Finite Group Databasebeta

← #13156508

Subgroups — #13156508

Order 1728.

734 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#11723517C211yesmaximal, characteristic
2C3×#63918C211yesmaximal, characteristic, complemented
3#11722075C211yesmaximal, characteristic, complemented
4#11722036C211yesmaximal, characteristic, complemented
5#11722035C211yesmaximal, characteristic, complemented
6C3×#63564C211yesmaximal, characteristic, complemented
7C3×Dic3×C3⋊C8C211yesmaximal, characteristic, complemented
8#10590878—31nomaximal
9#10590548—31nomaximal
10#10590281—31nomaximal
11C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
12C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C2×#6292—21no—
14C2×#6292—21no—
15C2×C3×#627—21no—
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18C32×C4⋊Dic3C2211yescharacteristic
19C32×C4×Dic3C2211yescharacteristic
20C2×C32×C3⋊C8C2211yescharacteristic
21C3×Dic3⋊Dic3—21no—
22#64083—31no—
23C3×#888—31no—
24C3×#869—31no—
25#63918D311yescharacteristic, complemented
26#63911—31no—
27#63911—31no—
28#63911—61no—
29#63648—31no—
30#63636—31no—
31C3×#821—31no—
32C3×C8×Dic3—31no—
33C3×#809—31no—
34C3×C4×C3⋊C8—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—
45Dic3×C3⋊C8D311yescharacteristic, 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×#627—42no—
60C3×C3⋊(C3⋊C8)—21no—
61C32×C3⋊C8—21no—
62#5088—91no—
63#4760—91no—
64#4694—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)—61no—
72C2×C4⋊(C3⋊D3)—61no—
73C2×C4⋊(C3⋊D3)—121no—
74C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
75C2×C3×D12—31no—
76C2×C3×D12—31no—
77C2×C3×D12—61no—
78C2×C3×Dic6—61no—
79C2×C3×Dic6—61no—
80C2×#627—21no—
81C2×#626—62no—
82C2×#626—62no—
83C2×#626—61no—
84C2×#626—61no—
85C2×#626—61no—
86C2×#626—61no—
87C2×C32×C8—31no—
88C32×C4⋊C4—31no—
89C32×C42—31no—
90C4⋊(C3⋊Dic3)C2×D311yescharacteristic
91C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
92C3×C4⋊Dic3—21no—
93C3×C4⋊Dic3—31no—
94C3×C4⋊Dic3—31no—
95C3×C4⋊Dic3—61no—
96C3×C4⋊Dic3—61no—
97C3×C4⋊Dic3C2×D311yescharacteristic
98C3×C4⋊Dic3C2×D311yescharacteristic
99C3×Dic3⋊C4—61no—
100C3×C4×Dic3—21no—
101C3×C4×Dic3C2×D311yescharacteristic
102C3×C4×Dic3C2×D311yescharacteristic
103C2×C3×C3⋊C8—21no—
104C2×C3×C3⋊C8—31no—
105C2×C3×C3⋊C8—31no—
106C2×C3×C3⋊C8—61no—
107C2×C3×C3⋊C8C2×D311yescharacteristic
108C2×C3×C3⋊C8C2×D311yescharacteristic
109Dic3⋊Dic3—21no—
110C22×C33C2×D411yescharacteristic
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—
116C2×C3×C3⋊D3—41no—
117C33×C4#4111yescharacteristic
118C33×C4#4111yescharacteristic
119C3×C3⋊Dic3—41no—
120C32×Dic3—41no—
121C32×Dic3D812yes—
122C32×Dic3SD1612yes—
123C3×C4⋊Q8—91no—
124#940—91no—
125#888—31no—
126#874—91no—
127#874—91no—
128#874—181no—
129#869—31no—
130C3×#102—91no—
131C3×C4×C8—91no—
132#832—91no—
133#821—31no—
134#821—91no—
135#821—91no—
136#821—91no—
137#821—91no—
138#821—181no—
139C8×Dic3—31no—
140#809—31no—
141#809—91no—
142#809—91no—
143#809—91no—
144#809—91no—
145#809—181no—
146C4×C3⋊C8—31no—
147C22×C3⋊D3—21no—
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—
155C22×C3×D3—62no—
156C22×C3×D3—61no—
157C22×C3×D3—61no—
158C2×C32×C4—21no—
159C2×C32×C4—21no—
160C2×C32×C4—21no—
161C2×C32×C4—31no—
162C2×C32×C4—31no—
163C2×C32×C4—41no—
164C2×C32×C4—61no—
165C2×C32×C4C22×D311yescharacteristic
166C2×C32×C4C22×D311yescharacteristic
167C2×C32×C4C22×D311yescharacteristic
168C2×C3⋊Dic3—21no—
169C4⋊(C3⋊D3)—21no—
170C4⋊(C3⋊D3)—21no—
171C4⋊(C3⋊D3)—61no—
172C4⋊(C3⋊D3)—61no—
173C4⋊(C3⋊D3)—62no—
174C4⋊(C3⋊D3)—62no—
175C4⋊(C3⋊D3)—61no—
176C4⋊(C3⋊D3)—61no—
177C4⋊(C3⋊D3)—61no—
178C4⋊(C3⋊D3)—61no—
179C4⋊(C3⋊D3)—121no—
180C4⋊(C3⋊D3)—121no—
181C4⋊(C3⋊D3)—121no—
182C4⋊(C3⋊D3)—121no—
183C4⋊(C3⋊D3)—122no—
184C4⋊(C3⋊D3)—122no—
185C4⋊(C3⋊D3)—122no—
186C4⋊(C3⋊D3)—122no—
187C2×C3×Dic3—21no—
188C2×C3×Dic3—21no—
189C2×C3×Dic3—21no—
190C2×C3×Dic3—21no—
191C2×C3×Dic3—41no—
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×D12—61no—
200C3×D12—61no—
201C3×D12—61no—
202C3×D12—61no—
203C3×D12—121no—
204C3×D12—121no—
205C3×Dic6—122no—
206C3×Dic6—122no—
207#627—42no—
208#626—124no—
209#626—124no—
210#626—122no—
211#626—122no—
212#626—122no—
213#626—122no—
214C32×C8—61no—
215C3⋊(C3⋊C8)—21no—
216C3×C3⋊C8—21no—
217C3×C3⋊C8—21no—
218C3×C3⋊C8—41no—
219C3×C3⋊C8—61no—
220C3×C3⋊C8—61no—
221C3×C3⋊C8—121no—
222#486—271noSylow
223C2×C33#12411yescharacteristic
224C2×C33C2×D811yescharacteristic
225C2×C33C2×SD1611yescharacteristic
226C3⋊(C3⋊D3)—81no—
227C3×C3⋊D3—81no—
228C2×C3×Q8—181no—
229C2×C3×D4—91no—
230C2×#67—181no—
231C2×#67—181no—
232C2×D12—31no—
233C2×D12—31no—
234C2×D12—61no—
235C2×D12—91no—
236C2×D12—91no—
237C2×D12—91no—
238C2×D12—181no—
239C2×D12—181no—
240C2×D12—181no—
241C2×D12—181no—
242C2×D12—181no—
243C2×D12—181no—
244C2×D12—181no—
245C2×D12—182no—
246C2×D12—182no—
247C2×D12—361no—
248C2×Dic6—61no—
249C2×Dic6—61no—
250C2×C3×C8—31no—
251C2×C3×C8—31no—
252C2×C3×C8—61no—
253C2×C3×C8—91no—
254C3×C4⋊C4—31no—
255C3×C4⋊C4—31no—
256C3×C4⋊C4—61no—
257C3×C4⋊C4—91no—
258C3×C4⋊C4—181no—
259C3×C42—31no—
260C3×C42—31no—
261C3×C42—61no—
262C4⋊Dic3—31no—
263C4⋊Dic3—31no—
264C4⋊Dic3—61no—
265C4⋊Dic3—61no—
266C4⋊Dic3D3211yescharacteristic
267Dic3⋊C4—61no—
268C4×Dic3D3211yescharacteristic
269C2×C3⋊C8—31no—
270C2×C3⋊C8—31no—
271C2×C3⋊C8—61no—
272C2×C3⋊C8D3211yescharacteristic
273C22×C32—21no—
274C22×C32—21no—
275C22×C32—21no—
276C22×C32—41no—
277C22×C32C2×D1211yescharacteristic
278C22×C32D3×D411yescharacteristic
279C22×C32C2×#6711yescharacteristic
280C2×C3⋊D3—41no—
281C2×C3⋊D3—41no—
282C2×C3⋊D3—41no—
283C2×C3⋊D3—122no—
284C2×C3⋊D3—122no—
285C2×C3⋊D3—122no—
286C2×C3⋊D3—124no—
287C2×C3⋊D3—121no—
288C2×C3⋊D3—121no—
289C2×C3⋊D3—121no—
290C2×C3⋊D3—121no—
291C2×C3⋊D3—121no—
292C2×C3⋊D3—122no—
293C2×C3⋊D3—122no—
294C2×C3⋊D3—121no—
295C2×C3⋊D3—121no—
296C2×C3⋊D3—121no—
297C2×C3⋊D3—121no—
298C2×C3⋊D3—241no—
299C2×C3⋊D3—241no—
300C2×C3⋊D3—241no—
301C2×C3⋊D3—241no—
302C2×C3⋊D3—242no—
303C2×C3⋊D3—242no—
304C2×C3×D3—122no—
305C2×C3×D3—122no—
306C2×C3×D3—122no—
307C2×C3×D3—121no—
308C2×C3×D3—121no—
309C2×C3×D3—121no—
310C2×C3×D3—121no—
311C2×C3×D3—121no—
312C2×C3×D3—121no—
313C32×C4—21no—
314C32×C4—21no—
315C32×C4—21no—
316C32×C4—21no—
317C32×C4—21no—
318C32×C4—21no—
319C32×C4—32no—
320C32×C4—32no—
321C32×C4—41no—
322C32×C4—41no—
323C32×C4—121no—
324C32×C4#23511yescharacteristic
325C32×C4#23511yescharacteristic
326C32×C4#23511yescharacteristic
327C32×C4#23711yescharacteristic
328C32×C4#23911yescharacteristic
329C32×C4#23911yescharacteristic
330C3⋊Dic3—41no—
331C3×Dic3—22no—
332C3×Dic3—22no—
333C3×Dic3—41no—
334C3×Dic3—41no—
335C3×Dic3—81no—
336C3×Dic3—121no—
337C3×Dic3—121no—
338C3×Dic3—241no—
339C3×Dic3#20412yes—
340C3×Dic3D2412yes—
341C3×Dic3#21312yes—
342C3×Dic3#21512yes—
343C4⋊Q8—91no—
344#127—271no—
345#102—91no—
346#102—271no—
347#102—271no—
348#102—271no—
349C4×C8—91no—
350C33#48611yescharacteristic, complemented, Sylow
351C23×C3—181no—
352C22×D3—62no—
353C22×D3—61no—
354C22×D3—61no—
355C22×D3—182no—
356C22×D3—181no—
357C22×D3—181no—
358C22×D3—181no—
359C22×D3—361no—
360C22×D3—361no—
361C22×D3—362no—
362C3×Q8—362no—
363C3×D4—181no—
364C3×D4—181no—
365C2×C3×C4—21no—
366C2×C3×C4—21no—
367C2×C3×C4—21no—
368C2×C3×C4—31no—
369C2×C3×C4—31no—
370C2×C3×C4—31no—
371C2×C3×C4—31no—
372C2×C3×C4—41no—
373C2×C3×C4—61no—
374C2×C3×C4—61no—
375C2×C3×C4—61no—
376C2×C3×C4—61no—
377C2×C3×C4—121no—
378C2×C3×C4—181no—
379C2×C3×C4C2×D3211yescharacteristic
380C2×C3×C4C2×D3211yescharacteristic
381C2×C3×C4C2×D3211yescharacteristic
382#67—362no—
383#67—362no—
384C2×Dic3—21no—
385C2×Dic3—61no—
386C2×Dic3—61no—
387C2×Dic3—121no—
388C2×Dic3#62611yescharacteristic
389C2×Dic3#62611yescharacteristic
390D12—61no—
391D12—61no—
392D12—61no—
393D12—61no—
394D12—121no—
395D12—121no—
396D12—182no—
397D12—182no—
398D12—181no—
399D12—181no—
400D12—181no—
401D12—181no—
402D12—181no—
403D12—181no—
404D12—361no—
405D12—361no—
406D12—361no—
407D12—361no—
408D12—362no—
409D12—362no—
410D12—362no—
411D12—362no—
412D12—362no—
413D12—362no—
414D12—364no—
415D12—364no—
416Dic6—122no—
417Dic6—122no—
418C3×C8—61no—
419C3×C8—61no—
420C3×C8—121no—
421C3×C8—181no—
422C3⋊C8—21no—
423C3⋊C8—61no—
424C3⋊C8—61no—
425C3⋊C8—121no—
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×C32#87511yescharacteristic
439C2×C32#88511yescharacteristic
440C2×C32C2×#20411yescharacteristic
441C2×C32C2×D2411yescharacteristic
442C2×C32D3×D811yescharacteristic
443C2×C32D3×SD1611yescharacteristic
444C2×C32C2×#21311yescharacteristic
445C2×C32C2×#21511yescharacteristic
446C2×C32#94711yescharacteristic
447C3⋊D3—81no—
448C3⋊D3—242no—
449C3⋊D3—242no—
450C3⋊D3—242no—
451C3⋊D3—244no—
452C3⋊D3—241no—
453C3⋊D3—241no—
454C3⋊D3—241no—
455C3×D3—242no—
456C3×D3—241no—
457C3×D3—241no—
458C2×Q8—181no—
459C2×D4—91no—
460C2×D4—271no—
461C2×D4—541no—
462C2×D4—541no—
463C2×C8—31no—
464C2×C8—91no—
465C4⋊C4—31no—
466C4⋊C4—91no—
467C4⋊C4—181no—
468C42—31no—
469C22×C3—21no—
470C22×C3—21no—
471C22×C3—21no—
472C22×C3—41no—
473C22×C3—361no—
474C22×C3—361no—
475C22×C3—361no—
476C22×C3D3×D1211yescharacteristic
477C22×C3C2×#62611yescharacteristic
478C22×C3D3×#6711yescharacteristic
479C2×D3—122no—
480C2×D3—122no—
481C2×D3—122no—
482C2×D3—121no—
483C2×D3—121no—
484C2×D3—121no—
485C2×D3—121no—
486C2×D3—121no—
487C2×D3—121no—
488C2×D3—362no—
489C2×D3—362no—
490C2×D3—362no—
491C2×D3—362no—
492C2×D3—362no—
493C2×D3—364no—
494C2×D3—361no—
495C2×D3—361no—
496C2×D3—361no—
497C2×D3—361no—
498C2×D3—361no—
499C2×D3—361no—
500C2×D3—361no—
501C2×D3—361no—
502C2×D3—361no—
503C2×D3—721no—
504C2×D3—721no—
505C2×D3—721no—
506C2×D3—721no—
507C2×D3—722no—
508C2×D3—722no—
509C3×C4—21no—
510C3×C4—21no—
511C3×C4—21no—
512C3×C4—21no—
513C3×C4—21no—
514C3×C4—21no—
515C3×C4—32no—
516C3×C4—32no—
517C3×C4—32no—
518C3×C4—32no—
519C3×C4—41no—
520C3×C4—41no—
521C3×C4—62no—
522C3×C4—62no—
523C3×C4—121no—
524C3×C4—121no—
525C3×C4—241no—
526C3×C4—361no—
527C3×C4#379611yescharacteristic
528C3×C4#379811yescharacteristic
529C3×C4#379911yescharacteristic
530C3×C4#379911yescharacteristic
531C3×C4#379911yescharacteristic
532C3×C4#380411yescharacteristic
533Dic3—41no—
534Dic3—121no—
535Dic3—121no—
536Dic3—241no—
537Dic3#371412yes—
538Dic3#371712yes—
539C32—21no—
540C32—21no—
541C32—21no—
542C32—41no—
543C32#469411yescharacteristic, complemented
544C32#476011yescharacteristic, complemented
545C32#508811yescharacteristic, complemented
546C23—181no—
547C23—541no—
548Q8—362no—
549D4—181no—
550D4—181no—
551D4—541no—
552D4—541no—
553D4—1082no—
554D4—1082no—
555C2×C4—31no—
556C2×C4—31no—
557C2×C4—61no—
558C2×C4—181no—
559C2×C4D3311yescharacteristic
560C8—61no—
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—721no—
575C2×C3D3×#20411yescharacteristic
576C2×C3D3×D2411yescharacteristic
577C2×C3C2×#371411yescharacteristic
578C2×C3C2×#371711yescharacteristic
579C2×C3#6385211yescharacteristic
580C2×C3#6386411yescharacteristic
581C2×C3#6390711yescharacteristic
582C2×C3D3×#21311yescharacteristic
583C2×C3D3×#21511yescharacteristic
584D3—242no—
585D3—241no—
586D3—241no—
587D3—722no—
588D3—722no—
589D3—722no—
590D3—724no—
591D3—721no—
592D3—721no—
593D3—721no—
594C22—361no—
595C22—361no—
596C22—361no—
597C22—1081no—
598C22—1081no—
599C22—1081no—
600C22D3×#62611yescharacteristic
601C4—32no—
602C4—32no—
603C4—121no—
604C4—361no—
605C4#8896211yescharacteristic
606C4#8896511yescharacteristic
607C3—21no—
608C3—21no—
609C3—21no—
610C3—41no—
611C3#1059028111yescharacteristic, complemented
612C3#1059054811yescharacteristic, complemented
613C3#1059087811yescharacteristic, complemented
614C2—721no—
615C2—2161no—
616C2D3×#371411yescharacteristic
617C2D3×#371711yescharacteristic
618C2#1172300311yescharacteristic
subgroupG/H
centerC22D3×#626
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radical#11723517C2
hypocenterC33#486
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×#63918C2semidirect
3. #11722075C2semidirect
4. #11722036C2semidirect
5. #11722035C2semidirect
6. C3×#63564C2semidirect
7. C3×Dic3×C3⋊C8C2semidirect
25. #63918D3semidirect
42. #63564D3semidirect
45. Dic3×C3⋊C8D3semidirect
350. C33#486semidirect
543. C32#4694semidirect
544. C32#4760semidirect
545. C32#5088semidirect
611. C3#10590281semidirect
612. C3#10590548semidirect
613. C3#10590878semidirect

Maximal subgroups

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

subgroupindex
#117235172
C3×#639182
#117220752
#117220362
#117220352
C3×#635642
C3×Dic3×C3⋊C82
#105908783
#105905483
#105902813