Finite Group Databasebeta

← #10694311

Subgroups — #10694311

Order 768.

856 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82700C211yesmaximal, characteristic
2C2×C3×#327C211yesmaximal, characteristic, complemented
3C2×#4537C211yesmaximal, characteristic, complemented
4C2×#4470C211yesmaximal, characteristic, complemented
5#69908C211yesmaximal, characteristic, complemented
6Dic3×#95C211yesmaximal, characteristic, complemented
7C2×#4451C211yesmaximal, characteristic, complemented
8#9712—31nomaximal, Sylow
9C2×C3×#121C2211yescharacteristic
10C2×#940C411yescharacteristic, complemented
11C2×#939—21no—
12C2×#939—21no—
13C2×#930C2211yescharacteristic
14C22×C4×Dic3—21no—
15C22×C4×Dic3C2211yescharacteristic
16C2×C3×C22⋊C8C2211yescharacteristic
17C2×C3×#95C2211yescharacteristic
18#5221C411yescharacteristic, complemented
19C2×C3⋊#95—21no—
20C2×C3⋊#95C2211yescharacteristic
21C2×#830C2211yescharacteristic
22C2×Dic3⋊C8—21no—
23#4988C411yescharacteristic, complemented
24#4975C411yescharacteristic, complemented
25#4957—21no—
26#4957—21no—
27C3×#327—22no—
28#4550—21no—
29#4537—22no—
30#4470—22no—
31#4451—22no—
32#2856—31no—
33#1797—31no—
34C2×#327—31no—
35C2×#327—31no—
36C2×#327—31no—
37C2×#327D311yescharacteristic, complemented
38C4×#95—31no—
39C22×C3×D4—21no—
40C22×C3×D4C2×C411yescharacteristic
41C23×C3×C4—21no—
42C23×C3×C4C2311yescharacteristic
43C22×#67—21no—
44C22×#67—21no—
45C22×#67—21no—
46C23×Dic3—21no—
47C23×Dic3D411yescharacteristic
48C23×Dic3D411yescharacteristic
49C22×D12C2×C411yescharacteristic
50C22×C3×C8—21no—
51C3×#121—21no—
52C3×#121—21no—
53C3×#121—21no—
54C3×#121—21no—
55C2×C3×C4⋊C4C2×C411yescharacteristic
56C2×C3×C22⋊C4—21no—
57C2×#217—21no—
58C2×#217—21no—
59#940—42no—
60#939—42no—
61#939—42no—
62#939—42no—
63#939—42no—
64#930—21no—
65#930—21no—
66#930—21no—
67#930—21no—
68C2×(C2×D3)⋊C4—21no—
69C2×(C2×D3)⋊C4—21no—
70C2×(C2×D3)⋊C4—21no—
71C2×C4⋊Dic3C2×C411yescharacteristic
72C2×Dic3⋊C4—21no—
73C2×Dic3⋊C4—21no—
74C2×C4×Dic3—24no—
75C2×C4×Dic3—42no—
76C2×C4×Dic3—42no—
77C2×C4×Dic3—42no—
78C2×C4×Dic3C2×C411yescharacteristic
79C2×C4×Dic3C2×C411yescharacteristic
80C2×C4×Dic3D411yescharacteristic
81C2×C4×Dic3D411yescharacteristic
82C22×C3⋊C8—21no—
83C3×C22⋊C8—21no—
84C3×C22⋊C8—21no—
85C3×#95—21no—
86C3×#95D412yescomplemented
87C3⋊#95—24no—
88C3⋊#95—21no—
89C3⋊#95D412yescomplemented
90#830—21no—
91#830—21no—
92Dic3⋊C8—42no—
93C2×#127—31no—
94C2×#121—31no—
95C2×#121—61no—
96C2×#121—61no—
97C2×#121C2×D311yescharacteristic
98C22×C42—31no—
99C22×C42—61no—
100C2×C4⋊C8—61no—
101C2×C22⋊C8—31no—
102C2×C22⋊C8C2×D311yescharacteristic
103#390—31no—
104#390—31no—
105#390—61no—
106#390—61no—
107#384—31no—
108C2×#95—31no—
109C2×#95—61no—
110C2×#95C2×D311yescharacteristic
111#336—61no—
112#327—22no—
113#327—62no—
114#327—62no—
115#327—62no—
116C24×C3—21no—
117C24×C3C2×D411yescharacteristic
118C23×D3—21no—
119C2×C3×D4—21no—
120C2×C3×D4—21no—
121C2×C3×D4—41no—
122C2×C3×D4—41no—
123C2×C3×D4—41no—
124C2×C3×D4—41no—
125C2×C3×D4—41no—
126C2×C3×D4—41no—
127C2×C3×D4—41no—
128C2×C3×D4—41no—
129C22×C3×C4—22no—
130C22×C3×C4—22no—
131C22×C3×C4—22no—
132C22×C3×C4—21no—
133C22×C3×C4—41no—
134C22×C3×C4—41no—
135C22×C3×C4—41no—
136C22×C3×C4—41no—
137C22×C3×C4—41no—
138C22×C3×C4—41no—
139C22×C3×C4C22×C411yescharacteristic
140C22×C3×C4C2×D411yescharacteristic
141C22×C3×C4C2×D411yescharacteristic
142C22×C3×C4C2×D411yescharacteristic
143C22×C3×C4#4111yescharacteristic
144C22×C3×C4#4111yescharacteristic
145C2×#67—42no—
146C2×#67—42no—
147C2×#67—41no—
148C2×#67—41no—
149C2×#67—41no—
150C2×#67—41no—
151C2×#67—41no—
152C2×#67—41no—
153C2×#67—42no—
154C2×#67—42no—
155C2×#67—42no—
156C2×#67—42no—
157C22×Dic3—22no—
158C22×Dic3—22no—
159C22×Dic3—22no—
160C22×Dic3—22no—
161C22×Dic3—22no—
162C22×Dic3—21no—
163C22×Dic3—21no—
164C22×Dic3—22no—
165C22×Dic3—21no—
166C22×Dic3—41no—
167C22×Dic3—41no—
168C22×Dic3—41no—
169C22×Dic3—41no—
170C22×Dic3—41no—
171C22×Dic3C22⋊C411yescharacteristic
172C22×Dic3C22⋊C411yescharacteristic
173C22×Dic3C22⋊C411yescharacteristic
174C22×Dic3C22⋊C411yescharacteristic
175C22×Dic3D812yes—
176C22×Dic3SD1612yes—
177C2×D12—21no—
178C2×D12—21no—
179C2×D12—41no—
180C2×D12—41no—
181C2×C3×C8—41no—
182C2×C3×C8—41no—
183C2×C3×C8—41no—
184C3×C4⋊C4—21no—
185C3×C4⋊C4—21no—
186C3×C22⋊C4—41no—
187C3×C22⋊C4—41no—
188#217—42no—
189#217—42no—
190(C2×D3)⋊C4—42no—
191(C2×D3)⋊C4—41no—
192(C2×D3)⋊C4—41no—
193(C2×D3)⋊C4—42no—
194C4⋊Dic3—21no—
195C4⋊Dic3—21no—
196Dic3⋊C4—42no—
197Dic3⋊C4—42no—
198C4×Dic3—22no—
199C4×Dic3—22no—
200C4×Dic3—41no—
201C4×Dic3—41no—
202C4×Dic3—48no—
203C2×C3⋊C8—41no—
204C2×C3⋊C8—41no—
205C2×C3⋊C8—41no—
206C22×D4—21no—
207C22×D4—31no—
208C22×D4—61no—
209C22×D4—61no—
210C22×D4—61no—
211C22×D4C4×D311yescharacteristic
212C23×C4—21no—
213C23×C4—31no—
214C23×C4—31no—
215C23×C4—61no—
216C23×C4C22×D311yescharacteristic
217C22×C8—21no—
218C22×C8—61no—
219#127—122no—
220#121—21no—
221#121—21no—
222#121—21no—
223#121—21no—
224#121—61no—
225#121—61no—
226#121—61no—
227#121—61no—
228#121—122no—
229#121—122no—
230#121—122no—
231#121—122no—
232C2×C4⋊C4—31no—
233C2×C4⋊C4—61no—
234C2×C4⋊C4—61no—
235C2×C4⋊C4C4×D311yescharacteristic
236C2×C22⋊C4—21no—
237C2×C22⋊C4—61no—
238C2×C22⋊C4—61no—
239C2×C22⋊C4—61no—
240C2×C22⋊C4—61no—
241C2×C22⋊C4—61no—
242C2×C42—31no—
243C2×C42—31no—
244C2×C42—31no—
245C2×C42—31no—
246C2×C42—64no—
247C2×C42—122no—
248C2×C42—122no—
249C2×C42—122no—
250C4⋊C8—122no—
251C22⋊C8—21no—
252C22⋊C8—21no—
253C22⋊C8—61no—
254C22⋊C8—61no—
255#95—21no—
256#95—32no—
257#95—64no—
258#95—61no—
259#95#6712yescomplemented
260C23×C3—21no—
261C23×C3—21no—
262C23×C3—21no—
263C23×C3—21no—
264C23×C3—41no—
265C23×C3—41no—
266C23×C3—41no—
267C23×C3—41no—
268C23×C3—41no—
269C23×C3—41no—
270C23×C3—41no—
271C23×C3C2×C22⋊C411yescharacteristic
272C23×C3#12111yescharacteristic
273C23×C3#12111yescharacteristic
274C23×C3C2×D811yescharacteristic
275C23×C3C2×SD1611yescharacteristic
276C23×C3AGL(1,ℤ/8)11yescharacteristic
277C23×C3AGL(1,ℤ/8)11yescharacteristic
278C22×D3—41no—
279C22×D3—41no—
280C22×D3—41no—
281C22×D3—41no—
282C22×D3—41no—
283C22×D3—41no—
284C22×D3—41no—
285C3×D4—81no—
286C3×D4—81no—
287C3×D4—81no—
288C3×D4—81no—
289C3×D4—81no—
290C3×D4—81no—
291C2×C3×C4—21no—
292C2×C3×C4—21no—
293C2×C3×C4—21no—
294C2×C3×C4—21no—
295C2×C3×C4—24no—
296C2×C3×C4—41no—
297C2×C3×C4—41no—
298C2×C3×C4—41no—
299C2×C3×C4—41no—
300C2×C3×C4—42no—
301C2×C3×C4—42no—
302C2×C3×C4—42no—
303C2×C3×C4—42no—
304C2×C3×C4—42no—
305C2×C3×C4—42no—
306C2×C3×C4—41no—
307C2×C3×C4—41no—
308C2×C3×C4—41no—
309C2×C3×C4C4×D411yescharacteristic
310C2×C3×C4C4×D411yescharacteristic
311C2×C3×C4#12411yescharacteristic
312C2×C3×C4#12711yescharacteristic
313#67—84no—
314#67—81no—
315#67—81no—
316#67—81no—
317#67—81no—
318#67—84no—
319C2×Dic3—21no—
320C2×Dic3—21no—
321C2×Dic3—21no—
322C2×Dic3—21no—
323C2×Dic3—24no—
324C2×Dic3—24no—
325C2×Dic3—24no—
326C2×Dic3—21no—
327C2×Dic3—21no—
328C2×Dic3—21no—
329C2×Dic3—42no—
330C2×Dic3—42no—
331C2×Dic3—42no—
332C2×Dic3—42no—
333C2×Dic3—42no—
334C2×Dic3—42no—
335C2×Dic3—42no—
336C2×Dic3—42no—
337C2×Dic3—41no—
338C2×Dic3—41no—
339C2×Dic3—41no—
340C2×Dic3#9912yes—
341C2×Dic3#10212yes—
342C2×Dic3#10212yes—
343C2×Dic3#10412yes—
344C2×Dic3#10412yes—
345D12—81no—
346D12—81no—
347C3×C8—81no—
348C3⋊C8—81no—
349C24—21no—
350C24—61no—
351C24D3×D411yescharacteristic
352C2×D4—21no—
353C2×D4—21no—
354C2×D4—41no—
355C2×D4—41no—
356C2×D4—41no—
357C2×D4—41no—
358C2×D4—41no—
359C2×D4—41no—
360C2×D4—41no—
361C2×D4—41no—
362C2×D4—61no—
363C2×D4—61no—
364C2×D4—121no—
365C2×D4—121no—
366C2×D4—122no—
367C2×D4—122no—
368C2×D4—122no—
369C2×D4—122no—
370C2×D4—122no—
371C2×D4—122no—
372C2×D4—121no—
373C2×D4—121no—
374C2×D4—121no—
375C2×D4—121no—
376C2×D4—121no—
377C2×D4—121no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—21no—
382C22×C4—31no—
383C22×C4—31no—
384C22×C4—31no—
385C22×C4—32no—
386C22×C4—32no—
387C22×C4—31no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—62no—
395C22×C4—62no—
396C22×C4—62no—
397C22×C4—62no—
398C22×C4—62no—
399C22×C4—62no—
400C22×C4—61no—
401C22×C4—61no—
402C22×C4—61no—
403C22×C4—121no—
404C22×C4—121no—
405C22×C4—121no—
406C22×C4—121no—
407C22×C4—121no—
408C22×C4C2×C4×D311yescharacteristic
409C22×C4#23511yescharacteristic
410C22×C4D3×D411yescharacteristic
411C22×C4D3×D411yescharacteristic
412C22×C4#23711yescharacteristic
413C22×C4C2×#6711yescharacteristic
414C2×C8—41no—
415C2×C8—41no—
416C2×C8—41no—
417C2×C8—121no—
418C2×C8—121no—
419C2×C8—121no—
420C4⋊C4—21no—
421C4⋊C4—21no—
422C4⋊C4—61no—
423C4⋊C4—61no—
424C4⋊C4—122no—
425C4⋊C4—122no—
426C22⋊C4—41no—
427C22⋊C4—41no—
428C22⋊C4—122no—
429C22⋊C4—122no—
430C22⋊C4—122no—
431C22⋊C4—122no—
432C22⋊C4—121no—
433C22⋊C4—121no—
434C42—62no—
435C42—62no—
436C42—121no—
437C42—121no—
438C42—128no—
439C22×C3—21no—
440C22×C3—21no—
441C22×C3—21no—
442C22×C3—21no—
443C22×C3—21no—
444C22×C3—21no—
445C22×C3—21no—
446C22×C3—21no—
447C22×C3—21no—
448C22×C3—21no—
449C22×C3—21no—
450C22×C3—21no—
451C22×C3—81no—
452C22×C3—81no—
453C22×C3—81no—
454C22×C3—81no—
455C22×C3—81no—
456C22×C3—81no—
457C22×C3—81no—
458C22×C3#39011yescharacteristic
459C22×C3C2×#9911yescharacteristic
460C22×C3#41011yescharacteristic
461C22×C3C2×#10211yescharacteristic
462C22×C3#41811yescharacteristic
463C22×C3C2×#10411yescharacteristic
464C22×C3C2×#10411yescharacteristic
465C22×C3#45911yescharacteristic
466C22×C3#45911yescharacteristic
467C22×C3#46011yescharacteristic
468C22×C3#46011yescharacteristic
469C2×D3—81no—
470C2×D3—81no—
471C2×D3—81no—
472C2×D3—81no—
473C2×D3—81no—
474C2×D3—81no—
475C2×D3—81no—
476C3×C4—21no—
477C3×C4—21no—
478C3×C4—21no—
479C3×C4—21no—
480C3×C4—44no—
481C3×C4—81no—
482Dic3—22no—
483Dic3—22no—
484Dic3—44no—
485Dic3—41no—
486Dic3—81no—
487Dic3#32714yescomplemented
488C23—21no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23—41no—
498C23—41no—
499C23—121no—
500C23—121no—
501C23—121no—
502C23—121no—
503C23—121no—
504C23—121no—
505C23—121no—
506C23D3×C22⋊C411yescharacteristic
507C23#88411yescharacteristic
508C23D3×D811yescharacteristic
509C23#91111yescharacteristic
510C23D3×SD1611yescharacteristic
511C23#91411yescharacteristic
512C23#93911yescharacteristic
513D4—81no—
514D4—81no—
515D4—81no—
516D4—81no—
517D4—81no—
518D4—81no—
519D4—241no—
520D4—241no—
521D4—244no—
522D4—244no—
523D4—241no—
524D4—241no—
525D4—241no—
526D4—241no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—24no—
532C2×C4—32no—
533C2×C4—32no—
534C2×C4—32no—
535C2×C4—32no—
536C2×C4—32no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—64no—
551C2×C4—64no—
552C2×C4—61no—
553C2×C4—61no—
554C2×C4—61no—
555C2×C4—61no—
556C2×C4—64no—
557C2×C4—61no—
558C2×C4—61no—
559C2×C4—61no—
560C2×C4—122no—
561C2×C4—122no—
562C2×C4—122no—
563C2×C4—122no—
564C2×C4—122no—
565C2×C4—122no—
566C2×C4—122no—
567C2×C4—122no—
568C2×C4—121no—
569C2×C4—121no—
570C2×C4—121no—
571C2×C4#67⋊C411yescharacteristic
572C2×C4#88511yescharacteristic
573C2×C4C4×#6711yescharacteristic
574C2×C4#94011yescharacteristic
575C8—81no—
576C8—241no—
577C2×C3—21no—
578C2×C3—21no—
579C2×C3—21no—
580C2×C3—21no—
581C2×C3—161no—
582C2×C3C2×#32711yescharacteristic
583C2×C3#150211yescharacteristic
584C2×C3#150511yescharacteristic
585C2×C3#150711yescharacteristic
586C2×C3#196211yescharacteristic
587C2×C3#196411yescharacteristic
588C2×C3#198211yescharacteristic
589D3—161no—
590C22—21no—
591C22—21no—
592C22—21no—
593C22—21no—
594C22—21no—
595C22—21no—
596C22—21no—
597C22—21no—
598C22—21no—
599C22—21no—
600C22—21no—
601C22—21no—
602C22—81no—
603C22—81no—
604C22—81no—
605C22—81no—
606C22—81no—
607C22—81no—
608C22—81no—
609C22—241no—
610C22—241no—
611C22—241no—
612C22—241no—
613C22—241no—
614C22—241no—
615C22—241no—
616C22#474011yescharacteristic
617C22D3×#9911yescharacteristic
618C22#476211yescharacteristic
619C22D3×#10211yescharacteristic
620C22#477011yescharacteristic
621C22#478611yescharacteristic
622C22D3×#10411yescharacteristic
623C22D3×#10411yescharacteristic
624C22#495711yescharacteristic
625C22#515011yescharacteristic
626C22#516311yescharacteristic
627C4—21no—
628C4—21no—
629C4—21no—
630C4—21no—
631C4—34no—
632C4—44no—
633C4—62no—
634C4—62no—
635C4—81no—
636C4—124no—
637C4—121no—
638C4—241no—
639C3#971211yescharacteristic, complemented, Sylow
640C2—21no—
641C2—21no—
642C2—21no—
643C2—21no—
644C2—161no—
645C2—481no—
646C2#6827711yescharacteristic
647C2#6827811yescharacteristic
648C2D3×#32711yescharacteristic
649C2#6830311yescharacteristic
650C2#7003011yescharacteristic
651C2#7016111yescharacteristic
652C2#7035611yescharacteristic
subgroupG/H
centerC23D3×C22⋊C4
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC2×C3×#327C2
socleC23×C3C2×C22⋊C4
prime radical#82700C2
cosocleC23×C3×C4C23
hypercenterC2×#327D3
hypocenterC3#9712
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#327

Complements

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

NH ≅ G/Ntype
2. C2×C3×#327C2semidirect
3. C2×#4537C2semidirect
4. C2×#4470C2semidirect
5. #69908C2semidirect
6. Dic3×#95C2semidirect
7. C2×#4451C2semidirect
10. C2×#940C4semidirect
18. #5221C4semidirect
23. #4988C4semidirect
24. #4975C4semidirect
37. C2×#327D3semidirect
86. C3×#95D4semidirect
89. C3⋊#95D4semidirect
259. #95#67semidirect
487. Dic3#327semidirect
639. C3#9712semidirect

Maximal subgroups

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

subgroupindex
#827002
C2×C3×#3272
C2×#45372
C2×#44702
#699082
Dic3×#952
C2×#44512
#97123