Finite Group Databasebeta

← #13175350

Subgroups — #13175350

Order 1728.

1804 conjugacy classes of subgroups.

The subgroup lattice has 546 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#11723532C211yesmaximal, characteristic, complemented
5#11723532C211yesmaximal, characteristic, complemented
6#11723512C212yesmaximal, complemented
7#11723503C211yesmaximal, characteristic, complemented
8#11723486C212yesmaximal, complemented
9#11723417C212yesmaximal, complemented
10#11723413C211yesmaximal, characteristic, complemented
11#11723408C212yesmaximal, complemented
12#10596166—31nomaximal
13#10593982—34nomaximal
14C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
15C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
16C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
17C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
18C2×C3×C4×C3⋊D3C2212yes—
19C2×#6292C2212yes—
20C2×#6292C2212yes—
21C2×#6289C2212yes—
22C33×C42C2211yescharacteristic
23#88859C2211yescharacteristic
24#88859C2211yescharacteristic
25C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
26C3×#3752C2212yes—
27C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
28C32×C4⋊Dic3C2211yescharacteristic
29C32×C4⋊Dic3C2211yescharacteristic
30C32×Dic3⋊C4C2212yes—
31C32×Dic3⋊C4C2212yes—
32#88806C2212yes—
33#88804C2212yes—
34#88803C2212yes—
35#88803C2212yes—
36Dic3×C3⋊Dic3C2212yes—
37C32×#125—31no—
38#64107—32no—
39#64104—31no—
40#64104—31no—
41#64104—32no—
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#63916—34no—
53#63916—34no—
54#63910—38no—
55#63904—38no—
56#63900—34no—
57#63861—38no—
58#63850—38no—
59#63849—34no—
60C22×C3⋊(C3⋊D3)C2311yescharacteristic
61C22×C3⋊(C3⋊D3)C2311yescharacteristic
62C22×C3×C3⋊D3C2312yes—
63C2×C33×C4C2311yescharacteristic
64C2×C33×C4C2311yescharacteristic
65C2×C33×C4C2311yescharacteristic
66C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
67C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
68C4⋊(C3⋊(C3⋊D3))—22no—
69C4×C3⋊(C3⋊D3)—22no—
70C4×C3⋊(C3⋊D3)—22no—
71C2×C3×C3⋊Dic3C2312yes—
72C3×C4⋊(C3⋊D3)—22no—
73C3×C4×C3⋊D3—42no—
74C2×C32×Dic3C2312yes—
75C2×C32×Dic3C2312yes—
76#6292—24no—
77#6292—42no—
78#6289—42no—
79#5695—94no—
80#5558—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—31no—
88C2×C4×C3⋊D3—31no—
89C2×C4×C3⋊D3—34no—
90C2×C4×C3⋊D3—34no—
91C2×C4×C3⋊D3—32no—
92C2×C4×C3⋊D3—64no—
93C2×C4×C3⋊D3—64no—
94C2×C4×C3⋊D3C2×D312yes—
95C2×C3×D12—34no—
96C2×C3×C4×D3—38no—
97C2×#626—38no—
98C2×#626—38no—
99C2×#624—38no—
100C32×C4⋊C4—31no—
101C32×C4⋊C4—31no—
102C32×C4⋊C4—32no—
103C32×C4⋊C4—32no—
104C32×C42—24no—
105C32×C42C2×D314yes—
106C32×C42C2×D311yescharacteristic
107#3752—32no—
108#3752—32no—
109#3752—31no—
110#3752—31no—
111#3752—34no—
112#3752—34no—
113#3752—32no—
114#3752—64no—
115#3752—64no—
116#3752C2×D312yes—
117C4⋊(C3⋊Dic3)—31no—
118C4⋊(C3⋊Dic3)—34no—
119C4⋊(C3⋊Dic3)—64no—
120C4⋊(C3⋊Dic3)C2×D311yescharacteristic
121#3750—32no—
122C4×C3⋊Dic3—32no—
123C3×(C2×D3)⋊C4—38no—
124C3×C4⋊Dic3—34no—
125C3×C4⋊Dic3C2×D314yes—
126C3×C4⋊Dic3C2×D314yes—
127C3×Dic3⋊C4C2×D318yes—
128C3×Dic3⋊C4C2×D318yes—
129#3724—38no—
130#3722—38no—
131(C2×D3)⋊Dic3—38no—
132(C2×D3)⋊Dic3—38no—
133Dic32—38no—
134C22×C33C2411yescharacteristic
135C2×C3⋊(C3⋊D3)—21no—
136C2×C3⋊(C3⋊D3)—21no—
137C2×C3⋊(C3⋊D3)—21no—
138C2×C3⋊(C3⋊D3)—21no—
139C2×C3⋊(C3⋊D3)—21no—
140C2×C3⋊(C3⋊D3)#4112yes—
141C2×C3×C3⋊D3—22no—
142C2×C3×C3⋊D3—22no—
143C2×C3×C3⋊D3—22no—
144C33×C4—21no—
145C33×C4—21no—
146C33×C4#4112yes—
147C3⋊(C3⋊Dic3)—21no—
148C3⋊(C3⋊Dic3)—21no—
149C3×C3⋊Dic3—22no—
150C32×Dic3—22no—
151C32×Dic3—22no—
152C3×#125—34no—
153C3×C4×D4—91no—
154#938—91no—
155#938—91no—
156#929—92no—
157C4×#67—92no—
158#898—98no—
159#895—98no—
160#895—94no—
161#895—94no—
162#894—98no—
163#893—98no—
164#892—94no—
165#892—91no—
166D3×C4⋊C4—94no—
167D3×C4⋊C4—91no—
168#884—92no—
169#879—92no—
170C4×D12—34no—
171C4×D12—94no—
172C4×D12—91no—
173C4×D12—184no—
174#870C3⋊D311yescharacteristic, complemented
175C22×C3⋊D3—31no—
176C22×C3⋊D3—31no—
177C22×C3⋊D3—34no—
178C22×C3⋊D3—34no—
179C22×C3⋊D3—64no—
180C22×C3⋊D3—64no—
181C22×C3⋊D3C22×D312yes—
182C22×C3×D3—38no—
183C2×C32×C4—24no—
184C2×C32×C4—24no—
185C2×C32×C4—24no—
186C2×C32×C4—32no—
187C2×C32×C4—32no—
188C2×C32×C4C22×D314yes—
189C2×C32×C4C22×D314yes—
190C2×C32×C4C22×D314yes—
191C2×C32×C4C22×D311yescharacteristic
192C2×C32×C4C22×D311yescharacteristic
193C2×C32×C4C22×D311yescharacteristic
194C2×C3⋊Dic3—31no—
195C2×C3⋊Dic3—31no—
196C2×C3⋊Dic3—34no—
197C2×C3⋊Dic3—34no—
198C2×C3⋊Dic3—64no—
199C2×C3⋊Dic3—64no—
200C2×C3⋊Dic3C22×D312yes—
201C4⋊(C3⋊D3)—22no—
202C4⋊(C3⋊D3)—62no—
203C4⋊(C3⋊D3)—68no—
204C4⋊(C3⋊D3)—616no—
205C4⋊(C3⋊D3)—64no—
206C4⋊(C3⋊D3)—122no—
207C4×C3⋊D3—42no—
208C4×C3⋊D3—62no—
209C4×C3⋊D3—62no—
210C4×C3⋊D3—68no—
211C4×C3⋊D3—68no—
212C4×C3⋊D3—128no—
213C4×C3⋊D3—128no—
214C4×C3⋊D3—122no—
215C2×C3×Dic3—38no—
216C2×C3×Dic3C22×D318yes—
217C2×C3×Dic3C22×D318yes—
218C3×D12—68no—
219C3×C4×D3—128no—
220#626—616no—
221#626—128no—
222#624—128no—
223#553—271noSylow
224C2×C33C2×#4111yescharacteristic
225C2×C33C2×#4111yescharacteristic
226C2×C33H(5,𝔽2)11yescharacteristic
227C3⋊(C3⋊D3)—22no—
228C3⋊(C3⋊D3)—41no—
229C3×C3⋊D3—42no—
230C2×C3×D4—91no—
231C22×C3×C4—92no—
232C2×#67—92no—
233C2×#67—92no—
234C2×D12—34no—
235C2×D12—98no—
236C2×D12—94no—
237C2×D12—91no—
238C2×D12—98no—
239C2×D12—184no—
240C2×C4×D3—38no—
241C2×C4×D3—94no—
242C2×C4×D3—94no—
243C2×C4×D3—91no—
244C2×C4×D3—91no—
245C2×C4×D3—98no—
246C2×C4×D3—92no—
247C2×C4×D3—184no—
248C2×C4×D3—184no—
249C3×C4⋊C4—34no—
250C3×C4⋊C4—34no—
251C3×C4⋊C4—38no—
252C3×C4⋊C4—38no—
253C3×C4⋊C4—91no—
254C3×C22⋊C4—92no—
255C3×C42—24no—
256C3×C42D3214yes—
257C3×C42C2×C3⋊D311yescharacteristic
258#217—92no—
259#217—92no—
260(C2×D3)⋊C4—38no—
261(C2×D3)⋊C4—98no—
262(C2×D3)⋊C4—98no—
263(C2×D3)⋊C4—98no—
264(C2×D3)⋊C4—94no—
265(C2×D3)⋊C4—94no—
266(C2×D3)⋊C4—91no—
267(C2×D3)⋊C4—91no—
268(C2×D3)⋊C4—92no—
269(C2×D3)⋊C4—184no—
270(C2×D3)⋊C4—184no—
271C4⋊Dic3—34no—
272C4⋊Dic3—94no—
273C4⋊Dic3—91no—
274C4⋊Dic3—184no—
275C4⋊Dic3C2×C3⋊D311yescharacteristic
276C4⋊Dic3C2×C3⋊D311yescharacteristic
277Dic3⋊C4—98no—
278Dic3⋊C4—92no—
279Dic3⋊C4C2×C3⋊D312yes—
280Dic3⋊C4C2×C3⋊D312yes—
281C4×Dic3—98no—
282C4×Dic3—92no—
283C22×C32—24no—
284C22×C32C23×D314yes—
285C22×C32C23×D311yescharacteristic
286C2×C3⋊D3—22no—
287C2×C3⋊D3—22no—
288C2×C3⋊D3—22no—
289C2×C3⋊D3—32no—
290C2×C3⋊D3—38no—
291C2×C3⋊D3—68no—
292C2×C3⋊D3—61no—
293C2×C3⋊D3—61no—
294C2×C3⋊D3—68no—
295C2×C3⋊D3—61no—
296C2×C3⋊D3—61no—
297C2×C3⋊D3—64no—
298C2×C3⋊D3—61no—
299C2×C3⋊D3—64no—
300C2×C3⋊D3—64no—
301C2×C3⋊D3—64no—
302C2×C3⋊D3—64no—
303C2×C3⋊D3—124no—
304C2×C3⋊D3—124no—
305C2×C3⋊D3—124no—
306C2×C3⋊D3—124no—
307C2×C3×D3—68no—
308C2×C3×D3—68no—
309C2×C3×D3—68no—
310C32×C4—28no—
311C32×C4—21no—
312C32×C4—24no—
313C32×C4—21no—
314C32×C4—28no—
315C32×C4—24no—
316C32×C4—44no—
317C32×C4—62no—
318C32×C4—62no—
319C32×C4#23512yes—
320C32×C4#23918yes—
321C3⋊Dic3—22no—
322C3⋊Dic3—61no—
323C3⋊Dic3—61no—
324C3⋊Dic3—64no—
325C3⋊Dic3—64no—
326C3⋊Dic3—124no—
327C3⋊Dic3—124no—
328C3×Dic3—28no—
329C3×Dic3—28no—
330C3×Dic3—68no—
331#126—272no—
332#125—31no—
333#123—272no—
334#121—272no—
335#121—271no—
336#121—271no—
337C4×D4—91no—
338C4×D4—272no—
339C4×D4—271no—
340#117—271no—
341C2×C4⋊C4—271no—
342C33#55311yescharacteristic, complemented, Sylow
343C23×C3—92no—
344C22×D3—38no—
345C22×D3—94no—
346C22×D3—91no—
347C22×D3—94no—
348C22×D3—91no—
349C22×D3—184no—
350C22×D3—184no—
351C3×D4—182no—
352C2×C3×C4—24no—
353C2×C3×C4—24no—
354C2×C3×C4—24no—
355C2×C3×C4—38no—
356C2×C3×C4—38no—
357C2×C3×C4—92no—
358C2×C3×C4—362no—
359C2×C3×C4C2×D3214yes—
360C2×C3×C4C2×D3214yes—
361C2×C3×C4C2×D3214yes—
362C2×C3×C4C22×C3⋊D311yescharacteristic
363C2×C3×C4C22×C3⋊D311yescharacteristic
364C2×C3×C4C22×C3⋊D311yescharacteristic
365#67—184no—
366#67—362no—
367C2×Dic3—38no—
368C2×Dic3—94no—
369C2×Dic3—94no—
370C2×Dic3—91no—
371C2×Dic3—91no—
372C2×Dic3—184no—
373C2×Dic3—184no—
374C2×Dic3C22×C3⋊D312yes—
375C2×Dic3C22×C3⋊D312yes—
376D12—68no—
377D12—188no—
378D12—182no—
379D12—1816no—
380D12—1816no—
381D12—368no—
382C4×D3—128no—
383C4×D3—188no—
384C4×D3—188no—
385C4×D3—182no—
386C4×D3—182no—
387C4×D3—368no—
388C4×D3—368no—
389C4×D3—362no—
390C4×D3—368no—
391C2×C32—24no—
392C2×C32—24no—
393C2×C32—24no—
394C2×C32C2×#23511yescharacteristic
395C2×C32#100411yescharacteristic
396C2×C32C2×#23914yes—
397C2×C32D3×#4114yes—
398C2×C32D3×#4111yescharacteristic
399C2×C32#100914yes—
400C3⋊D3—42no—
401C3⋊D3—62no—
402C3⋊D3—68no—
403C3⋊D3—128no—
404C3⋊D3—128no—
405C3⋊D3—124no—
406C3⋊D3—121no—
407C3×D3—128no—
408C2×D4—91no—
409C2×D4—272no—
410C2×D4—272no—
411C2×D4—271no—
412C22×C4—92no—
413C22×C4—272no—
414C22×C4—271no—
415C22×C4—271no—
416C4⋊C4—31no—
417C4⋊C4—31no—
418C4⋊C4—32no—
419C4⋊C4—32no—
420C4⋊C4—91no—
421C4⋊C4—272no—
422C4⋊C4—271no—
423C22⋊C4—92no—
424C22⋊C4—272no—
425C22⋊C4—272no—
426C22⋊C4—272no—
427C22⋊C4—271no—
428C22⋊C4—271no—
429C42—272no—
430C42D3×C3⋊D311yescharacteristic
431C22×C3—24no—
432C22×C3—182no—
433C22×C3—182no—
434C22×C3—182no—
435C22×C3C22×D3214yes—
436C22×C3C23×C3⋊D311yescharacteristic
437C2×D3—68no—
438C2×D3—68no—
439C2×D3—68no—
440C2×D3—98no—
441C2×D3—92no—
442C2×D3—188no—
443C2×D3—184no—
444C2×D3—184no—
445C2×D3—181no—
446C2×D3—181no—
447C2×D3—188no—
448C2×D3—184no—
449C2×D3—184no—
450C2×D3—184no—
451C2×D3—181no—
452C2×D3—181no—
453C2×D3—181no—
454C2×D3—364no—
455C2×D3—364no—
456C2×D3—364no—
457C2×D3—364no—
458C3×C4—28no—
459C3×C4—24no—
460C3×C4—24no—
461C3×C4—21no—
462C3×C4—21no—
463C3×C4—28no—
464C3×C4—44no—
465C3×C4—68no—
466C3×C4—68no—
467C3×C4—182no—
468C3×C4#379918yes—
469C3×C4#383212yes—
470Dic3—22no—
471Dic3—22no—
472Dic3—68no—
473Dic3—184no—
474Dic3—184no—
475Dic3—181no—
476Dic3—181no—
477Dic3—364no—
478Dic3—364no—
479C32—24no—
480C32#555811yescharacteristic, complemented
481C32#569514yescomplemented
482C23—92no—
483C23—271no—
484C23—271no—
485D4—182no—
486D4—542no—
487D4—544no—
488D4—1082no—
489C2×C4—32no—
490C2×C4—32no—
491C2×C4—92no—
492C2×C4—271no—
493C2×C4—271no—
494C2×C4—362no—
495C2×C4—542no—
496C2×C4—542no—
497C2×C4—1082no—
498C2×C4C2×D3×C3⋊D311yescharacteristic
499C2×C4C2×D3×C3⋊D311yescharacteristic
500C2×C4C2×D3×C3⋊D311yescharacteristic
501C2×C3—24no—
502C2×C3—24no—
503C2×C3—24no—
504C2×C3—362no—
505C2×C3C2×#379914yes—
506C2×C3#6430614yes—
507C2×C3#6430714yes—
508C2×C3C2×#383211yescharacteristic
509C2×C3#41×C3⋊D311yescharacteristic
510C2×C3#6436611yescharacteristic
511D3—128no—
512D3—182no—
513D3—188no—
514D3—368no—
515D3—368no—
516D3—361no—
517D3—364no—
518C22—182no—
519C22—182no—
520C22—182no—
521C22—272no—
522C22—541no—
523C22—541no—
524C22—541no—
525C22—541no—
526C22—541no—
527C22C22×D3×C3⋊D311yescharacteristic
528C4—21no—
529C4—21no—
530C4—62no—
531C4—62no—
532C4—182no—
533C4—541no—
534C4—541no—
535C4#8901812yes—
536C3—24no—
537C3#1059398214yescomplemented
538C3#1059616611yescharacteristic, complemented
539C2—362no—
540C2—542no—
541C2—1081no—
542C2C2×#8901811yescharacteristic
543C2#1172457811yescharacteristic
544C2#1172458011yescharacteristic
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. #11723532C2semidirect
5. #11723532C2semidirect
6. #11723512C2semidirect
7. #11723503C2semidirect
8. #11723486C2semidirect
9. #11723417C2semidirect
10. #11723413C2semidirect
11. #11723408C2semidirect
49. C4×C4⋊(C3⋊D3)D3semidirect
51. C3×#870D3semidirect
174. #870C3⋊D3semidirect
342. C33#553semidirect
480. C32#5558semidirect
481. C32#5695semidirect
537. C3#10593982semidirect
538. 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
#117235322
#117235122
#117235032
#117234862
#117234172
#117234132
#117234082
#105961663
#105939823