Finite Group Databasebeta

← #13155409

Subgroups — #13155409

Order 1728.

774 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×D3×Dic6C211yesmaximal, characteristic
2C2×#88786C211yesmaximal, characteristic
3C3×#63841C211yesmaximal, characteristic
4#11722076C211yesmaximal, characteristic, complemented
5#11722039C211yesmaximal, characteristic, complemented
6#11722037C211yesmaximal, characteristic, complemented
7C3×#63565C211yesmaximal, characteristic, complemented
8#10591557—31nomaximal
9#10591099—31nomaximal
10#10590600—31nomaximal
11C2×C3×#634C2211yescharacteristic
12C2×C32×C4×D3C2211yescharacteristic
13C2×C32×Dic6C2211yescharacteristic
14C2×C3×#627—21no—
15C2×C3×D3×Dic3—21no—
16C3×D3×Dic6—22no—
17C3×D3×Dic6C414yescomplemented
18C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
19C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
20C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
21C32×C4⋊Dic3C2211yescharacteristic
22#88786—22no—
23C3×(C2×D3)⋊Dic3—21no—
24C3×Dic32—21no—
25C2×C3×D3×Q8—31no—
26C22×C3×Dic6—31no—
27C2×D3×Dic6D311yescharacteristic
28C2×#3748—31no—
29C3×#926—31no—
30C3×#892—31no—
31#63841D311yescharacteristic
32C2×#3711—31no—
33C2×#3711—31no—
34C2×#3711—61no—
35#63662—31no—
36#63637—31no—
37C3×#835—31no—
38C3×#810—31no—
39#63567—31no—
40#63566—31no—
41#63565—31no—
42#63565—31no—
43#63565—31no—
44#63565—31no—
45#63565—31no—
46#63565—61no—
47#63565—61no—
48#63565D311yescharacteristic, complemented
49C22×C32×D3D411yescharacteristic
50C2×C33×C4C2311yescharacteristic
51C2×C3×C3⋊Dic3—21no—
52C2×C3×C3⋊Dic3—21no—
53C3×#634—21no—
54C3×#634C2×C412yes—
55C2×C32×Dic3—21no—
56C2×C32×Dic3—21no—
57C2×C32×Dic3D411yescharacteristic
58C32×C4×D3C2×C412yes—
59C32×C4×D3D412yes—
60C32×Dic6—21no—
61C32×Dic6C2×C412yes—
62C3×#627—24no—
63C3×D3×Dic3—24no—
64C3⋊(C3⋊(C3⋊C8))—21no—
65C3×C3⋊(C3⋊C8)—21no—
66#5225—91no—
67#5004—91no—
68#4802—91no—
69C2×C32×Q8—31no—
70C22×C32×C4—31no—
71C2×#634C2×D311yescharacteristic
72C22×C3×Dic3—61no—
73C3×D3×Q8—34no—
74C3×D3×Q8—62no—
75C2×C3×C4×D3—21no—
76C2×C3×C4×D3—61no—
77C2×C3×C4×D3C2×D311yescharacteristic
78C2×C3×C4×D3C2×D311yescharacteristic
79C2×C3×Dic6—21no—
80C2×C3×Dic6—31no—
81C2×C3×Dic6—31no—
82C2×C3×Dic6—34no—
83C2×C3×Dic6—62no—
84C2×C3×Dic6—61no—
85C2×C3×Dic6—61no—
86C2×C3×Dic6—61no—
87C2×C3×Dic6C2×D311yescharacteristic
88C2×C3×Dic6C2×D311yescharacteristic
89C2×#627—21no—
90C2×D3×Dic3—21no—
91D3×Dic6—22no—
92D3×Dic6Dic314yescomplemented
93C32×C4⋊C4—31no—
94C4⋊(C3⋊Dic3)C2×D311yescharacteristic
95#3748—62no—
96C2×C3⋊(C3⋊C8)—31no—
97C2×C3⋊(C3⋊C8)—31no—
98C2×C3⋊(C3⋊C8)—31no—
99C2×C3⋊(C3⋊C8)—61no—
100C2×C3⋊(C3⋊C8)—61no—
101C2×C3⋊(C3⋊C8)—61no—
102C2×C3⋊(C3⋊C8)—121no—
103C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
104C3×#217—61no—
105C3×(C2×D3)⋊C4—61no—
106C3×C4⋊Dic3—21no—
107C3×C4⋊Dic3—31no—
108C3×C4⋊Dic3—31no—
109C3×C4⋊Dic3—61no—
110C3×C4⋊Dic3C2×D311yescharacteristic
111C3×C4⋊Dic3C2×D311yescharacteristic
112C3×C4×Dic3—61no—
113C3×C4×Dic3—61no—
114C2×C3×C3⋊C8—31no—
115C2×C3×C3⋊C8—31no—
116C2×C3×C3⋊C8—61no—
117(C2×D3)⋊Dic3—21no—
118Dic32—21no—
119#3711—64no—
120#3711—62no—
121#3711—62no—
122C22×C33C2×D411yescharacteristic
123C2×C32×D3—21no—
124C2×C32×D3—21no—
125C2×C32×D3C22⋊C412yes—
126C33×C4C22×C411yescharacteristic
127C33×C4C2×D411yescharacteristic
128C3×C3⋊Dic3—22no—
129C3×C3⋊Dic3—41no—
130C32×Dic3—22no—
131C32×Dic3—41no—
132C32×Dic3C22⋊C412yes—
133C22×C3×Q8—91no—
134C2×D3×Q8—31no—
135C22×Dic6—31no—
136C3×#117—91no—
137#926—31no—
138C2×#208—91no—
139C2×#208—91no—
140C2×#208—181no—
141C2×ΓL(1,𝔽25)—91no—
142#892—31no—
143C3×#103—91no—
144#835—31no—
145#835—91no—
146#835—91no—
147#835—91no—
148#835—91no—
149#835—181no—
150#816—91no—
151#810—31no—
152#810—91no—
153#810—91no—
154#810—91no—
155#810—91no—
156#810—181no—
157C23×C32—31no—
158C22×C3×D3—21no—
159C22×C3×D3#6711yescharacteristic
160C22×C3×D3#6711yescharacteristic
161C32×Q8—32no—
162C32×Q8—61no—
163C2×C32×C4—21no—
164C2×C32×C4—21no—
165C2×C32×C4—21no—
166C2×C32×C4—31no—
167C2×C32×C4—32no—
168C2×C32×C4—32no—
169C2×C32×C4—41no—
170C2×C32×C4—61no—
171C2×C32×C4—61no—
172C2×C32×C4C22×D311yescharacteristic
173C2×C32×C4C22×D311yescharacteristic
174C2×C32×C4C22×D311yescharacteristic
175C2×C3⋊Dic3—21no—
176C2×C3⋊Dic3—21no—
177#634—21no—
178#634C2×Dic312yes—
179C2×C3×Dic3—21no—
180C2×C3×Dic3—21no—
181C2×C3×Dic3—21no—
182C2×C3×Dic3—21no—
183C2×C3×Dic3—21no—
184C2×C3×Dic3—41no—
185C2×C3×Dic3—41no—
186C2×C3×Dic3—64no—
187C2×C3×Dic3—61no—
188C2×C3×Dic3—61no—
189C2×C3×Dic3—61no—
190C2×C3×Dic3—61no—
191C2×C3×Dic3—121no—
192C2×C3×Dic3—121no—
193C2×C3×Dic3#6711yescharacteristic
194C2×C3×Dic3#6711yescharacteristic
195C3×C4×D3—22no—
196C3×C4×D3—22no—
197C3×C4×D3—64no—
198C3×C4×D3C4×D312yes—
199C3×C4×D3D1212yes—
200C3×C4×D3C2×Dic312yes—
201C3×C4×D3#6712yes—
202C3×Dic6—21no—
203C3×Dic6—21no—
204C3×Dic6—22no—
205C3×Dic6—32no—
206C3×Dic6—32no—
207C3×Dic6—41no—
208C3×Dic6—62no—
209C3×Dic6—64no—
210C3×Dic6—61no—
211C3×Dic6—61no—
212C3×Dic6—64no—
213C3×Dic6—121no—
214C3×Dic6C4×D312yes—
215C3×Dic6C2×Dic312yes—
216#627—24no—
217D3×Dic3—24no—
218C3⋊(C3⋊C8)—21no—
219C3⋊(C3⋊C8)—61no—
220C3⋊(C3⋊C8)—62no—
221C3⋊(C3⋊C8)—61no—
222C3⋊(C3⋊C8)—61no—
223C3⋊(C3⋊C8)—121no—
224C3⋊(C3⋊C8)—121no—
225C3⋊(C3⋊C8)—122no—
226C3×C3⋊C8—62no—
227C3×C3⋊C8—61no—
228C3×C3⋊C8—61no—
229#419—271noSylow
230C2×C33C2×C22⋊C411yescharacteristic
231C2×C33#13711yescharacteristic
232C2×C33#13711yescharacteristic
233C32×D3—22no—
234C2×C3×Q8—31no—
235C2×C3×Q8—31no—
236C2×C3×Q8—61no—
237C2×C3×Q8—94no—
238C2×C3×Q8—91no—
239C2×C3×Q8—181no—
240C2×C3×Q8—182no—
241C22×C3×C4—31no—
242C22×C3×C4—31no—
243C22×C3×C4—61no—
244C22×C3×C4—181no—
245C22×Dic3—61no—
246D3×Q8—34no—
247D3×Q8—62no—
248C2×C4×D3—61no—
249C2×C4×D3D3211yescharacteristic
250C2×Dic6—31no—
251C2×Dic6—31no—
252C2×Dic6—34no—
253C2×Dic6—61no—
254C2×Dic6—62no—
255C2×Dic6—61no—
256C2×Dic6—61no—
257C2×Dic6D3211yescharacteristic
258C2×C3×C8—91no—
259C3×C4⋊C4—31no—
260C3×C4⋊C4—31no—
261C3×C4⋊C4—61no—
262C3×C4⋊C4—91no—
263C3×C22⋊C4—181no—
264C3×C42—181no—
265#217—61no—
266(C2×D3)⋊C4—61no—
267C4⋊Dic3—31no—
268C4⋊Dic3—31no—
269C4⋊Dic3—61no—
270C4⋊Dic3D3211yescharacteristic
271C4×Dic3—61no—
272C4×Dic3—61no—
273#208—182no—
274#208—182no—
275#208—184no—
276C2×C3⋊C8—31no—
277C2×C3⋊C8—31no—
278C2×C3⋊C8—61no—
279C2×C3⋊C8—91no—
280C2×C3⋊C8—91no—
281C2×C3⋊C8—91no—
282C2×C3⋊C8—181no—
283C2×C3⋊C8—181no—
284C2×C3⋊C8—181no—
285C2×C3⋊C8—361no—
286ΓL(1,𝔽25)—182no—
287C22×C32—21no—
288C22×C32—21no—
289C22×C32—21no—
290C22×C32—32no—
291C22×C32—41no—
292C22×C32—61no—
293C22×C32—61no—
294C22×C32D3×D411yescharacteristic
295C22×C32C2×#6711yescharacteristic
296C22×C32C2×#6711yescharacteristic
297C2×C3×D3—21no—
298C2×C3×D3—21no—
299C2×C3×D3—21no—
300C2×C3×D3—21no—
301C2×C3×D3—22no—
302C2×C3×D3—41no—
303C2×C3×D3—41no—
304C2×C3×D3(C2×D3)⋊C412yes—
305C2×C3×D3#21712yes—
306C32×C4—21no—
307C32×C4—21no—
308C32×C4—21no—
309C32×C4—21no—
310C32×C4—21no—
311C32×C4—21no—
312C32×C4—32no—
313C32×C4—41no—
314C32×C4—41no—
315C32×C4—62no—
316C32×C4—121no—
317C32×C4C2×C4×D311yescharacteristic
318C32×C4C2×C4×D311yescharacteristic
319C32×C4C2×D1211yescharacteristic
320C32×C4D3×D411yescharacteristic
321C32×C4C22×Dic311yescharacteristic
322C32×C4C2×#6711yescharacteristic
323C3⋊Dic3—22no—
324C3⋊Dic3—41no—
325C3×Dic3—22no—
326C3×Dic3—22no—
327C3×Dic3—22no—
328C3×Dic3—42no—
329C3×Dic3—41no—
330C3×Dic3—41no—
331C3×Dic3—62no—
332C3×Dic3—62no—
333C3×Dic3—81no—
334C3×Dic3—122no—
335C3×Dic3—121no—
336C3×Dic3—121no—
337C3×Dic3—241no—
338C3×Dic3(C2×D3)⋊C412yes—
339C3×Dic3#21712yes—
340C22×Q8—91no—
341C2×M16—271no—
342#117—91no—
343#103—91no—
344#103—271no—
345#103—271no—
346#103—271no—
347C33#41911yescharacteristic, complemented, Sylow
348C23×C3—31no—
349C23×C3—31no—
350C23×C3—61no—
351C22×D3#62511yescharacteristic
352C3×Q8—32no—
353C3×Q8—32no—
354C3×Q8—62no—
355C3×Q8—61no—
356C3×Q8—61no—
357C3×Q8—92no—
358C3×Q8—121no—
359C3×Q8—184no—
360C3×Q8—181no—
361C2×C3×C4—21no—
362C2×C3×C4—21no—
363C2×C3×C4—21no—
364C2×C3×C4—31no—
365C2×C3×C4—31no—
366C2×C3×C4—32no—
367C2×C3×C4—32no—
368C2×C3×C4—32no—
369C2×C3×C4—32no—
370C2×C3×C4—41no—
371C2×C3×C4—62no—
372C2×C3×C4—62no—
373C2×C3×C4—61no—
374C2×C3×C4—61no—
375C2×C3×C4—61no—
376C2×C3×C4—61no—
377C2×C3×C4—61no—
378C2×C3×C4—121no—
379C2×C3×C4—121no—
380C2×C3×C4—184no—
381C2×C3×C4—181no—
382C2×C3×C4—181no—
383C2×C3×C4C2×D3211yescharacteristic
384C2×C3×C4C2×D3211yescharacteristic
385C2×C3×C4C2×D3211yescharacteristic
386C2×Dic3—21no—
387C2×Dic3—21no—
388C2×Dic3—64no—
389C2×Dic3—61no—
390C2×Dic3—61no—
391C2×Dic3—61no—
392C2×Dic3—61no—
393C2×Dic3—121no—
394C2×Dic3—121no—
395C2×Dic3#62511yescharacteristic
396C4×D3—64no—
397C4×D3D3×Dic312yes—
398C4×D3#62612yes—
399Dic6—21no—
400Dic6—32no—
401Dic6—32no—
402Dic6—62no—
403Dic6—64no—
404Dic6—61no—
405Dic6—61no—
406Dic6—64no—
407Dic6—121no—
408Dic6D3×Dic312yes—
409C3×C8—181no—
410C3⋊C8—62no—
411C3⋊C8—61no—
412C3⋊C8—61no—
413C3⋊C8—182no—
414C3⋊C8—181no—
415C3⋊C8—181no—
416C3⋊C8—181no—
417C3⋊C8—361no—
418C3⋊C8—361no—
419C3⋊C8—362no—
420C2×C32—21no—
421C2×C32—21no—
422C2×C32—21no—
423C2×C32—21no—
424C2×C32—21no—
425C2×C32—21no—
426C2×C32—21no—
427C2×C32—21no—
428C2×C32—21no—
429C2×C32—41no—
430C2×C32—41no—
431C2×C32—41no—
432C2×C32—62no—
433C2×C32D3×C22⋊C411yescharacteristic
434C2×C32#91511yescharacteristic
435C2×C32#91811yescharacteristic
436C2×C32C2×(C2×D3)⋊C411yescharacteristic
437C2×C32#94211yescharacteristic
438C2×C32#94211yescharacteristic
439C2×C32#95111yescharacteristic
440C2×C32#95111yescharacteristic
441C2×C32C2×#21711yescharacteristic
442C3×D3—22no—
443C3×D3—22no—
444C3×D3—42no—
445C2×Q8—31no—
446C2×Q8—94no—
447C2×Q8—91no—
448C2×Q8—181no—
449C2×Q8—182no—
450C22×C4—31no—
451C22×C4—181no—
452M16—542no—
453C2×C8—91no—
454C2×C8—271no—
455C4⋊C4—31no—
456C4⋊C4—91no—
457C22⋊C4—181no—
458C42—181no—
459C22×C3—21no—
460C22×C3—21no—
461C22×C3—21no—
462C22×C3—32no—
463C22×C3—32no—
464C22×C3—41no—
465C22×C3—62no—
466C22×C3—61no—
467C22×C3—61no—
468C22×C3—61no—
469C22×C3—61no—
470C22×C3—121no—
471C22×C3—121no—
472C22×C3C2×#62511yescharacteristic
473C22×C3D3×#6711yescharacteristic
474C22×C3D3×#6711yescharacteristic
475C2×D3—21no—
476C2×D3—21no—
477C2×D3(C2×D3)⋊Dic312yes—
478C3×C4—21no—
479C3×C4—21no—
480C3×C4—21no—
481C3×C4—21no—
482C3×C4—21no—
483C3×C4—21no—
484C3×C4—32no—
485C3×C4—32no—
486C3×C4—41no—
487C3×C4—41no—
488C3×C4—62no—
489C3×C4—62no—
490C3×C4—62no—
491C3×C4—122no—
492C3×C4—121no—
493C3×C4—121no—
494C3×C4—182no—
495C3×C4—241no—
496C3×C4—361no—
497C3×C4C4×D3211yescharacteristic
498C3×C4D3×D1211yescharacteristic
499C3×C4C2×D3×Dic311yescharacteristic
500C3×C4C2×D3×Dic311yescharacteristic
501C3×C4C2×#62611yescharacteristic
502C3×C4D3×#6711yescharacteristic
503Dic3—22no—
504Dic3—41no—
505Dic3—62no—
506Dic3—62no—
507Dic3—122no—
508Dic3—121no—
509Dic3—121no—
510Dic3—241no—
511Dic3(C2×D3)⋊Dic312yes—
512C32—21no—
513C32—21no—
514C32—21no—
515C32—41no—
516C32#480211yescharacteristic, complemented
517C32#500411yescharacteristic, complemented
518C32#522511yescharacteristic, complemented
519C23—31no—
520Q8—32no—
521Q8—61no—
522Q8—92no—
523Q8—184no—
524Q8—181no—
525C2×C4—31no—
526C2×C4—32no—
527C2×C4—32no—
528C2×C4—61no—
529C2×C4—61no—
530C2×C4—184no—
531C2×C4—181no—
532C2×C4—181no—
533C2×C4D3311yescharacteristic
534C8—181no—
535C8—541no—
536C2×C3—21no—
537C2×C3—21no—
538C2×C3—21no—
539C2×C3—21no—
540C2×C3—21no—
541C2×C3—21no—
542C2×C3—21no—
543C2×C3—21no—
544C2×C3—21no—
545C2×C3—41no—
546C2×C3—41no—
547C2×C3—41no—
548C2×C3—62no—
549C2×C3—62no—
550C2×C3—122no—
551C2×C3#6382711yescharacteristic
552C2×C3#6382711yescharacteristic
553C2×C3D3×(C2×D3)⋊C411yescharacteristic
554C2×C3#6392911yescharacteristic
555C2×C3#6394311yescharacteristic
556C2×C3#6394611yescharacteristic
557C2×C3#6394811yescharacteristic
558C2×C3C2×(C2×D3)⋊Dic311yescharacteristic
559C2×C3D3×#21711yescharacteristic
560D3—22no—
561C22—32no—
562C22—61no—
563C22—61no—
564C22D3×#62511yescharacteristic
565C4—32no—
566C4—62no—
567C4—121no—
568C4—182no—
569C4—361no—
570C4D32×Dic311yescharacteristic
571C4D3×#62611yescharacteristic
572C3—21no—
573C3—21no—
574C3—21no—
575C3—41no—
576C3#1059060011yescharacteristic, complemented
577C3#1059109911yescharacteristic, complemented
578C3#1059155711yescharacteristic, complemented
579C2—62no—
580C2#1172288211yescharacteristic
581C2#1172288311yescharacteristic
582C2D3×(C2×D3)⋊Dic311yescharacteristic
subgroupG/H
centerC22D3×#625
commutatorC33×C4C22×C4
Frattini, hypercenterC2×C4D33
Fitting, cosocleC2×C33×C4C23
socleC22×C33C2×D4
prime radicalC22×C32×D3D4
hypocenterC33#419
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
4. #11722076C2semidirect
5. #11722039C2semidirect
6. #11722037C2semidirect
7. C3×#63565C2semidirect
17. C3×D3×Dic6C4semidirect
48. #63565D3semidirect
92. D3×Dic6Dic3semidirect
347. C33#419semidirect
516. C32#4802semidirect
517. C32#5004semidirect
518. C32#5225semidirect
576. C3#10590600semidirect
577. C3#10591099semidirect
578. C3#10591557semidirect

Maximal subgroups

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

subgroupindex
C2×C3×D3×Dic62
C2×#887862
C3×#638412
#117220762
#117220392
#117220372
C3×#635652
#105915573
#105910993
#105906003