Finite Group Databasebeta

← #13161461

Subgroups — #13161461

Order 1728.

1740 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6293C211yesmaximal, characteristic, complemented
2C2×#88864C211yesmaximal, characteristic
3C2×C3×#3757C211yesmaximal, characteristic
4C2×C32×#217C211yesmaximal, characteristic
5#11722061C211yesmaximal, characteristic, complemented
6#11722061C211yesmaximal, characteristic, complemented
7#11722061C211yesmaximal, characteristic, complemented
8#10592646—31nomaximal
9#10591692—34nomaximal
10C24×C33C2211yescharacteristic
11C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
12C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
13C22×C3×C3⋊Dic3C2211yescharacteristic
14C22×C3×C3⋊Dic3C2211yescharacteristic
15C22×C32×Dic3C2211yescharacteristic
16C22×C32×Dic3C2211yescharacteristic
17C2×#6293—22no—
18C2×#6293—22no—
19C2×#6293—22no—
20#88864—22no—
21C3×#3757—22no—
22C32×#217—22no—
23C22×#634—31no—
24C22×#627—34no—
25C2×C32×C22⋊C4—31no—
26C2×#3757—31no—
27C2×#3757—34no—
28C2×#3757—64no—
29C2×#3757D311yescharacteristic
30C2×C3×#217—34no—
31C2×C3×#217D314yes—
32#63658—31no—
33#63658—31no—
34#63658—31no—
35#63579—34no—
36#63579—34no—
37#63579—34no—
38C23×C33—21no—
39C23×C33—21no—
40C23×C33—21no—
41C23×C33—21no—
42C23×C33—21no—
43C23×C33—21no—
44C23×C33Q812yescomplemented
45C23×C33C2311yescharacteristic
46C2×C3⋊(C3⋊Dic3)—21no—
47C2×C3⋊(C3⋊Dic3)—21no—
48C2×C3⋊(C3⋊Dic3)—21no—
49C2×C3⋊(C3⋊Dic3)—21no—
50C2×C3⋊(C3⋊Dic3)—21no—
51C2×C3⋊(C3⋊Dic3)D412yes—
52C2×C3×C3⋊Dic3—21no—
53C2×C3×C3⋊Dic3—21no—
54C2×C3×C3⋊Dic3—21no—
55C2×C3×C3⋊Dic3—21no—
56C2×C3×C3⋊Dic3—21no—
57C2×C3×C3⋊Dic3D412yes—
58C2×C32×Dic3—21no—
59C2×C32×Dic3—21no—
60C2×C32×Dic3—21no—
61C2×C32×Dic3—21no—
62C2×C32×Dic3—21no—
63C2×C32×Dic3D412yes—
64#6293—28no—
65#4947—94no—
66#4947—91no—
67C24×C32—24no—
68C24×C32C2×D314yes—
69C24×C32C2×D311yescharacteristic
70C22×C32×C4—31no—
71C22×C32×C4—31no—
72C22×C3⋊Dic3—31no—
73C22×C3⋊Dic3—34no—
74C22×C3⋊Dic3—31no—
75C22×C3⋊Dic3—34no—
76C22×C3⋊Dic3—64no—
77C22×C3⋊Dic3—64no—
78C22×C3⋊Dic3C2×D311yescharacteristic
79C22×C3⋊Dic3C2×D311yescharacteristic
80C2×#634—62no—
81C2×#634—62no—
82C2×#634—62no—
83C22×C3×Dic3—34no—
84C22×C3×Dic3—34no—
85C22×C3×Dic3C2×D314yes—
86C22×C3×Dic3C2×D314yes—
87C2×#627—68no—
88C2×#627—68no—
89C2×#627—68no—
90C32×C22⋊C4—62no—
91#3757—22no—
92#3757—62no—
93#3757—68no—
94#3757—616no—
95C3×#217—28no—
96C3×#217—68no—
97C22×C33—22no—
98C22×C33—22no—
99C22×C33—22no—
100C22×C33—41no—
101C22×C33—41no—
102C22×C33—41no—
103C22×C33—41no—
104C22×C33C2×D411yescharacteristic
105C22×C33C2×D411yescharacteristic
106C22×C33C2×D411yescharacteristic
107C22×C33C2×Q811yescharacteristic
108C22×C33#4111yescharacteristic
109C22×C33#4111yescharacteristic
110C22×C33#4111yescharacteristic
111C3⋊(C3⋊Dic3)—22no—
112C3⋊(C3⋊Dic3)—41no—
113C3×C3⋊Dic3—22no—
114C3×C3⋊Dic3—41no—
115C32×Dic3—22no—
116C32×Dic3—41no—
117C22×Dic6—94no—
118C22×Dic6—91no—
119C2×C3×C22⋊C4—34no—
120C2×C3×C22⋊C4—91no—
121C2×#217—34no—
122C2×#217—94no—
123C2×#217—91no—
124C2×#217—184no—
125C2×#217C3⋊D311yescharacteristic
126C3⋊#95—94no—
127C3⋊#95—94no—
128C3⋊#95—94no—
129C3⋊#95—91no—
130C3⋊#95—91no—
131C3⋊#95—91no—
132C23×C32—28no—
133C23×C32—28no—
134C23×C32—21no—
135C23×C32—24no—
136C23×C32—21no—
137C23×C32—21no—
138C23×C32—24no—
139C23×C32—21no—
140C23×C32—21no—
141C23×C32—21no—
142C23×C32—28no—
143C23×C32—24no—
144C23×C32—24no—
145C23×C32—24no—
146C23×C32—24no—
147C23×C32—24no—
148C23×C32—44no—
149C23×C32—44no—
150C23×C32—44no—
151C23×C32—44no—
152C23×C32Dic618yescomplemented
153C23×C32Dic612yescomplemented
154C23×C32C22×D314yes—
155C23×C32C22×D311yescharacteristic
156C2×C32×C4—32no—
157C2×C32×C4—61no—
158C2×C32×C4—61no—
159C2×C32×C4—61no—
160C2×C32×C4—61no—
161C2×C32×C4—61no—
162C2×C3⋊Dic3—21no—
163C2×C3⋊Dic3—21no—
164C2×C3⋊Dic3—21no—
165C2×C3⋊Dic3—21no—
166C2×C3⋊Dic3—21no—
167C2×C3⋊Dic3—32no—
168C2×C3⋊Dic3—38no—
169C2×C3⋊Dic3—68no—
170C2×C3⋊Dic3—61no—
171C2×C3⋊Dic3—61no—
172C2×C3⋊Dic3—68no—
173C2×C3⋊Dic3—61no—
174C2×C3⋊Dic3—64no—
175C2×C3⋊Dic3—61no—
176C2×C3⋊Dic3—61no—
177C2×C3⋊Dic3—64no—
178C2×C3⋊Dic3—64no—
179C2×C3⋊Dic3—64no—
180C2×C3⋊Dic3—64no—
181C2×C3⋊Dic3—124no—
182C2×C3⋊Dic3—124no—
183C2×C3⋊Dic3—124no—
184C2×C3⋊Dic3—124no—
185C2×C3⋊Dic3#6712yes—
186#634—68no—
187C2×C3×Dic3—24no—
188C2×C3×Dic3—24no—
189C2×C3×Dic3—24no—
190C2×C3×Dic3—24no—
191C2×C3×Dic3—24no—
192C2×C3×Dic3—38no—
193C2×C3×Dic3—64no—
194C2×C3×Dic3—64no—
195C2×C3×Dic3—64no—
196C2×C3×Dic3—64no—
197C2×C3×Dic3—64no—
198C2×C3×Dic3#6718yes—
199#627—632no—
200#393—271noSylow
201C2×C33—42no—
202C2×C33#12011yescharacteristic
203C2×C33C22⋊Q811yescharacteristic
204C2×C33C22⋊Q811yescharacteristic
205C2×C33C22⋊Q811yescharacteristic
206C2×C33#12411yescharacteristic
207C2×C33#12411yescharacteristic
208C2×C33#12411yescharacteristic
209C24×C3—24no—
210C24×C3D3214yes—
211C24×C3C2×C3⋊D311yescharacteristic
212C22×C3×C4—34no—
213C22×C3×C4—34no—
214C22×C3×C4—91no—
215C22×C3×C4—91no—
216C22×Dic3—34no—
217C22×Dic3—34no—
218C22×Dic3—94no—
219C22×Dic3—91no—
220C22×Dic3—94no—
221C22×Dic3—91no—
222C22×Dic3—184no—
223C22×Dic3—184no—
224C22×Dic3C2×C3⋊D311yescharacteristic
225C22×Dic3C2×C3⋊D311yescharacteristic
226C2×Dic6—182no—
227C2×Dic6—182no—
228C2×Dic6—188no—
229C2×Dic6—188no—
230C2×Dic6—188no—
231C2×Dic6—182no—
232C3×C22⋊C4—68no—
233C3×C22⋊C4—182no—
234#217—22no—
235#217—68no—
236#217—188no—
237#217—182no—
238#217—1816no—
239C22×C32—216no—
240C22×C32—28no—
241C22×C32—22no—
242C22×C32—22no—
243C22×C32—22no—
244C22×C32—24no—
245C22×C32—24no—
246C22×C32—24no—
247C22×C32—24no—
248C22×C32—24no—
249C22×C32—24no—
250C22×C32—24no—
251C22×C32—28no—
252C22×C32—28no—
253C22×C32—48no—
254C22×C32—48no—
255C22×C32—48no—
256C22×C32—48no—
257C22×C32—48no—
258C22×C32—48no—
259C22×C32—44no—
260C22×C32—44no—
261C22×C32—41no—
262C22×C32—41no—
263C22×C32—44no—
264C22×C32—41no—
265C22×C32—44no—
266C22×C32—41no—
267C22×C32C2×Dic614yes—
268C22×C32C2×Dic611yescharacteristic
269C22×C32#23514yes—
270C22×C32#23511yescharacteristic
271C22×C32D3×D414yes—
272C22×C32D3×D414yes—
273C22×C32D3×D411yescharacteristic
274C22×C32D3×D411yescharacteristic
275C22×C32#23714yes—
276C22×C32#23714yes—
277C22×C32#23711yescharacteristic
278C22×C32#23711yescharacteristic
279C22×C32C2×#6714yes—
280C22×C32C2×#6711yescharacteristic
281C32×C4—62no—
282C32×C4—121no—
283C3⋊Dic3—22no—
284C3⋊Dic3—41no—
285C3⋊Dic3—62no—
286C3⋊Dic3—68no—
287C3⋊Dic3—128no—
288C3⋊Dic3—128no—
289C3⋊Dic3—124no—
290C3⋊Dic3—121no—
291C3×Dic3—28no—
292C3×Dic3—44no—
293C3×Dic3—68no—
294C3×Dic3—124no—
295C22×Q8—271no—
296C2×C22⋊C4—31no—
297C2×C22⋊C4—91no—
298C2×C22⋊C4—271no—
299#95—271no—
300#95—271no—
301#95—271no—
302C33#39311yescharacteristic, complemented, Sylow
303C23×C3—28no—
304C23×C3—28no—
305C23×C3—24no—
306C23×C3—24no—
307C23×C3—24no—
308C23×C3—24no—
309C23×C3—24no—
310C23×C3—24no—
311C23×C3—21no—
312C23×C3—21no—
313C23×C3—21no—
314C23×C3—21no—
315C23×C3—21no—
316C23×C3—21no—
317C23×C3—28no—
318C23×C3—24no—
319C23×C3—44no—
320C23×C3—44no—
321C23×C3—44no—
322C23×C3—44no—
323C23×C3#62718yescomplemented
324C23×C3#63412yescomplemented
325C23×C3C2×D3214yes—
326C23×C3C22×C3⋊D311yescharacteristic
327C2×C3×C4—38no—
328C2×C3×C4—64no—
329C2×C3×C4—64no—
330C2×C3×C4—64no—
331C2×C3×C4—64no—
332C2×C3×C4—64no—
333C2×C3×C4—92no—
334C2×C3×C4—181no—
335C2×C3×C4—181no—
336C2×C3×C4—181no—
337C2×C3×C4—181no—
338C2×C3×C4—181no—
339C2×Dic3—21no—
340C2×Dic3—21no—
341C2×Dic3—21no—
342C2×Dic3—21no—
343C2×Dic3—21no—
344C2×Dic3—38no—
345C2×Dic3—64no—
346C2×Dic3—64no—
347C2×Dic3—64no—
348C2×Dic3—64no—
349C2×Dic3—64no—
350C2×Dic3—98no—
351C2×Dic3—92no—
352C2×Dic3—188no—
353C2×Dic3—184no—
354C2×Dic3—184no—
355C2×Dic3—181no—
356C2×Dic3—181no—
357C2×Dic3—188no—
358C2×Dic3—184no—
359C2×Dic3—184no—
360C2×Dic3—184no—
361C2×Dic3—181no—
362C2×Dic3—181no—
363C2×Dic3—181no—
364C2×Dic3—364no—
365C2×Dic3—364no—
366C2×Dic3—364no—
367C2×Dic3—364no—
368C2×Dic3#63812yes—
369Dic6—188no—
370Dic6—1832no—
371C2×C32—24no—
372C2×C32—24no—
373C2×C32—24no—
374C2×C32—24no—
375C2×C32—24no—
376C2×C32—24no—
377C2×C32—24no—
378C2×C32—416no—
379C2×C32—48no—
380C2×C32—42no—
381C2×C32#87814yes—
382C2×C32#87814yes—
383C2×C32#87811yescharacteristic
384C2×C32#87811yescharacteristic
385C2×C32#88514yes—
386C2×C32#88514yes—
387C2×C32#88511yescharacteristic
388C2×C32#88511yescharacteristic
389C2×C32#92414yes—
390C2×C32#92411yescharacteristic
391C2×C32#93614yes—
392C2×C32#93611yescharacteristic
393C2×C32#93714yes—
394C2×C32#93711yescharacteristic
395C24D3×C3⋊D311yescharacteristic
396C2×Q8—542no—
397C2×Q8—542no—
398C2×Q8—542no—
399C22×C4—31no—
400C22×C4—31no—
401C22×C4—91no—
402C22×C4—91no—
403C22×C4—271no—
404C22×C4—271no—
405C22⋊C4—62no—
406C22⋊C4—182no—
407C22⋊C4—542no—
408C22×C3—216no—
409C22×C3—22no—
410C22×C3—22no—
411C22×C3—22no—
412C22×C3—28no—
413C22×C3—28no—
414C22×C3—28no—
415C22×C3—24no—
416C22×C3—24no—
417C22×C3—24no—
418C22×C3—24no—
419C22×C3—24no—
420C22×C3—24no—
421C22×C3—24no—
422C22×C3—48no—
423C22×C3—48no—
424C22×C3—48no—
425C22×C3—48no—
426C22×C3—48no—
427C22×C3—48no—
428C22×C3—41no—
429C22×C3—41no—
430C22×C3—41no—
431C22×C3—41no—
432C22×C3—44no—
433C22×C3—44no—
434C22×C3—44no—
435C22×C3—44no—
436C22×C3#380414yes—
437C22×C3#380414yes—
438C22×C3#380514yes—
439C22×C3C2×#62714yes—
440C22×C3D3×#6714yes—
441C22×C3D3×#6714yes—
442C22×C3#381114yes—
443C22×C3C2×#63411yescharacteristic
444C22×C3#382811yescharacteristic
445C22×C3D4×C3⋊D311yescharacteristic
446C22×C3D4×C3⋊D311yescharacteristic
447C22×C3#383011yescharacteristic
448C22×C3#383011yescharacteristic
449C22×C3C2×#63811yescharacteristic
450C3×C4—68no—
451C3×C4—124no—
452C3×C4—182no—
453C3×C4—361no—
454Dic3—22no—
455Dic3—41no—
456Dic3—68no—
457Dic3—124no—
458Dic3—182no—
459Dic3—188no—
460Dic3—368no—
461Dic3—368no—
462Dic3—361no—
463Dic3—364no—
464C32—24no—
465C32#494714yescomplemented
466C32#494711yescharacteristic, complemented
467C23—21no—
468C23—21no—
469C23—21no—
470C23—21no—
471C23—21no—
472C23—21no—
473C23#629312yescomplemented
474C23C2×D3×C3⋊D311yescharacteristic
475Q8—548no—
476C2×C4—32no—
477C2×C4—61no—
478C2×C4—61no—
479C2×C4—61no—
480C2×C4—61no—
481C2×C4—61no—
482C2×C4—92no—
483C2×C4—181no—
484C2×C4—181no—
485C2×C4—181no—
486C2×C4—181no—
487C2×C4—181no—
488C2×C4—272no—
489C2×C4—541no—
490C2×C4—541no—
491C2×C4—541no—
492C2×C4—541no—
493C2×C4—541no—
494C2×C3—24no—
495C2×C3—24no—
496C2×C3—24no—
497C2×C3—24no—
498C2×C3—24no—
499C2×C3—24no—
500C2×C3—24no—
501C2×C3—416no—
502C2×C3—42no—
503C2×C3—48no—
504C2×C3#6395314yes—
505C2×C3#6395914yes—
506C2×C3#6395914yes—
507C2×C3#6396414yes—
508C2×C3#6396414yes—
509C2×C3#6398214yes—
510C2×C3#6398314yes—
511C2×C3#6408711yescharacteristic
512C2×C3#6408711yescharacteristic
513C2×C3#6409411yescharacteristic
514C2×C3#6409411yescharacteristic
515C2×C3#6413311yescharacteristic
516C2×C3#6414511yescharacteristic
517C2×C3#6414611yescharacteristic
518C22—22no—
519C22—22no—
520C22—22no—
521C22—41no—
522C22—41no—
523C22—41no—
524C22—41no—
525C22#8902811yescharacteristic
526C22#8902911yescharacteristic
527C22#8903111yescharacteristic
528C22C2×#629311yescharacteristic
529C22D3×#63811yescharacteristic
530C22C3⋊D3×#6711yescharacteristic
531C22#8903911yescharacteristic
532C4—62no—
533C4—121no—
534C4—182no—
535C4—361no—
536C4—542no—
537C4—1081no—
538C3—24no—
539C3#1059169214yescomplemented
540C3#1059264611yescharacteristic, complemented
541C2—42no—
542C2#1172359111yescharacteristic
543C2#1172359211yescharacteristic
544C2#1172360011yescharacteristic
545C2#1172364011yescharacteristic
546C2#1172364111yescharacteristic
547C2#1172364211yescharacteristic
548C2#1172364311yescharacteristic
subgroupG/H
center, FrattiniC23C2×D3×C3⋊D3
commutator, socle, cosocleC23×C33C23
Fitting, prime radicalC24×C33C22
hypercenterC24D3×C3⋊D3
hypocenterC33#393
LCSG ▷ C23×C33 ▷ C33
UCSC1 ◁ C23 ◁ C24

Complements

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

NH ≅ G/Ntype
1. C22×#6293C2semidirect
5. #11722061C2semidirect
6. #11722061C2semidirect
7. #11722061C2semidirect
44. C23×C33Q8semidirect
152. C23×C32Dic6semidirect
153. C23×C32Dic6semidirect
302. C33#393semidirect
323. C23×C3#627semidirect
324. C23×C3#634semidirect
465. C32#4947semidirect
466. C32#4947semidirect
473. C23#6293semidirect
539. C3#10591692semidirect
540. C3#10592646semidirect

Maximal subgroups

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

subgroupindex
C22×#62932
C2×#888642
C2×C3×#37572
C2×C32×#2172
#117220612
#105926463
#105916923