Finite Group Databasebeta

← C2×#11722076 #13161815

Subgroups — C2×#11722076 #13161815

Order 1728.

1026 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#634C211yesmaximal, characteristic
2C2×C3×C4⋊(C3⋊Dic3)C211yesmaximal, characteristic
3C22×C3×C3⋊(C3⋊C8)C211yesmaximal, characteristic
4#11722076C214yesmaximal, direct product factor
5C2×#63567—31nomaximal
6C2×#63566—31nomaximal
7C2×#63565—31nomaximal
8C22×C33×C4C2211yescharacteristic
9C22×C3×C3⋊Dic3—21no—
10C22×C3×C3⋊Dic3—21no—
11C2×C3×#634—21no—
12C2×C3×#634—22no—
13C2×C3×#634C414yescomplemented
14C2×C3×#634C2212yes—
15C3×C4⋊(C3⋊Dic3)—21no—
16C3×C4⋊(C3⋊Dic3)C2212yes—
17C2×C3×C3⋊(C3⋊C8)—22no—
18C2×C3×C3⋊(C3⋊C8)C2212yes—
19C22×#634D311yescharacteristic
20C22×C3×Dic6—31no—
21C22×C3×Dic6—31no—
22C22×C3×Dic6—61no—
23C2×C4⋊(C3⋊Dic3)D311yescharacteristic
24C22×C3⋊(C3⋊C8)D311yescharacteristic
25C2×C3×C4⋊Dic3—31no—
26C2×C3×C4⋊Dic3—31no—
27C2×C3×C4⋊Dic3—61no—
28C22×C3×C3⋊C8—31no—
29C22×C3×C3⋊C8—31no—
30C22×C3×C3⋊C8—61no—
31#63567—34no—
32#63566—34no—
33#63565—34no—
34C23×C33D411yescharacteristic
35C2×C33×C4C2×C412yes—
36C2×C33×C4D411yescharacteristic
37C2×C33×C4D412yes—
38C2×C33×C4C2311yescharacteristic
39C2×C3×C3⋊Dic3—24no—
40C2×C3×C3⋊Dic3—22no—
41C2×C3×C3⋊Dic3—22no—
42C2×C3×C3⋊Dic3—42no—
43C3×#634—24no—
44C3×#634—22no—
45C3×#634C2×C414yescomplemented
46C3×C3⋊(C3⋊C8)—22no—
47C2×#835—91no—
48C2×#816—91no—
49C2×#810—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—21no—
59C2×#634—21no—
60C2×#634—22no—
61C2×#634Dic314yescomplemented
62C2×#634C2×D312yes—
63C22×C3×Dic3—62no—
64C22×C3×Dic3—61no—
65C22×C3×Dic3—61no—
66C22×C3×Dic3—62no—
67C22×C3×Dic3—61no—
68C22×C3×Dic3—61no—
69C2×C3×Dic6—34no—
70C2×C3×Dic6—34no—
71C2×C3×Dic6—32no—
72C2×C3×Dic6—32no—
73C2×C3×Dic6—64no—
74C2×C3×Dic6—62no—
75C2×C3×Dic6—62no—
76C2×C3×Dic6—64no—
77C2×C3×Dic6—61no—
78C2×C3×Dic6—62no—
79C2×C3×Dic6—61no—
80C2×C3×Dic6—62no—
81C4⋊(C3⋊Dic3)—21no—
82C4⋊(C3⋊Dic3)C2×D312yes—
83C2×C3⋊(C3⋊C8)—22no—
84C2×C3⋊(C3⋊C8)C2×D312yes—
85C3×C4⋊Dic3—32no—
86C3×C4⋊Dic3—32no—
87C3×C4⋊Dic3—62no—
88C3×C4⋊Dic3—62no—
89C3×C4⋊Dic3—61no—
90C3×C4⋊Dic3—61no—
91C2×C3×C3⋊C8—32no—
92C2×C3×C3⋊C8—32no—
93C2×C3×C3⋊C8—62no—
94C2×C3×C3⋊C8—64no—
95C2×C3×C3⋊C8—62no—
96C2×C3×C3⋊C8—62no—
97C22×C33C22⋊C412yes—
98C22×C33SD1612yes—
99C22×C33Q1612yes—
100C22×C33C2×D411yescharacteristic
101C33×C4C22⋊C412yes—
102C33×C4C22×C411yescharacteristic
103C33×C4C2×D411yescharacteristic
104C3×C3⋊Dic3—24no—
105C3×C3⋊Dic3—42no—
106C22×C3×Q8—91no—
107C22×Dic6—31no—
108C22×Dic6—31no—
109C22×Dic6—61no—
110C22×C3×C8—91no—
111C2×C3×C4⋊C4—91no—
112C2×C4⋊Dic3—31no—
113C2×C4⋊Dic3—31no—
114C2×C4⋊Dic3—61no—
115C22×C3⋊C8—31no—
116C22×C3⋊C8—31no—
117C22×C3⋊C8—61no—
118#835—94no—
119#816—94no—
120#810—94no—
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—22no—
133C2×C32×C4—22no—
134C2×C32×C4—21no—
135C2×C32×C4—21no—
136C2×C32×C4—22no—
137C2×C32×C4—21no—
138C2×C32×C4—21no—
139C2×C32×C4—21no—
140C2×C32×C4—42no—
141C2×C32×C4—42no—
142C2×C32×C4—41no—
143C2×C32×C4—41no—
144C2×C32×C4C4×D312yes—
145C2×C32×C4C4×D312yes—
146C2×C32×C4D1211yescharacteristic
147C2×C32×C4D1212yes—
148C2×C32×C4C2×Dic312yes—
149C2×C32×C4#6711yescharacteristic
150C2×C32×C4#6712yes—
151C2×C32×C4#6711yescharacteristic
152C2×C32×C4#6712yes—
153C2×C32×C4C22×D311yescharacteristic
154C2×C32×C4C22×D311yescharacteristic
155C2×C32×C4C22×D311yescharacteristic
156C2×C3⋊Dic3—24no—
157C2×C3⋊Dic3—22no—
158C2×C3⋊Dic3—22no—
159C2×C3⋊Dic3—42no—
160#634—24no—
161#634—22no—
162#634C2×Dic314yescomplemented
163C2×C3×Dic3—68no—
164C2×C3×Dic3—64no—
165C2×C3×Dic3—64no—
166C2×C3×Dic3—64no—
167C2×C3×Dic3—62no—
168C2×C3×Dic3—62no—
169C2×C3×Dic3—64no—
170C2×C3×Dic3—62no—
171C2×C3×Dic3—62no—
172C2×C3×Dic3—124no—
173C2×C3×Dic3—122no—
174C2×C3×Dic3—122no—
175C3×Dic6—34no—
176C3×Dic6—34no—
177C3×Dic6—64no—
178C3×Dic6—68no—
179C3×Dic6—64no—
180C3×Dic6—64no—
181C3×Dic6—64no—
182C3×Dic6—62no—
183C3×Dic6—62no—
184C3⋊(C3⋊C8)—22no—
185C3×C3⋊C8—64no—
186C3×C3⋊C8—62no—
187C3×C3⋊C8—62no—
188C2×#103—271noSylow
189C2×C33#10314yescomplemented
190C2×C33C2×C22⋊C411yescharacteristic
191C2×C33C2×SD1611yescharacteristic
192C2×C33C2×Q1611yescharacteristic
193C2×C3×Q8—94no—
194C2×C3×Q8—92no—
195C2×C3×Q8—182no—
196C2×C3×Q8—181no—
197C22×C3×C4—21no—
198C22×C3×C4—21no—
199C22×C3×C4—21no—
200C22×C3×C4—41no—
201C22×C3×C4—181no—
202C22×C3×C4—181no—
203C22×C3×C4D3211yescharacteristic
204C22×C3×C4D3211yescharacteristic
205C22×C3×C4D3211yescharacteristic
206C22×Dic3—62no—
207C22×Dic3—61no—
208C22×Dic3—61no—
209C22×Dic3—62no—
210C22×Dic3—61no—
211C22×Dic3—61no—
212C2×Dic6—34no—
213C2×Dic6—34no—
214C2×Dic6—32no—
215C2×Dic6—32no—
216C2×Dic6—64no—
217C2×Dic6—62no—
218C2×Dic6—62no—
219C2×Dic6—64no—
220C2×Dic6—61no—
221C2×Dic6—62no—
222C2×Dic6—61no—
223C2×Dic6—62no—
224C2×C3×C8—92no—
225C2×C3×C8—182no—
226C3×C4⋊C4—92no—
227C3×C4⋊C4—181no—
228C4⋊Dic3—32no—
229C4⋊Dic3—32no—
230C4⋊Dic3—62no—
231C4⋊Dic3—62no—
232C4⋊Dic3—61no—
233C4⋊Dic3—61no—
234C2×C3⋊C8—32no—
235C2×C3⋊C8—32no—
236C2×C3⋊C8—62no—
237C2×C3⋊C8—64no—
238C2×C3⋊C8—62no—
239C2×C3⋊C8—62no—
240C22×C32—22no—
241C22×C32—22no—
242C22×C32—22no—
243C22×C32—22no—
244C22×C32—22no—
245C22×C32—22no—
246C22×C32—22no—
247C22×C32—22no—
248C22×C32—22no—
249C22×C32—21no—
250C22×C32—21no—
251C22×C32—21no—
252C22×C32—42no—
253C22×C32—42no—
254C22×C32—42no—
255C22×C32—41no—
256C22×C32#20412yes—
257C22×C32Dic1212yes—
258C22×C32(C2×D3)⋊C412yes—
259C22×C32(C2×D3)⋊C412yes—
260C22×C32#21412yes—
261C22×C32#21512yes—
262C22×C32#21612yes—
263C22×C32#21612yes—
264C22×C32#21712yes—
265C22×C32C2×D1211yescharacteristic
266C22×C32C2×#6711yescharacteristic
267C22×C32C2×#6711yescharacteristic
268C32×C4—22no—
269C32×C4—22no—
270C32×C4—22no—
271C32×C4—21no—
272C32×C4—21no—
273C32×C4—21no—
274C32×C4—21no—
275C32×C4—21no—
276C32×C4—21no—
277C32×C4—42no—
278C32×C4—41no—
279C32×C4—41no—
280C32×C4(C2×D3)⋊C412yes—
281C32×C4(C2×D3)⋊C412yes—
282C32×C4#21712yes—
283C32×C4C2×C4×D311yescharacteristic
284C32×C4C2×C4×D311yescharacteristic
285C32×C4C2×D1211yescharacteristic
286C32×C4C22×Dic311yescharacteristic
287C32×C4C2×#6711yescharacteristic
288C32×C4C2×#6711yescharacteristic
289C3⋊Dic3—24no—
290C3⋊Dic3—42no—
291C3×Dic3—68no—
292C3×Dic3—64no—
293C3×Dic3—64no—
294C3×Dic3—124no—
295C3×Dic3—122no—
296C3×Dic3—122no—
297C22×Q8—91no—
298C22×C8—91no—
299C2×C4⋊C4—91no—
300#103—274no—
301C33C2×#10311yescharacteristic, complemented, Sylow
302C23×C3—21no—
303C23×C3—21no—
304C23×C3—21no—
305C23×C3—41no—
306C23×C3#62511yescharacteristic
307C23×C3#62611yescharacteristic
308C23×C3#62611yescharacteristic
309C3×Q8—94no—
310C3×Q8—184no—
311C3×Q8—182no—
312C2×C3×C4—22no—
313C2×C3×C4—22no—
314C2×C3×C4—22no—
315C2×C3×C4—22no—
316C2×C3×C4—21no—
317C2×C3×C4—22no—
318C2×C3×C4—21no—
319C2×C3×C4—22no—
320C2×C3×C4—21no—
321C2×C3×C4—21no—
322C2×C3×C4—21no—
323C2×C3×C4—21no—
324C2×C3×C4—42no—
325C2×C3×C4—42no—
326C2×C3×C4—41no—
327C2×C3×C4—41no—
328C2×C3×C4—184no—
329C2×C3×C4—182no—
330C2×C3×C4—182no—
331C2×C3×C4—362no—
332C2×C3×C4D3×Dic312yes—
333C2×C3×C4D3×Dic312yes—
334C2×C3×C4#62412yes—
335C2×C3×C4#62511yescharacteristic
336C2×C3×C4#62512yes—
337C2×C3×C4#62611yescharacteristic
338C2×C3×C4#62612yes—
339C2×C3×C4#62611yescharacteristic
340C2×C3×C4#62612yes—
341C2×C3×C4C2×D3211yescharacteristic
342C2×C3×C4C2×D3211yescharacteristic
343C2×C3×C4C2×D3211yescharacteristic
344C2×Dic3—64no—
345C2×Dic3—68no—
346C2×Dic3—64no—
347C2×Dic3—64no—
348C2×Dic3—62no—
349C2×Dic3—62no—
350C2×Dic3—64no—
351C2×Dic3—62no—
352C2×Dic3—62no—
353C2×Dic3—124no—
354C2×Dic3—122no—
355C2×Dic3—122no—
356Dic6—34no—
357Dic6—34no—
358Dic6—64no—
359Dic6—64no—
360Dic6—68no—
361Dic6—64no—
362Dic6—64no—
363Dic6—62no—
364Dic6—62no—
365C3×C8—182no—
366C3⋊C8—64no—
367C3⋊C8—62no—
368C3⋊C8—62no—
369C2×C32—24no—
370C2×C32—24no—
371C2×C32—24no—
372C2×C32—21no—
373C2×C32—21no—
374C2×C32—21no—
375C2×C32—21no—
376C2×C32—21no—
377C2×C32—21no—
378C2×C32—21no—
379C2×C32—21no—
380C2×C32—21no—
381C2×C32—44no—
382C2×C32—41no—
383C2×C32—41no—
384C2×C32—41no—
385C2×C32#81014yescomplemented
386C2×C32#81614yescomplemented
387C2×C32#83514yescomplemented
388C2×C32C2×#20411yescharacteristic
389C2×C32C2×Dic1211yescharacteristic
390C2×C32C2×(C2×D3)⋊C411yescharacteristic
391C2×C32C2×(C2×D3)⋊C411yescharacteristic
392C2×C32C2×#21411yescharacteristic
393C2×C32C2×#21511yescharacteristic
394C2×C32C2×#21611yescharacteristic
395C2×C32C2×#21611yescharacteristic
396C2×C32C2×#21711yescharacteristic
397C2×Q8—94no—
398C2×Q8—92no—
399C2×Q8—181no—
400C2×Q8—182no—
401C22×C4—181no—
402C22×C4—181no—
403C22×C4#111311yescharacteristic
404C2×C8—92no—
405C2×C8—182no—
406C4⋊C4—92no—
407C4⋊C4—181no—
408C22×C3—21no—
409C22×C3—21no—
410C22×C3—21no—
411C22×C3—22no—
412C22×C3—22no—
413C22×C3—22no—
414C22×C3—22no—
415C22×C3—22no—
416C22×C3—22no—
417C22×C3—22no—
418C22×C3—22no—
419C22×C3—22no—
420C22×C3—41no—
421C22×C3—42no—
422C22×C3—42no—
423C22×C3—42no—
424C22×C3#371512yes—
425C22×C3#371612yes—
426C22×C3#371712yes—
427C22×C3#371812yes—
428C22×C3#371912yes—
429C22×C3#371912yes—
430C22×C3(C2×D3)⋊Dic312yes—
431C22×C3(C2×D3)⋊Dic312yes—
432C22×C3#372212yes—
433C22×C3C2×#62511yescharacteristic
434C22×C3C2×#62611yescharacteristic
435C22×C3C2×#62611yescharacteristic
436C3×C4—21no—
437C3×C4—21no—
438C3×C4—21no—
439C3×C4—21no—
440C3×C4—21no—
441C3×C4—21no—
442C3×C4—22no—
443C3×C4—22no—
444C3×C4—22no—
445C3×C4—41no—
446C3×C4—41no—
447C3×C4—42no—
448C3×C4—184no—
449C3×C4—362no—
450C3×C4(C2×D3)⋊Dic312yes—
451C3×C4(C2×D3)⋊Dic312yes—
452C3×C4#372212yes—
453C3×C4C2×D3×Dic311yescharacteristic
454C3×C4C2×D3×Dic311yescharacteristic
455C3×C4C2×#62411yescharacteristic
456C3×C4C2×#62511yescharacteristic
457C3×C4C2×#62611yescharacteristic
458C3×C4C2×#62611yescharacteristic
459Dic3—68no—
460Dic3—64no—
461Dic3—64no—
462Dic3—124no—
463Dic3—122no—
464Dic3—122no—
465C32—21no—
466C32—21no—
467C32—21no—
468C32—41no—
469C32C2×#81011yescharacteristic, complemented
470C32C2×#81611yescharacteristic, complemented
471C32C2×#83511yescharacteristic, complemented
472C23#629511yescharacteristic
473Q8—94no—
474Q8—182no—
475Q8—184no—
476C2×C4—182no—
477C2×C4—184no—
478C2×C4—182no—
479C2×C4—362no—
480C2×C4#629412yes—
481C2×C4#629512yes—
482C2×C4#629511yescharacteristic
483C2×C4C2×#111311yescharacteristic
484C8—182no—
485C2×C3—24no—
486C2×C3—24no—
487C2×C3—24no—
488C2×C3—21no—
489C2×C3—21no—
490C2×C3—21no—
491C2×C3—21no—
492C2×C3—21no—
493C2×C3—21no—
494C2×C3—21no—
495C2×C3—21no—
496C2×C3—21no—
497C2×C3—44no—
498C2×C3—41no—
499C2×C3—41no—
500C2×C3—41no—
501C2×C3#6356514yescomplemented
502C2×C3#6356614yescomplemented
503C2×C3#6356714yescomplemented
504C2×C3C2×#371511yescharacteristic
505C2×C3C2×#371611yescharacteristic
506C2×C3C2×#371711yescharacteristic
507C2×C3C2×#371811yescharacteristic
508C2×C3C2×#371911yescharacteristic
509C2×C3C2×#371911yescharacteristic
510C2×C3C2×(C2×D3)⋊Dic311yescharacteristic
511C2×C3C2×(C2×D3)⋊Dic311yescharacteristic
512C2×C3C2×#372211yescharacteristic
513C22#8881112yes—
514C22#8881212yes—
515C22#8881412yes—
516C22C2×#629511yescharacteristic
517C4—184no—
518C4—362no—
519C4#8881412yes—
520C4C2×#629411yescharacteristic
521C4C2×#629511yescharacteristic
522C3—21no—
523C3—21no—
524C3—21no—
525C3—41no—
526C3C2×#6356511yescharacteristic, complemented
527C3C2×#6356611yescharacteristic, complemented
528C3C2×#6356711yescharacteristic, complemented
529C2#1172207614yesdirect product factor
530C2C2×#8881111yescharacteristic
531C2C2×#8881211yescharacteristic
532C2C2×#8881411yescharacteristic
subgroupG/H
centerC23#6295
commutatorC33×C4C22×C4
FrattiniC2×C4C2×#1113
FittingC22×C33×C4C22
socle, prime radicalC23×C33D4
cosocleC2×C33×C4C23
hypercenterC22×C4#1113
hypocenterC33C2×#103
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
4. #11722076C2direct (N × H)
13. C2×C3×#634C4semidirect
45. C3×#634C2×C4semidirect
61. C2×#634Dic3semidirect
162. #634C2×Dic3semidirect
189. C2×C33#103semidirect
301. C33C2×#103semidirect
385. C2×C32#810semidirect
386. C2×C32#816semidirect
387. C2×C32#835semidirect
469. C32C2×#810semidirect
470. C32C2×#816semidirect
471. C32C2×#835semidirect
501. C2×C3#63565semidirect
502. C2×C3#63566semidirect
503. C2×C3#63567semidirect
526. C3C2×#63565semidirect
527. C3C2×#63566semidirect
528. C3C2×#63567semidirect
529. C2#11722076direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#6342
C2×C3×C4⋊(C3⋊Dic3)2
C22×C3×C3⋊(C3⋊C8)2
#117220762
C2×#635673
C2×#635663
C2×#635653