Finite Group Databasebeta

← #13156038

Subgroups — #13156038

Order 1728.

752 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723441C211yesmaximal, characteristic, complemented
2C2×#88790C211yesmaximal, characteristic
3C3×Dic3×D12C211yesmaximal, characteristic, complemented
4C2×C3×#3716C211yesmaximal, characteristic, complemented
5#11722076C211yesmaximal, characteristic, complemented
6#11722036C211yesmaximal, characteristic, complemented
7C3×Dic3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
8#10591471—31nomaximal
9#10590754—31nomaximal
10#10590366—31nomaximal
11C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C2×C32×D12C2211yescharacteristic
14C2×C3×D3×Dic3—21no—
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C32×C4×Dic3C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19#88804—21no—
20#88804—21no—
21#88790—22no—
22C3×(C2×D3)⋊Dic3—21no—
23C3×#3716—22no—
24C2×#3753—31no—
25#64084—31no—
26C3×D4×Dic3—31no—
27C2×C3×#214—31no—
28C2×C3×#204—31no—
29C3×C4×D12—31no—
30Dic3×D12D311yescharacteristic, complemented
31#63864—31no—
32#63864—31no—
33#63864—61no—
34C2×#3716D311yescharacteristic, complemented
35C2×#3714—31no—
36C2×#3714—31no—
37C2×#3714—61no—
38#63648—31no—
39C3×Dic3⋊C8—31no—
40C3×C4⋊(C3⋊C8)—31no—
41Dic3⋊(C3⋊C8)D311yescharacteristic, complemented
42#63567—31no—
43#63566—31no—
44#63565—31no—
45#63564—31no—
46#63564—31no—
47#63564—61no—
48C22×C3⋊(C3⋊D3)—21no—
49C22×C32×D3—21no—
50C2×C33×C4C2311yescharacteristic
51C4⋊(C3⋊(C3⋊D3))—21no—
52C4⋊(C3⋊(C3⋊D3))—21no—
53C2×C3×C3⋊Dic3—21no—
54C2×C3×C3⋊Dic3—21no—
55C3×#634—21no—
56C3×#634—21no—
57C2×C32×Dic3D411yescharacteristic
58C2×C32×Dic3D411yescharacteristic
59C32×D12—21no—
60C32×D12D412yes—
61C3×D3×Dic3—42no—
62C3×C3⋊(C3⋊C8)—21no—
63C32×C3⋊C8—21no—
64#5165—91no—
65#5016—91no—
66#4844—91no—
67C2×C32×D4—31no—
68C2×C4⋊(C3⋊D3)—31no—
69C2×C4⋊(C3⋊D3)—31no—
70C2×C4⋊(C3⋊D3)—31no—
71C2×C4⋊(C3⋊D3)—61no—
72C2×C4⋊(C3⋊D3)—61no—
73C2×C4⋊(C3⋊D3)—61no—
74C2×C4⋊(C3⋊D3)—121no—
75C2×#634C2×D311yescharacteristic
76C22×C3×Dic3—61no—
77C2×C3×D12—21no—
78C2×C3×D12C2×D311yescharacteristic
79C2×C3×D12C2×D311yescharacteristic
80C2×C3×C4×D3—61no—
81C2×C3×Dic6—31no—
82C2×C3×Dic6—31no—
83C2×C3×Dic6—61no—
84C2×D3×Dic3—21no—
85C2×C32×C8—31no—
86C32×C42—31no—
87#3753—62no—
88#3752—61no—
89#3752—61no—
90C4⋊(C3⋊Dic3)C2×D311yescharacteristic
91C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
92C3×#217—61no—
93C3×#214—62no—
94C3×(C2×D3)⋊C4—61no—
95C3×C4⋊Dic3—31no—
96C3×C4⋊Dic3—31no—
97C3×C4⋊Dic3—61no—
98C3×C4×Dic3—21no—
99C3×C4×Dic3C2×D311yescharacteristic
100C3×C4×Dic3C2×D311yescharacteristic
101C2×C3×C3⋊C8—21no—
102C2×C3×C3⋊C8—31no—
103C2×C3×C3⋊C8—31no—
104C2×C3×C3⋊C8—61no—
105C2×C3×C3⋊C8C2×D311yescharacteristic
106C2×C3×C3⋊C8C2×D311yescharacteristic
107C3×#204—62no—
108#3722—62no—
109#3722—62no—
110#3722—61no—
111#3722—61no—
112#3722—61no—
113#3722—61no—
114(C2×D3)⋊Dic3—21no—
115#3716—22no—
116#3714—64no—
117#3714—62no—
118#3714—62no—
119C22×C33C2×D411yescharacteristic
120C2×C3⋊(C3⋊D3)—41no—
121C2×C3⋊(C3⋊D3)—41no—
122C2×C3⋊(C3⋊D3)—41no—
123C2×C32×D3—22no—
124C2×C32×D3—41no—
125C2×C32×D3—41no—
126C33×C4C2×D411yescharacteristic
127C33×C4#4111yescharacteristic
128C3×C3⋊Dic3—41no—
129C3×C3⋊Dic3—41no—
130C32×Dic3—21no—
131C32×Dic3—21no—
132C2×C3×SD16—91no—
133C3×C4×D4—91no—
134#947—91no—
135D4×Dic3—31no—
136C2×#214—31no—
137C2×#213—91no—
138C2×#213—91no—
139C2×#213—181no—
140C2×D24—91no—
141C2×#204—31no—
142#875—91no—
143#875—91no—
144#875—181no—
145C4×D12—31no—
146C3×C4⋊C8—91no—
147#835—91no—
148#821—91no—
149#821—91no—
150#821—181no—
151#816—91no—
152Dic3⋊C8—31no—
153#810—91no—
154#809—91no—
155C4⋊(C3⋊C8)—31no—
156C23×C32—61no—
157C22×C3⋊D3—61no—
158C22×C3⋊D3—62no—
159C22×C3⋊D3—61no—
160C22×C3⋊D3—61no—
161C22×C3⋊D3—121no—
162C22×C3⋊D3—121no—
163C22×C3⋊D3—122no—
164C22×C3×D3—21no—
165C22×C3×D3—21no—
166C22×C3×D3—41no—
167C32×D4—32no—
168C32×D4—61no—
169C2×C32×C4—21no—
170C2×C32×C4—21no—
171C2×C32×C4—21no—
172C2×C32×C4—31no—
173C2×C32×C4—31no—
174C2×C32×C4—41no—
175C2×C32×C4C22×D311yescharacteristic
176C2×C32×C4C22×D311yescharacteristic
177C2×C32×C4C22×D311yescharacteristic
178C2×C3⋊Dic3—21no—
179C2×C3⋊Dic3—21no—
180C4⋊(C3⋊D3)—61no—
181C4⋊(C3⋊D3)—61no—
182C4⋊(C3⋊D3)—62no—
183C4⋊(C3⋊D3)—62no—
184C4⋊(C3⋊D3)—61no—
185C4⋊(C3⋊D3)—61no—
186C4⋊(C3⋊D3)—61no—
187C4⋊(C3⋊D3)—61no—
188C4⋊(C3⋊D3)—121no—
189C4⋊(C3⋊D3)—121no—
190C4⋊(C3⋊D3)—121no—
191C4⋊(C3⋊D3)—121no—
192C4⋊(C3⋊D3)—122no—
193C4⋊(C3⋊D3)—122no—
194#634—21no—
195#634—21no—
196C2×C3×Dic3—21no—
197C2×C3×Dic3—21no—
198C2×C3×Dic3—62no—
199C2×C3×Dic3—61no—
200C2×C3×Dic3—61no—
201C2×C3×Dic3—61no—
202C2×C3×Dic3—61no—
203C2×C3×Dic3—121no—
204C2×C3×Dic3—122no—
205C2×C3×Dic3D1211yescharacteristic
206C2×C3×Dic3D1211yescharacteristic
207C2×C3×Dic3#6711yescharacteristic
208C2×C3×Dic3#6711yescharacteristic
209C3×D12—21no—
210C3×D12—21no—
211C3×D12—22no—
212C3×D12—41no—
213C3×D12D1212yes—
214C3×D12#6712yes—
215C3×C4×D3—122no—
216C3×Dic6—61no—
217C3×Dic6—61no—
218C3×Dic6—61no—
219C3×Dic6—61no—
220C3×Dic6—121no—
221C3×Dic6—121no—
222D3×Dic3—42no—
223C32×C8—61no—
224C3⋊(C3⋊C8)—21no—
225C3×C3⋊C8—21no—
226C3×C3⋊C8—21no—
227C3×C3⋊C8—41no—
228C3×C3⋊C8—62no—
229C3×C3⋊C8—61no—
230C3×C3⋊C8—61no—
231#463—271noSylow
232C2×C33#12111yescharacteristic
233C2×C33#13511yescharacteristic
234C2×C33AGL(1,ℤ/8)11yescharacteristic
235C3⋊(C3⋊D3)—81no—
236C32×D3—42no—
237C2×C3×Q8—91no—
238C2×C3×D4—31no—
239C2×C3×D4—31no—
240C2×C3×D4—61no—
241C22×C3×C4—181no—
242C22×Dic3—61no—
243C2×D12—91no—
244C2×D12—91no—
245C2×D12—91no—
246C2×D12—181no—
247C2×D12—181no—
248C2×D12—181no—
249C2×D12—361no—
250C2×D12D3211yescharacteristic
251C2×C4×D3—61no—
252C2×Dic6—31no—
253C2×Dic6—31no—
254C2×Dic6—61no—
255C3×SD16—182no—
256C2×C3×C8—31no—
257C2×C3×C8—31no—
258C2×C3×C8—61no—
259C2×C3×C8—91no—
260C3×C4⋊C4—91no—
261C3×C22⋊C4—181no—
262C3×C42—31no—
263C3×C42—31no—
264C3×C42—61no—
265#217—61no—
266#214—62no—
267#213—182no—
268#213—182no—
269#213—184no—
270(C2×D3)⋊C4—61no—
271(C2×D3)⋊C4—181no—
272(C2×D3)⋊C4—181no—
273(C2×D3)⋊C4—181no—
274(C2×D3)⋊C4—181no—
275(C2×D3)⋊C4—182no—
276(C2×D3)⋊C4—182no—
277(C2×D3)⋊C4—181no—
278(C2×D3)⋊C4—181no—
279C4⋊Dic3—31no—
280C4⋊Dic3—31no—
281C4⋊Dic3—61no—
282C4×Dic3D3211yescharacteristic
283C2×C3⋊C8—31no—
284C2×C3⋊C8—31no—
285C2×C3⋊C8—61no—
286C2×C3⋊C8D3211yescharacteristic
287D24—182no—
288#204—62no—
289C22×C32—21no—
290C22×C32—21no—
291C22×C32—21no—
292C22×C32—41no—
293C22×C32—62no—
294C22×C32—121no—
295C22×C32—121no—
296C22×C32C2×D1211yescharacteristic
297C22×C32D3×D411yescharacteristic
298C22×C32C2×#6711yescharacteristic
299C2×C3⋊D3—122no—
300C2×C3⋊D3—122no—
301C2×C3⋊D3—122no—
302C2×C3⋊D3—124no—
303C2×C3⋊D3—121no—
304C2×C3⋊D3—121no—
305C2×C3⋊D3—121no—
306C2×C3⋊D3—121no—
307C2×C3⋊D3—121no—
308C2×C3⋊D3—122no—
309C2×C3⋊D3—122no—
310C2×C3⋊D3—121no—
311C2×C3⋊D3—121no—
312C2×C3⋊D3—121no—
313C2×C3⋊D3—121no—
314C2×C3⋊D3—241no—
315C2×C3⋊D3—241no—
316C2×C3⋊D3—241no—
317C2×C3⋊D3—241no—
318C2×C3⋊D3—242no—
319C2×C3⋊D3—242no—
320C2×C3×D3—22no—
321C2×C3×D3—22no—
322C2×C3×D3—42no—
323C2×C3×D3—41no—
324C2×C3×D3—41no—
325C2×C3×D3—41no—
326C2×C3×D3—41no—
327C2×C3×D3—81no—
328C2×C3×D3—81no—
329C32×C4—21no—
330C32×C4—21no—
331C32×C4—21no—
332C32×C4—21no—
333C32×C4—21no—
334C32×C4—21no—
335C32×C4—41no—
336C32×C4—41no—
337C32×C4—61no—
338C32×C4—61no—
339C32×C4C2×D1211yescharacteristic
340C32×C4#23511yescharacteristic
341C32×C4D3×D411yescharacteristic
342C32×C4#23711yescharacteristic
343C32×C4#23911yescharacteristic
344C32×C4C2×#6711yescharacteristic
345C3⋊Dic3—41no—
346C3⋊Dic3—41no—
347C3×Dic3—21no—
348C3×Dic3—21no—
349C3×Dic3—21no—
350C3×Dic3—21no—
351C3×Dic3—41no—
352C3×Dic3—41no—
353C3×Dic3—122no—
354C3×Dic3—121no—
355C3×Dic3—121no—
356C3×Dic3—121no—
357C3×Dic3—121no—
358C3×Dic3—241no—
359C2×SD16—91no—
360C2×D8—271no—
361#124—271no—
362C4×D4—91no—
363C4⋊C8—91no—
364#103—271no—
365#102—271no—
366C33#46311yescharacteristic, complemented, Sylow
367C23×C3—61no—
368C23×C3—61no—
369C23×C3—121no—
370C22×D3—21no—
371C22×D3—182no—
372C22×D3—181no—
373C22×D3—181no—
374C22×D3—181no—
375C22×D3—361no—
376C22×D3—361no—
377C22×D3—362no—
378C3×Q8—181no—
379C3×Q8—181no—
380C3×D4—32no—
381C3×D4—32no—
382C3×D4—62no—
383C3×D4—61no—
384C3×D4—61no—
385C3×D4—121no—
386C2×C3×C4—21no—
387C2×C3×C4—21no—
388C2×C3×C4—21no—
389C2×C3×C4—31no—
390C2×C3×C4—31no—
391C2×C3×C4—31no—
392C2×C3×C4—31no—
393C2×C3×C4—41no—
394C2×C3×C4—61no—
395C2×C3×C4—61no—
396C2×C3×C4—181no—
397C2×C3×C4—181no—
398C2×C3×C4—362no—
399C2×C3×C4C2×D3211yescharacteristic
400C2×C3×C4C2×D3211yescharacteristic
401C2×C3×C4C2×D3211yescharacteristic
402C2×Dic3—62no—
403C2×Dic3—61no—
404C2×Dic3—61no—
405C2×Dic3—61no—
406C2×Dic3—61no—
407C2×Dic3—121no—
408C2×Dic3—122no—
409C2×Dic3#62611yescharacteristic
410C2×Dic3#62611yescharacteristic
411D12—21no—
412D12—182no—
413D12—182no—
414D12—181no—
415D12—181no—
416D12—181no—
417D12—181no—
418D12—181no—
419D12—181no—
420D12—361no—
421D12—361no—
422D12—361no—
423D12—361no—
424D12—362no—
425D12—362no—
426D12#62612yes—
427C4×D3—122no—
428Dic6—61no—
429Dic6—61no—
430Dic6—61no—
431Dic6—61no—
432Dic6—121no—
433Dic6—121no—
434C3×C8—61no—
435C3×C8—61no—
436C3×C8—121no—
437C3×C8—181no—
438C3⋊C8—21no—
439C3⋊C8—62no—
440C3⋊C8—61no—
441C3⋊C8—61no—
442C2×C32—21no—
443C2×C32—21no—
444C2×C32—21no—
445C2×C32—21no—
446C2×C32—21no—
447C2×C32—21no—
448C2×C32—21no—
449C2×C32—21no—
450C2×C32—21no—
451C2×C32—41no—
452C2×C32—41no—
453C2×C32—41no—
454C2×C32—122no—
455C2×C32#89511yescharacteristic
456C2×C32#90411yescharacteristic
457C2×C32#90811yescharacteristic
458C2×C32#91411yescharacteristic
459C2×C32#91611yescharacteristic
460C2×C32#93011yescharacteristic
461C2×C32#93211yescharacteristic
462C2×C32#93911yescharacteristic
463C2×C32#95011yescharacteristic
464C3⋊D3—242no—
465C3⋊D3—242no—
466C3⋊D3—242no—
467C3⋊D3—244no—
468C3⋊D3—241no—
469C3⋊D3—241no—
470C3⋊D3—241no—
471C3×D3—42no—
472C3×D3—42no—
473C3×D3—82no—
474C2×Q8—91no—
475C2×D4—31no—
476C2×D4—271no—
477C22×C4—181no—
478SD16—182no—
479D8—542no—
480C2×C8—31no—
481C2×C8—91no—
482C4⋊C4—91no—
483C22⋊C4—181no—
484C22⋊C4—541no—
485C22⋊C4—541no—
486C42—31no—
487C22×C3—21no—
488C22×C3—21no—
489C22×C3—21no—
490C22×C3—41no—
491C22×C3—62no—
492C22×C3—62no—
493C22×C3—122no—
494C22×C3—121no—
495C22×C3—121no—
496C22×C3—121no—
497C22×C3—121no—
498C22×C3—241no—
499C22×C3—241no—
500C22×C3D3×D1211yescharacteristic
501C22×C3C2×#62611yescharacteristic
502C22×C3D3×#6711yescharacteristic
503C2×D3—22no—
504C2×D3—41no—
505C2×D3—41no—
506C2×D3—362no—
507C2×D3—362no—
508C2×D3—362no—
509C2×D3—362no—
510C2×D3—362no—
511C2×D3—364no—
512C2×D3—361no—
513C2×D3—361no—
514C2×D3—361no—
515C2×D3—361no—
516C2×D3—361no—
517C2×D3—361no—
518C2×D3—361no—
519C2×D3—361no—
520C2×D3—361no—
521C2×D3—721no—
522C2×D3—721no—
523C2×D3—721no—
524C2×D3—721no—
525C2×D3—722no—
526C2×D3—722no—
527C3×C4—21no—
528C3×C4—21no—
529C3×C4—21no—
530C3×C4—21no—
531C3×C4—21no—
532C3×C4—21no—
533C3×C4—41no—
534C3×C4—41no—
535C3×C4—61no—
536C3×C4—61no—
537C3×C4—61no—
538C3×C4—61no—
539C3×C4—121no—
540C3×C4—121no—
541C3×C4—361no—
542C3×C4—361no—
543C3×C4#379611yescharacteristic
544C3×C4#379911yescharacteristic
545C3×C4D3×D1211yescharacteristic
546C3×C4#380411yescharacteristic
547C3×C4C2×#62611yescharacteristic
548C3×C4D3×#6711yescharacteristic
549Dic3—21no—
550Dic3—21no—
551Dic3—122no—
552Dic3—121no—
553Dic3—121no—
554Dic3—121no—
555Dic3—121no—
556Dic3—241no—
557C32—21no—
558C32—21no—
559C32—21no—
560C32—41no—
561C32#484411yescharacteristic, complemented
562C32#501611yescharacteristic, complemented
563C32#516511yescharacteristic, complemented
564C23—61no—
565C23—541no—
566Q8—181no—
567Q8—181no—
568D4—32no—
569D4—61no—
570D4—541no—
571D4—541no—
572C2×C4—31no—
573C2×C4—31no—
574C2×C4—181no—
575C2×C4—181no—
576C2×C4—362no—
577C2×C4D3311yescharacteristic
578C8—61no—
579C8—181no—
580C2×C3—21no—
581C2×C3—21no—
582C2×C3—21no—
583C2×C3—21no—
584C2×C3—21no—
585C2×C3—21no—
586C2×C3—21no—
587C2×C3—21no—
588C2×C3—21no—
589C2×C3—41no—
590C2×C3—41no—
591C2×C3—41no—
592C2×C3—122no—
593C2×C3—122no—
594C2×C3—242no—
595C2×C3#6379411yescharacteristic
596C2×C3#6380611yescharacteristic
597C2×C3#6382911yescharacteristic
598C2×C3#6383011yescharacteristic
599C2×C3#6388611yescharacteristic
600C2×C3#6390911yescharacteristic
601C2×C3#6393411yescharacteristic
602C2×C3#6393511yescharacteristic
603C2×C3#6397811yescharacteristic
604D3—42no—
605D3—722no—
606D3—722no—
607D3—722no—
608D3—724no—
609D3—721no—
610D3—721no—
611D3—721no—
612C22—62no—
613C22—121no—
614C22—121no—
615C22—1081no—
616C22—1081no—
617C22—1081no—
618C22D3×#62611yescharacteristic
619C4—61no—
620C4—61no—
621C4—361no—
622C4—361no—
623C4D3×#62611yescharacteristic
624C4#8896211yescharacteristic
625C3—21no—
626C3—21no—
627C3—21no—
628C3—41no—
629C3#1059036611yescharacteristic, complemented
630C3#1059075411yescharacteristic, complemented
631C3#1059147111yescharacteristic, complemented
632C2—122no—
633C2—2161no—
634C2#1172285311yescharacteristic
635C2#1172286811yescharacteristic
636C2#1172295911yescharacteristic
subgroupG/H
centerC22D3×#626
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radicalC2×#88790C2
hypocenterC33#463
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. #11723441C2semidirect
3. C3×Dic3×D12C2semidirect
4. C2×C3×#3716C2semidirect
5. #11722076C2semidirect
6. #11722036C2semidirect
7. C3×Dic3⋊(C3⋊C8)C2semidirect
30. Dic3×D12D3semidirect
34. C2×#3716D3semidirect
41. Dic3⋊(C3⋊C8)D3semidirect
366. C33#463semidirect
561. C32#4844semidirect
562. C32#5016semidirect
563. C32#5165semidirect
629. C3#10590366semidirect
630. C3#10590754semidirect
631. C3#10591471semidirect

Maximal subgroups

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

subgroupindex
#117234412
C2×#887902
C3×Dic3×D122
C2×C3×#37162
#117220762
#117220362
C3×Dic3⋊(C3⋊C8)2
#105914713
#105907543
#105903663