Finite Group Databasebeta

← #13156591

Subgroups — #13156591

Order 1728.

962 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88810C211yesmaximal, characteristic, complemented
2#11723517C211yesmaximal, characteristic, complemented
3C2×#88791C211yesmaximal, characteristic, complemented
4C2×#88790C211yesmaximal, characteristic, complemented
5C3×#63911C211yesmaximal, characteristic, complemented
6C2×C3×#3714C211yesmaximal, characteristic, complemented
7C3×Dic3×C3⋊C8C211yesmaximal, characteristic, complemented
8#10591482—31nomaximal
9#10590880—31nomaximal
10#10590350—31nomaximal
11C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
12C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
14C2×C32×D12C2211yescharacteristic
15C2×#6292—22no—
16C2×C3×#626—22no—
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18C32×C4×Dic3C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20#88810—22no—
21#88791—22no—
22#88790—22no—
23C3×#3714—22no—
24C2×#3753—31no—
25C2×#3745—31no—
26#64083—31no—
27C3×#940—31no—
28C2×C3×#213—31no—
29C2×C3×D24—31no—
30C3×#874—31no—
31#63911—31no—
32#63911—31no—
33#63911—61no—
34#63911D311yescharacteristic, complemented
35C2×#3714—31no—
36C2×#3714—31no—
37C2×#3714—31no—
38C2×#3714—31no—
39C2×#3714—31no—
40C2×#3714—31no—
41C2×#3714—61no—
42C2×#3714—61no—
43C2×#3714D311yescharacteristic, complemented
44C2×#3713—31no—
45C3×C8×Dic3—31no—
46C3×C4×C3⋊C8—31no—
47Dic3×C3⋊C8D311yescharacteristic, complemented
48C22×C3⋊(C3⋊D3)—21no—
49C22×C3×C3⋊D3—21no—
50C22×C3×C3⋊D3—21no—
51C22×C32×D3—21no—
52C2×C33×C4C2311yescharacteristic
53C4⋊(C3⋊(C3⋊D3))—21no—
54C4⋊(C3⋊(C3⋊D3))—21no—
55C3×C4⋊(C3⋊D3)—21no—
56C3×C4⋊(C3⋊D3)—21no—
57C3×C4⋊(C3⋊D3)—21no—
58C3×C4⋊(C3⋊D3)—21no—
59C2×C32×Dic3D412yes—
60C32×D12—21no—
61C32×D12—21no—
62#6292—44no—
63C3×#626—44no—
64C3×C3⋊(C3⋊C8)D412yescomplemented
65C32×C3⋊C8D412yescomplemented
66#5151—91no—
67#5073—91no—
68#4695—91no—
69C2×C32×D4—31no—
70C2×C4⋊(C3⋊D3)—31no—
71C2×C4⋊(C3⋊D3)—31no—
72C2×C4⋊(C3⋊D3)—31no—
73C2×C4⋊(C3⋊D3)—61no—
74C2×C4⋊(C3⋊D3)—61no—
75C2×C4⋊(C3⋊D3)—61no—
76C2×C4⋊(C3⋊D3)—62no—
77C2×C4⋊(C3⋊D3)—121no—
78C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
79C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
80C2×C3×#67—62no—
81C2×C3×D12—21no—
82C2×C3×D12—31no—
83C2×C3×D12—31no—
84C2×C3×D12—31no—
85C2×C3×D12—31no—
86C2×C3×D12—61no—
87C2×C3×D12—61no—
88C2×C3×D12—62no—
89C2×C3×D12C2×D311yescharacteristic
90C2×C3×D12C2×D311yescharacteristic
91C2×#626—22no—
92C2×#626—64no—
93C2×#626—62no—
94C2×#626—62no—
95C2×C32×C8—31no—
96C32×C42—31no—
97#3753—62no—
98C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
99#3745—62no—
100C3×#213—62no—
101C3×C4×Dic3—21no—
102C3×C4×Dic3C2×D311yescharacteristic
103C3×C4×Dic3C2×D311yescharacteristic
104C2×C3×C3⋊C8—21no—
105C2×C3×C3⋊C8—31no—
106C2×C3×C3⋊C8—31no—
107C2×C3×C3⋊C8—61no—
108C2×C3×C3⋊C8C2×D311yescharacteristic
109C2×C3×C3⋊C8C2×D311yescharacteristic
110C3×D24—62no—
111#3714—22no—
112#3714—64no—
113#3714—62no—
114#3714—62no—
115#3714—62no—
116#3714—62no—
117#3714—64no—
118#3714—62no—
119#3714—62no—
120#3713—62no—
121C22×C33C2×D411yescharacteristic
122C2×C3⋊(C3⋊D3)—41no—
123C2×C3⋊(C3⋊D3)—42no—
124C2×C3×C3⋊D3—41no—
125C2×C3×C3⋊D3—41no—
126C2×C3×C3⋊D3—42no—
127C2×C3×C3⋊D3—42no—
128C2×C32×D3—41no—
129C2×C32×D3—42no—
130C33×C4C2×D411yescharacteristic
131C33×C4C2×D411yescharacteristic
132C32×Dic3D814yescomplemented
133C2×C3×D8—91no—
134C3×#127—91no—
135#940—31no—
136#940—91no—
137C2×#213—31no—
138C2×#213—91no—
139C2×#213—91no—
140C2×#213—91no—
141C2×#213—91no—
142C2×#213—91no—
143C2×#213—181no—
144C2×D24—31no—
145C2×D24—91no—
146C2×D24—91no—
147C2×D24—91no—
148C2×D24—91no—
149C2×D24—181no—
150#874—31no—
151#874—91no—
152#874—91no—
153#874—181no—
154C3×C4×C8—91no—
155C8×Dic3—31no—
156C4×C3⋊C8—31no—
157C23×C32—61no—
158C22×C3⋊D3—21no—
159C22×C3⋊D3—21no—
160C22×C3⋊D3—61no—
161C22×C3⋊D3—62no—
162C22×C3⋊D3—61no—
163C22×C3⋊D3—61no—
164C22×C3⋊D3—121no—
165C22×C3⋊D3—121no—
166C22×C3⋊D3—122no—
167C22×C3×D3—21no—
168C22×C3×D3—21no—
169C22×C3×D3—41no—
170C22×C3×D3—62no—
171C22×C3×D3—61no—
172C22×C3×D3—61no—
173C22×C3×D3—61no—
174C22×C3×D3—61no—
175C22×C3×D3—121no—
176C32×D4—61no—
177C32×D4—61no—
178C2×C32×C4—21no—
179C2×C32×C4—21no—
180C2×C32×C4—21no—
181C2×C32×C4—32no—
182C2×C32×C4—41no—
183C2×C32×C4C22×D311yescharacteristic
184C2×C32×C4C22×D311yescharacteristic
185C2×C32×C4C22×D311yescharacteristic
186C4⋊(C3⋊D3)—21no—
187C4⋊(C3⋊D3)—21no—
188C4⋊(C3⋊D3)—21no—
189C4⋊(C3⋊D3)—21no—
190C4⋊(C3⋊D3)—61no—
191C4⋊(C3⋊D3)—61no—
192C4⋊(C3⋊D3)—62no—
193C4⋊(C3⋊D3)—62no—
194C4⋊(C3⋊D3)—61no—
195C4⋊(C3⋊D3)—61no—
196C4⋊(C3⋊D3)—61no—
197C4⋊(C3⋊D3)—61no—
198C4⋊(C3⋊D3)—121no—
199C4⋊(C3⋊D3)—121no—
200C4⋊(C3⋊D3)—121no—
201C4⋊(C3⋊D3)—121no—
202C4⋊(C3⋊D3)—122no—
203C4⋊(C3⋊D3)—122no—
204C4⋊(C3⋊D3)—124no—
205C3×#67—124no—
206C2×C3×Dic3—22no—
207C2×C3×Dic3D1212yes—
208C2×C3×Dic3#6712yes—
209C3×D12—22no—
210C3×D12—21no—
211C3×D12—21no—
212C3×D12—21no—
213C3×D12—21no—
214C3×D12—41no—
215C3×D12—62no—
216C3×D12—61no—
217C3×D12—61no—
218C3×D12—61no—
219C3×D12—61no—
220C3×D12—61no—
221C3×D12—61no—
222C3×D12—61no—
223C3×D12—61no—
224C3×D12—121no—
225C3×D12—121no—
226C3×D12—121no—
227C3×D12—124no—
228#626—44no—
229#626—128no—
230#626—124no—
231#626—124no—
232C32×C8—32no—
233C3⋊(C3⋊C8)D1212yescomplemented
234C3×C3⋊C8—22no—
235C3×C3⋊C8—32no—
236C3×C3⋊C8—32no—
237C3×C3⋊C8—62no—
238C3×C3⋊C8D1212yescomplemented
239C3×C3⋊C8#6712yescomplemented
240#493—271noSylow
241C2×C33#12711yescharacteristic
242C2×C33C2×D812yes—
243C3⋊(C3⋊D3)—81no—
244C3×C3⋊D3—81no—
245C3×C3⋊D3—81no—
246C32×D3—81no—
247C2×C3×D4—31no—
248C2×C3×D4—31no—
249C2×C3×D4—61no—
250C2×C3×D4—91no—
251C2×C3×D4—91no—
252C2×C3×D4—182no—
253C2×#67—62no—
254C2×#67—182no—
255C2×D12—31no—
256C2×D12—31no—
257C2×D12—31no—
258C2×D12—31no—
259C2×D12—61no—
260C2×D12—61no—
261C2×D12—62no—
262C2×D12—91no—
263C2×D12—91no—
264C2×D12—91no—
265C2×D12—181no—
266C2×D12—181no—
267C2×D12—181no—
268C2×D12—182no—
269C2×D12—182no—
270C2×D12—184no—
271C2×D12—361no—
272C2×D12D3211yescharacteristic
273C3×D8—182no—
274C2×C3×C8—31no—
275C2×C3×C8—31no—
276C2×C3×C8—61no—
277C2×C3×C8—91no—
278C3×C42—31no—
279C3×C42—31no—
280C3×C42—61no—
281#213—62no—
282#213—182no—
283#213—182no—
284#213—182no—
285#213—182no—
286#213—184no—
287#213—182no—
288C4×Dic3D3211yescharacteristic
289C2×C3⋊C8—31no—
290C2×C3⋊C8—31no—
291C2×C3⋊C8—61no—
292C2×C3⋊C8D3211yescharacteristic
293D24—62no—
294D24—182no—
295D24—182no—
296D24—184no—
297D24—182no—
298D24—182no—
299C22×C32—21no—
300C22×C32—21no—
301C22×C32—21no—
302C22×C32—41no—
303C22×C32—121no—
304C22×C32—122no—
305C22×C32C2×D1211yescharacteristic
306C22×C32D3×D411yescharacteristic
307C22×C32C2×#6711yescharacteristic
308C2×C3⋊D3—42no—
309C2×C3⋊D3—41no—
310C2×C3⋊D3—41no—
311C2×C3⋊D3—42no—
312C2×C3⋊D3—122no—
313C2×C3⋊D3—122no—
314C2×C3⋊D3—122no—
315C2×C3⋊D3—124no—
316C2×C3⋊D3—121no—
317C2×C3⋊D3—122no—
318C2×C3⋊D3—121no—
319C2×C3⋊D3—121no—
320C2×C3⋊D3—124no—
321C2×C3⋊D3—122no—
322C2×C3⋊D3—122no—
323C2×C3⋊D3—242no—
324C2×C3⋊D3—242no—
325C2×C3⋊D3—244no—
326C2×C3×D3—42no—
327C2×C3×D3—42no—
328C2×C3×D3—42no—
329C2×C3×D3—41no—
330C2×C3×D3—41no—
331C2×C3×D3—82no—
332C2×C3×D3—124no—
333C2×C3×D3—122no—
334C2×C3×D3—122no—
335C2×C3×D3—121no—
336C2×C3×D3—121no—
337C2×C3×D3—122no—
338C2×C3×D3—122no—
339C2×C3×D3—122no—
340C2×C3×D3—122no—
341C2×C3×D3—121no—
342C2×C3×D3—121no—
343C2×C3×D3—242no—
344C32×C4—21no—
345C32×C4—21no—
346C32×C4—21no—
347C32×C4—21no—
348C32×C4—21no—
349C32×C4—21no—
350C32×C4—34no—
351C32×C4—41no—
352C32×C4—41no—
353C32×C4C2×D1211yescharacteristic
354C32×C4C2×D1211yescharacteristic
355C32×C4D3×D411yescharacteristic
356C32×C4D3×D411yescharacteristic
357C32×C4D3×D411yescharacteristic
358C32×C4C2×#6711yescharacteristic
359C3×Dic3—24no—
360C3×Dic3D2414yescomplemented
361C3×Dic3#21314yescomplemented
362C2×D8—91no—
363C2×D8—271no—
364C2×D8—271no—
365C2×D8—271no—
366#127—91no—
367#127—271no—
368C4×C8—91no—
369C33#49311yescharacteristic, complemented, Sylow
370C23×C3—61no—
371C23×C3—61no—
372C23×C3—121no—
373C23×C3—181no—
374C23×C3—181no—
375C22×D3—21no—
376C22×D3—62no—
377C22×D3—61no—
378C22×D3—61no—
379C22×D3—61no—
380C22×D3—61no—
381C22×D3—121no—
382C22×D3—182no—
383C22×D3—181no—
384C22×D3—181no—
385C22×D3—181no—
386C22×D3—361no—
387C22×D3—361no—
388C22×D3—362no—
389C3×D4—62no—
390C3×D4—61no—
391C3×D4—61no—
392C3×D4—61no—
393C3×D4—61no—
394C3×D4—121no—
395C3×D4—181no—
396C3×D4—181no—
397C3×D4—181no—
398C3×D4—181no—
399C3×D4—364no—
400C2×C3×C4—21no—
401C2×C3×C4—21no—
402C2×C3×C4—21no—
403C2×C3×C4—32no—
404C2×C3×C4—32no—
405C2×C3×C4—41no—
406C2×C3×C4—62no—
407C2×C3×C4C2×D3211yescharacteristic
408C2×C3×C4C2×D3211yescharacteristic
409C2×C3×C4C2×D3211yescharacteristic
410#67—124no—
411#67—364no—
412C2×Dic3#62612yes—
413D12—21no—
414D12—21no—
415D12—62no—
416D12—61no—
417D12—61no—
418D12—61no—
419D12—61no—
420D12—61no—
421D12—61no—
422D12—61no—
423D12—61no—
424D12—121no—
425D12—121no—
426D12—121no—
427D12—124no—
428D12—182no—
429D12—182no—
430D12—181no—
431D12—181no—
432D12—181no—
433D12—181no—
434D12—181no—
435D12—181no—
436D12—361no—
437D12—361no—
438D12—361no—
439D12—361no—
440D12—362no—
441D12—362no—
442D12—364no—
443D12—364no—
444D12—368no—
445C3×C8—32no—
446C3×C8—32no—
447C3×C8—62no—
448C3×C8—92no—
449C3⋊C8—32no—
450C3⋊C8—32no—
451C3⋊C8—62no—
452C3⋊C8#62612yescomplemented
453C2×C32—22no—
454C2×C32—22no—
455C2×C32—21no—
456C2×C32—21no—
457C2×C32—22no—
458C2×C32—21no—
459C2×C32—41no—
460C2×C32—42no—
461C2×C32—241no—
462C2×C32#87411yescharacteristic
463C2×C32C2×D2412yes—
464C2×C32D3×D812yes—
465C2×C32C2×#21312yes—
466C2×C32#94011yescharacteristic
467C2×C32#94011yescharacteristic
468C3⋊D3—81no—
469C3⋊D3—81no—
470C3⋊D3—242no—
471C3⋊D3—242no—
472C3⋊D3—242no—
473C3⋊D3—244no—
474C3⋊D3—241no—
475C3⋊D3—241no—
476C3⋊D3—241no—
477C3×D3—82no—
478C3×D3—81no—
479C3×D3—81no—
480C3×D3—242no—
481C3×D3—242no—
482C3×D3—241no—
483C3×D3—241no—
484C3×D3—241no—
485C3×D3—241no—
486C2×D4—31no—
487C2×D4—91no—
488C2×D4—91no—
489C2×D4—182no—
490C2×D4—271no—
491C2×D4—542no—
492D8—182no—
493D8—542no—
494D8—542no—
495D8—542no—
496C2×C8—31no—
497C2×C8—91no—
498C42—31no—
499C22×C3—21no—
500C22×C3—21no—
501C22×C3—21no—
502C22×C3—41no—
503C22×C3—122no—
504C22×C3—122no—
505C22×C3—122no—
506C22×C3—121no—
507C22×C3—121no—
508C22×C3—242no—
509C22×C3—362no—
510C22×C3—362no—
511C22×C3—361no—
512C22×C3—361no—
513C22×C3D3×D1211yescharacteristic
514C22×C3C2×#62611yescharacteristic
515C22×C3D3×#6711yescharacteristic
516C2×D3—42no—
517C2×D3—41no—
518C2×D3—122no—
519C2×D3—122no—
520C2×D3—124no—
521C2×D3—122no—
522C2×D3—122no—
523C2×D3—121no—
524C2×D3—121no—
525C2×D3—122no—
526C2×D3—122no—
527C2×D3—121no—
528C2×D3—121no—
529C2×D3—242no—
530C2×D3—362no—
531C2×D3—362no—
532C2×D3—362no—
533C2×D3—364no—
534C2×D3—364no—
535C2×D3—362no—
536C2×D3—362no—
537C2×D3—361no—
538C2×D3—361no—
539C2×D3—362no—
540C2×D3—361no—
541C2×D3—722no—
542C2×D3—722no—
543C2×D3—724no—
544C3×C4—21no—
545C3×C4—21no—
546C3×C4—21no—
547C3×C4—21no—
548C3×C4—21no—
549C3×C4—21no—
550C3×C4—34no—
551C3×C4—34no—
552C3×C4—41no—
553C3×C4—41no—
554C3×C4—64no—
555C3×C4D3×D1211yescharacteristic
556C3×C4D3×D1211yescharacteristic
557C3×C4D3×D1211yescharacteristic
558C3×C4C2×#62611yescharacteristic
559C3×C4D3×#6711yescharacteristic
560C3×C4#381111yescharacteristic
561Dic3#371414yescomplemented
562C32—21no—
563C32—21no—
564C32—21no—
565C32—41no—
566C32#469511yescharacteristic, complemented
567C32#507311yescharacteristic, complemented
568C32#515111yescharacteristic, complemented
569C23—61no—
570C23—181no—
571C23—181no—
572C23—541no—
573D4—61no—
574D4—61no—
575D4—181no—
576D4—181no—
577D4—181no—
578D4—181no—
579D4—364no—
580D4—541no—
581D4—541no—
582D4—1084no—
583C2×C4—32no—
584C2×C4D3311yescharacteristic
585C8—32no—
586C8—92no—
587C2×C3—21no—
588C2×C3—21no—
589C2×C3—21no—
590C2×C3—22no—
591C2×C3—22no—
592C2×C3—22no—
593C2×C3—41no—
594C2×C3—42no—
595C2×C3—242no—
596C2×C3—241no—
597C2×C3—241no—
598C2×C3—721no—
599C2×C3—721no—
600C2×C3D3×D2412yes—
601C2×C3C2×#371412yes—
602C2×C3#6391111yescharacteristic
603C2×C3#6391111yescharacteristic
604C2×C3D3×#21312yes—
605C2×C3#6397911yescharacteristic
606D3—81no—
607D3—242no—
608D3—242no—
609D3—241no—
610D3—241no—
611D3—241no—
612D3—241no—
613D3—722no—
614D3—722no—
615D3—722no—
616D3—724no—
617D3—721no—
618D3—721no—
619D3—721no—
620C22—121no—
621C22—122no—
622C22—361no—
623C22—361no—
624C22—362no—
625C22—362no—
626C22—1081no—
627C22—1082no—
628C22D3×#62611yescharacteristic
629C4—34no—
630C4D3×#62611yescharacteristic
631C4#8895511yescharacteristic
632C3—21no—
633C3—21no—
634C3—21no—
635C3—41no—
636C3#1059035011yescharacteristic, complemented
637C3#1059088011yescharacteristic, complemented
638C3#1059148211yescharacteristic, complemented
639C2—241no—
640C2—721no—
641C2—721no—
642C2—2161no—
643C2D3×#371412yes—
644C2#1172303011yescharacteristic
subgroupG/H
centerC22D3×#626
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
hypocenterC33#493
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
1. C2×#88810C2semidirect
2. #11723517C2semidirect
3. C2×#88791C2semidirect
4. C2×#88790C2semidirect
5. C3×#63911C2semidirect
6. C2×C3×#3714C2semidirect
7. C3×Dic3×C3⋊C8C2semidirect
34. #63911D3semidirect
43. C2×#3714D3semidirect
47. Dic3×C3⋊C8D3semidirect
64. C3×C3⋊(C3⋊C8)D4semidirect
65. C32×C3⋊C8D4semidirect
132. C32×Dic3D8semidirect
233. C3⋊(C3⋊C8)D12semidirect
238. C3×C3⋊C8D12semidirect
239. C3×C3⋊C8#67semidirect
360. C3×Dic3D24semidirect
361. C3×Dic3#213semidirect
369. C33#493semidirect
452. C3⋊C8#626semidirect
561. Dic3#3714semidirect
566. C32#4695semidirect
567. C32#5073semidirect
568. C32#5151semidirect
636. C3#10590350semidirect
637. C3#10590880semidirect
638. C3#10591482semidirect

Maximal subgroups

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

subgroupindex
C2×#888102
#117235172
C2×#887912
C2×#887902
C3×#639112
C2×C3×#37142
C3×Dic3×C3⋊C82
#105914823
#105908803
#105903503