Finite Group Databasebeta

← #13156560

Subgroups — #13156560

Order 1728.

702 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723517C211yesmaximal, characteristic
2C3×#63869C211yesmaximal, characteristic, complemented
3#11722075C211yesmaximal, characteristic, complemented
4#11722036C211yesmaximal, characteristic, complemented
5#11722035C211yesmaximal, characteristic, complemented
6C3×#63564C211yesmaximal, characteristic, complemented
7C3×#63554C211yesmaximal, characteristic, complemented
8#10590882—31nomaximal
9#10590550—31nomaximal
10#10590282—31nomaximal
11C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
12C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C2×#6292—21no—
14C2×#6292—21no—
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C32×C4⋊Dic3C2211yescharacteristic
18C32×C4×Dic3C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20C3×#3724—21no—
21C3×Dic3⋊Dic3—21no—
22#64083—31no—
23C3×#890—31no—
24C3×#870—31no—
25#63911—31no—
26#63911—31no—
27#63911—61no—
28#63869D311yescharacteristic, complemented
29#63648—31no—
30#63636—31no—
31C3×#821—31no—
32C3×#815—31no—
33C3×#809—31no—
34C3×#803—31no—
35#63564—31no—
36#63564—31no—
37#63564—31no—
38#63564—31no—
39#63564—31no—
40#63564—61no—
41#63564—61no—
42#63564D311yescharacteristic, complemented
43#63563—31no—
44#63562—31no—
45#63554D311yescharacteristic, complemented
46C22×C3⋊(C3⋊D3)—21no—
47C22×C3×C3⋊D3—21no—
48C2×C33×C4C2311yescharacteristic
49C4⋊(C3⋊(C3⋊D3))—21no—
50C4⋊(C3⋊(C3⋊D3))—21no—
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
57#6292—42no—
58#6292—42no—
59C3×C3⋊(C3⋊C8)—21no—
60C32×C3⋊C8—21no—
61#5067—91no—
62#4764—91no—
63#4713—91no—
64C2×C4⋊(C3⋊D3)—31no—
65C2×C4⋊(C3⋊D3)—31no—
66C2×C4⋊(C3⋊D3)—31no—
67C2×C4⋊(C3⋊D3)—61no—
68C2×C4⋊(C3⋊D3)—61no—
69C2×C4⋊(C3⋊D3)—61no—
70C2×C4⋊(C3⋊D3)—61no—
71C2×C4⋊(C3⋊D3)—61no—
72C2×C4⋊(C3⋊D3)—121no—
73C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
74C2×C3×D12—31no—
75C2×C3×D12—31no—
76C2×C3×D12—61no—
77C2×#626—62no—
78C2×#626—62no—
79C2×#626—61no—
80C2×#626—61no—
81C2×#626—61no—
82C2×#626—61no—
83C2×C32×C8—31no—
84C32×C4⋊C4—31no—
85C32×C42—31no—
86C4⋊(C3⋊Dic3)C2×D311yescharacteristic
87C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
88C3×C4⋊Dic3—21no—
89C3×C4⋊Dic3—31no—
90C3×C4⋊Dic3—31no—
91C3×C4⋊Dic3—61no—
92C3×C4⋊Dic3—61no—
93C3×C4⋊Dic3C2×D311yescharacteristic
94C3×C4⋊Dic3C2×D311yescharacteristic
95C3×Dic3⋊C4—61no—
96C3×Dic3⋊C4—61no—
97C3×Dic3⋊C4—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
107#3724—21no—
108Dic3⋊Dic3—21no—
109C22×C33C2×D411yescharacteristic
110C2×C3⋊(C3⋊D3)—41no—
111C2×C3⋊(C3⋊D3)—41no—
112C2×C3⋊(C3⋊D3)—41no—
113C2×C3×C3⋊D3—41no—
114C2×C3×C3⋊D3—41no—
115C2×C3×C3⋊D3—41no—
116C33×C4#4111yescharacteristic
117C33×C4#4111yescharacteristic
118C3×C3⋊Dic3—41no—
119C32×Dic3—21no—
120C32×Dic3—21no—
121C32×Dic3—41no—
122C3×#125—91no—
123#940—91no—
124#890—31no—
125#874—91no—
126#874—91no—
127#874—181no—
128#870—31no—
129C3×#102—91no—
130C3×#97—91no—
131#832—91no—
132#821—31no—
133#821—91no—
134#821—91no—
135#821—91no—
136#821—91no—
137#821—181no—
138#815—31no—
139#809—31no—
140#809—91no—
141#809—91no—
142#809—91no—
143#809—91no—
144#809—181no—
145#803—31no—
146C22×C3⋊D3—21no—
147C22×C3⋊D3—61no—
148C22×C3⋊D3—62no—
149C22×C3⋊D3—61no—
150C22×C3⋊D3—61no—
151C22×C3⋊D3—121no—
152C22×C3⋊D3—121no—
153C22×C3⋊D3—122no—
154C22×C3×D3—62no—
155C22×C3×D3—61no—
156C22×C3×D3—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—
168C4⋊(C3⋊D3)—21no—
169C4⋊(C3⋊D3)—21no—
170C4⋊(C3⋊D3)—61no—
171C4⋊(C3⋊D3)—61no—
172C4⋊(C3⋊D3)—62no—
173C4⋊(C3⋊D3)—62no—
174C4⋊(C3⋊D3)—61no—
175C4⋊(C3⋊D3)—61no—
176C4⋊(C3⋊D3)—61no—
177C4⋊(C3⋊D3)—61no—
178C4⋊(C3⋊D3)—121no—
179C4⋊(C3⋊D3)—121no—
180C4⋊(C3⋊D3)—121no—
181C4⋊(C3⋊D3)—121no—
182C4⋊(C3⋊D3)—122no—
183C4⋊(C3⋊D3)—122no—
184C4⋊(C3⋊D3)—122no—
185C4⋊(C3⋊D3)—122no—
186C2×C3×Dic3—21no—
187C2×C3×Dic3—21no—
188C2×C3×Dic3—21no—
189C2×C3×Dic3—21no—
190C2×C3×Dic3—41no—
191C2×C3×Dic3—61no—
192C2×C3×Dic3—61no—
193C2×C3×Dic3—121no—
194C2×C3×Dic3D1211yescharacteristic
195C2×C3×Dic3D1211yescharacteristic
196C2×C3×Dic3#6711yescharacteristic
197C2×C3×Dic3#6711yescharacteristic
198C3×D12—61no—
199C3×D12—61no—
200C3×D12—61no—
201C3×D12—61no—
202C3×D12—121no—
203C3×D12—121no—
204#626—124no—
205#626—124no—
206#626—122no—
207#626—122no—
208#626—122no—
209#626—122no—
210C32×C8—61no—
211C3⋊(C3⋊C8)—21no—
212C3×C3⋊C8—21no—
213C3×C3⋊C8—21no—
214C3×C3⋊C8—41no—
215C3×C3⋊C8—61no—
216C3×C3⋊C8—61no—
217C3×C3⋊C8—121no—
218#490—271noSylow
219C2×C33#12411yescharacteristic
220C2×C33AGL(1,ℤ/8)11yescharacteristic
221C2×C33AGL(1,ℤ/8)11yescharacteristic
222C3⋊(C3⋊D3)—81no—
223C3×C3⋊D3—81no—
224C2×C3×D4—91no—
225C2×#67—181no—
226C2×#67—181no—
227C2×D12—31no—
228C2×D12—31no—
229C2×D12—61no—
230C2×D12—91no—
231C2×D12—91no—
232C2×D12—91no—
233C2×D12—181no—
234C2×D12—181no—
235C2×D12—181no—
236C2×D12—181no—
237C2×D12—181no—
238C2×D12—181no—
239C2×D12—181no—
240C2×D12—182no—
241C2×D12—182no—
242C2×D12—361no—
243C2×C3×C8—31no—
244C2×C3×C8—31no—
245C2×C3×C8—61no—
246C2×C3×C8—91no—
247C3×C4⋊C4—31no—
248C3×C4⋊C4—31no—
249C3×C4⋊C4—61no—
250C3×C4⋊C4—91no—
251C3×C4⋊C4—181no—
252C3×C4⋊C4—181no—
253C3×C42—31no—
254C3×C42—31no—
255C3×C42—61no—
256C4⋊Dic3—31no—
257C4⋊Dic3—31no—
258C4⋊Dic3—61no—
259C4⋊Dic3—61no—
260C4⋊Dic3D3211yescharacteristic
261Dic3⋊C4—61no—
262Dic3⋊C4—61no—
263Dic3⋊C4—61no—
264C4×Dic3D3211yescharacteristic
265C2×C3⋊C8—31no—
266C2×C3⋊C8—31no—
267C2×C3⋊C8—61no—
268C2×C3⋊C8D3211yescharacteristic
269C22×C32—21no—
270C22×C32—21no—
271C22×C32—21no—
272C22×C32—41no—
273C22×C32C2×D1211yescharacteristic
274C22×C32D3×D411yescharacteristic
275C22×C32C2×#6711yescharacteristic
276C2×C3⋊D3—41no—
277C2×C3⋊D3—41no—
278C2×C3⋊D3—41no—
279C2×C3⋊D3—122no—
280C2×C3⋊D3—122no—
281C2×C3⋊D3—122no—
282C2×C3⋊D3—124no—
283C2×C3⋊D3—121no—
284C2×C3⋊D3—121no—
285C2×C3⋊D3—121no—
286C2×C3⋊D3—121no—
287C2×C3⋊D3—121no—
288C2×C3⋊D3—122no—
289C2×C3⋊D3—122no—
290C2×C3⋊D3—121no—
291C2×C3⋊D3—121no—
292C2×C3⋊D3—121no—
293C2×C3⋊D3—121no—
294C2×C3⋊D3—241no—
295C2×C3⋊D3—241no—
296C2×C3⋊D3—241no—
297C2×C3⋊D3—241no—
298C2×C3⋊D3—242no—
299C2×C3⋊D3—242no—
300C2×C3×D3—122no—
301C2×C3×D3—122no—
302C2×C3×D3—122no—
303C2×C3×D3—121no—
304C2×C3×D3—121no—
305C2×C3×D3—121no—
306C2×C3×D3—121no—
307C2×C3×D3—121no—
308C2×C3×D3—121no—
309C32×C4—21no—
310C32×C4—21no—
311C32×C4—21no—
312C32×C4—21no—
313C32×C4—21no—
314C32×C4—21no—
315C32×C4—41no—
316C32×C4—41no—
317C32×C4—61no—
318C32×C4—61no—
319C32×C4—121no—
320C32×C4#23511yescharacteristic
321C32×C4#23511yescharacteristic
322C32×C4#23511yescharacteristic
323C32×C4#23711yescharacteristic
324C32×C4#23911yescharacteristic
325C32×C4#23911yescharacteristic
326C3⋊Dic3—41no—
327C3×Dic3—22no—
328C3×Dic3—22no—
329C3×Dic3—21no—
330C3×Dic3—21no—
331C3×Dic3—21no—
332C3×Dic3—21no—
333C3×Dic3—41no—
334C3×Dic3—41no—
335C3×Dic3—81no—
336C3×Dic3—121no—
337C3×Dic3—121no—
338C3×Dic3—241no—
339#127—271no—
340#125—91no—
341#102—91no—
342#102—271no—
343#102—271no—
344#102—271no—
345#97—91no—
346C33#49011yescharacteristic, complemented, Sylow
347C23×C3—181no—
348C22×D3—62no—
349C22×D3—61no—
350C22×D3—61no—
351C22×D3—182no—
352C22×D3—181no—
353C22×D3—181no—
354C22×D3—181no—
355C22×D3—361no—
356C22×D3—361no—
357C22×D3—362no—
358C3×D4—181no—
359C3×D4—181no—
360C2×C3×C4—21no—
361C2×C3×C4—21no—
362C2×C3×C4—21no—
363C2×C3×C4—31no—
364C2×C3×C4—31no—
365C2×C3×C4—31no—
366C2×C3×C4—31no—
367C2×C3×C4—41no—
368C2×C3×C4—61no—
369C2×C3×C4—61no—
370C2×C3×C4—61no—
371C2×C3×C4—61no—
372C2×C3×C4—121no—
373C2×C3×C4—181no—
374C2×C3×C4C2×D3211yescharacteristic
375C2×C3×C4C2×D3211yescharacteristic
376C2×C3×C4C2×D3211yescharacteristic
377#67—362no—
378#67—362no—
379C2×Dic3—21no—
380C2×Dic3—61no—
381C2×Dic3—61no—
382C2×Dic3—121no—
383C2×Dic3#62611yescharacteristic
384C2×Dic3#62611yescharacteristic
385D12—61no—
386D12—61no—
387D12—61no—
388D12—61no—
389D12—121no—
390D12—121no—
391D12—182no—
392D12—182no—
393D12—181no—
394D12—181no—
395D12—181no—
396D12—181no—
397D12—181no—
398D12—181no—
399D12—361no—
400D12—361no—
401D12—361no—
402D12—361no—
403D12—362no—
404D12—362no—
405D12—364no—
406D12—364no—
407D12—362no—
408D12—362no—
409D12—362no—
410D12—362no—
411C3×C8—61no—
412C3×C8—61no—
413C3×C8—121no—
414C3×C8—181no—
415C3⋊C8—21no—
416C3⋊C8—61no—
417C3⋊C8—61no—
418C3⋊C8—121no—
419C2×C32—21no—
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—41no—
429C2×C32—41no—
430C2×C32—41no—
431C2×C32#87511yescharacteristic
432C2×C32#88511yescharacteristic
433C2×C32#90811yescharacteristic
434C2×C32#90811yescharacteristic
435C2×C32#91111yescharacteristic
436C2×C32#91411yescharacteristic
437C2×C32#94711yescharacteristic
438C2×C32#94911yescharacteristic
439C2×C32#94911yescharacteristic
440C3⋊D3—81no—
441C3⋊D3—242no—
442C3⋊D3—242no—
443C3⋊D3—242no—
444C3⋊D3—244no—
445C3⋊D3—241no—
446C3⋊D3—241no—
447C3⋊D3—241no—
448C3×D3—242no—
449C3×D3—241no—
450C3×D3—241no—
451C2×D4—91no—
452C2×D4—271no—
453C2×D4—541no—
454C2×D4—541no—
455C2×C8—31no—
456C2×C8—91no—
457C4⋊C4—31no—
458C4⋊C4—91no—
459C4⋊C4—181no—
460C4⋊C4—181no—
461C42—31no—
462C22×C3—21no—
463C22×C3—21no—
464C22×C3—21no—
465C22×C3—41no—
466C22×C3—361no—
467C22×C3—361no—
468C22×C3—361no—
469C22×C3D3×D1211yescharacteristic
470C22×C3C2×#62611yescharacteristic
471C22×C3D3×#6711yescharacteristic
472C2×D3—122no—
473C2×D3—122no—
474C2×D3—122no—
475C2×D3—121no—
476C2×D3—121no—
477C2×D3—121no—
478C2×D3—121no—
479C2×D3—121no—
480C2×D3—121no—
481C2×D3—362no—
482C2×D3—362no—
483C2×D3—362no—
484C2×D3—362no—
485C2×D3—362no—
486C2×D3—364no—
487C2×D3—361no—
488C2×D3—361no—
489C2×D3—361no—
490C2×D3—361no—
491C2×D3—361no—
492C2×D3—361no—
493C2×D3—361no—
494C2×D3—361no—
495C2×D3—361no—
496C2×D3—721no—
497C2×D3—721no—
498C2×D3—721no—
499C2×D3—721no—
500C2×D3—722no—
501C2×D3—722no—
502C3×C4—21no—
503C3×C4—21no—
504C3×C4—21no—
505C3×C4—21no—
506C3×C4—21no—
507C3×C4—21no—
508C3×C4—41no—
509C3×C4—41no—
510C3×C4—62no—
511C3×C4—62no—
512C3×C4—61no—
513C3×C4—61no—
514C3×C4—61no—
515C3×C4—61no—
516C3×C4—121no—
517C3×C4—121no—
518C3×C4—241no—
519C3×C4—361no—
520C3×C4#379611yescharacteristic
521C3×C4#379811yescharacteristic
522C3×C4#379911yescharacteristic
523C3×C4#379911yescharacteristic
524C3×C4#379911yescharacteristic
525C3×C4#380411yescharacteristic
526Dic3—21no—
527Dic3—21no—
528Dic3—41no—
529Dic3—121no—
530Dic3—121no—
531Dic3—241no—
532C32—21no—
533C32—21no—
534C32—21no—
535C32—41no—
536C32#471311yescharacteristic, complemented
537C32#476411yescharacteristic, complemented
538C32#506711yescharacteristic, complemented
539C23—181no—
540C23—541no—
541D4—181no—
542D4—181no—
543D4—541no—
544D4—541no—
545D4—1082no—
546D4—1082no—
547C2×C4—31no—
548C2×C4—31no—
549C2×C4—61no—
550C2×C4—181no—
551C2×C4D3311yescharacteristic
552C8—61no—
553C8—181no—
554C2×C3—21no—
555C2×C3—21no—
556C2×C3—21no—
557C2×C3—21no—
558C2×C3—21no—
559C2×C3—21no—
560C2×C3—21no—
561C2×C3—21no—
562C2×C3—21no—
563C2×C3—41no—
564C2×C3—41no—
565C2×C3—41no—
566C2×C3—721no—
567C2×C3#6379411yescharacteristic
568C2×C3#6379511yescharacteristic
569C2×C3#6382511yescharacteristic
570C2×C3#6382511yescharacteristic
571C2×C3#6385211yescharacteristic
572C2×C3#6386411yescharacteristic
573C2×C3#6390711yescharacteristic
574C2×C3#6392511yescharacteristic
575C2×C3#6393911yescharacteristic
576D3—242no—
577D3—241no—
578D3—241no—
579D3—722no—
580D3—722no—
581D3—722no—
582D3—724no—
583D3—721no—
584D3—721no—
585D3—721no—
586C22—361no—
587C22—361no—
588C22—361no—
589C22—1081no—
590C22—1081no—
591C22—1081no—
592C22D3×#62611yescharacteristic
593C4—61no—
594C4—61no—
595C4—121no—
596C4—361no—
597C4#8896211yescharacteristic
598C4#8896511yescharacteristic
599C3—21no—
600C3—21no—
601C3—21no—
602C3—41no—
603C3#1059028211yescharacteristic, complemented
604C3#1059055011yescharacteristic, complemented
605C3#1059088211yescharacteristic, complemented
606C2—721no—
607C2—2161no—
608C2#1172287011yescharacteristic
609C2#1172287111yescharacteristic
610C2#1172300311yescharacteristic
subgroupG/H
centerC22D3×#626
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radical#11723517C2
hypocenterC33#490
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×#63869C2semidirect
3. #11722075C2semidirect
4. #11722036C2semidirect
5. #11722035C2semidirect
6. C3×#63564C2semidirect
7. C3×#63554C2semidirect
28. #63869D3semidirect
42. #63564D3semidirect
45. #63554D3semidirect
346. C33#490semidirect
536. C32#4713semidirect
537. C32#4764semidirect
538. C32#5067semidirect
603. C3#10590282semidirect
604. C3#10590550semidirect
605. C3#10590882semidirect

Maximal subgroups

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

subgroupindex
#117235172
C3×#638692
#117220752
#117220362
#117220352
C3×#635642
C3×#635542
#105908823
#105905503
#105902823