Finite Group Databasebeta

← C2×#11722769 #13167270

Subgroups — C2×#11722769 #13167270

Order 1728.

1240 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C3⋊#157C211yesmaximal, characteristic
2C22×#6294C211yesmaximal, characteristic
3C22×C3×#624C211yesmaximal, characteristic
4#11722769C214yesmaximal, direct product factor
5C2×#63737—31nomaximal
6C23×C3×C3⋊D3C2211yescharacteristic
7C22×C3⋊#157—22no—
8C22×C3⋊#157—22no—
9C22×C3⋊#157C414yescomplemented
10C22×C3⋊#157C2212yes—
11C22×C3×C3⋊Dic3—21no—
12C22×C32×Dic3—21no—
13C2×#6294—21no—
14C2×#6294—22no—
15C2×#6294C414yescomplemented
16C2×#6294C2212yes—
17C2×C3×#624—21no—
18C2×C3×#624—22no—
19C2×C3×#624C414yescomplemented
20C2×C3×#624C2212yes—
21C23×#157—31no—
22C22×C3×C4×D3—61no—
23C22×#624—31no—
24C22×#624D311yescharacteristic
25C22×D3×Dic3—61no—
26#63737—34no—
27C23×C33D411yescharacteristic
28C22×C3×C3⋊D3C2×C412yes—
29C22×C3×C3⋊D3C2×C412yes—
30C22×C3×C3⋊D3C2×C412yes—
31C22×C3×C3⋊D3D411yescharacteristic
32C22×C3×C3⋊D3D412yes—
33C22×C3×C3⋊D3D412yes—
34C22×C3×C3⋊D3Q812yes—
35C22×C3×C3⋊D3C2311yescharacteristic
36C2×C3⋊#157—28no—
37C2×C3⋊#157—22no—
38C2×C3⋊#157—22no—
39C2×C3⋊#157C2×C414yescomplemented
40C2×C3×C3⋊Dic3—24no—
41C2×C3×C3⋊Dic3—22no—
42C2×C32×Dic3—24no—
43C2×C32×Dic3—22no—
44#6294—24no—
45#6294—22no—
46#6294C2×C414yescomplemented
47C3×#624—24no—
48C3×#624—22no—
49C3×#624C2×C414yescomplemented
50C2×C3⋊#95—91nomaximal
51C23×C3⋊D3C2×D311yescharacteristic
52C23×C3×D3—61no—
53C23×C3×D3—61no—
54C22×#157—34no—
55C22×#157—32no—
56C22×#157—62no—
57C22×#157—62no—
58C22×C32×C4—61no—
59C22×C3⋊Dic3—21no—
60C22×C3×Dic3—21no—
61C22×C3×Dic3—21no—
62C22×C3×Dic3—41no—
63C22×C3×Dic3—61no—
64C22×C3×Dic3—61no—
65C22×C3×Dic3—121no—
66C2×C3×C4×D3—64no—
67C2×C3×C4×D3—62no—
68C2×C3×C4×D3—62no—
69C2×C3×C4×D3—64no—
70C2×#624—21no—
71C2×#624—22no—
72C2×#624—34no—
73C2×#624—32no—
74C2×#624—62no—
75C2×#624—61no—
76C2×#624Dic314yescomplemented
77C2×#624C2×D312yes—
78C2×D3×Dic3—64no—
79C2×D3×Dic3—62no—
80C2×D3×Dic3—62no—
81C2×D3×Dic3—64no—
82C22×C33C22⋊C412yes—
83C22×C33C22⋊C412yes—
84C22×C33C4⋊C412yes—
85C22×C33C2×D411yescharacteristic
86C2×C3×C3⋊D3C4214yes—
87C2×C3×C3⋊D3C22⋊C412yes—
88C2×C3×C3⋊D3C22⋊C412yes—
89C2×C3×C3⋊D3C22⋊C414yes—
90C2×C3×C3⋊D3C4⋊C412yes—
91C2×C3×C3⋊D3C4⋊C414yes—
92C2×C3×C3⋊D3C4⋊C414yes—
93C2×C3×C3⋊D3C22×C411yescharacteristic
94C2×C3×C3⋊D3C22×C411yescharacteristic
95C2×C3×C3⋊D3C22×C411yescharacteristic
96C2×C3×C3⋊D3C2×D411yescharacteristic
97C2×C3×C3⋊D3C2×D411yescharacteristic
98C2×C3×C3⋊D3C2×Q811yescharacteristic
99C3⋊#157—24no—
100C3×C3⋊Dic3—24no—
101C32×Dic3—24no—
102C23×C3×C4—91no—
103C23×Dic3—91no—
104C23×Dic3—91no—
105C22×C4×D3—61no—
106C22×C4×D3—181no—
107C3⋊#95—94no—
108C23×C32—21no—
109C23×C32—21no—
110C23×C32—41no—
111C23×C32—41no—
112C23×C32#6711yescharacteristic
113C22×C3⋊D3Dic612yes—
114C22×C3⋊D3C4×D312yes—
115C22×C3⋊D3C4×D312yes—
116C22×C3⋊D3D1212yes—
117C22×C3⋊D3C2×Dic312yes—
118C22×C3⋊D3#6711yescharacteristic
119C22×C3⋊D3#6712yes—
120C22×C3⋊D3C22×D311yescharacteristic
121C22×C3×D3—62no—
122C22×C3×D3—62no—
123C22×C3×D3—62no—
124C22×C3×D3—62no—
125C22×C3×D3—62no—
126C22×C3×D3—62no—
127C22×C3×D3—62no—
128C22×C3×D3—62no—
129C22×C3×D3—62no—
130C22×C3×D3—61no—
131C22×C3×D3—61no—
132C22×C3×D3—62no—
133C22×C3×D3—62no—
134C22×C3×D3—62no—
135C22×C3×D3—61no—
136C22×C3×D3—61no—
137C2×#157—34no—
138C2×#157—68no—
139C2×#157—62no—
140C2×#157—62no—
141C2×C32×C4—62no—
142C2×C32×C4—64no—
143C2×C3⋊Dic3—22no—
144C2×C3⋊Dic3—24no—
145C2×C3×Dic3—22no—
146C2×C3×Dic3—24no—
147C2×C3×Dic3—24no—
148C2×C3×Dic3—22no—
149C2×C3×Dic3—44no—
150C2×C3×Dic3—42no—
151C2×C3×Dic3—62no—
152C2×C3×Dic3—64no—
153C2×C3×Dic3—64no—
154C2×C3×Dic3—62no—
155C2×C3×Dic3—124no—
156C2×C3×Dic3—122no—
157C3×C4×D3—64no—
158C3×C4×D3—68no—
159C3×C4×D3—64no—
160#624—24no—
161#624—22no—
162#624—34no—
163#624—64no—
164#624—62no—
165#624C2×Dic314yescomplemented
166D3×Dic3—64no—
167D3×Dic3—68no—
168D3×Dic3—64no—
169C2×#95—271noSylow
170C2×C33#9514yescomplemented
171C2×C33C2×C22⋊C411yescharacteristic
172C2×C33C2×C22⋊C411yescharacteristic
173C2×C33C2×C4⋊C411yescharacteristic
174C3×C3⋊D3#9514yescomplemented
175C3×C3⋊D3C2×C4211yescharacteristic
176C3×C3⋊D3C2×C22⋊C411yescharacteristic
177C3×C3⋊D3C2×C4⋊C411yescharacteristic
178C3×C3⋊D3C2×C4⋊C411yescharacteristic
179C24×C3—91no—
180C23×D3—61no—
181C23×D3—61no—
182C22×C3×C4—61no—
183C22×C3×C4—61no—
184C22×C3×C4—92no—
185C22×C3×C4—94no—
186C22×C3×C4—121no—
187C22×C3×C4—181no—
188C22×C3×C4—182no—
189C22×C3×C4—181no—
190C22×Dic3—21no—
191C22×Dic3—61no—
192C22×Dic3—61no—
193C22×Dic3—92no—
194C22×Dic3—94no—
195C22×Dic3—94no—
196C22×Dic3—92no—
197C22×Dic3—121no—
198C22×Dic3—181no—
199C22×Dic3—182no—
200C22×Dic3—182no—
201C22×Dic3—182no—
202C2×C4×D3—64no—
203C2×C4×D3—62no—
204C2×C4×D3—62no—
205C2×C4×D3—64no—
206C2×C4×D3—182no—
207C2×C4×D3—184no—
208C2×C4×D3—184no—
209C2×C4×D3—182no—
210C22×C32—21no—
211C22×C32—21no—
212C22×C32—22no—
213C22×C32—22no—
214C22×C32—22no—
215C22×C32—22no—
216C22×C32—22no—
217C22×C32—22no—
218C22×C32—42no—
219C22×C32—42no—
220C22×C32—42no—
221C22×C32—42no—
222C22×C32—42no—
223C22×C32—42no—
224C22×C32—41no—
225C22×C32—41no—
226C22×C32Dic3⋊C412yes—
227C22×C32(C2×D3)⋊C412yes—
228C22×C32#21712yes—
229C22×C32C2×#6711yescharacteristic
230C2×C3⋊D3C4×Dic314yes—
231C2×C3⋊D3Dic3⋊C412yes—
232C2×C3⋊D3Dic3⋊C414yes—
233C2×C3⋊D3C4⋊Dic314yes—
234C2×C3⋊D3(C2×D3)⋊C412yes—
235C2×C3⋊D3(C2×D3)⋊C414yes—
236C2×C3⋊D3#21712yes—
237C2×C3⋊D3C2×Dic611yescharacteristic
238C2×C3⋊D3C2×C4×D311yescharacteristic
239C2×C3⋊D3C2×C4×D311yescharacteristic
240C2×C3⋊D3C2×D1211yescharacteristic
241C2×C3⋊D3C22×Dic311yescharacteristic
242C2×C3⋊D3C2×#6711yescharacteristic
243C2×C3×D3—64no—
244C2×C3×D3—64no—
245C2×C3×D3—62no—
246C2×C3×D3—62no—
247C2×C3×D3—62no—
248C2×C3×D3—64no—
249C2×C3×D3—64no—
250C2×C3×D3—62no—
251C2×C3×D3—62no—
252C2×C3×D3—62no—
253C2×C3×D3—64no—
254C2×C3×D3—64no—
255C2×C3×D3—64no—
256C2×C3×D3—64no—
257C2×C3×D3—61no—
258C2×C3×D3—61no—
259C2×C3×D3—61no—
260C2×C3×D3—61no—
261C2×C3×D3—61no—
262C2×C3×D3—61no—
263C2×C3×D3—61no—
264C2×C3×D3—61no—
265C2×C3×D3—61no—
266C2×C3×D3—61no—
267C2×C3×D3—61no—
268C2×C3×D3—61no—
269#157—64no—
270C32×C4—64no—
271C3⋊Dic3—24no—
272C3×Dic3—24no—
273C3×Dic3—24no—
274C3×Dic3—44no—
275C3×Dic3—64no—
276C3×Dic3—64no—
277C3×Dic3—124no—
278C23×C4—91no—
279C23×C4—271no—
280C23×C4—271no—
281#95—274no—
282C33C2×#9511yescharacteristic, complemented, Sylow
283C23×C3—21no—
284C23×C3—21no—
285C23×C3—41no—
286C23×C3—41no—
287C23×C3—91no—
288C23×C3—91no—
289C23×C3—92no—
290C23×C3—92no—
291C23×C3—92no—
292C23×C3—92no—
293C23×C3—92no—
294C23×C3—92no—
295C23×C3GO+(4,𝔽2)11yescharacteristic
296C22×D3—62no—
297C22×D3—62no—
298C22×D3—62no—
299C22×D3—62no—
300C22×D3—62no—
301C22×D3—62no—
302C22×D3—62no—
303C22×D3—62no—
304C22×D3—62no—
305C22×D3—62no—
306C22×D3—61no—
307C22×D3—62no—
308C22×D3—61no—
309C22×D3—62no—
310C22×D3—61no—
311C22×D3—61no—
312C2×C3×C4—62no—
313C2×C3×C4—64no—
314C2×C3×C4—62no—
315C2×C3×C4—64no—
316C2×C3×C4—94no—
317C2×C3×C4—122no—
318C2×C3×C4—124no—
319C2×C3×C4—182no—
320C2×C3×C4—184no—
321C2×C3×C4—182no—
322C2×C3×C4—184no—
323C2×Dic3—22no—
324C2×Dic3—24no—
325C2×Dic3—62no—
326C2×Dic3—64no—
327C2×Dic3—62no—
328C2×Dic3—64no—
329C2×Dic3—94no—
330C2×Dic3—94no—
331C2×Dic3—122no—
332C2×Dic3—124no—
333C2×Dic3—182no—
334C2×Dic3—184no—
335C2×Dic3—182no—
336C2×Dic3—182no—
337C2×Dic3—188no—
338C4×D3—64no—
339C4×D3—68no—
340C4×D3—64no—
341C4×D3—184no—
342C4×D3—188no—
343C4×D3—184no—
344C2×C32—21no—
345C2×C32—21no—
346C2×C32—21no—
347C2×C32—21no—
348C2×C32—21no—
349C2×C32—21no—
350C2×C32—24no—
351C2×C32—24no—
352C2×C32—44no—
353C2×C32—44no—
354C2×C32—41no—
355C2×C32—41no—
356C2×C32—41no—
357C2×C32—41no—
358C2×C32—41no—
359C2×C32—41no—
360C2×C32C3⋊#9514yescomplemented
361C2×C32C2×Dic3⋊C411yescharacteristic
362C2×C32C2×(C2×D3)⋊C411yescharacteristic
363C2×C32C2×#21711yescharacteristic
364C3⋊D3C3⋊#9514yescomplemented
365C3⋊D3C2×C4×Dic311yescharacteristic
366C3⋊D3C2×Dic3⋊C411yescharacteristic
367C3⋊D3C2×C4⋊Dic311yescharacteristic
368C3⋊D3C2×(C2×D3)⋊C411yescharacteristic
369C3×D3—64no—
370C3×D3—64no—
371C3×D3—61no—
372C3×D3—61no—
373C3×D3—61no—
374C3×D3—61no—
375C3×D3—61no—
376C3×D3—61no—
377C3×D3—61no—
378C3×D3—61no—
379C24—91no—
380C22×C4—61no—
381C22×C4—92no—
382C22×C4—94no—
383C22×C4—181no—
384C22×C4—182no—
385C22×C4—181no—
386C22×C4—272no—
387C22×C4—274no—
388C22×C4—272no—
389C22×C4—274no—
390C22×C4—541no—
391C22×C4—542no—
392C22×C4—542no—
393C22×C4—542no—
394C22×C3—21no—
395C22×C3—21no—
396C22×C3—22no—
397C22×C3—22no—
398C22×C3—22no—
399C22×C3—22no—
400C22×C3—22no—
401C22×C3—22no—
402C22×C3—41no—
403C22×C3—41no—
404C22×C3—42no—
405C22×C3—42no—
406C22×C3—42no—
407C22×C3—42no—
408C22×C3—42no—
409C22×C3—42no—
410C22×C3—91no—
411C22×C3—91no—
412C22×C3—91no—
413C22×C3—92no—
414C22×C3—91no—
415C22×C3—92no—
416C22×C3—91no—
417C22×C3—91no—
418C22×C3—92no—
419C22×C3—94no—
420C22×C3—94no—
421C22×C3—94no—
422C22×C3—94no—
423C22×C3D32⋊C412yes—
424C22×C3D32⋊C412yes—
425C22×C3#377312yes—
426C22×C3C2×GO+(4,𝔽2)11yescharacteristic
427C2×D3—62no—
428C2×D3—61no—
429C2×D3—61no—
430C2×D3—62no—
431C2×D3—61no—
432C2×D3—61no—
433C2×D3—64no—
434C2×D3—64no—
435C2×D3—64no—
436C2×D3—64no—
437C2×D3—62no—
438C2×D3—62no—
439C2×D3—64no—
440C2×D3—64no—
441C2×D3—64no—
442C2×D3—64no—
443C2×D3—62no—
444C2×D3—62no—
445C2×D3—61no—
446C2×D3—61no—
447C2×D3—61no—
448C2×D3—61no—
449C2×D3—61no—
450C2×D3—61no—
451C2×D3—61no—
452C2×D3—61no—
453C3×C4—64no—
454C3×C4—64no—
455C3×C4—124no—
456C3×C4—184no—
457Dic3—24no—
458Dic3—64no—
459Dic3—64no—
460Dic3—124no—
461Dic3—184no—
462C32—21no—
463C32—21no—
464C32—41no—
465C32—41no—
466C32C2×C3⋊#9511yescharacteristic, complemented
467C23—91no—
468C23—91no—
469C23—92no—
470C23—92no—
471C23—92no—
472C23—92no—
473C23—92no—
474C23—92no—
475C23D32⋊D311yescharacteristic
476C2×C4—62no—
477C2×C4—64no—
478C2×C4—94no—
479C2×C4—182no—
480C2×C4—184no—
481C2×C4—182no—
482C2×C4—184no—
483C2×C4—274no—
484C2×C4—274no—
485C2×C4—542no—
486C2×C4—544no—
487C2×C4—548no—
488C2×C4—542no—
489C2×C4—542no—
490C2×C3—24no—
491C2×C3—24no—
492C2×C3—21no—
493C2×C3—21no—
494C2×C3—21no—
495C2×C3—21no—
496C2×C3—21no—
497C2×C3—21no—
498C2×C3—44no—
499C2×C3—44no—
500C2×C3—41no—
501C2×C3—41no—
502C2×C3—41no—
503C2×C3—41no—
504C2×C3—41no—
505C2×C3—41no—
506C2×C3—91no—
507C2×C3—91no—
508C2×C3—91no—
509C2×C3—91no—
510C2×C3—94no—
511C2×C3#6373714yescomplemented
512C2×C3C2×D32⋊C411yescharacteristic
513C2×C3C2×D32⋊C411yescharacteristic
514C2×C3C2×#377311yescharacteristic
515D3—64no—
516D3—64no—
517D3—61no—
518D3—61no—
519D3—61no—
520D3—61no—
521D3—61no—
522D3—61no—
523D3—61no—
524D3—61no—
525C22—94no—
526C22—94no—
527C22—94no—
528C22—94no—
529C22—91no—
530C22—91no—
531C22—92no—
532C22—91no—
533C22—91no—
534C22—92no—
535C22—91no—
536C22—91no—
537C22—92no—
538C22#8893212yes—
539C22D32⋊Dic312yes—
540C22#8893412yes—
541C22C2×D32⋊D311yescharacteristic
542C4—64no—
543C4—184no—
544C4—544no—
545C3—21no—
546C3—21no—
547C3—41no—
548C3—41no—
549C3C2×#6373711yescharacteristic, complemented
550C2—91no—
551C2—94no—
552C2—91no—
553C2—91no—
554C2—91no—
555C2#1172276914yesdirect product factor
556C2C2×#8893211yescharacteristic
557C2C2×D32⋊Dic311yescharacteristic
558C2C2×#8893411yescharacteristic
subgroupG/H
center, hypercenterC23D32⋊D3
commutatorC3×C3⋊D3C2×C42
FrattiniC22C2×D32⋊D3
Fitting, socleC23×C33D4
prime radicalC23×C3×C3⋊D3C22
cosocleC22×C3×C3⋊D3C23
hypocenterC33C2×#95
LCSG ▷ C3×C3⋊D3 ▷ C33
UCSC1 ◁ C23

Complements

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

NH ≅ G/Ntype
4. #11722769C2direct (N × H)
9. C22×C3⋊#157C4semidirect
15. C2×#6294C4semidirect
19. C2×C3×#624C4semidirect
39. C2×C3⋊#157C2×C4semidirect
46. #6294C2×C4semidirect
49. C3×#624C2×C4semidirect
76. C2×#624Dic3semidirect
165. #624C2×Dic3semidirect
170. C2×C33#95semidirect
174. C3×C3⋊D3#95semidirect
282. C33C2×#95semidirect
360. C2×C32C3⋊#95semidirect
364. C3⋊D3C3⋊#95semidirect
466. C32C2×C3⋊#95semidirect
511. C2×C3#63737semidirect
549. C3C2×#63737semidirect
555. C2#11722769direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3⋊#1572
C22×#62942
C22×C3×#6242
#117227692
C2×#637373
C2×C3⋊#959