Finite Group Databasebeta

← #13161829

Subgroups — #13161829

Order 1728.

762 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64135C211yesmaximal, characteristic
2C2×C3×C4⋊(C3⋊Dic3)C211yesmaximal, characteristic
3C2×C3×#3748C211yesmaximal, characteristic
4#11722080C212yesmaximal, complemented
5#11722079C212yesmaximal, complemented
6#10591139—31nomaximal
7#10591131—31nomaximal
8#10591130—31nomaximal
9C22×C33×C4C2211yescharacteristic
10C22×C3×C3⋊Dic3—21no—
11C3×#3757—21no—
12C3×C4⋊(C3⋊Dic3)—21no—
13C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
14C3×C4⋊(C3⋊Dic3)C2212yes—
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C3×C4×C3⋊Dic3—21no—
17C3×#3748C414yescomplemented
18C2×C3×C3⋊(C3⋊C8)C2212yes—
19#64135D311yescharacteristic
20C2×C4⋊(C3⋊Dic3)D311yescharacteristic
21C2×#3748D311yescharacteristic
22C3×#926—31no—
23C3×#926—31no—
24C3×#926—61no—
25C2×C3×C4⋊Dic3—31no—
26C2×C3×C4⋊Dic3—31no—
27C2×C3×C4⋊Dic3—61no—
28C2×C3×#208—31no—
29C2×C3×#208—31no—
30C2×C3×#208—61no—
31#63576—32no—
32#63575—32no—
33#63575—32no—
34#63574—32no—
35#63573—32no—
36#63573—32no—
37C23×C33D411yescharacteristic
38C2×C33×C4C2×C412yes—
39C2×C33×C4D411yescharacteristic
40C2×C33×C4Q812yes—
41C2×C33×C4C2311yescharacteristic
42C2×C3×C3⋊Dic3—21no—
43C2×C3×C3⋊Dic3—21no—
44C2×C3×C3⋊Dic3—22no—
45C2×C3×C3⋊Dic3—42no—
46C3×C3⋊(C3⋊C8)C2×C414yescomplemented
47#5121—91no—
48#4995—91no—
49#4964—91no—
50C22×C32×C4—21no—
51C22×C32×C4—21no—
52C22×C32×C4—21no—
53C22×C32×C4—41no—
54C22×C32×C4C2×D311yescharacteristic
55C22×C32×C4C2×D311yescharacteristic
56C22×C32×C4C2×D311yescharacteristic
57C22×C3⋊Dic3—21no—
58C22×C3×Dic3—62no—
59C22×C3×Dic3—61no—
60C22×C3×Dic3—61no—
61#3757—21no—
62C4⋊(C3⋊Dic3)—21no—
63C4⋊(C3⋊Dic3)C2×D311yescharacteristic
64C4⋊(C3⋊Dic3)C2×D312yes—
65C4⋊(C3⋊Dic3)C2×D311yescharacteristic
66C4×C3⋊Dic3—21no—
67#3748Dic314yescomplemented
68C2×C3⋊(C3⋊C8)C2×D312yes—
69C3×#217—62no—
70C3×#217—61no—
71C3×#217—61no—
72C3×C4⋊Dic3—32no—
73C3×C4⋊Dic3—32no—
74C3×C4⋊Dic3—31no—
75C3×C4⋊Dic3—31no—
76C3×C4⋊Dic3—31no—
77C3×C4⋊Dic3—31no—
78C3×C4⋊Dic3—62no—
79C3×C4⋊Dic3—61no—
80C3×C4⋊Dic3—61no—
81C3×C4⋊Dic3—62no—
82C3×C4⋊Dic3—61no—
83C3×C4⋊Dic3—61no—
84C3×C4×Dic3—62no—
85C3×C4×Dic3—61no—
86C3×C4×Dic3—61no—
87C3×#208—34no—
88C3×#208—34no—
89C3×#208—64no—
90C2×C3×C3⋊C8—32no—
91C2×C3×C3⋊C8—32no—
92C2×C3×C3⋊C8—62no—
93C22×C33—21no—
94C22×C33—21no—
95C22×C33C4⋊C412yes—
96C22×C33C2×D411yescharacteristic
97C33×C4C4⋊C412yes—
98C33×C4C22×C411yescharacteristic
99C33×C4C2×Q811yescharacteristic
100C3×C3⋊Dic3—41no—
101C3×C3⋊Dic3—42no—
102C3×C3⋊Dic3—41no—
103C2×C3×M16—91no—
104C3×#117—91no—
105C2×C3×C4⋊C4—91no—
106#926—31no—
107#926—31no—
108#926—61no—
109C2×C4⋊Dic3—31no—
110C2×C4⋊Dic3—31no—
111C2×C4⋊Dic3—61no—
112C2×#208—31no—
113C2×#208—31no—
114C2×#208—61no—
115#818—92no—
116#817—92no—
117#808—92no—
118#808—92no—
119#807—92no—
120#807—92no—
121C23×C32—21no—
122C23×C32—21no—
123C23×C32—21no—
124C23×C32—41no—
125C23×C32D1211yescharacteristic
126C23×C32#6711yescharacteristic
127C23×C32#6711yescharacteristic
128C2×C32×C4—22no—
129C2×C32×C4—22no—
130C2×C32×C4—22no—
131C2×C32×C4—21no—
132C2×C32×C4—21no—
133C2×C32×C4—21no—
134C2×C32×C4—22no—
135C2×C32×C4—22no—
136C2×C32×C4—22no—
137C2×C32×C4—21no—
138C2×C32×C4—21no—
139C2×C32×C4—21no—
140C2×C32×C4—42no—
141C2×C32×C4—41no—
142C2×C32×C4—42no—
143C2×C32×C4—41no—
144C2×C32×C4Dic612yes—
145C2×C32×C4Dic612yes—
146C2×C32×C4Dic612yes—
147C2×C32×C4C4×D312yes—
148C2×C32×C4C4×D312yes—
149C2×C32×C4D1211yescharacteristic
150C2×C32×C4C2×Dic312yes—
151C2×C32×C4#6711yescharacteristic
152C2×C32×C4#6711yescharacteristic
153C2×C32×C4C22×D311yescharacteristic
154C2×C32×C4C22×D311yescharacteristic
155C2×C32×C4C22×D311yescharacteristic
156C2×C3⋊Dic3—21no—
157C2×C3⋊Dic3—21no—
158C2×C3⋊Dic3—22no—
159C2×C3⋊Dic3—42no—
160C2×C3×Dic3—62no—
161C2×C3×Dic3—62no—
162C2×C3×Dic3—64no—
163C2×C3×Dic3—61no—
164C2×C3×Dic3—61no—
165C2×C3×Dic3—62no—
166C2×C3×Dic3—61no—
167C2×C3×Dic3—61no—
168C2×C3×Dic3—62no—
169C2×C3×Dic3—124no—
170C2×C3×Dic3—122no—
171C2×C3×Dic3—122no—
172C3⋊(C3⋊C8)C2×Dic314yescomplemented
173C3×C3⋊C8—34no—
174C3×C3⋊C8—34no—
175C3×C3⋊C8—64no—
176#428—271noSylow
177C2×C33—22no—
178C2×C33C2×C4⋊C411yescharacteristic
179C2×C33AGL(1,ℤ/8)11yescharacteristic
180C2×C33#13711yescharacteristic
181C22×C3×C4—21no—
182C22×C3×C4—21no—
183C22×C3×C4—21no—
184C22×C3×C4—41no—
185C22×C3×C4—181no—
186C22×C3×C4D3211yescharacteristic
187C22×C3×C4D3211yescharacteristic
188C22×C3×C4D3211yescharacteristic
189C22×Dic3—62no—
190C22×Dic3—61no—
191C22×Dic3—61no—
192C3×M16—94no—
193C2×C3×C8—92no—
194C3×C4⋊C4—91no—
195C3×C4⋊C4—92no—
196C3×C4⋊C4—91no—
197C3×C4⋊C4—181no—
198C3×C22⋊C4—181no—
199C3×C42—181no—
200#217—62no—
201#217—61no—
202#217—61no—
203C4⋊Dic3—31no—
204C4⋊Dic3—32no—
205C4⋊Dic3—31no—
206C4⋊Dic3—31no—
207C4⋊Dic3—32no—
208C4⋊Dic3—31no—
209C4⋊Dic3—62no—
210C4⋊Dic3—61no—
211C4⋊Dic3—61no—
212C4⋊Dic3—62no—
213C4⋊Dic3—61no—
214C4⋊Dic3—61no—
215C4×Dic3—62no—
216C4×Dic3—61no—
217C4×Dic3—61no—
218#208—34no—
219#208—34no—
220#208—64no—
221C2×C3⋊C8—32no—
222C2×C3⋊C8—32no—
223C2×C3⋊C8—62no—
224C22×C32—21no—
225C22×C32—21no—
226C22×C32—22no—
227C22×C32—22no—
228C22×C32—21no—
229C22×C32—21no—
230C22×C32—21no—
231C22×C32—21no—
232C22×C32—22no—
233C22×C32—22no—
234C22×C32—22no—
235C22×C32—21no—
236C22×C32—21no—
237C22×C32—21no—
238C22×C32—42no—
239C22×C32—41no—
240C22×C32—41no—
241C22×C32—41no—
242C22×C32—41no—
243C22×C32—41no—
244C22×C32—42no—
245C22×C32—42no—
246C22×C32Dic3⋊C412yes—
247C22×C32Dic3⋊C412yes—
248C22×C32C4⋊Dic312yes—
249C22×C32C2×D1211yescharacteristic
250C22×C32C2×#6711yescharacteristic
251C22×C32C2×#6711yescharacteristic
252C32×C4—22no—
253C32×C4—22no—
254C32×C4—22no—
255C32×C4—21no—
256C32×C4—21no—
257C32×C4—21no—
258C32×C4—21no—
259C32×C4—21no—
260C32×C4—21no—
261C32×C4—42no—
262C32×C4—41no—
263C32×C4—41no—
264C32×C4Dic3⋊C412yes—
265C32×C4Dic3⋊C412yes—
266C32×C4C4⋊Dic312yes—
267C32×C4C2×Dic611yescharacteristic
268C32×C4C2×Dic611yescharacteristic
269C32×C4C2×Dic611yescharacteristic
270C32×C4C2×C4×D311yescharacteristic
271C32×C4C2×C4×D311yescharacteristic
272C32×C4C22×Dic311yescharacteristic
273C3⋊Dic3—42no—
274C3⋊Dic3—41no—
275C3⋊Dic3—41no—
276C3×Dic3—122no—
277C3×Dic3—122no—
278C3×Dic3—124no—
279C3×Dic3—121no—
280C3×Dic3—122no—
281C3×Dic3—121no—
282C3×Dic3—121no—
283C3×Dic3—121no—
284C3×Dic3—122no—
285C2×M16—91no—
286#117—91no—
287C2×C4⋊C4—91no—
288#107—272no—
289#106—272no—
290C33#42811yescharacteristic, complemented, Sylow
291C23×C3—21no—
292C23×C3—21no—
293C23×C3—21no—
294C23×C3—41no—
295C23×C3#62511yescharacteristic
296C23×C3#62611yescharacteristic
297C23×C3#62611yescharacteristic
298C2×C3×C4—22no—
299C2×C3×C4—22no—
300C2×C3×C4—22no—
301C2×C3×C4—21no—
302C2×C3×C4—21no—
303C2×C3×C4—21no—
304C2×C3×C4—22no—
305C2×C3×C4—22no—
306C2×C3×C4—22no—
307C2×C3×C4—21no—
308C2×C3×C4—21no—
309C2×C3×C4—21no—
310C2×C3×C4—42no—
311C2×C3×C4—41no—
312C2×C3×C4—42no—
313C2×C3×C4—41no—
314C2×C3×C4—181no—
315C2×C3×C4—181no—
316C2×C3×C4—182no—
317C2×C3×C4—362no—
318C2×C3×C4D3×Dic312yes—
319C2×C3×C4D3×Dic312yes—
320C2×C3×C4#62412yes—
321C2×C3×C4#62511yescharacteristic
322C2×C3×C4#62611yescharacteristic
323C2×C3×C4#62611yescharacteristic
324C2×C3×C4#62712yes—
325C2×C3×C4#62712yes—
326C2×C3×C4#62712yes—
327C2×C3×C4C2×D3211yescharacteristic
328C2×C3×C4C2×D3211yescharacteristic
329C2×C3×C4C2×D3211yescharacteristic
330C2×Dic3—62no—
331C2×Dic3—62no—
332C2×Dic3—64no—
333C2×Dic3—61no—
334C2×Dic3—61no—
335C2×Dic3—62no—
336C2×Dic3—61no—
337C2×Dic3—61no—
338C2×Dic3—62no—
339C2×Dic3—124no—
340C2×Dic3—122no—
341C2×Dic3—122no—
342C3×C8—94no—
343C3⋊C8—34no—
344C3⋊C8—34no—
345C3⋊C8—64no—
346C2×C32—22no—
347C2×C32—24no—
348C2×C32—22no—
349C2×C32—22no—
350C2×C32—21no—
351C2×C32—21no—
352C2×C32—21no—
353C2×C32—21no—
354C2×C32—21no—
355C2×C32—21no—
356C2×C32—21no—
357C2×C32—21no—
358C2×C32—21no—
359C2×C32—41no—
360C2×C32—41no—
361C2×C32—41no—
362C2×C32—42no—
363C2×C32—42no—
364C2×C32—44no—
365C2×C32#90811yescharacteristic
366C2×C32#90911yescharacteristic
367C2×C32C2×Dic3⋊C411yescharacteristic
368C2×C32C2×Dic3⋊C411yescharacteristic
369C2×C32C2×C4⋊Dic311yescharacteristic
370C2×C32#93211yescharacteristic
371C2×C32#94211yescharacteristic
372C2×C32#94911yescharacteristic
373C2×C32#95111yescharacteristic
374C22×C4—181no—
375C22×C4#111311yescharacteristic
376M16—94no—
377C2×C8—92no—
378C4⋊C4—91no—
379C4⋊C4—92no—
380C4⋊C4—91no—
381C4⋊C4—181no—
382C22⋊C4—181no—
383C42—181no—
384C22×C3—22no—
385C22×C3—22no—
386C22×C3—21no—
387C22×C3—21no—
388C22×C3—21no—
389C22×C3—21no—
390C22×C3—21no—
391C22×C3—21no—
392C22×C3—21no—
393C22×C3—21no—
394C22×C3—21no—
395C22×C3—22no—
396C22×C3—22no—
397C22×C3—22no—
398C22×C3—41no—
399C22×C3—41no—
400C22×C3—41no—
401C22×C3—41no—
402C22×C3—42no—
403C22×C3—42no—
404C22×C3—41no—
405C22×C3—42no—
406C22×C3Dic3⋊Dic312yes—
407C22×C3Dic3⋊Dic312yes—
408C22×C3#372412yes—
409C22×C3C2×#62511yescharacteristic
410C22×C3C2×#62611yescharacteristic
411C22×C3C2×#62611yescharacteristic
412C3×C4—21no—
413C3×C4—21no—
414C3×C4—21no—
415C3×C4—21no—
416C3×C4—21no—
417C3×C4—21no—
418C3×C4—22no—
419C3×C4—22no—
420C3×C4—22no—
421C3×C4—41no—
422C3×C4—41no—
423C3×C4—42no—
424C3×C4—362no—
425C3×C4—361no—
426C3×C4—361no—
427C3×C4Dic3⋊Dic312yes—
428C3×C4Dic3⋊Dic312yes—
429C3×C4#372412yes—
430C3×C4C2×D3×Dic311yescharacteristic
431C3×C4C2×D3×Dic311yescharacteristic
432C3×C4C2×#62411yescharacteristic
433C3×C4C2×#62711yescharacteristic
434C3×C4C2×#62711yescharacteristic
435C3×C4C2×#62711yescharacteristic
436Dic3—124no—
437Dic3—122no—
438Dic3—122no—
439Dic3—122no—
440Dic3—121no—
441Dic3—121no—
442Dic3—121no—
443Dic3—122no—
444Dic3—121no—
445C32—21no—
446C32—21no—
447C32—21no—
448C32—41no—
449C32#496411yescharacteristic, complemented
450C32#499511yescharacteristic, complemented
451C32#512111yescharacteristic, complemented
452C23#629511yescharacteristic
453C2×C4—181no—
454C2×C4—181no—
455C2×C4—182no—
456C2×C4—362no—
457C2×C4#629412yes—
458C2×C4#629511yescharacteristic
459C2×C4#629612yes—
460C2×C4C2×#111311yescharacteristic
461C8—94no—
462C2×C3—24no—
463C2×C3—22no—
464C2×C3—22no—
465C2×C3—22no—
466C2×C3—21no—
467C2×C3—21no—
468C2×C3—21no—
469C2×C3—21no—
470C2×C3—21no—
471C2×C3—21no—
472C2×C3—21no—
473C2×C3—21no—
474C2×C3—21no—
475C2×C3—42no—
476C2×C3—42no—
477C2×C3—44no—
478C2×C3—41no—
479C2×C3—41no—
480C2×C3—41no—
481C2×C3#6382311yescharacteristic
482C2×C3#6382511yescharacteristic
483C2×C3#6382711yescharacteristic
484C2×C3#6383011yescharacteristic
485C2×C3#6383111yescharacteristic
486C2×C3#6383311yescharacteristic
487C2×C3C2×Dic3⋊Dic311yescharacteristic
488C2×C3C2×Dic3⋊Dic311yescharacteristic
489C2×C3C2×#372411yescharacteristic
490C22—21no—
491C22—21no—
492C22#8881512yes—
493C22C2×#629511yescharacteristic
494C4—362no—
495C4—361no—
496C4—361no—
497C4#8881512yes—
498C4C2×#629411yescharacteristic
499C4C2×#629611yescharacteristic
500C3—21no—
501C3—21no—
502C3—21no—
503C3—41no—
504C3#1059113011yescharacteristic, complemented
505C3#1059113111yescharacteristic, complemented
506C3#1059113911yescharacteristic, complemented
507C2—22no—
508C2#1172366111yescharacteristic
509C2#1172366311yescharacteristic
510C2C2×#8881511yescharacteristic
subgroupG/H
centerC22C2×#6295
commutatorC33×C4C22×C4
FrattiniC2×C4C2×#1113
FittingC22×C33×C4C22
socleC22×C33C2×D4
prime radicalC23×C33D4
cosocleC2×C33×C4C23
hypercenterC22×C4#1113
hypocenterC33#428
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
4. #11722080C2semidirect
5. #11722079C2semidirect
17. C3×#3748C4semidirect
46. C3×C3⋊(C3⋊C8)C2×C4semidirect
67. #3748Dic3semidirect
172. C3⋊(C3⋊C8)C2×Dic3semidirect
290. C33#428semidirect
449. C32#4964semidirect
450. C32#4995semidirect
451. C32#5121semidirect
504. C3#10591130semidirect
505. C3#10591131semidirect
506. C3#10591139semidirect

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⋊Dic3)2
C2×C3×#37482
#117220802
#117220792
#105911393
#105911313
#105911303