Finite Group Databasebeta

← #13156032

Subgroups — #13156032

Order 1728.

674 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88789C211yesmaximal, characteristic
2C3×Dic3×D12C211yesmaximal, characteristic, complemented
3C3×#63864C211yesmaximal, characteristic, complemented
4C2×C3×#3715C211yesmaximal, characteristic, complemented
5#11722075C211yesmaximal, characteristic, complemented
6Dic3⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic, complemented
7#11722037C211yesmaximal, characteristic, complemented
8#10591567—31nomaximal
9#10591471—31nomaximal
10#10590348—31nomaximal
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C32×D12C2211yescharacteristic
13C2×C32×Dic6C2211yescharacteristic
14C2×C3×D3×Dic3—21no—
15C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18C32×C4×Dic3C2211yescharacteristic
19#88789—22no—
20C3×#3722—21no—
21C3×#3722—21no—
22C3×(C2×D3)⋊Dic3—21no—
23C3×#3715—22no—
24C2×#3753—31no—
25C3×#947—31no—
26C2×C3×#215—31no—
27C3×D4×Dic3—31no—
28C2×C3×#214—31no—
29C3×#875—31no—
30C3×C4×D12—31no—
31Dic3×D12D311yescharacteristic, complemented
32#63864D311yescharacteristic, complemented
33C2×#3715D311yescharacteristic, complemented
34C2×#3713—31no—
35C2×#3713—31no—
36C2×#3713—61no—
37#63662—31no—
38C4⋊(C3⋊(C3⋊C8))—31no—
39Dic3⋊(C3⋊C8)—31no—
40Dic3⋊(C3⋊C8)—31no—
41Dic3⋊(C3⋊C8)—61no—
42#63565—31no—
43#63565—31no—
44#63565—61no—
45#63564—31no—
46#63563—31no—
47#63562—31no—
48C22×C3×C3⋊D3—21no—
49C22×C32×D3—21no—
50C2×C33×C4C2311yescharacteristic
51C2×C3×C3⋊Dic3—21no—
52C3×C4⋊(C3⋊D3)—21no—
53C3×C4⋊(C3⋊D3)—21no—
54C2×C32×Dic3—21no—
55C2×C32×Dic3D411yescharacteristic
56C2×C32×Dic3D411yescharacteristic
57C32×D12—21no—
58C32×D12D412yes—
59C32×Dic6—21no—
60C32×Dic6—21no—
61C3×D3×Dic3—42no—
62C3⋊(C3⋊(C3⋊C8))—21no—
63C3×C3⋊(C3⋊C8)—21no—
64#5165—91no—
65#5057—91no—
66#5016—91no—
67C2×C32×Q8—31no—
68C2×C32×D4—31no—
69C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
70C22×C3×Dic3—61no—
71C2×C3×D12—21no—
72C2×C3×D12—31no—
73C2×C3×D12—31no—
74C2×C3×D12—61no—
75C2×C3×D12C2×D311yescharacteristic
76C2×C3×D12C2×D311yescharacteristic
77C2×C3×C4×D3—61no—
78C2×C3×Dic6—21no—
79C2×C3×Dic6C2×D311yescharacteristic
80C2×C3×Dic6C2×D311yescharacteristic
81C2×D3×Dic3—21no—
82C32×C42—31no—
83#3753—62no—
84C4⋊(C3⋊Dic3)C2×D311yescharacteristic
85C2×C3⋊(C3⋊C8)—31no—
86C2×C3⋊(C3⋊C8)—31no—
87C2×C3⋊(C3⋊C8)—31no—
88C2×C3⋊(C3⋊C8)—61no—
89C2×C3⋊(C3⋊C8)—61no—
90C2×C3⋊(C3⋊C8)—61no—
91C2×C3⋊(C3⋊C8)—121no—
92C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
93C3×#217—61no—
94C3×#215—62no—
95C3×#214—62no—
96C3×(C2×D3)⋊C4—61no—
97C3×(C2×D3)⋊C4—61no—
98C3×(C2×D3)⋊C4—61no—
99C3×(C2×D3)⋊C4—61no—
100C3×(C2×D3)⋊C4—61no—
101C3×C4⋊Dic3—31no—
102C3×C4⋊Dic3—31no—
103C3×C4⋊Dic3—61no—
104C3×C4×Dic3—21no—
105C3×C4×Dic3C2×D311yescharacteristic
106C3×C4×Dic3C2×D311yescharacteristic
107C2×C3×C3⋊C8—31no—
108C2×C3×C3⋊C8—31no—
109C2×C3×C3⋊C8—61no—
110#3722—21no—
111#3722—21no—
112(C2×D3)⋊Dic3—21no—
113#3715—22no—
114#3713—64no—
115#3713—62no—
116#3713—62no—
117C22×C33C2×D411yescharacteristic
118C2×C3×C3⋊D3—41no—
119C2×C3×C3⋊D3—41no—
120C2×C3×C3⋊D3—41no—
121C2×C32×D3—22no—
122C2×C32×D3—41no—
123C2×C32×D3—41no—
124C33×C4C2×D411yescharacteristic
125C33×C4#4111yescharacteristic
126C3×C3⋊Dic3—41no—
127C32×Dic3—21no—
128C32×Dic3—21no—
129C32×Dic3—41no—
130C2×C3×SD16—91no—
131C3×#124—91no—
132C3×C4×D4—91no—
133#947—31no—
134C2×#215—31no—
135D4×Dic3—31no—
136C2×#214—31no—
137C2×#213—91no—
138C2×#213—91no—
139C2×#213—91no—
140C2×#213—181no—
141#875—31no—
142C4×D12—31no—
143#835—91no—
144#835—91no—
145#835—181no—
146#832—91no—
147#821—91no—
148Dic3⋊C8—91no—
149#810—91no—
150#809—91no—
151C4⋊(C3⋊C8)—91no—
152C4⋊(C3⋊C8)—91no—
153C4⋊(C3⋊C8)—181no—
154C23×C32—61no—
155C22×C3⋊D3—21no—
156C22×C3×D3—21no—
157C22×C3×D3—21no—
158C22×C3×D3—41no—
159C22×C3×D3—61no—
160C22×C3×D3—61no—
161C22×C3×D3—121no—
162C32×Q8—61no—
163C32×Q8—61no—
164C32×D4—32no—
165C32×D4—61no—
166C2×C32×C4—21no—
167C2×C32×C4—21no—
168C2×C32×C4—21no—
169C2×C32×C4—31no—
170C2×C32×C4—31no—
171C2×C32×C4—41no—
172C2×C32×C4—61no—
173C2×C32×C4C22×D311yescharacteristic
174C2×C32×C4C22×D311yescharacteristic
175C2×C32×C4C22×D311yescharacteristic
176C2×C3⋊Dic3—21no—
177C4⋊(C3⋊D3)—21no—
178C4⋊(C3⋊D3)—21no—
179C2×C3×Dic3—21no—
180C2×C3×Dic3—21no—
181C2×C3×Dic3—21no—
182C2×C3×Dic3—21no—
183C2×C3×Dic3—41no—
184C2×C3×Dic3—61no—
185C2×C3×Dic3—61no—
186C2×C3×Dic3—121no—
187C2×C3×Dic3—122no—
188C2×C3×Dic3#6711yescharacteristic
189C2×C3×Dic3#6711yescharacteristic
190C2×C3×Dic3#6711yescharacteristic
191C2×C3×Dic3#6711yescharacteristic
192C3×D12—21no—
193C3×D12—21no—
194C3×D12—22no—
195C3×D12—41no—
196C3×D12—62no—
197C3×D12—61no—
198C3×D12—61no—
199C3×D12—61no—
200C3×D12—61no—
201C3×D12—121no—
202C3×D12D1212yes—
203C3×D12#6712yes—
204C3×C4×D3—122no—
205C3×Dic6—22no—
206C3×Dic6—21no—
207C3×Dic6—21no—
208C3×Dic6—21no—
209C3×Dic6—21no—
210C3×Dic6—41no—
211D3×Dic3—42no—
212C3⋊(C3⋊C8)—21no—
213C3⋊(C3⋊C8)—61no—
214C3⋊(C3⋊C8)—62no—
215C3⋊(C3⋊C8)—61no—
216C3⋊(C3⋊C8)—61no—
217C3⋊(C3⋊C8)—121no—
218C3⋊(C3⋊C8)—121no—
219C3⋊(C3⋊C8)—122no—
220C3×C3⋊C8—61no—
221C3×C3⋊C8—61no—
222C3×C3⋊C8—121no—
223#463—271noSylow
224C2×C33#12111yescharacteristic
225C2×C33#13511yescharacteristic
226C2×C33AGL(1,ℤ/8)11yescharacteristic
227C3×C3⋊D3—81no—
228C32×D3—42no—
229C2×C3×Q8—31no—
230C2×C3×Q8—31no—
231C2×C3×Q8—61no—
232C2×C3×D4—31no—
233C2×C3×D4—31no—
234C2×C3×D4—61no—
235C2×C3×D4—91no—
236C22×C3×C4—181no—
237C22×Dic3—61no—
238C2×D12—31no—
239C2×D12—31no—
240C2×D12—61no—
241C2×D12D3211yescharacteristic
242C2×C4×D3—61no—
243C2×Dic6D3211yescharacteristic
244C3×SD16—182no—
245C2×C3×C8—91no—
246C3×C4⋊C4—91no—
247C3×C22⋊C4—181no—
248C3×C22⋊C4—181no—
249C3×C22⋊C4—181no—
250C3×C42—31no—
251C3×C42—31no—
252C3×C42—61no—
253#217—61no—
254#215—62no—
255#214—62no—
256#213—182no—
257#213—182no—
258#213—184no—
259#213—182no—
260(C2×D3)⋊C4—61no—
261(C2×D3)⋊C4—61no—
262(C2×D3)⋊C4—61no—
263(C2×D3)⋊C4—61no—
264(C2×D3)⋊C4—61no—
265C4⋊Dic3—31no—
266C4⋊Dic3—31no—
267C4⋊Dic3—61no—
268C4×Dic3D3211yescharacteristic
269C2×C3⋊C8—31no—
270C2×C3⋊C8—31no—
271C2×C3⋊C8—61no—
272C2×C3⋊C8—91no—
273C2×C3⋊C8—91no—
274C2×C3⋊C8—91no—
275C2×C3⋊C8—181no—
276C2×C3⋊C8—181no—
277C2×C3⋊C8—181no—
278C2×C3⋊C8—361no—
279C22×C32—21no—
280C22×C32—21no—
281C22×C32—21no—
282C22×C32—41no—
283C22×C32—62no—
284C22×C32—121no—
285C22×C32—121no—
286C22×C32D3×D411yescharacteristic
287C22×C32C2×#6711yescharacteristic
288C22×C32C2×#6711yescharacteristic
289C2×C3⋊D3—41no—
290C2×C3⋊D3—41no—
291C2×C3⋊D3—41no—
292C2×C3×D3—22no—
293C2×C3×D3—22no—
294C2×C3×D3—42no—
295C2×C3×D3—41no—
296C2×C3×D3—41no—
297C2×C3×D3—41no—
298C2×C3×D3—41no—
299C2×C3×D3—81no—
300C2×C3×D3—81no—
301C2×C3×D3—122no—
302C2×C3×D3—121no—
303C2×C3×D3—121no—
304C2×C3×D3—121no—
305C2×C3×D3—121no—
306C2×C3×D3—121no—
307C2×C3×D3—121no—
308C2×C3×D3—241no—
309C2×C3×D3—241no—
310C32×C4—21no—
311C32×C4—21no—
312C32×C4—21no—
313C32×C4—21no—
314C32×C4—21no—
315C32×C4—21no—
316C32×C4—41no—
317C32×C4—41no—
318C32×C4—61no—
319C32×C4—61no—
320C32×C4—121no—
321C32×C4C2×D1211yescharacteristic
322C32×C4#23511yescharacteristic
323C32×C4D3×D411yescharacteristic
324C32×C4#23711yescharacteristic
325C32×C4#23711yescharacteristic
326C32×C4C2×#6711yescharacteristic
327C3⋊Dic3—41no—
328C3×Dic3—22no—
329C3×Dic3—21no—
330C3×Dic3—21no—
331C3×Dic3—21no—
332C3×Dic3—21no—
333C3×Dic3—41no—
334C3×Dic3—42no—
335C3×Dic3—41no—
336C3×Dic3—41no—
337C3×Dic3—121no—
338C3×Dic3—121no—
339C3×Dic3—241no—
340C2×SD16—91no—
341C2×D8—271no—
342#124—91no—
343C4×D4—91no—
344C4⋊C8—271no—
345#103—271no—
346#102—271no—
347C33#46311yescharacteristic, complemented, Sylow
348C23×C3—61no—
349C23×C3—61no—
350C23×C3—121no—
351C23×C3—181no—
352C22×D3—21no—
353C22×D3—61no—
354C22×D3—61no—
355C22×D3—121no—
356C3×Q8—62no—
357C3×Q8—61no—
358C3×Q8—61no—
359C3×Q8—61no—
360C3×Q8—61no—
361C3×Q8—121no—
362C3×D4—32no—
363C3×D4—32no—
364C3×D4—62no—
365C3×D4—61no—
366C3×D4—61no—
367C3×D4—121no—
368C3×D4—181no—
369C3×D4—181no—
370C2×C3×C4—21no—
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—31no—
374C2×C3×C4—31no—
375C2×C3×C4—31no—
376C2×C3×C4—31no—
377C2×C3×C4—41no—
378C2×C3×C4—61no—
379C2×C3×C4—61no—
380C2×C3×C4—61no—
381C2×C3×C4—61no—
382C2×C3×C4—121no—
383C2×C3×C4—181no—
384C2×C3×C4—362no—
385C2×C3×C4C2×D3211yescharacteristic
386C2×C3×C4C2×D3211yescharacteristic
387C2×C3×C4C2×D3211yescharacteristic
388C2×Dic3—21no—
389C2×Dic3—61no—
390C2×Dic3—61no—
391C2×Dic3—121no—
392C2×Dic3—122no—
393C2×Dic3#62511yescharacteristic
394C2×Dic3#62511yescharacteristic
395D12—21no—
396D12—62no—
397D12—61no—
398D12—61no—
399D12—61no—
400D12—61no—
401D12—121no—
402D12#62612yes—
403C4×D3—122no—
404Dic6—21no—
405Dic6—21no—
406C3×C8—181no—
407C3⋊C8—61no—
408C3⋊C8—61no—
409C3⋊C8—121no—
410C3⋊C8—182no—
411C3⋊C8—181no—
412C3⋊C8—181no—
413C3⋊C8—181no—
414C3⋊C8—361no—
415C3⋊C8—361no—
416C3⋊C8—362no—
417C2×C32—21no—
418C2×C32—21no—
419C2×C32—21no—
420C2×C32—21no—
421C2×C32—21no—
422C2×C32—21no—
423C2×C32—21no—
424C2×C32—21no—
425C2×C32—21no—
426C2×C32—41no—
427C2×C32—41no—
428C2×C32—41no—
429C2×C32—122no—
430C2×C32#91411yescharacteristic
431C2×C32#91611yescharacteristic
432C2×C32#93011yescharacteristic
433C2×C32#93211yescharacteristic
434C2×C32#93811yescharacteristic
435C2×C32#93911yescharacteristic
436C2×C32#94911yescharacteristic
437C2×C32#95011yescharacteristic
438C2×C32#95011yescharacteristic
439C3⋊D3—81no—
440C3×D3—42no—
441C3×D3—42no—
442C3×D3—82no—
443C3×D3—242no—
444C3×D3—241no—
445C3×D3—241no—
446C2×Q8—31no—
447C2×D4—31no—
448C2×D4—91no—
449C22×C4—181no—
450SD16—182no—
451D8—542no—
452C2×C8—91no—
453C2×C8—271no—
454C4⋊C4—91no—
455C22⋊C4—181no—
456C22⋊C4—181no—
457C22⋊C4—181no—
458C42—31no—
459C22×C3—21no—
460C22×C3—21no—
461C22×C3—21no—
462C22×C3—41no—
463C22×C3—62no—
464C22×C3—62no—
465C22×C3—122no—
466C22×C3—121no—
467C22×C3—121no—
468C22×C3—121no—
469C22×C3—121no—
470C22×C3—241no—
471C22×C3—241no—
472C22×C3—361no—
473C22×C3—361no—
474C22×C3—361no—
475C22×C3C2×#62511yescharacteristic
476C22×C3D3×#6711yescharacteristic
477C22×C3D3×#6711yescharacteristic
478C2×D3—22no—
479C2×D3—41no—
480C2×D3—41no—
481C2×D3—122no—
482C2×D3—121no—
483C2×D3—121no—
484C2×D3—121no—
485C2×D3—121no—
486C2×D3—121no—
487C2×D3—121no—
488C2×D3—241no—
489C2×D3—241no—
490C3×C4—21no—
491C3×C4—21no—
492C3×C4—21no—
493C3×C4—21no—
494C3×C4—21no—
495C3×C4—21no—
496C3×C4—41no—
497C3×C4—41no—
498C3×C4—62no—
499C3×C4—61no—
500C3×C4—61no—
501C3×C4—61no—
502C3×C4—61no—
503C3×C4—121no—
504C3×C4—122no—
505C3×C4—121no—
506C3×C4—121no—
507C3×C4—361no—
508C3×C4#379511yescharacteristic
509C3×C4D3×D1211yescharacteristic
510C3×C4#380411yescharacteristic
511C3×C4#380511yescharacteristic
512C3×C4C2×#62611yescharacteristic
513C3×C4D3×#6711yescharacteristic
514Dic3—21no—
515Dic3—21no—
516Dic3—41no—
517Dic3—121no—
518Dic3—121no—
519Dic3—241no—
520C32—21no—
521C32—21no—
522C32—21no—
523C32—41no—
524C32#501611yescharacteristic, complemented
525C32#505711yescharacteristic, complemented
526C32#516511yescharacteristic, complemented
527C23—61no—
528C23—181no—
529Q8—61no—
530Q8—61no—
531D4—32no—
532D4—61no—
533D4—181no—
534D4—181no—
535C2×C4—31no—
536C2×C4—31no—
537C2×C4—61no—
538C2×C4—181no—
539C2×C4—362no—
540C2×C4D3311yescharacteristic
541C8—181no—
542C8—541no—
543C2×C3—21no—
544C2×C3—21no—
545C2×C3—21no—
546C2×C3—21no—
547C2×C3—21no—
548C2×C3—21no—
549C2×C3—21no—
550C2×C3—21no—
551C2×C3—21no—
552C2×C3—41no—
553C2×C3—41no—
554C2×C3—41no—
555C2×C3—122no—
556C2×C3—122no—
557C2×C3—242no—
558C2×C3—721no—
559C2×C3#6382211yescharacteristic
560C2×C3#6382311yescharacteristic
561C2×C3#6390811yescharacteristic
562C2×C3#6393411yescharacteristic
563C2×C3#6393511yescharacteristic
564C2×C3#6393911yescharacteristic
565C2×C3#6394711yescharacteristic
566C2×C3#6397011yescharacteristic
567C2×C3#6397811yescharacteristic
568D3—42no—
569D3—242no—
570D3—241no—
571D3—241no—
572C22—62no—
573C22—121no—
574C22—121no—
575C22—361no—
576C22—361no—
577C22—361no—
578C22D3×#62511yescharacteristic
579C4—61no—
580C4—61no—
581C4—121no—
582C4—361no—
583C4D3×#62611yescharacteristic
584C4#8896111yescharacteristic
585C3—21no—
586C3—21no—
587C3—21no—
588C3—41no—
589C3#1059034811yescharacteristic, complemented
590C3#1059147111yescharacteristic, complemented
591C3#1059156711yescharacteristic, complemented
592C2—122no—
593C2—721no—
594C2#1172285111yescharacteristic
595C2#1172286711yescharacteristic
596C2#1172295611yescharacteristic
subgroupG/H
centerC22D3×#625
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radicalC2×#88789C2
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
2. C3×Dic3×D12C2semidirect
3. C3×#63864C2semidirect
4. C2×C3×#3715C2semidirect
5. #11722075C2semidirect
6. Dic3⋊(C3⋊(C3⋊C8))C2semidirect
7. #11722037C2semidirect
31. Dic3×D12D3semidirect
32. #63864D3semidirect
33. C2×#3715D3semidirect
347. C33#463semidirect
524. C32#5016semidirect
525. C32#5057semidirect
526. C32#5165semidirect
589. C3#10590348semidirect
590. C3#10591471semidirect
591. C3#10591567semidirect

Maximal subgroups

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

subgroupindex
C2×#887892
C3×Dic3×D122
C3×#638642
C2×C3×#37152
#117220752
Dic3⋊(C3⋊(C3⋊C8))2
#117220372
#105915673
#105914713
#105903483