Finite Group Databasebeta

← #13155721

Subgroups — #13155721

Order 1728.

802 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723484C211yesmaximal, characteristic, complemented
2C2×#88791C211yesmaximal, characteristic
3C3×#63894C211yesmaximal, characteristic, complemented
4#11722075C211yesmaximal, characteristic, complemented
5#11722035C211yesmaximal, characteristic, complemented
6C3×#63573C211yesmaximal, characteristic, complemented
7C3×#63554C211yesmaximal, characteristic, complemented
8#10590768—31nomaximal
9#10590589—31nomaximal
10#10590346—31nomaximal
11C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
12C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C2×#6289—21no—
14C2×C3×#624—21no—
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C32×C4⋊Dic3C2211yescharacteristic
18C32×C4×Dic3C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20#88804—21no—
21#88791C414yescomplemented
22C3×#3722—21no—
23C2×#3745—31no—
24C4×C4⋊(C3⋊D3)—31no—
25C3×#893—31no—
26C3×C4×D12—31no—
27#63894—31no—
28#63894—31no—
29#63894—61no—
30#63894D311yescharacteristic, complemented
31C2×#3714—31no—
32C2×#3714—31no—
33C2×#3714—61no—
34#63636—31no—
35C3×#817—31no—
36C3×#815—31no—
37C3×#808—31no—
38C3×#803—31no—
39#63573D311yescharacteristic, complemented
40#63564—31no—
41#63564—31no—
42#63564—31no—
43#63564—61no—
44#63563—31no—
45#63562—31no—
46#63554D311yescharacteristic, complemented
47C22×C3⋊(C3⋊D3)—21no—
48C22×C3×C3⋊D3—21no—
49C2×C33×C4C2311yescharacteristic
50C4⋊(C3⋊(C3⋊D3))—21no—
51C4⋊(C3⋊(C3⋊D3))C2×C412yescomplemented
52C2×C3×C3⋊Dic3—21no—
53C3×C4⋊(C3⋊D3)—21no—
54C3×C4⋊(C3⋊D3)C2×C412yescomplemented
55C2×C32×Dic3—21no—
56C2×C32×Dic3D411yescharacteristic
57C2×C32×Dic3D411yescharacteristic
58#6289—42no—
59C3×#624—42no—
60C3×C3⋊(C3⋊C8)—21no—
61C32×C3⋊C8C2×C412yescomplemented
62#5014—91no—
63#4869—91no—
64#4711—91no—
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)C2×D311yescharacteristic
73C2×C4×C3⋊D3—61no—
74C2×C3×D12—31no—
75C2×C3×D12—31no—
76C2×C3×D12—61no—
77C2×C3×C4×D3—61no—
78C2×C3×C4×D3—61no—
79C2×#624—21no—
80C2×#624—62no—
81C2×#624—61no—
82C2×#624—61no—
83C2×C32×C8—31no—
84C32×C4⋊C4—31no—
85C32×C42—31no—
86#3752—61no—
87C4⋊(C3⋊Dic3)C2×D311yescharacteristic
88C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
89#3745—34no—
90C3×(C2×D3)⋊C4—61no—
91C3×(C2×D3)⋊C4—61no—
92C3×C4⋊Dic3—21no—
93C3×C4⋊Dic3—31no—
94C3×C4⋊Dic3—31no—
95C3×C4⋊Dic3—61no—
96C3×C4⋊Dic3C2×D311yescharacteristic
97C3×C4⋊Dic3C2×D311yescharacteristic
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
107#3722—21no—
108#3722—62no—
109#3722—61no—
110#3722—61no—
111#3714—34no—
112#3714—34no—
113#3714—64no—
114C22×C33C2×D411yescharacteristic
115C2×C3⋊(C3⋊D3)—22no—
116C2×C3⋊(C3⋊D3)—41no—
117C2×C3⋊(C3⋊D3)—41no—
118C2×C3×C3⋊D3—22no—
119C2×C3×C3⋊D3—41no—
120C2×C3×C3⋊D3—41no—
121C33×C4C22×C411yescharacteristic
122C33×C4#4111yescharacteristic
123C3×C3⋊Dic3—41no—
124C32×Dic3—21no—
125C32×Dic3—21no—
126C32×Dic3—41no—
127C3×C4×D4—91no—
128C2×#213—91no—
129C2×D24—91no—
130C2×D24—91no—
131C2×D24—181no—
132#893—31no—
133#893—91no—
134C4×D12—31no—
135C4×D12—91no—
136C4×D12—91no—
137C4×D12—181no—
138C3×#106—91no—
139C3×#97—91no—
140#832—91no—
141#821—91no—
142#821—91no—
143#817—31no—
144#815—31no—
145#809—91no—
146#809—91no—
147#809—91no—
148#809—181no—
149#808—31no—
150#803—31no—
151C22×C3⋊D3—21no—
152C22×C3⋊D3—61no—
153C22×C3⋊D3—62no—
154C22×C3⋊D3—61no—
155C22×C3⋊D3—61no—
156C22×C3⋊D3—121no—
157C22×C3⋊D3—121no—
158C22×C3⋊D3—122no—
159C22×C3×D3—61no—
160C22×C3×D3—61no—
161C22×C3×D3—121no—
162C2×C32×C4—21no—
163C2×C32×C4—21no—
164C2×C32×C4—21no—
165C2×C32×C4—31no—
166C2×C32×C4—31no—
167C2×C32×C4—41no—
168C2×C32×C4—61no—
169C2×C32×C4C22×D311yescharacteristic
170C2×C32×C4C22×D311yescharacteristic
171C2×C32×C4C22×D311yescharacteristic
172C2×C3⋊Dic3—21no—
173C4⋊(C3⋊D3)—21no—
174C4⋊(C3⋊D3)—32no—
175C4⋊(C3⋊D3)—32no—
176C4⋊(C3⋊D3)—32no—
177C4⋊(C3⋊D3)—62no—
178C4⋊(C3⋊D3)—62no—
179C4⋊(C3⋊D3)—62no—
180C4⋊(C3⋊D3)—61no—
181C4⋊(C3⋊D3)—62no—
182C4⋊(C3⋊D3)—61no—
183C4⋊(C3⋊D3)—61no—
184C4⋊(C3⋊D3)—122no—
185C4⋊(C3⋊D3)—121no—
186C4⋊(C3⋊D3)—121no—
187C4⋊(C3⋊D3)—122no—
188C4⋊(C3⋊D3)C4×D312yescomplemented
189C4×C3⋊D3—122no—
190C2×C3×Dic3—21no—
191C2×C3×Dic3—21no—
192C2×C3×Dic3—21no—
193C2×C3×Dic3—21no—
194C2×C3×Dic3—41no—
195C2×C3×Dic3—62no—
196C2×C3×Dic3—61no—
197C2×C3×Dic3—61no—
198C2×C3×Dic3D1211yescharacteristic
199C2×C3×Dic3D1211yescharacteristic
200C2×C3×Dic3#6711yescharacteristic
201C2×C3×Dic3#6711yescharacteristic
202C3×D12—32no—
203C3×D12—32no—
204C3×D12—62no—
205C3×D12—61no—
206C3×D12—61no—
207C3×D12—121no—
208C3×C4×D3—122no—
209C3×C4×D3—122no—
210#624—42no—
211#624—124no—
212#624—122no—
213#624—122no—
214C32×C8—32no—
215C3⋊(C3⋊C8)—21no—
216C3×C3⋊C8—22no—
217C3×C3⋊C8—61no—
218C3×C3⋊C8—61no—
219C3×C3⋊C8—121no—
220C3×C3⋊C8C4×D312yescomplemented
221C3×C3⋊C8C4×D312yescomplemented
222#442—271noSylow
223C2×C33C4×D411yescharacteristic
224C2×C33AGL(1,ℤ/8)11yescharacteristic
225C2×C33AGL(1,ℤ/8)11yescharacteristic
226C3⋊(C3⋊D3)—42no—
227C3×C3⋊D3—42no—
228C2×C3×D4—91no—
229C22×C3×C4—181no—
230C2×D12—31no—
231C2×D12—31no—
232C2×D12—61no—
233C2×D12—91no—
234C2×D12—91no—
235C2×D12—91no—
236C2×D12—181no—
237C2×D12—181no—
238C2×D12—181no—
239C2×D12—361no—
240C2×C4×D3—61no—
241C2×C4×D3—61no—
242C2×C4×D3—181no—
243C2×C4×D3—181no—
244C2×C4×D3—182no—
245C2×C4×D3—181no—
246C2×C3×C8—31no—
247C2×C3×C8—31no—
248C2×C3×C8—61no—
249C2×C3×C8—91no—
250C3×C4⋊C4—31no—
251C3×C4⋊C4—31no—
252C3×C4⋊C4—61no—
253C3×C4⋊C4—91no—
254C3×C22⋊C4—181no—
255C3×C42—31no—
256C3×C42—31no—
257C3×C42—61no—
258#213—94no—
259(C2×D3)⋊C4—61no—
260(C2×D3)⋊C4—61no—
261(C2×D3)⋊C4—181no—
262(C2×D3)⋊C4—181no—
263(C2×D3)⋊C4—182no—
264(C2×D3)⋊C4—181no—
265C4⋊Dic3—31no—
266C4⋊Dic3—31no—
267C4⋊Dic3—61no—
268C4⋊Dic3D3211yescharacteristic
269C4×Dic3D3211yescharacteristic
270C2×C3⋊C8—31no—
271C2×C3⋊C8—31no—
272C2×C3⋊C8—61no—
273C2×C3⋊C8D3211yescharacteristic
274D24—94no—
275D24—94no—
276D24—184no—
277C22×C32—21no—
278C22×C32—21no—
279C22×C32—21no—
280C22×C32—41no—
281C22×C32C2×D1211yescharacteristic
282C22×C32D3×D411yescharacteristic
283C22×C32C2×#6711yescharacteristic
284C2×C3⋊D3—22no—
285C2×C3⋊D3—41no—
286C2×C3⋊D3—41no—
287C2×C3⋊D3—62no—
288C2×C3⋊D3—64no—
289C2×C3⋊D3—62no—
290C2×C3⋊D3—62no—
291C2×C3⋊D3—122no—
292C2×C3⋊D3—122no—
293C2×C3⋊D3—124no—
294C2×C3⋊D3—121no—
295C2×C3⋊D3—121no—
296C2×C3⋊D3—122no—
297C2×C3⋊D3—122no—
298C2×C3⋊D3—121no—
299C2×C3⋊D3—121no—
300C2×C3⋊D3—121no—
301C2×C3⋊D3—121no—
302C2×C3⋊D3—241no—
303C2×C3⋊D3—241no—
304C2×C3⋊D3—241no—
305C2×C3⋊D3—241no—
306C2×C3⋊D3—242no—
307C2×C3⋊D3—242no—
308C2×C3×D3—62no—
309C2×C3×D3—62no—
310C2×C3×D3—122no—
311C2×C3×D3—121no—
312C2×C3×D3—121no—
313C2×C3×D3—121no—
314C2×C3×D3—121no—
315C2×C3×D3—241no—
316C2×C3×D3—241no—
317C32×C4—21no—
318C32×C4—21no—
319C32×C4—21no—
320C32×C4—21no—
321C32×C4—21no—
322C32×C4—21no—
323C32×C4—41no—
324C32×C4—41no—
325C32×C4—61no—
326C32×C4—61no—
327C32×C4—121no—
328C32×C4C2×C4×D311yescharacteristic
329C32×C4C2×C4×D311yescharacteristic
330C32×C4C2×C4×D311yescharacteristic
331C32×C4#23511yescharacteristic
332C32×C4#23511yescharacteristic
333C32×C4#23911yescharacteristic
334C3⋊Dic3—41no—
335C3×Dic3—22no—
336C3×Dic3—21no—
337C3×Dic3—21no—
338C3×Dic3—21no—
339C3×Dic3—21no—
340C3×Dic3—41no—
341C3×Dic3—41no—
342C3×Dic3—41no—
343C3×Dic3—81no—
344C3×Dic3—122no—
345C3×Dic3—121no—
346C3×Dic3—121no—
347C2×D8—271no—
348C4×D4—91no—
349C4×D4—271no—
350#106—91no—
351#102—271no—
352#102—271no—
353#97—91no—
354C33#44211yescharacteristic, complemented, Sylow
355C23×C3—181no—
356C22×D3—61no—
357C22×D3—61no—
358C22×D3—121no—
359C22×D3—182no—
360C22×D3—181no—
361C22×D3—181no—
362C22×D3—181no—
363C22×D3—361no—
364C22×D3—361no—
365C22×D3—362no—
366C3×D4—92no—
367C3×D4—181no—
368C2×C3×C4—21no—
369C2×C3×C4—21no—
370C2×C3×C4—21no—
371C2×C3×C4—31no—
372C2×C3×C4—31no—
373C2×C3×C4—31no—
374C2×C3×C4—31no—
375C2×C3×C4—41no—
376C2×C3×C4—61no—
377C2×C3×C4—61no—
378C2×C3×C4—61no—
379C2×C3×C4—61no—
380C2×C3×C4—121no—
381C2×C3×C4—181no—
382C2×C3×C4—362no—
383C2×C3×C4C2×D3211yescharacteristic
384C2×C3×C4C2×D3211yescharacteristic
385C2×C3×C4C2×D3211yescharacteristic
386C2×Dic3—21no—
387C2×Dic3—62no—
388C2×Dic3—61no—
389C2×Dic3—61no—
390C2×Dic3#62611yescharacteristic
391C2×Dic3#62611yescharacteristic
392D12—32no—
393D12—32no—
394D12—62no—
395D12—61no—
396D12—61no—
397D12—92no—
398D12—92no—
399D12—92no—
400D12—121no—
401D12—182no—
402D12—182no—
403D12—182no—
404D12—182no—
405D12—181no—
406D12—181no—
407D12—181no—
408D12—362no—
409D12—361no—
410D12—361no—
411D12—362no—
412C4×D3—122no—
413C4×D3—122no—
414C4×D3—362no—
415C4×D3—364no—
416C4×D3—362no—
417C4×D3—362no—
418C3×C8—32no—
419C3×C8—32no—
420C3×C8—62no—
421C3×C8—181no—
422C3⋊C8—61no—
423C3⋊C8—61no—
424C3⋊C8—121no—
425C3⋊C8#62412yescomplemented
426C2×C32—21no—
427C2×C32—21no—
428C2×C32—21no—
429C2×C32—21no—
430C2×C32—21no—
431C2×C32—21no—
432C2×C32—21no—
433C2×C32—21no—
434C2×C32—21no—
435C2×C32—41no—
436C2×C32—41no—
437C2×C32—41no—
438C2×C32C4×D1211yescharacteristic
439C2×C32#89311yescharacteristic
440C2×C32#90811yescharacteristic
441C2×C32#90811yescharacteristic
442C2×C32#91411yescharacteristic
443C2×C32#91411yescharacteristic
444C2×C32C4×#6711yescharacteristic
445C2×C32#93211yescharacteristic
446C2×C32#94911yescharacteristic
447C3⋊D3—42no—
448C3⋊D3—122no—
449C3⋊D3—124no—
450C3⋊D3—122no—
451C3⋊D3—122no—
452C3⋊D3—242no—
453C3⋊D3—242no—
454C3⋊D3—244no—
455C3×D3—122no—
456C3×D3—122no—
457C3×D3—242no—
458C2×D4—91no—
459C2×D4—271no—
460C22×C4—181no—
461C22×C4—541no—
462D8—274no—
463C2×C8—31no—
464C2×C8—91no—
465C4⋊C4—31no—
466C4⋊C4—91no—
467C22⋊C4—181no—
468C22⋊C4—541no—
469C42—31no—
470C22×C3—21no—
471C22×C3—21no—
472C22×C3—21no—
473C22×C3—41no—
474C22×C3—182no—
475C22×C3—361no—
476C22×C3—361no—
477C22×C3D3×D1211yescharacteristic
478C22×C3C2×#62611yescharacteristic
479C22×C3D3×#6711yescharacteristic
480C2×D3—62no—
481C2×D3—62no—
482C2×D3—122no—
483C2×D3—121no—
484C2×D3—121no—
485C2×D3—121no—
486C2×D3—121no—
487C2×D3—184no—
488C2×D3—182no—
489C2×D3—182no—
490C2×D3—182no—
491C2×D3—241no—
492C2×D3—241no—
493C2×D3—362no—
494C2×D3—362no—
495C2×D3—364no—
496C2×D3—362no—
497C2×D3—362no—
498C2×D3—361no—
499C2×D3—361no—
500C2×D3—361no—
501C2×D3—361no—
502C2×D3—361no—
503C2×D3—361no—
504C2×D3—721no—
505C2×D3—721no—
506C2×D3—721no—
507C2×D3—721no—
508C2×D3—722no—
509C2×D3—722no—
510C3×C4—21no—
511C3×C4—21no—
512C3×C4—21no—
513C3×C4—21no—
514C3×C4—21no—
515C3×C4—21no—
516C3×C4—41no—
517C3×C4—41no—
518C3×C4—62no—
519C3×C4—61no—
520C3×C4—61no—
521C3×C4—61no—
522C3×C4—61no—
523C3×C4—121no—
524C3×C4—121no—
525C3×C4—121no—
526C3×C4—241no—
527C3×C4—361no—
528C3×C4#379811yescharacteristic
529C3×C4#379911yescharacteristic
530C3×C4#379911yescharacteristic
531C3×C4C4×D3211yescharacteristic
532C3×C4C4×D3211yescharacteristic
533C3×C4C2×#62411yescharacteristic
534Dic3—21no—
535Dic3—21no—
536Dic3—41no—
537Dic3—122no—
538Dic3—121no—
539Dic3—121no—
540C32—21no—
541C32—21no—
542C32—21no—
543C32—41no—
544C32#471111yescharacteristic, complemented
545C32#486911yescharacteristic, complemented
546C32#501411yescharacteristic, complemented
547C23—181no—
548C23—541no—
549D4—92no—
550D4—181no—
551D4—272no—
552D4—541no—
553C2×C4—31no—
554C2×C4—31no—
555C2×C4—61no—
556C2×C4—181no—
557C2×C4—362no—
558C2×C4—1082no—
559C2×C4D3311yescharacteristic
560C8—32no—
561C8—181no—
562C2×C3—21no—
563C2×C3—21no—
564C2×C3—21no—
565C2×C3—21no—
566C2×C3—21no—
567C2×C3—21no—
568C2×C3—21no—
569C2×C3—21no—
570C2×C3—21no—
571C2×C3—41no—
572C2×C3—41no—
573C2×C3—41no—
574C2×C3—362no—
575C2×C3#6379411yescharacteristic
576C2×C3#6379411yescharacteristic
577C2×C3#6382511yescharacteristic
578C2×C3#6383011yescharacteristic
579C2×C3#6389411yescharacteristic
580C2×C3C4×#62611yescharacteristic
581C2×C3#6390411yescharacteristic
582C2×C3#6393411yescharacteristic
583C2×C3#6393911yescharacteristic
584D3—122no—
585D3—122no—
586D3—242no—
587D3—364no—
588D3—362no—
589D3—362no—
590D3—362no—
591D3—722no—
592D3—722no—
593D3—724no—
594C22—182no—
595C22—361no—
596C22—361no—
597C22—542no—
598C22—1081no—
599C22—1081no—
600C22D3×#62611yescharacteristic
601C4—61no—
602C4—61no—
603C4—121no—
604C4—361no—
605C4D3×#62411yescharacteristic
606C4#8896511yescharacteristic
607C3—21no—
608C3—21no—
609C3—21no—
610C3—41no—
611C3#1059034611yescharacteristic, complemented
612C3#1059058911yescharacteristic, complemented
613C3#1059076811yescharacteristic, complemented
614C2—362no—
615C2—1082no—
616C2#1172286811yescharacteristic
617C2#1172287111yescharacteristic
618C2#1172293311yescharacteristic
subgroupG/H
centerC22D3×#626
commutatorC33×C4C22×C4
Frattini, hypercenterC2×C4D33
Fitting, cosocleC2×C33×C4C23
socleC22×C33C2×D4
prime radicalC2×#88791C2
hypocenterC33#442
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. #11723484C2semidirect
3. C3×#63894C2semidirect
4. #11722075C2semidirect
5. #11722035C2semidirect
6. C3×#63573C2semidirect
7. C3×#63554C2semidirect
21. #88791C4semidirect
30. #63894D3semidirect
39. #63573D3semidirect
46. #63554D3semidirect
51. C4⋊(C3⋊(C3⋊D3))C2×C4semidirect
54. C3×C4⋊(C3⋊D3)C2×C4semidirect
61. C32×C3⋊C8C2×C4semidirect
188. C4⋊(C3⋊D3)C4×D3semidirect
220. C3×C3⋊C8C4×D3semidirect
221. C3×C3⋊C8C4×D3semidirect
354. C33#442semidirect
425. C3⋊C8#624semidirect
544. C32#4711semidirect
545. C32#4869semidirect
546. C32#5014semidirect
611. C3#10590346semidirect
612. C3#10590589semidirect
613. C3#10590768semidirect

Maximal subgroups

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

subgroupindex
#117234842
C2×#887912
C3×#638942
#117220752
#117220352
C3×#635732
C3×#635542
#105907683
#105905893
#105903463