Finite Group Databasebeta

← #13155641

Subgroups — #13155641

Order 1728.

754 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723484C211yesmaximal, characteristic
2C2×#88795C211yesmaximal, characteristic
3C3×Dic3×Dic6C211yesmaximal, characteristic, complemented
4#11722079C211yesmaximal, characteristic, complemented
5#11722036C211yesmaximal, characteristic
6C3×#63566C211yesmaximal, characteristic, complemented
7C3×Dic3×C3⋊C8C211yesmaximal, characteristic, complemented
8#10591536—31nomaximal
9#10590792—31nomaximal
10C4×#3717—31nomaximal
11C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
12C2×C32×Dic6C2211yescharacteristic
13C2×#6289—21no—
14C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C32×C4×Dic3C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19#88804—21no—
20#88795C414yescomplemented
21C3×Dic3⋊Dic3—21no—
22C3×Dic32—21no—
23C2×#3755—31no—
24C4×C4⋊(C3⋊D3)—31no—
25C3×Q8×Dic3—31no—
26C3×C4×Dic6—31no—
27#63894—31no—
28#63894—31no—
29#63894—61no—
30Dic3×Dic6D311yescharacteristic, complemented
31C2×#3717—31no—
32C2×#3717—31no—
33C2×#3717—61no—
34#63648—31no—
35C3×#835—31no—
36C3×#816—31no—
37C3×C8×Dic3—31no—
38C3×C4×C3⋊C8—31no—
39#63574—31no—
40#63573—31no—
41#63573—31no—
42#63566D311yescharacteristic, complemented
43#63564—31no—
44#63564—31no—
45#63564—61no—
46Dic3×C3⋊C8D311yescharacteristic, complemented
47C22×C3⋊(C3⋊D3)—21no—
48C2×C33×C4C2311yescharacteristic
49C4⋊(C3⋊(C3⋊D3))—21no—
50C4⋊(C3⋊(C3⋊D3))C2×C412yes—
51C2×C3×C3⋊Dic3—21no—
52C2×C3×C3⋊Dic3—21no—
53C2×C32×Dic3—21no—
54C2×C32×Dic3D411yescharacteristic
55C2×C32×Dic3D411yescharacteristic
56C32×Dic6—21no—
57C32×Dic6C2×C412yescomplemented
58#6289—42no—
59C3×C3⋊(C3⋊C8)C2×C412yescomplemented
60C32×C3⋊C8—21no—
61C4×#215—91no—
62#4788—91no—
63C4×#204—91no—
64C2×C32×Q8—31no—
65C2×C4⋊(C3⋊D3)—31no—
66C2×C4⋊(C3⋊D3)—31no—
67C2×C4⋊(C3⋊D3)—31no—
68C2×C4⋊(C3⋊D3)—61no—
69C2×C4⋊(C3⋊D3)—61no—
70C2×C4⋊(C3⋊D3)—61no—
71C2×C4⋊(C3⋊D3)—121no—
72C2×C4×C3⋊D3—61no—
73C2×C3×Dic6—21no—
74C2×C3×Dic6C2×D311yescharacteristic
75C2×C3×Dic6C2×D311yescharacteristic
76C2×#624—62no—
77C2×#624—61no—
78C2×#624—61no—
79C2×C32×C8—31no—
80C32×C42—31no—
81#3755—34no—
82#3752—61no—
83C4⋊(C3⋊Dic3)C2×D311yescharacteristic
84C4⋊(C3⋊Dic3)C2×D311yescharacteristic
85C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
86C3×C4⋊Dic3—31no—
87C3×C4⋊Dic3—31no—
88C3×C4⋊Dic3—31no—
89C3×C4⋊Dic3—31no—
90C3×C4⋊Dic3—61no—
91C3×C4⋊Dic3—61no—
92C3×C4⋊Dic3—61no—
93C3×Dic3⋊C4—61no—
94C3×C4×Dic3—21no—
95C3×C4×Dic3—61no—
96C3×C4×Dic3—61no—
97C3×C4×Dic3C2×D311yescharacteristic
98C3×C4×Dic3C2×D311yescharacteristic
99C2×C3×C3⋊C8—21no—
100C2×C3×C3⋊C8—31no—
101C2×C3×C3⋊C8—31no—
102C2×C3×C3⋊C8—61no—
103C2×C3×C3⋊C8C2×D311yescharacteristic
104C2×C3×C3⋊C8C2×D311yescharacteristic
105Dic3⋊Dic3—21no—
106#3722—62no—
107#3722—61no—
108#3722—61no—
109Dic32—21no—
110#3717—34no—
111#3717—34no—
112#3717—64no—
113C22×C33C2×D411yescharacteristic
114C2×C3⋊(C3⋊D3)—22no—
115C2×C3⋊(C3⋊D3)—41no—
116C2×C3⋊(C3⋊D3)—41no—
117C33×C4C22×C411yescharacteristic
118C33×C4#4111yescharacteristic
119C3×C3⋊Dic3—41no—
120C3×C3⋊Dic3—41no—
121C32×Dic3—22no—
122C32×Dic3—21no—
123C32×Dic3SD1612yescomplemented
124C3×C4×Q8—91no—
125Q8×Dic3—31no—
126C2×#215—91no—
127C2×#215—91no—
128C2×#215—181no—
129C2×#204—91no—
130#893—91no—
131C4×D12—91no—
132C4×D12—91no—
133C4×D12—181no—
134C4×Dic6—31no—
135C3×#103—91no—
136C3×C4×C8—91no—
137#835—31no—
138#821—91no—
139#821—91no—
140#821—181no—
141#817—91no—
142#816—31no—
143C8×Dic3—31no—
144#809—91no—
145#808—91no—
146#808—91no—
147C4×C3⋊C8—31no—
148C22×C3⋊D3—61no—
149C22×C3⋊D3—62no—
150C22×C3⋊D3—61no—
151C22×C3⋊D3—61no—
152C22×C3⋊D3—121no—
153C22×C3⋊D3—121no—
154C22×C3⋊D3—122no—
155C32×Q8—32no—
156C32×Q8—61no—
157C2×C32×C4—21no—
158C2×C32×C4—21no—
159C2×C32×C4—21no—
160C2×C32×C4—31no—
161C2×C32×C4—31no—
162C2×C32×C4—41no—
163C2×C32×C4—61no—
164C2×C32×C4C22×D311yescharacteristic
165C2×C32×C4C22×D311yescharacteristic
166C2×C32×C4C22×D311yescharacteristic
167C2×C3⋊Dic3—21no—
168C2×C3⋊Dic3—21no—
169C4⋊(C3⋊D3)—32no—
170C4⋊(C3⋊D3)—32no—
171C4⋊(C3⋊D3)—32no—
172C4⋊(C3⋊D3)—62no—
173C4⋊(C3⋊D3)—62no—
174C4⋊(C3⋊D3)—62no—
175C4⋊(C3⋊D3)—61no—
176C4⋊(C3⋊D3)—62no—
177C4⋊(C3⋊D3)—61no—
178C4⋊(C3⋊D3)—61no—
179C4⋊(C3⋊D3)—122no—
180C4⋊(C3⋊D3)—121no—
181C4⋊(C3⋊D3)—121no—
182C4⋊(C3⋊D3)—122no—
183C4×C3⋊D3—122no—
184C2×C3×Dic3—21no—
185C2×C3×Dic3—21no—
186C2×C3×Dic3—21no—
187C2×C3×Dic3—21no—
188C2×C3×Dic3—41no—
189C2×C3×Dic3—62no—
190C2×C3×Dic3—61no—
191C2×C3×Dic3—61no—
192C2×C3×Dic3—61no—
193C2×C3×Dic3—61no—
194C2×C3×Dic3—121no—
195C2×C3×Dic3D1211yescharacteristic
196C2×C3×Dic3D1211yescharacteristic
197C2×C3×Dic3#6711yescharacteristic
198C2×C3×Dic3#6711yescharacteristic
199C3×Dic6—21no—
200C3×Dic6—21no—
201C3×Dic6—22no—
202C3×Dic6—41no—
203C3×Dic6C4×D312yescomplemented
204C3×Dic6C4×D312yescomplemented
205#624—124no—
206#624—122no—
207#624—122no—
208C32×C8—61no—
209C3⋊(C3⋊C8)C4×D312yescomplemented
210C3×C3⋊C8—21no—
211C3×C3⋊C8—21no—
212C3×C3⋊C8—32no—
213C3×C3⋊C8—32no—
214C3×C3⋊C8—41no—
215C3×C3⋊C8—62no—
216C4×SD16—271noSylow
217C2×C33C4×D411yescharacteristic
218C2×C33C2×SD1611yescharacteristic
219C2×C33#13511yescharacteristic
220C3⋊(C3⋊D3)—42no—
221C2×C3×Q8—31no—
222C2×C3×Q8—31no—
223C2×C3×Q8—61no—
224C2×D12—91no—
225C2×D12—91no—
226C2×D12—91no—
227C2×D12—181no—
228C2×D12—181no—
229C2×D12—181no—
230C2×D12—361no—
231C2×C4×D3—181no—
232C2×C4×D3—181no—
233C2×C4×D3—182no—
234C2×C4×D3—181no—
235C2×Dic6D3211yescharacteristic
236C2×C3×C8—31no—
237C2×C3×C8—31no—
238C2×C3×C8—61no—
239C2×C3×C8—91no—
240C3×C4⋊C4—91no—
241C3×C4⋊C4—91no—
242C3×C4⋊C4—181no—
243C3×C42—31no—
244C3×C42—31no—
245C3×C42—61no—
246C3×C42—181no—
247#215—94no—
248#215—94no—
249#215—184no—
250(C2×D3)⋊C4—181no—
251(C2×D3)⋊C4—181no—
252(C2×D3)⋊C4—182no—
253(C2×D3)⋊C4—181no—
254C4⋊Dic3—31no—
255C4⋊Dic3—31no—
256C4⋊Dic3—31no—
257C4⋊Dic3—31no—
258C4⋊Dic3—61no—
259C4⋊Dic3—61no—
260C4⋊Dic3—61no—
261Dic3⋊C4—61no—
262C4×Dic3—61no—
263C4×Dic3—61no—
264C4×Dic3D3211yescharacteristic
265C2×C3⋊C8—31no—
266C2×C3⋊C8—31no—
267C2×C3⋊C8—61no—
268C2×C3⋊C8D3211yescharacteristic
269#204—94no—
270C22×C32—21no—
271C22×C32—21no—
272C22×C32—21no—
273C22×C32—41no—
274C22×C32C2×D1211yescharacteristic
275C22×C32D3×D411yescharacteristic
276C22×C32C2×#6711yescharacteristic
277C2×C3⋊D3—62no—
278C2×C3⋊D3—64no—
279C2×C3⋊D3—62no—
280C2×C3⋊D3—62no—
281C2×C3⋊D3—122no—
282C2×C3⋊D3—122no—
283C2×C3⋊D3—124no—
284C2×C3⋊D3—121no—
285C2×C3⋊D3—121no—
286C2×C3⋊D3—122no—
287C2×C3⋊D3—122no—
288C2×C3⋊D3—121no—
289C2×C3⋊D3—121no—
290C2×C3⋊D3—121no—
291C2×C3⋊D3—121no—
292C2×C3⋊D3—241no—
293C2×C3⋊D3—241no—
294C2×C3⋊D3—241no—
295C2×C3⋊D3—241no—
296C2×C3⋊D3—242no—
297C2×C3⋊D3—242no—
298C32×C4—21no—
299C32×C4—21no—
300C32×C4—21no—
301C32×C4—21no—
302C32×C4—21no—
303C32×C4—21no—
304C32×C4—32no—
305C32×C4—41no—
306C32×C4—41no—
307C32×C4—62no—
308C32×C4—61no—
309C32×C4C2×C4×D311yescharacteristic
310C32×C4C2×C4×D311yescharacteristic
311C32×C4C2×C4×D311yescharacteristic
312C32×C4#23511yescharacteristic
313C32×C4#23511yescharacteristic
314C32×C4#23711yescharacteristic
315C3⋊Dic3—41no—
316C3⋊Dic3—41no—
317C3×Dic3—22no—
318C3×Dic3—21no—
319C3×Dic3—22no—
320C3×Dic3—21no—
321C3×Dic3—22no—
322C3×Dic3—42no—
323C3×Dic3—41no—
324C3×Dic3—122no—
325C3×Dic3—121no—
326C3×Dic3—121no—
327C3×Dic3—121no—
328C3×Dic3—121no—
329C3×Dic3—241no—
330C3×Dic3#20412yescomplemented
331C3×Dic3#21512yescomplemented
332C2×SD16—271no—
333C4×Q8—91no—
334C4×D4—271no—
335#106—271no—
336#103—91no—
337#102—271no—
338C4×C8—91no—
339C33C4×SD1611yescharacteristic, complemented, Sylow
340C22×D3—182no—
341C22×D3—181no—
342C22×D3—181no—
343C22×D3—181no—
344C22×D3—361no—
345C22×D3—361no—
346C22×D3—362no—
347C3×Q8—32no—
348C3×Q8—32no—
349C3×Q8—62no—
350C3×Q8—61no—
351C3×Q8—61no—
352C3×Q8—121no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4—21no—
356C2×C3×C4—31no—
357C2×C3×C4—31no—
358C2×C3×C4—31no—
359C2×C3×C4—31no—
360C2×C3×C4—41no—
361C2×C3×C4—61no—
362C2×C3×C4—61no—
363C2×C3×C4—61no—
364C2×C3×C4—61no—
365C2×C3×C4—121no—
366C2×C3×C4—181no—
367C2×C3×C4—181no—
368C2×C3×C4C2×D3211yescharacteristic
369C2×C3×C4C2×D3211yescharacteristic
370C2×C3×C4C2×D3211yescharacteristic
371C2×Dic3—21no—
372C2×Dic3—62no—
373C2×Dic3—61no—
374C2×Dic3—61no—
375C2×Dic3—61no—
376C2×Dic3—61no—
377C2×Dic3—121no—
378C2×Dic3#62611yescharacteristic
379C2×Dic3#62611yescharacteristic
380D12—92no—
381D12—92no—
382D12—92no—
383D12—182no—
384D12—182no—
385D12—182no—
386D12—182no—
387D12—181no—
388D12—181no—
389D12—181no—
390D12—362no—
391D12—361no—
392D12—361no—
393D12—362no—
394C4×D3—362no—
395C4×D3—362no—
396C4×D3—364no—
397C4×D3—362no—
398Dic6—21no—
399Dic6#62412yescomplemented
400C3×C8—61no—
401C3×C8—61no—
402C3×C8—92no—
403C3×C8—121no—
404C3⋊C8—21no—
405C3⋊C8—32no—
406C3⋊C8—32no—
407C3⋊C8—62no—
408C2×C32—21no—
409C2×C32—21no—
410C2×C32—21no—
411C2×C32—21no—
412C2×C32—21no—
413C2×C32—21no—
414C2×C32—21no—
415C2×C32—21no—
416C2×C32—21no—
417C2×C32—41no—
418C2×C32—41no—
419C2×C32—41no—
420C2×C32C4×D1211yescharacteristic
421C2×C32#67⋊C411yescharacteristic
422C2×C32C2×#20411yescharacteristic
423C2×C32#90411yescharacteristic
424C2×C32D3×SD1611yescharacteristic
425C2×C32#91911yescharacteristic
426C2×C32C4×#6711yescharacteristic
427C2×C32C2×#21511yescharacteristic
428C2×C32#95011yescharacteristic
429C3⋊D3—122no—
430C3⋊D3—124no—
431C3⋊D3—122no—
432C3⋊D3—122no—
433C3⋊D3—242no—
434C3⋊D3—242no—
435C3⋊D3—244no—
436C2×Q8—31no—
437C2×D4—271no—
438C22×C4—541no—
439SD16—274no—
440C2×C8—31no—
441C2×C8—91no—
442C4⋊C4—91no—
443C4⋊C4—91no—
444C4⋊C4—181no—
445C22⋊C4—541no—
446C42—31no—
447C42—181no—
448C22×C3—21no—
449C22×C3—21no—
450C22×C3—21no—
451C22×C3—41no—
452C22×C3D3×D1211yescharacteristic
453C22×C3C2×#62611yescharacteristic
454C22×C3D3×#6711yescharacteristic
455C2×D3—184no—
456C2×D3—182no—
457C2×D3—182no—
458C2×D3—182no—
459C2×D3—362no—
460C2×D3—362no—
461C2×D3—364no—
462C2×D3—362no—
463C2×D3—362no—
464C2×D3—361no—
465C2×D3—361no—
466C2×D3—361no—
467C2×D3—361no—
468C2×D3—361no—
469C2×D3—361no—
470C2×D3—721no—
471C2×D3—721no—
472C2×D3—721no—
473C2×D3—721no—
474C2×D3—722no—
475C2×D3—722no—
476C3×C4—21no—
477C3×C4—21no—
478C3×C4—21no—
479C3×C4—21no—
480C3×C4—21no—
481C3×C4—21no—
482C3×C4—32no—
483C3×C4—32no—
484C3×C4—41no—
485C3×C4—41no—
486C3×C4—62no—
487C3×C4—61no—
488C3×C4—62no—
489C3×C4—62no—
490C3×C4—61no—
491C3×C4—121no—
492C3×C4—122no—
493C3×C4—361no—
494C3×C4—361no—
495C3×C4#379811yescharacteristic
496C3×C4C4×D3211yescharacteristic
497C3×C4C4×D3211yescharacteristic
498C3×C4#380411yescharacteristic
499C3×C4#380411yescharacteristic
500C3×C4C2×#62411yescharacteristic
501Dic3—21no—
502Dic3—22no—
503Dic3—122no—
504Dic3—121no—
505Dic3—121no—
506Dic3—121no—
507Dic3—121no—
508Dic3—241no—
509Dic3#371712yescomplemented
510C32—21no—
511C32—21no—
512C32—21no—
513C32—41no—
514C32C4×#20411yescharacteristic, complemented
515C32#478811yescharacteristic, complemented
516C32C4×#21511yescharacteristic, complemented
517C23—541no—
518Q8—32no—
519Q8—61no—
520D4—272no—
521D4—541no—
522C2×C4—31no—
523C2×C4—31no—
524C2×C4—61no—
525C2×C4—181no—
526C2×C4—181no—
527C2×C4—1082no—
528C2×C4D3311yescharacteristic
529C8—61no—
530C8—92no—
531C2×C3—21no—
532C2×C3—21no—
533C2×C3—21no—
534C2×C3—21no—
535C2×C3—21no—
536C2×C3—21no—
537C2×C3—21no—
538C2×C3—21no—
539C2×C3—21no—
540C2×C3—41no—
541C2×C3—41no—
542C2×C3—41no—
543C2×C3D3×#20411yescharacteristic
544C2×C3#6380811yescharacteristic
545C2×C3#6382911yescharacteristic
546C2×C3C2×#371711yescharacteristic
547C2×C3#6388011yescharacteristic
548C2×C3C4×#62611yescharacteristic
549C2×C3D3×#21511yescharacteristic
550C2×C3#6394911yescharacteristic
551C2×C3#6395211yescharacteristic
552D3—364no—
553D3—362no—
554D3—362no—
555D3—362no—
556D3—722no—
557D3—722no—
558D3—724no—
559C22—542no—
560C22—1081no—
561C22—1081no—
562C22D3×#62611yescharacteristic
563C4—32no—
564C4—61no—
565C4—62no—
566C4—361no—
567C4—361no—
568C4D3×#62411yescharacteristic
569C4#8895911yescharacteristic
570C3—21no—
571C3—21no—
572C3—21no—
573C3—41no—
574C3C4×#371711yescharacteristic, complemented
575C3#1059079211yescharacteristic, complemented
576C3#1059153611yescharacteristic, complemented
577C2—1082no—
578C2D3×#371711yescharacteristic
579C2#1172285611yescharacteristic
580C2#1172293211yescharacteristic
subgroupG/H
centerC22D3×#626
commutatorC33×C4C22×C4
Frattini, hypercenterC2×C4D33
Fitting, cosocleC2×C33×C4C23
socleC22×C33C2×D4
prime radicalC2×C4⋊(C3⋊(C3⋊D3))C22
hypocenterC33C4×SD16
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
3. C3×Dic3×Dic6C2semidirect
4. #11722079C2semidirect
6. C3×#63566C2semidirect
7. C3×Dic3×C3⋊C8C2semidirect
20. #88795C4semidirect
30. Dic3×Dic6D3semidirect
42. #63566D3semidirect
46. Dic3×C3⋊C8D3semidirect
57. C32×Dic6C2×C4semidirect
59. C3×C3⋊(C3⋊C8)C2×C4semidirect
123. C32×Dic3SD16semidirect
203. C3×Dic6C4×D3semidirect
204. C3×Dic6C4×D3semidirect
209. C3⋊(C3⋊C8)C4×D3semidirect
330. C3×Dic3#204semidirect
331. C3×Dic3#215semidirect
339. C33C4×SD16semidirect
399. Dic6#624semidirect
509. Dic3#3717semidirect
514. C32C4×#204semidirect
515. C32#4788semidirect
516. C32C4×#215semidirect
574. C3C4×#3717semidirect
575. C3#10590792semidirect
576. C3#10591536semidirect

Maximal subgroups

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

subgroupindex
#117234842
C2×#887952
C3×Dic3×Dic62
#117220792
#117220362
C3×#635662
C3×Dic3×C3⋊C82
#105915363
#105907923
C4×#37173