Finite Group Databasebeta

← #13145734

Subgroups — #13145734

Order 1728.

746 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64135C211yesmaximal, characteristic
2C2×C3×C4×C3⋊Dic3C211yesmaximal, characteristic
3C2×C3×#3748C211yesmaximal, characteristic
4#10589162—31nomaximal
5#10589161—31nomaximal
6#10589160—31nomaximal
7C22×C33×C4C2211yescharacteristic
8C22×C3×C3⋊Dic3—21no—
9C3×#3757—21no—
10C3×C4⋊(C3⋊Dic3)C412yescomplemented
11C3×C4×C3⋊Dic3—21no—
12C3×C4×C3⋊Dic3—21no—
13C3×C4×C3⋊Dic3C412yescomplemented
14C3×#3748—21no—
15C3×#3748C412yescomplemented
16C2×C3×C3⋊(C3⋊C8)—21no—
17#64135D311yescharacteristic
18C2×C4×C3⋊Dic3D311yescharacteristic
19C2×#3748D311yescharacteristic
20C3×#926—31no—
21C3×#926—31no—
22C3×#926—61no—
23C2×C3×C4×Dic3—31no—
24C2×C3×C4×Dic3—31no—
25C2×C3×C4×Dic3—61no—
26C2×C3×#208—31no—
27C2×C3×#208—31no—
28C2×C3×#208—61no—
29C23×C33D411yescharacteristic
30C2×C33×C4C2×C411yescharacteristic
31C2×C33×C4C2×C411yescharacteristic
32C2×C33×C4C2×C411yescharacteristic
33C2×C33×C4D411yescharacteristic
34C2×C33×C4D411yescharacteristic
35C2×C33×C4Q811yescharacteristic
36C2×C3×C3⋊Dic3—24no—
37C2×C3×C3⋊Dic3—22no—
38C2×C3×C3⋊Dic3—22no—
39C3×C3⋊(C3⋊C8)—22no—
40#4555—91no—
41#4532—91no—
42#4523—91no—
43C22×C32×C4—21no—
44C22×C32×C4—21no—
45C22×C32×C4—21no—
46C22×C32×C4—41no—
47C22×C32×C4C2×D311yescharacteristic
48C22×C32×C4C2×D311yescharacteristic
49C22×C32×C4C2×D311yescharacteristic
50C22×C3⋊Dic3—21no—
51C22×C3×Dic3—62no—
52C22×C3×Dic3—61no—
53C22×C3×Dic3—61no—
54#3757—21no—
55C4⋊(C3⋊Dic3)Dic312yescomplemented
56C4×C3⋊Dic3—21no—
57C4×C3⋊Dic3—21no—
58C4×C3⋊Dic3Dic312yescomplemented
59#3748—21no—
60#3748Dic312yescomplemented
61C2×C3⋊(C3⋊C8)—21no—
62C3×#217—62no—
63C3×#217—61no—
64C3×#217—61no—
65C3×C4⋊Dic3—32no—
66C3×C4⋊Dic3—32no—
67C3×C4⋊Dic3—62no—
68C3×C4×Dic3—32no—
69C3×C4×Dic3—32no—
70C3×C4×Dic3—62no—
71C3×C4×Dic3—62no—
72C3×C4×Dic3—62no—
73C3×C4×Dic3—61no—
74C3×C4×Dic3—61no—
75C3×C4×Dic3—61no—
76C3×C4×Dic3—61no—
77C3×#208—32no—
78C3×#208—32no—
79C3×#208—62no—
80C3×#208—62no—
81C3×#208—61no—
82C3×#208—61no—
83C2×C3×C3⋊C8—62no—
84C2×C3×C3⋊C8—61no—
85C2×C3×C3⋊C8—61no—
86C22×C33—22no—
87C22×C33C22⋊C411yescharacteristic
88C22×C33C22⋊C411yescharacteristic
89C22×C33C4⋊C411yescharacteristic
90C33×C4C4211yescharacteristic
91C33×C4C22⋊C411yescharacteristic
92C33×C4C4⋊C411yescharacteristic
93C33×C4C4⋊C411yescharacteristic
94C3×C3⋊Dic3—24no—
95C3×C3⋊Dic3—42no—
96C2×C3×M16—91no—
97C3×#117—91no—
98C2×C3×C42—91no—
99#926—31no—
100#926—31no—
101#926—61no—
102C2×C4×Dic3—31no—
103C2×C4×Dic3—31no—
104C2×C4×Dic3—61no—
105C2×#208—31no—
106C2×#208—31no—
107C2×#208—61no—
108C23×C32—21no—
109C23×C32—21no—
110C23×C32—21no—
111C23×C32—41no—
112C23×C32D1211yescharacteristic
113C23×C32#6711yescharacteristic
114C23×C32#6711yescharacteristic
115C2×C32×C4—21no—
116C2×C32×C4—21no—
117C2×C32×C4—21no—
118C2×C32×C4—21no—
119C2×C32×C4—21no—
120C2×C32×C4—21no—
121C2×C32×C4—21no—
122C2×C32×C4—21no—
123C2×C32×C4—21no—
124C2×C32×C4—21no—
125C2×C32×C4—21no—
126C2×C32×C4—21no—
127C2×C32×C4—21no—
128C2×C32×C4—21no—
129C2×C32×C4—21no—
130C2×C32×C4—21no—
131C2×C32×C4—21no—
132C2×C32×C4—21no—
133C2×C32×C4—41no—
134C2×C32×C4—41no—
135C2×C32×C4—41no—
136C2×C32×C4—41no—
137C2×C32×C4—41no—
138C2×C32×C4—41no—
139C2×C32×C4Dic611yescharacteristic
140C2×C32×C4Dic611yescharacteristic
141C2×C32×C4Dic611yescharacteristic
142C2×C32×C4C4×D311yescharacteristic
143C2×C32×C4C4×D311yescharacteristic
144C2×C32×C4C4×D311yescharacteristic
145C2×C32×C4C4×D311yescharacteristic
146C2×C32×C4C4×D311yescharacteristic
147C2×C32×C4C4×D311yescharacteristic
148C2×C32×C4D1211yescharacteristic
149C2×C32×C4D1211yescharacteristic
150C2×C32×C4C2×Dic311yescharacteristic
151C2×C32×C4C2×Dic311yescharacteristic
152C2×C32×C4C2×Dic311yescharacteristic
153C2×C32×C4#6711yescharacteristic
154C2×C32×C4#6711yescharacteristic
155C2×C32×C4#6711yescharacteristic
156C2×C32×C4#6711yescharacteristic
157C2×C3⋊Dic3—24no—
158C2×C3⋊Dic3—22no—
159C2×C3⋊Dic3—22no—
160C2×C3×Dic3—64no—
161C2×C3×Dic3—68no—
162C2×C3×Dic3—64no—
163C2×C3×Dic3—64no—
164C2×C3×Dic3—62no—
165C2×C3×Dic3—62no—
166C2×C3×Dic3—64no—
167C2×C3×Dic3—62no—
168C2×C3×Dic3—62no—
169C3⋊(C3⋊C8)—22no—
170C3×C3⋊C8—64no—
171C3×C3⋊C8—62no—
172C3×C3⋊C8—62no—
173#339—271noSylow
174C2×C33—21no—
175C2×C33—21no—
176C2×C33#9511yescharacteristic
177C2×C33#10412yes—
178C22×C3×C4—21no—
179C22×C3×C4—21no—
180C22×C3×C4—21no—
181C22×C3×C4—41no—
182C22×C3×C4—181no—
183C22×C3×C4D3211yescharacteristic
184C22×C3×C4D3211yescharacteristic
185C22×C3×C4D3211yescharacteristic
186C22×Dic3—62no—
187C22×Dic3—61no—
188C22×Dic3—61no—
189C3×M16—92no—
190C3×M16—181no—
191C2×C3×C8—181no—
192C3×C4⋊C4—92no—
193C3×C22⋊C4—181no—
194C3×C42—92no—
195C3×C42—181no—
196C3×C42—181no—
197#217—62no—
198#217—61no—
199#217—61no—
200C4⋊Dic3—32no—
201C4⋊Dic3—32no—
202C4⋊Dic3—62no—
203C4×Dic3—32no—
204C4×Dic3—32no—
205C4×Dic3—62no—
206C4×Dic3—62no—
207C4×Dic3—62no—
208C4×Dic3—61no—
209C4×Dic3—61no—
210C4×Dic3—61no—
211C4×Dic3—61no—
212#208—32no—
213#208—32no—
214#208—62no—
215#208—62no—
216#208—61no—
217#208—61no—
218C2×C3⋊C8—62no—
219C2×C3⋊C8—61no—
220C2×C3⋊C8—61no—
221C22×C32—22no—
222C22×C32—24no—
223C22×C32—22no—
224C22×C32—22no—
225C22×C32—21no—
226C22×C32—21no—
227C22×C32—21no—
228C22×C32—21no—
229C22×C32—21no—
230C22×C32—21no—
231C22×C32—21no—
232C22×C32—21no—
233C22×C32—21no—
234C22×C32—41no—
235C22×C32—41no—
236C22×C32—41no—
237C22×C32—42no—
238C22×C32—42no—
239C22×C32—44no—
240C22×C32Dic3⋊C411yescharacteristic
241C22×C32Dic3⋊C411yescharacteristic
242C22×C32C4⋊Dic311yescharacteristic
243C22×C32(C2×D3)⋊C411yescharacteristic
244C22×C32(C2×D3)⋊C411yescharacteristic
245C22×C32(C2×D3)⋊C411yescharacteristic
246C22×C32(C2×D3)⋊C411yescharacteristic
247C22×C32#21711yescharacteristic
248C22×C32#21711yescharacteristic
249C32×C4—21no—
250C32×C4—21no—
251C32×C4—21no—
252C32×C4—21no—
253C32×C4—21no—
254C32×C4—21no—
255C32×C4—21no—
256C32×C4—21no—
257C32×C4—21no—
258C32×C4—21no—
259C32×C4—21no—
260C32×C4—21no—
261C32×C4—41no—
262C32×C4—41no—
263C32×C4—41no—
264C32×C4—41no—
265C32×C4C4×Dic311yescharacteristic
266C32×C4C4×Dic311yescharacteristic
267C32×C4C4×Dic311yescharacteristic
268C32×C4Dic3⋊C411yescharacteristic
269C32×C4Dic3⋊C411yescharacteristic
270C32×C4Dic3⋊C411yescharacteristic
271C32×C4Dic3⋊C411yescharacteristic
272C32×C4C4⋊Dic311yescharacteristic
273C32×C4C4⋊Dic311yescharacteristic
274C32×C4(C2×D3)⋊C411yescharacteristic
275C32×C4(C2×D3)⋊C411yescharacteristic
276C32×C4#21711yescharacteristic
277C3⋊Dic3—24no—
278C3⋊Dic3—42no—
279C3×Dic3—68no—
280C3×Dic3—64no—
281C3×Dic3—64no—
282C3×Dic3—124no—
283C3×Dic3—122no—
284C3×Dic3—122no—
285C2×M16—91no—
286#117—91no—
287C2×C42—91no—
288C33#33911yescharacteristic, complemented, Sylow
289C23×C3—21no—
290C23×C3—21no—
291C23×C3—21no—
292C23×C3—41no—
293C23×C3#62511yescharacteristic
294C23×C3#62611yescharacteristic
295C23×C3#62611yescharacteristic
296C2×C3×C4—21no—
297C2×C3×C4—21no—
298C2×C3×C4—21no—
299C2×C3×C4—21no—
300C2×C3×C4—21no—
301C2×C3×C4—21no—
302C2×C3×C4—21no—
303C2×C3×C4—21no—
304C2×C3×C4—21no—
305C2×C3×C4—21no—
306C2×C3×C4—21no—
307C2×C3×C4—21no—
308C2×C3×C4—21no—
309C2×C3×C4—21no—
310C2×C3×C4—21no—
311C2×C3×C4—21no—
312C2×C3×C4—21no—
313C2×C3×C4—21no—
314C2×C3×C4—41no—
315C2×C3×C4—41no—
316C2×C3×C4—41no—
317C2×C3×C4—41no—
318C2×C3×C4—41no—
319C2×C3×C4—41no—
320C2×C3×C4—184no—
321C2×C3×C4—182no—
322C2×C3×C4—182no—
323C2×C3×C4D3×Dic311yescharacteristic
324C2×C3×C4D3×Dic311yescharacteristic
325C2×C3×C4D3×Dic311yescharacteristic
326C2×C3×C4D3×Dic311yescharacteristic
327C2×C3×C4D3×Dic311yescharacteristic
328C2×C3×C4D3×Dic311yescharacteristic
329C2×C3×C4#62411yescharacteristic
330C2×C3×C4#62411yescharacteristic
331C2×C3×C4#62411yescharacteristic
332C2×C3×C4#62511yescharacteristic
333C2×C3×C4#62511yescharacteristic
334C2×C3×C4#62611yescharacteristic
335C2×C3×C4#62611yescharacteristic
336C2×C3×C4#62611yescharacteristic
337C2×C3×C4#62611yescharacteristic
338C2×C3×C4#62711yescharacteristic
339C2×C3×C4#62711yescharacteristic
340C2×C3×C4#62711yescharacteristic
341C2×Dic3—64no—
342C2×Dic3—68no—
343C2×Dic3—64no—
344C2×Dic3—64no—
345C2×Dic3—62no—
346C2×Dic3—62no—
347C2×Dic3—64no—
348C2×Dic3—62no—
349C2×Dic3—62no—
350C3×C8—182no—
351C3⋊C8—64no—
352C3⋊C8—62no—
353C3⋊C8—62no—
354C2×C32—21no—
355C2×C32—21no—
356C2×C32—22no—
357C2×C32—22no—
358C2×C32—21no—
359C2×C32—21no—
360C2×C32—21no—
361C2×C32—21no—
362C2×C32—22no—
363C2×C32—22no—
364C2×C32—22no—
365C2×C32—21no—
366C2×C32—21no—
367C2×C32—21no—
368C2×C32—41no—
369C2×C32—42no—
370C2×C32—41no—
371C2×C32—41no—
372C2×C32—41no—
373C2×C32—41no—
374C2×C32—42no—
375C2×C32—42no—
376C2×C32#80512yes—
377C2×C32#82512yes—
378C2×C32C3⋊#9511yescharacteristic
379C2×C32C3⋊#9511yescharacteristic
380C2×C32C3⋊#9511yescharacteristic
381C2×C32#83712yes—
382C22×C4—181no—
383C22×C4#111311yescharacteristic
384M16—92no—
385M16—181no—
386C2×C8—181no—
387C4⋊C4—92no—
388C22⋊C4—181no—
389C42—92no—
390C42—181no—
391C42—181no—
392C22×C3—24no—
393C22×C3—22no—
394C22×C3—22no—
395C22×C3—22no—
396C22×C3—21no—
397C22×C3—21no—
398C22×C3—21no—
399C22×C3—21no—
400C22×C3—21no—
401C22×C3—21no—
402C22×C3—21no—
403C22×C3—21no—
404C22×C3—21no—
405C22×C3—42no—
406C22×C3—42no—
407C22×C3—44no—
408C22×C3—41no—
409C22×C3—41no—
410C22×C3—41no—
411C22×C3(C2×D3)⋊Dic311yescharacteristic
412C22×C3(C2×D3)⋊Dic311yescharacteristic
413C22×C3(C2×D3)⋊Dic311yescharacteristic
414C22×C3(C2×D3)⋊Dic311yescharacteristic
415C22×C3#372211yescharacteristic
416C22×C3#372211yescharacteristic
417C22×C3Dic3⋊Dic311yescharacteristic
418C22×C3Dic3⋊Dic311yescharacteristic
419C22×C3#372411yescharacteristic
420C3×C4—21no—
421C3×C4—21no—
422C3×C4—21no—
423C3×C4—21no—
424C3×C4—21no—
425C3×C4—21no—
426C3×C4—21no—
427C3×C4—21no—
428C3×C4—21no—
429C3×C4—21no—
430C3×C4—21no—
431C3×C4—21no—
432C3×C4—41no—
433C3×C4—41no—
434C3×C4—41no—
435C3×C4—41no—
436C3×C4—184no—
437C3×C4—362no—
438C3×C4Dic3211yescharacteristic
439C3×C4Dic3211yescharacteristic
440C3×C4Dic3211yescharacteristic
441C3×C4(C2×D3)⋊Dic311yescharacteristic
442C3×C4(C2×D3)⋊Dic311yescharacteristic
443C3×C4#372211yescharacteristic
444C3×C4Dic3⋊Dic311yescharacteristic
445C3×C4Dic3⋊Dic311yescharacteristic
446C3×C4Dic3⋊Dic311yescharacteristic
447C3×C4Dic3⋊Dic311yescharacteristic
448C3×C4#372411yescharacteristic
449C3×C4#372411yescharacteristic
450Dic3—68no—
451Dic3—64no—
452Dic3—64no—
453Dic3—124no—
454Dic3—122no—
455Dic3—122no—
456C32—21no—
457C32—21no—
458C32—21no—
459C32—41no—
460C32#452311yescharacteristic, complemented
461C32#453211yescharacteristic, complemented
462C32#455511yescharacteristic, complemented
463C23#629511yescharacteristic
464C2×C4—184no—
465C2×C4—182no—
466C2×C4—182no—
467C2×C4#629411yescharacteristic
468C2×C4#629411yescharacteristic
469C2×C4#629411yescharacteristic
470C2×C4#629511yescharacteristic
471C2×C4#629511yescharacteristic
472C2×C4#629611yescharacteristic
473C8—182no—
474C2×C3—22no—
475C2×C3—22no—
476C2×C3—21no—
477C2×C3—21no—
478C2×C3—21no—
479C2×C3—21no—
480C2×C3—21no—
481C2×C3—21no—
482C2×C3—22no—
483C2×C3—22no—
484C2×C3—22no—
485C2×C3—21no—
486C2×C3—21no—
487C2×C3—21no—
488C2×C3—41no—
489C2×C3—41no—
490C2×C3—41no—
491C2×C3—41no—
492C2×C3—42no—
493C2×C3—42no—
494C2×C3—42no—
495C2×C3—41no—
496C2×C3#6356812yes—
497C2×C3#6356912yes—
498C2×C3#6357012yes—
499C2×C3#6357911yescharacteristic
500C2×C3#6357911yescharacteristic
501C2×C3#6357911yescharacteristic
502C22—22no—
503C22#8881411yescharacteristic
504C22#8881411yescharacteristic
505C22#8881511yescharacteristic
506C4—184no—
507C4—362no—
508C4#8881311yescharacteristic
509C4#8881411yescharacteristic
510C4#8881511yescharacteristic
511C4#8881511yescharacteristic
512C3—21no—
513C3—21no—
514C3—21no—
515C3—41no—
516C3#1058916011yescharacteristic, complemented
517C3#1058916111yescharacteristic, complemented
518C3#1058916211yescharacteristic, complemented
519C2—21no—
520C2—21no—
521C2#1172207712yes—
522C2#1172208211yescharacteristic
subgroupG/H
centerC2×C4#6295
commutatorC33×C4C42
Frattini, hypercenterC22×C4#1113
Fitting, cosocleC22×C33×C4C22
socleC22×C33C22⋊C4
prime radicalC23×C33D4
hypocenterC33#339
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2×C4 ◁ C22×C4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
10. C3×C4⋊(C3⋊Dic3)C4semidirect
13. C3×C4×C3⋊Dic3C4semidirect
15. C3×#3748C4semidirect
55. C4⋊(C3⋊Dic3)Dic3semidirect
58. C4×C3⋊Dic3Dic3semidirect
60. #3748Dic3semidirect
288. C33#339semidirect
460. C32#4523semidirect
461. C32#4532semidirect
462. C32#4555semidirect
516. C3#10589160semidirect
517. C3#10589161semidirect
518. C3#10589162semidirect

Maximal subgroups

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

subgroupindex
C3×#641352
C2×C3×C4×C3⋊Dic32
C2×C3×#37482
#105891623
#105891613
#105891603