—
404C22×C3—64no—
405C22×C3—64no—
406C22×C3C22×D3214yes—
407C22×C3C23×C3⋊D311yescharacteristic
408C2×D3—21no—
409C2×D3—21no—
410C2×D3—1816no—
411C2×D3—188no—
412C2×D3—188no—
413C2×D3—188no—
414C2×D3—182no—
415C2×D3—182no—
416C2×D3—182no—
417C2×D3—368no—
418C2×D3—368no—
419C2×D3#382812yes—
420C3×C4—24no—
421C3×C4—21no—
422C3×C4—28no—
423C3×C4—28no—
424C3×C4—44no—
425C3×C4—64no—
426C3×C4—64no—
427C3×C4—64no—
428C3×C4—182no—
429C3×C4—182no—
430C3×C4#379818yes—
431C3×C4#379918yes—
432C3×C4#382812yes—
433C3×C4#382812yes—
434Dic3—21no—
435Dic3—21no—
436Dic3—21no—
437Dic3—68no—
438Dic3—68no—
439Dic3—188no—
440Dic3—182no—
441Dic3—368no—
442C32—24no—
443C32#547914yescomplemented
444C32#558011yescharacteristic, complemented
445C23—31no—
446C23—272no—
447Q8—182no—
448D4—544no—
449D4—542no—
450C2×C4—31no—
451C2×C4—31no—
452C2×C4—31no—
453C2×C4—62no—
454C2×C4—62no—
455C2×C4—62no—
456C2×C4—92no—
457C2×C4—92no—
458C2×C4—272no—
459C2×C4—544no—
460C2×C4C2×D3×C3⋊D311yescharacteristic
461C2×C4C2×D3×C3⋊D311yescharacteristic
462C2×C4C2×D3×C3⋊D311yescharacteristic
463C2×C3—24no—
464C2×C3—24no—
465C2×C3—24no—
466C2×C3—68no—
467C2×C3C2×#379814yes—
468C2×C3C2×#379914yes—
469C2×C3D3×#23514yes—
470C2×C3C2×#382811yescharacteristic
471C2×C3C2×#382811yescharacteristic
472C2×C3C2×#382811yescharacteristic
473D3—22no—
474D3—3616no—
475D3—368no—
476D3—362no—
477C22—32no—
478C22—61no—
479C22—61no—
480C22—542no—
481C22—542no—
482C22—542no—
483C22C22×D3×C3⋊D311yescharacteristic
484C4—21no—
485C4—61no—
486C4—61no—
487C4—61no—
488C4—182no—
489C4—182no—
490C4—542no—
491C4#8902012yes—
492C4#8902212yes—
493C3—24no—
494C3#1059387014yescomplemented
495C3#1059595011yescharacteristic, complemented
496C2—62no—
497C2—1082no—
498C2C2×#8902011yescharacteristic
499C2C2×#8902211yescharacteristic
500C2D3×#382811yescharacteristic
subgroupG/H
centerC2×C4C2×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FrattiniC22C22×D3×C3⋊D3
FittingC33×C42C22
prime radical#11723516C2
hypercenterC42D3×C3⋊D3
hypocenterC33#529
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C2×C4 ◁ 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×#634C2semidirect
3. C32×C42×D3C2semidirect
5. C4×#6291C2semidirect
6. #11723454C2semidirect
7. #11723453C2semidirect
8. #11723452C2semidirect
9. #11723441C2semidirect
10. #11723425C2semidirect
11. #11723414C2semidirect
49. C4×#634D3semidirect
50. C3×C42×D3D3semidirect
167. C42×D3C3⋊D3semidirect
321. C33#529semidirect
443. C32#5479semidirect
444. C32#5580semidirect
494. C3#10593870semidirect
495. C3#10595950semidirect

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×#6342
C32×C42×D32
#117235162
C4×#62912
#117234542
#117234532
#117234522
#117234412
#117234252
#117234142
#105959503
#105938703