Finite Group Databasebeta

← #13175349

Subgroups — #13175349

Order 1728.

1836 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C42×C3⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
2C3×C4×C4⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
3C32×C4×Dic6C211yesmaximal, characteristic, complemented
4#11723532C211yesmaximal, characteristic
5C4×#6292C212yesmaximal, complemented
6#11723456C211yesmaximal, characteristic, complemented
7#11723450C211yesmaximal, characteristic, complemented
8#11723437C212yesmaximal, complemented
9#11723436C211yesmaximal, characteristic, complemented
10#11723433C212yesmaximal, complemented
11#11723421C212yesmaximal, complemented
12#10596051—31nomaximal
13#10593875—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×C32×Dic6C2211yescharacteristic
20C2×#6292C2212yes—
21C33×C42C2211yescharacteristic
22C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
23C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
24C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
25C3×#3752C2212yes—
26C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
27C32×C4⋊Dic3C2211yescharacteristic
28C32×Dic3⋊C4C2212yes—
29C32×C4×Dic3C2212yes—
30#88807C2212yes—
31#88807C2212yes—
32Dic3⋊(C3⋊Dic3)C2212yes—
33#88804C2212yes—
34#88803C2212yes—
35#88803C2212yes—
36#88803C2212yes—
37C32×C4×Q8—31no—
38#64156—31no—
39#64155—31no—
40#64107—32no—
41#64104—31no—
42#64103—32no—
43#64099—31no—
44C4×C4⋊(C3⋊D3)—32no—
45C4×C4⋊(C3⋊D3)D311yescharacteristic, complemented
46#64081—32no—
47C42×C3⋊D3—31no—
48C42×C3⋊D3—34no—
49C42×C3⋊D3—64no—
50C3×C4×D12—34no—
51C3×C4×Dic6D314yescomplemented
52#63916—34no—
53C4×#626—38no—
54#63872—34no—
55#63871—34no—
56#63863—34no—
57#63862—38no—
58#63855—38no—
59#63853—38no—
60C22×C3⋊(C3⋊D3)C2311yescharacteristic
61C22×C3×C3⋊D3C2312yes—
62C2×C33×C4C2311yescharacteristic
63C2×C33×C4C2311yescharacteristic
64C2×C33×C4C2311yescharacteristic
65C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
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—24no—
74C2×C32×Dic3C2312yes—
75C2×C32×Dic3C2312yes—
76C32×Dic6—22no—
77#6292—24no—
78#5580—94no—
79#5557—91no—
80C2×C32×Q8—31no—
81C2×C4⋊(C3⋊D3)—32no—
82C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
83C2×C4×C3⋊D3—31no—
84C2×C4×C3⋊D3—31no—
85C2×C4×C3⋊D3—31no—
86C2×C4×C3⋊D3—34no—
87C2×C4×C3⋊D3—34no—
88C2×C4×C3⋊D3—34no—
89C2×C4×C3⋊D3—64no—
90C2×C4×C3⋊D3—64no—
91C2×C4×C3⋊D3—64no—
92C2×C4×C3⋊D3C2×D312yes—
93C2×C3×D12—34no—
94C2×C3×C4×D3—38no—
95C2×C3×Dic6C2×D314yes—
96C2×#626—38no—
97C32×C4⋊C4—31no—
98C32×C4⋊C4—32no—
99C32×C42—24no—
100C32×C42—32no—
101C32×C42C2×D314yes—
102C32×C42C2×D311yescharacteristic
103#3752—32no—
104#3752—32no—
105#3752—32no—
106#3752—32no—
107#3752C2×D312yes—
108C4⋊(C3⋊Dic3)—32no—
109C4⋊(C3⋊Dic3)C2×D311yescharacteristic
110#3750—32no—
111#3750—32no—
112C4×C3⋊Dic3—31no—
113C4×C3⋊Dic3—31no—
114C4×C3⋊Dic3—31no—
115C4×C3⋊Dic3—34no—
116C4×C3⋊Dic3—34no—
117C4×C3⋊Dic3—34no—
118C4×C3⋊Dic3—64no—
119C4×C3⋊Dic3—64no—
120C4×C3⋊Dic3—64no—
121C3×(C2×D3)⋊C4—38no—
122C3×C4⋊Dic3—34no—
123C3×C4⋊Dic3C2×D314yes—
124C3×Dic3⋊C4C2×D318yes—
125C3×C4×Dic3C2×D318yes—
126Dic3⋊Dic3—38no—
127Dic3⋊Dic3—38no—
128Dic3⋊Dic3—38no—
129#3722—38no—
130(C2×D3)⋊Dic3—38no—
131(C2×D3)⋊Dic3—38no—
132(C2×D3)⋊Dic3—38no—
133C22×C33C2411yescharacteristic
134C2×C3⋊(C3⋊D3)—21no—
135C2×C3⋊(C3⋊D3)—21no—
136C2×C3⋊(C3⋊D3)#4112yes—
137C2×C3×C3⋊D3—22no—
138C2×C3×C3⋊D3—22no—
139C2×C3×C3⋊D3—22no—
140C33×C4—21no—
141C33×C4#4112yes—
142C33×C4#4112yes—
143C3⋊(C3⋊Dic3)—21no—
144C3⋊(C3⋊Dic3)—21no—
145C3⋊(C3⋊Dic3)—21no—
146C3×C3⋊Dic3—22no—
147C32×Dic3—22no—
148C32×Dic3—22no—
149C3×C4×Q8—34no—
150C3×C4×D4—91no—
151#947—94no—
152#946—94no—
153#938—91no—
154#936—91no—
155C4×#67—92no—
156#926—92no—
157#898—98no—
158#897—91no—
159#895—94no—
160#894—98no—
161#890—94no—
162#890—91no—
163#883—92no—
164#879—92no—
165C4×D12—34no—
166C4×D12—98no—
167#872—98no—
168C42×D3—94no—
169C42×D3—91no—
170C42×D3—184no—
171C4×Dic6C3⋊D311yescharacteristic, complemented
172C22×C3⋊D3—31no—
173C22×C3⋊D3—34no—
174C22×C3⋊D3—64no—
175C22×C3⋊D3C22×D312yes—
176C22×C3×D3—38no—
177C32×Q8—62no—
178C2×C32×C4—24no—
179C2×C32×C4—24no—
180C2×C32×C4—24no—
181C2×C32×C4—32no—
182C2×C32×C4—32no—
183C2×C32×C4C22×D314yes—
184C2×C32×C4C22×D314yes—
185C2×C32×C4C22×D314yes—
186C2×C32×C4C22×D311yescharacteristic
187C2×C32×C4C22×D311yescharacteristic
188C2×C32×C4C22×D311yescharacteristic
189C2×C3⋊Dic3—31no—
190C2×C3⋊Dic3—31no—
191C2×C3⋊Dic3—31no—
192C2×C3⋊Dic3—34no—
193C2×C3⋊Dic3—34no—
194C2×C3⋊Dic3—34no—
195C2×C3⋊Dic3—64no—
196C2×C3⋊Dic3—64no—
197C2×C3⋊Dic3—64no—
198C2×C3⋊Dic3C22×D312yes—
199C4⋊(C3⋊D3)—22no—
200C4⋊(C3⋊D3)—64no—
201C4×C3⋊D3—24no—
202C4×C3⋊D3—62no—
203C4×C3⋊D3—68no—
204C4×C3⋊D3—616no—
205C4×C3⋊D3—616no—
206C4×C3⋊D3—616no—
207C4×C3⋊D3—68no—
208C4×C3⋊D3—62no—
209C4×C3⋊D3—62no—
210C4×C3⋊D3—68no—
211C2×C3×Dic3—38no—
212C2×C3×Dic3C22×D318yes—
213C2×C3×Dic3C22×D318yes—
214C3×D12—68no—
215C3×C4×D3—616no—
216C3×Dic6—28no—
217#626—616no—
218#529—271noSylow
219C2×C33C2×#4111yescharacteristic
220C2×C33C2×#4111yescharacteristic
221C2×C33C2×#4111yescharacteristic
222C3⋊(C3⋊D3)—22no—
223C3×C3⋊D3—42no—
224C2×C3×Q8—34no—
225C2×C3×D4—91no—
226C22×C3×C4—92no—
227C2×#67—92no—
228C2×D12—34no—
229C2×D12—98no—
230C2×C4×D3—38no—
231C2×C4×D3—94no—
232C2×C4×D3—94no—
233C2×C4×D3—94no—
234C2×C4×D3—91no—
235C2×C4×D3—91no—
236C2×C4×D3—91no—
237C2×C4×D3—184no—
238C2×C4×D3—184no—
239C2×C4×D3—184no—
240C2×Dic6C2×C3⋊D311yescharacteristic
241C3×C4⋊C4—34no—
242C3×C4⋊C4—38no—
243C3×C4⋊C4—91no—
244C3×C22⋊C4—92no—
245C3×C42—24no—
246C3×C42—38no—
247C3×C42D3214yes—
248C3×C42C2×C3⋊D311yescharacteristic
249#217—92no—
250#217—92no—
251#217—92no—
252(C2×D3)⋊C4—38no—
253(C2×D3)⋊C4—98no—
254(C2×D3)⋊C4—98no—
255(C2×D3)⋊C4—98no—
256(C2×D3)⋊C4—98no—
257(C2×D3)⋊C4—92no—
258C4⋊Dic3—34no—
259C4⋊Dic3—98no—
260C4⋊Dic3—92no—
261C4⋊Dic3—92no—
262C4⋊Dic3C2×C3⋊D311yescharacteristic
263Dic3⋊C4—98no—
264Dic3⋊C4—98no—
265Dic3⋊C4—92no—
266Dic3⋊C4C2×C3⋊D312yes—
267C4×Dic3—94no—
268C4×Dic3—94no—
269C4×Dic3—94no—
270C4×Dic3—91no—
271C4×Dic3—91no—
272C4×Dic3—91no—
273C4×Dic3—184no—
274C4×Dic3—184no—
275C4×Dic3—184no—
276C4×Dic3C2×C3⋊D312yes—
277C22×C32—24no—
278C22×C32C23×D314yes—
279C22×C32C23×D311yescharacteristic
280C2×C3⋊D3—22no—
281C2×C3⋊D3—22no—
282C2×C3⋊D3—22no—
283C2×C3⋊D3—32no—
284C2×C3⋊D3—38no—
285C2×C3⋊D3—68no—
286C2×C3⋊D3—68no—
287C2×C3⋊D3—68no—
288C2×C3⋊D3—64no—
289C2×C3⋊D3—61no—
290C2×C3⋊D3—61no—
291C2×C3⋊D3—64no—
292C2×C3×D3—68no—
293C2×C3×D3—68no—
294C2×C3×D3—68no—
295C32×C4—28no—
296C32×C4—24no—
297C32×C4—21no—
298C32×C4—28no—
299C32×C4—28no—
300C32×C4—62no—
301C32×C4—62no—
302C32×C4#23518yes—
303C32×C4#23512yes—
304C32×C4#23712yes—
305C32×C4#23918yes—
306C3⋊Dic3—22no—
307C3⋊Dic3—68no—
308C3⋊Dic3—68no—
309C3⋊Dic3—68no—
310C3⋊Dic3—64no—
311C3⋊Dic3—61no—
312C3⋊Dic3—61no—
313C3⋊Dic3—64no—
314C3⋊Dic3—64no—
315C3⋊Dic3—61no—
316C3×Dic3—28no—
317C3×Dic3—28no—
318C3×Dic3—68no—
319#126—272no—
320#125—271no—
321#124—271no—
322#123—272no—
323C22⋊Q8—271no—
324#121—271no—
325C4×Q8—31no—
326C4×D4—91no—
327C4×D4—272no—
328#117—272no—
329C2×C42—271no—
330C33#52911yescharacteristic, complemented, Sylow
331C23×C3—92no—
332C22×D3—38no—
333C22×D3—94no—
334C22×D3—91no—
335C22×D3—184no—
336C3×Q8—68no—
337C3×D4—182no—
338C2×C3×C4—24no—
339C2×C3×C4—24no—
340C2×C3×C4—24no—
341C2×C3×C4—38no—
342C2×C3×C4—38no—
343C2×C3×C4—92no—
344C2×C3×C4—184no—
345C2×C3×C4C2×D3214yes—
346C2×C3×C4C2×D3214yes—
347C2×C3×C4C2×D3214yes—
348C2×C3×C4C22×C3⋊D311yescharacteristic
349C2×C3×C4C22×C3⋊D311yescharacteristic
350C2×C3×C4C22×C3⋊D311yescharacteristic
351#67—184no—
352C2×Dic3—38no—
353C2×Dic3—94no—
354C2×Dic3—94no—
355C2×Dic3—94no—
356C2×Dic3—91no—
357C2×Dic3—91no—
358C2×Dic3—91no—
359C2×Dic3—184no—
360C2×Dic3—184no—
361C2×Dic3—184no—
362C2×Dic3C22×C3⋊D312yes—
363C2×Dic3C22×C3⋊D312yes—
364D12—68no—
365D12—1816no—
366C4×D3—616no—
367C4×D3—188no—
368C4×D3—182no—
369C4×D3—1816no—
370C4×D3—1816no—
371C4×D3—1816no—
372C4×D3—188no—
373C4×D3—188no—
374C4×D3—182no—
375C4×D3—182no—
376Dic6—22no—
377C2×C32—24no—
378C2×C32—24no—
379C2×C32—24no—
380C2×C32C2×#23514yes—
381C2×C32C2×#23511yescharacteristic
382C2×C32C2×#23711yescharacteristic
383C2×C32C2×#23914yes—
384C2×C32D3×#4114yes—
385C2×C32D3×#4111yescharacteristic
386C3⋊D3—42no—
387C3⋊D3—616no—
388C3⋊D3—62no—
389C3⋊D3—68no—
390C3×D3—128no—
391C2×Q8—31no—
392C2×D4—91no—
393C2×D4—272no—
394C22×C4—92no—
395C22×C4—271no—
396C22×C4—271no—
397C22×C4—271no—
398C4⋊C4—31no—
399C4⋊C4—32no—
400C4⋊C4—91no—
401C4⋊C4—272no—
402C4⋊C4—272no—
403C4⋊C4—272no—
404C22⋊C4—92no—
405C22⋊C4—272no—
406C22⋊C4—272no—
407C22⋊C4—272no—
408C22⋊C4—272no—
409C42—32no—
410C42—271no—
411C42—271no—
412C42—271no—
413C42D3×C3⋊D311yescharacteristic
414C22×C3—24no—
415C22×C3—182no—
416C22×C3—182no—
417C22×C3—182no—
418C22×C3C22×D3214yes—
419C22×C3C23×C3⋊D311yescharacteristic
420C2×D3—68no—
421C2×D3—68no—
422C2×D3—68no—
423C2×D3—98no—
424C2×D3—92no—
425C2×D3—188no—
426C2×D3—188no—
427C2×D3—188no—
428C2×D3—184no—
429C2×D3—184no—
430C2×D3—181no—
431C2×D3—181no—
432C3×C4—28no—
433C3×C4—24no—
434C3×C4—21no—
435C3×C4—28no—
436C3×C4—28no—
437C3×C4—68no—
438C3×C4—68no—
439C3×C4—182no—
440C3×C4#379618yes—
441C3×C4#379818yes—
442C3×C4#382812yes—
443C3×C4#383212yes—
444Dic3—22no—
445Dic3—22no—
446Dic3—68no—
447Dic3—188no—
448Dic3—188no—
449Dic3—188no—
450Dic3—184no—
451Dic3—184no—
452Dic3—184no—
453Dic3—181no—
454Dic3—181no—
455Dic3—181no—
456C32—24no—
457C32#555711yescharacteristic, complemented
458C32#558014yescomplemented
459C23—92no—
460C23—271no—
461Q8—62no—
462D4—182no—
463D4—544no—
464C2×C4—32no—
465C2×C4—32no—
466C2×C4—92no—
467C2×C4—184no—
468C2×C4—271no—
469C2×C4—271no—
470C2×C4—271no—
471C2×C4—542no—
472C2×C4—542no—
473C2×C4—542no—
474C2×C4C2×D3×C3⋊D311yescharacteristic
475C2×C4C2×D3×C3⋊D311yescharacteristic
476C2×C4C2×D3×C3⋊D311yescharacteristic
477C2×C3—24no—
478C2×C3—24no—
479C2×C3—24no—
480C2×C3—362no—
481C2×C3C2×#379614yes—
482C2×C3C2×#379814yes—
483C2×C3#6430614yes—
484C2×C3C2×#382811yescharacteristic
485C2×C3C2×#383211yescharacteristic
486C2×C3#41×C3⋊D311yescharacteristic
487D3—128no—
488D3—1816no—
489D3—188no—
490D3—182no—
491C22—182no—
492C22—182no—
493C22—182no—
494C22—272no—
495C22—541no—
496C22—541no—
497C22C22×D3×C3⋊D311yescharacteristic
498C4—21no—
499C4—62no—
500C4—62no—
501C4—182no—
502C4—541no—
503C4—541no—
504C4—541no—
505C4#8901712yes—
506C4#8902012yes—
507C3—24no—
508C3#1059387514yescomplemented
509C3#1059605111yescharacteristic, complemented
510C2—362no—
511C2—542no—
512C2C2×#8901711yescharacteristic
513C2C2×#8902011yescharacteristic
514C2#1172457811yescharacteristic
subgroupG/H
centerC2×C4C2×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FrattiniC22C22×D3×C3⋊D3
FittingC33×C42C22
prime radical#11723532C2
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. C42×C3⋊(C3⋊D3)C2semidirect
2. C3×C4×C4⋊(C3⋊D3)C2semidirect
3. C32×C4×Dic6C2semidirect
5. C4×#6292C2semidirect
6. #11723456C2semidirect
7. #11723450C2semidirect
8. #11723437C2semidirect
9. #11723436C2semidirect
10. #11723433C2semidirect
11. #11723421C2semidirect
45. C4×C4⋊(C3⋊D3)D3semidirect
51. C3×C4×Dic6D3semidirect
171. C4×Dic6C3⋊D3semidirect
330. C33#529semidirect
457. C32#5557semidirect
458. C32#5580semidirect
508. C3#10593875semidirect
509. C3#10596051semidirect

Maximal subgroups

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

subgroupindex
C42×C3⋊(C3⋊D3)2
C3×C4×C4⋊(C3⋊D3)2
C32×C4×Dic62
#117235322
C4×#62922
#117234562
#117234502
#117234372
#117234362
#117234332
#117234212
#105960513
#105938753