Finite Group Databasebeta

← #10690749

Subgroups — #10690749

Order 768.

1062 conjugacy classes of subgroups.

The subgroup lattice has 744 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82723C211yesmaximal, characteristic
2C2×D3×C22⋊C8C211yesmaximal, characteristic, complemented
3C2×C3×#327C211yesmaximal, characteristic, complemented
4C2×#4537C211yesmaximal, characteristic, complemented
5(C2×D3)⋊#95C211yesmaximal, characteristic, complemented
6#67675C211yesmaximal, characteristic, complemented
7#67612C211yesmaximal, characteristic, complemented
8#9113—31nomaximal, Sylow
9C23×C4×D3C2211yescharacteristic
10C2×C3×#121C2211yescharacteristic
11C2×#938C411yescharacteristic, complemented
12C2×#938C411yescharacteristic, complemented
13C2×#937—21no—
14C22×(C2×D3)⋊C4—21no—
15C22×D3×C8—21no—
16C2×C3×C22⋊C8C2211yescharacteristic
17C2×C3×#95C2211yescharacteristic
18#5220—21no—
19#5212C2211yescharacteristic
20C2×C3⋊#95—21no—
21C2×C3⋊#95C2211yescharacteristic
22C2×#830C2211yescharacteristic
23C2×(C2×D3)⋊C8—21no—
24#4987C411yescharacteristic, complemented
25#4987C411yescharacteristic, complemented
26#4956—21no—
27#4956—21no—
28D3×C22⋊C8—22no—
29D3×C22⋊C8—22no—
30C3×#327—22no—
31#4537—22no—
32C22×C22⋊C8—31no—
33#2847—31no—
34C2×#327—31no—
35C2×#327D311yescharacteristic, complemented
36C22⋊#95—31no—
37#1294—31no—
38#1294—31no—
39C24×D3D411yescharacteristic
40C22×C3×D4—21no—
41C22×C3×D4C2×C411yescharacteristic
42C23×C3×C4—21no—
43C23×C3×C4C2311yescharacteristic
44C22×#67—21no—
45C22×#67—21no—
46C23×Dic3—21no—
47C23×Dic3D411yescharacteristic
48C22×C4×D3—21no—
49C22×C4×D3—21no—
50C22×C4×D3—21no—
51C22×C4×D3—21no—
52C22×C4×D3—42no—
53C22×C4×D3—42no—
54C22×C4×D3C2×C411yescharacteristic
55C22×C4×D3C2×C411yescharacteristic
56C22×C4×D3D411yescharacteristic
57C22×C4×D3D411yescharacteristic
58C22×C3×C8—21no—
59C3×#121—21no—
60C3×#121—21no—
61C3×#121—21no—
62C3×#121—21no—
63C2×C3×C4⋊C4C2×C411yescharacteristic
64C2×C3×C22⋊C4—21no—
65C2×#217—21no—
66C2×#217—21no—
67C2×#217—21no—
68#938—42no—
69#938—42no—
70#937—42no—
71#937—42no—
72C2×(C2×D3)⋊C4—24no—
73C2×(C2×D3)⋊C4—21no—
74C2×(C2×D3)⋊C4—21no—
75C2×(C2×D3)⋊C4—42no—
76C2×(C2×D3)⋊C4—42no—
77C2×C4⋊Dic3C2×C411yescharacteristic
78C2×C4⋊Dic3C2×C411yescharacteristic
79C22×C3⋊C8—21no—
80C2×D3×C8—24no—
81C2×D3×C8—42no—
82C2×D3×C8—42no—
83C3×C22⋊C8—21no—
84C3×C22⋊C8D412yescomplemented
85C3×#95—21no—
86C3×#95—21no—
87C3⋊#95—21no—
88C3⋊#95—21no—
89C3⋊#95—21no—
90C3⋊#95—21no—
91C3⋊#95—21no—
92C3⋊#95—42no—
93#830—21no—
94#830D412yescomplemented
95(C2×D3)⋊C8—24no—
96C24×C4—31no—
97C23×C8—61no—
98C2×#121—31no—
99C2×#121—31no—
100C2×#121C2×D311yescharacteristic
101C2×#120—61no—
102C22×C22⋊C4—61no—
103C2×C22⋊C8—31no—
104C2×C22⋊C8—62no—
105C2×C22⋊C8—61no—
106C2×C22⋊C8—62no—
107C2×C22⋊C8C2×D311yescharacteristic
108#386—61no—
109#386—61no—
110#386—61no—
111#380—31no—
112#380—31no—
113#380—31no—
114C2×#95—31no—
115C2×#95—61no—
116C2×#95C2×D311yescharacteristic
117#327—22no—
118#327—62no—
119C24×C3—21no—
120C24×C3C2×D411yescharacteristic
121C23×D3—21no—
122C23×D3—21no—
123C23×D3—21no—
124C23×D3—21no—
125C23×D3—22no—
126C23×D3—22no—
127C23×D3—41no—
128C23×D3—41no—
129C23×D3C22⋊C411yescharacteristic
130C23×D3C22⋊C411yescharacteristic
131C23×D3D812yes—
132C23×D3SD1612yes—
133C2×C3×D4—21no—
134C2×C3×D4—21no—
135C2×C3×D4—41no—
136C2×C3×D4—41no—
137C2×C3×D4—41no—
138C2×C3×D4—41no—
139C2×C3×D4—41no—
140C2×C3×D4—41no—
141C2×C3×D4—41no—
142C2×C3×D4—41no—
143C22×C3×C4—22no—
144C22×C3×C4—21no—
145C22×C3×C4—41no—
146C22×C3×C4—41no—
147C22×C3×C4—41no—
148C22×C3×C4—41no—
149C22×C3×C4—41no—
150C22×C3×C4—41no—
151C22×C3×C4—41no—
152C22×C3×C4—41no—
153C22×C3×C4C22×C411yescharacteristic
154C22×C3×C4C2×D411yescharacteristic
155C22×C3×C4C2×D411yescharacteristic
156C22×C3×C4C2×D411yescharacteristic
157C22×C3×C4#4111yescharacteristic
158C22×C3×C4#4111yescharacteristic
159C2×#67—42no—
160C2×#67—42no—
161C2×#67—42no—
162C2×#67—42no—
163C2×#67—42no—
164C2×#67—42no—
165C22×Dic3—21no—
166C22×Dic3—21no—
167C22×Dic3—22no—
168C22×Dic3—22no—
169C22×Dic3—21no—
170C22×Dic3—21no—
171C22×Dic3—41no—
172C22×Dic3—41no—
173C22×Dic3—41no—
174C22×Dic3—41no—
175C22×Dic3—41no—
176C22×Dic3—41no—
177C22×Dic3—41no—
178C22×Dic3C22⋊C411yescharacteristic
179C22×Dic3C22⋊C411yescharacteristic
180C2×C4×D3—22no—
181C2×C4×D3—21no—
182C2×C4×D3—21no—
183C2×C4×D3—22no—
184C2×C4×D3—22no—
185C2×C4×D3—21no—
186C2×C4×D3—21no—
187C2×C4×D3—21no—
188C2×C4×D3—21no—
189C2×C4×D3—21no—
190C2×C4×D3—21no—
191C2×C4×D3—22no—
192C2×C4×D3—44no—
193C2×C4×D3—44no—
194C2×C4×D3—44no—
195C2×C4×D3—44no—
196C2×C4×D3—41no—
197C2×C4×D3—41no—
198C2×C4×D3—41no—
199C2×C4×D3—41no—
200C2×C3×C8—22no—
201C2×C3×C8—41no—
202C2×C3×C8—41no—
203C3×C4⋊C4—21no—
204C3×C4⋊C4—21no—
205C3×C22⋊C4—41no—
206C3×C22⋊C4—41no—
207#217—42no—
208#217—41no—
209#217—41no—
210#217—42no—
211(C2×D3)⋊C4—48no—
212(C2×D3)⋊C4—42no—
213(C2×D3)⋊C4—42no—
214C4⋊Dic3—41no—
215C4⋊Dic3—41no—
216C2×C3⋊C8—22no—
217C2×C3⋊C8—41no—
218C2×C3⋊C8—41no—
219D3×C8—48no—
220C25—31no—
221C22×D4—21no—
222C22×D4—61no—
223C22×D4—61no—
224C22×D4C4×D311yescharacteristic
225C23×C4—21no—
226C23×C4—31no—
227C23×C4—31no—
228C23×C4—31no—
229C23×C4—31no—
230C23×C4—31no—
231C23×C4—61no—
232C23×C4—61no—
233C23×C4—61no—
234C23×C4—61no—
235C23×C4—61no—
236C23×C4—122no—
237C23×C4—122no—
238C23×C4C22×D311yescharacteristic
239C22×C8—21no—
240C22×C8—64no—
241C22×C8—61no—
242C22×C8—122no—
243C22×C8—122no—
244#121—21no—
245#121—21no—
246#121—21no—
247#121—21no—
248#121—122no—
249#121—122no—
250#120—122no—
251#120—122no—
252C2×C4⋊C4—31no—
253C2×C4⋊C4—31no—
254C2×C4⋊C4C4×D311yescharacteristic
255C2×C22⋊C4—21no—
256C2×C22⋊C4—61no—
257C2×C22⋊C4—61no—
258C2×C22⋊C4—64no—
259C2×C22⋊C4—61no—
260C2×C22⋊C4—61no—
261C2×C22⋊C4—61no—
262C2×C22⋊C4—122no—
263C2×C22⋊C4—122no—
264C22⋊C8—21no—
265C22⋊C8—32no—
266C22⋊C8—64no—
267C22⋊C8—61no—
268C22⋊C8#6712yescomplemented
269#95—21no—
270#95—21no—
271#95—61no—
272#95—61no—
273#95—61no—
274#95—61no—
275#95—61no—
276#95—122no—
277C23×C3—21no—
278C23×C3—21no—
279C23×C3—21no—
280C23×C3—21no—
281C23×C3—41no—
282C23×C3—41no—
283C23×C3—41no—
284C23×C3—41no—
285C23×C3—41no—
286C23×C3—41no—
287C23×C3—41no—
288C23×C3C2×C22⋊C411yescharacteristic
289C23×C3#12111yescharacteristic
290C23×C3#12111yescharacteristic
291C23×C3C2×D811yescharacteristic
292C23×C3C2×SD1611yescharacteristic
293C23×C3#13511yescharacteristic
294C23×C3#13511yescharacteristic
295C22×D3—21no—
296C22×D3—21no—
297C22×D3—21no—
298C22×D3—21no—
299C22×D3—21no—
300C22×D3—21no—
301C22×D3—21no—
302C22×D3—21no—
303C22×D3—21no—
304C22×D3—21no—
305C22×D3—22no—
306C22×D3—22no—
307C22×D3—22no—
308C22×D3—22no—
309C22×D3—24no—
310C22×D3—24no—
311C22×D3—42no—
312C22×D3—42no—
313C22×D3—42no—
314C22×D3—42no—
315C22×D3—42no—
316C22×D3—42no—
317C22×D3—42no—
318C22×D3—42no—
319C22×D3—42no—
320C22×D3—42no—
321C22×D3#9912yes—
322C22×D3#10212yes—
323C22×D3#10212yes—
324C22×D3#10412yes—
325C3×D4—81no—
326C3×D4—81no—
327C3×D4—81no—
328C3×D4—81no—
329C3×D4—81no—
330C3×D4—81no—
331C2×C3×C4—21no—
332C2×C3×C4—21no—
333C2×C3×C4—21no—
334C2×C3×C4—21no—
335C2×C3×C4—41no—
336C2×C3×C4—41no—
337C2×C3×C4—41no—
338C2×C3×C4—41no—
339C2×C3×C4—42no—
340C2×C3×C4—42no—
341C2×C3×C4—42no—
342C2×C3×C4—41no—
343C2×C3×C4—41no—
344C2×C3×C4—41no—
345C2×C3×C4—81no—
346C2×C3×C4—81no—
347C2×C3×C4—81no—
348C2×C3×C4—81no—
349C2×C3×C4C4×D411yescharacteristic
350C2×C3×C4C4×D411yescharacteristic
351C2×C3×C4#12011yescharacteristic
352C2×C3×C4#12311yescharacteristic
353#67—84no—
354#67—84no—
355C2×Dic3—21no—
356C2×Dic3—21no—
357C2×Dic3—21no—
358C2×Dic3—21no—
359C2×Dic3—21no—
360C2×Dic3—42no—
361C2×Dic3—42no—
362C2×Dic3—42no—
363C2×Dic3—42no—
364C2×Dic3—42no—
365C2×Dic3—41no—
366C2×Dic3—41no—
367C2×Dic3—41no—
368C2×Dic3—41no—
369C2×Dic3—41no—
370C2×Dic3—41no—
371C2×Dic3—81no—
372C2×Dic3—81no—
373C2×Dic3—81no—
374C2×Dic3—81no—
375C2×Dic3#9912yes—
376C4×D3—24no—
377C4×D3—24no—
378C4×D3—42no—
379C4×D3—42no—
380C4×D3—42no—
381C4×D3—42no—
382C4×D3—42no—
383C4×D3—42no—
384C3×C8—42no—
385C3⋊C8—42no—
386C24—21no—
387C24—31no—
388C24—31no—
389C24—32no—
390C24—32no—
391C24—62no—
392C24—62no—
393C24—61no—
394C24—61no—
395C24—61no—
396C24—61no—
397C24—121no—
398C24—121no—
399C24D3×D411yescharacteristic
400C2×D4—21no—
401C2×D4—21no—
402C2×D4—41no—
403C2×D4—41no—
404C2×D4—41no—
405C2×D4—41no—
406C2×D4—41no—
407C2×D4—41no—
408C2×D4—41no—
409C2×D4—41no—
410C2×D4—122no—
411C2×D4—122no—
412C2×D4—122no—
413C2×D4—122no—
414C2×D4—122no—
415C2×D4—122no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—31no—
419C22×C4—31no—
420C22×C4—41no—
421C22×C4—41no—
422C22×C4—41no—
423C22×C4—41no—
424C22×C4—41no—
425C22×C4—41no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—62no—
429C22×C4—62no—
430C22×C4—62no—
431C22×C4—62no—
432C22×C4—62no—
433C22×C4—61no—
434C22×C4—61no—
435C22×C4—61no—
436C22×C4—61no—
437C22×C4—61no—
438C22×C4—61no—
439C22×C4—61no—
440C22×C4—61no—
441C22×C4—61no—
442C22×C4—61no—
443C22×C4—62no—
444C22×C4—61no—
445C22×C4—61no—
446C22×C4—121no—
447C22×C4—121no—
448C22×C4—121no—
449C22×C4—121no—
450C22×C4—124no—
451C22×C4—124no—
452C22×C4—124no—
453C22×C4—124no—
454C22×C4—121no—
455C22×C4—121no—
456C22×C4—121no—
457C22×C4—121no—
458C22×C4—121no—
459C22×C4—121no—
460C22×C4—121no—
461C22×C4C2×C4×D311yescharacteristic
462C22×C4#23511yescharacteristic
463C22×C4D3×D411yescharacteristic
464C22×C4D3×D411yescharacteristic
465C22×C4#23711yescharacteristic
466C22×C4C2×#6711yescharacteristic
467C2×C8—22no—
468C2×C8—41no—
469C2×C8—41no—
470C2×C8—62no—
471C2×C8—128no—
472C2×C8—121no—
473C2×C8—121no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—121no—
477C4⋊C4—121no—
478C22⋊C4—41no—
479C22⋊C4—41no—
480C22⋊C4—122no—
481C22⋊C4—122no—
482C22⋊C4—122no—
483C22⋊C4—128no—
484C22⋊C4—122no—
485C22⋊C4—121no—
486C22⋊C4—121no—
487C22×C3—21no—
488C22×C3—21no—
489C22×C3—21no—
490C22×C3—21no—
491C22×C3—21no—
492C22×C3—21no—
493C22×C3—21no—
494C22×C3—21no—
495C22×C3—21no—
496C22×C3—21no—
497C22×C3—21no—
498C22×C3—21no—
499C22×C3—81no—
500C22×C3—81no—
501C22×C3—81no—
502C22×C3—81no—
503C22×C3—81no—
504C22×C3—81no—
505C22×C3—81no—
506C22×C3#38611yescharacteristic
507C22×C3C2×#9911yescharacteristic
508C22×C3C2×#9911yescharacteristic
509C22×C3C2×#10211yescharacteristic
510C22×C3#41611yescharacteristic
511C22×C3C2×#10411yescharacteristic
512C22×C3#42111yescharacteristic
513C22×C3#46511yescharacteristic
514C22×C3#46511yescharacteristic
515C22×C3#46611yescharacteristic
516C22×C3#46611yescharacteristic
517C2×D3—22no—
518C2×D3—22no—
519C2×D3—22no—
520C2×D3—22no—
521C2×D3—22no—
522C2×D3—22no—
523C2×D3—22no—
524C2×D3—22no—
525C2×D3—22no—
526C2×D3—44no—
527C2×D3—44no—
528C2×D3—44no—
529C2×D3—44no—
530C2×D3—41no—
531C2×D3—41no—
532C2×D3—41no—
533C2×D3—41no—
534C2×D3#32714yescomplemented
535C3×C4—21no—
536C3×C4—21no—
537C3×C4—21no—
538C3×C4—21no—
539C3×C4—82no—
540C3×C4—81no—
541Dic3—22no—
542Dic3—41no—
543Dic3—82no—
544Dic3—81no—
545Dic3—81no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—32no—
551C23—32no—
552C23—32no—
553C23—32no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23—41no—
558C23—41no—
559C23—41no—
560C23—41no—
561C23—64no—
562C23—64no—
563C23—61no—
564C23—61no—
565C23—61no—
566C23—61no—
567C23—62no—
568C23—62no—
569C23—61no—
570C23—61no—
571C23—61no—
572C23—61no—
573C23—62no—
574C23—62no—
575C23—61no—
576C23—61no—
577C23—122no—
578C23—122no—
579C23—122no—
580C23—122no—
581C23—122no—
582C23—122no—
583C23—122no—
584C23—122no—
585C23—122no—
586C23—122no—
587C23D3×C22⋊C411yescharacteristic
588C23#88411yescharacteristic
589C23D3×D811yescharacteristic
590C23#91211yescharacteristic
591C23D3×SD1611yescharacteristic
592C23#91611yescharacteristic
593C23#93811yescharacteristic
594D4—81no—
595D4—81no—
596D4—81no—
597D4—81no—
598D4—81no—
599D4—81no—
600D4—244no—
601D4—244no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—32no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—64no—
618C2×C4—64no—
619C2×C4—61no—
620C2×C4—61no—
621C2×C4—61no—
622C2×C4—61no—
623C2×C4—61no—
624C2×C4—81no—
625C2×C4—81no—
626C2×C4—81no—
627C2×C4—81no—
628C2×C4—122no—
629C2×C4—122no—
630C2×C4—122no—
631C2×C4—122no—
632C2×C4—122no—
633C2×C4—122no—
634C2×C4—122no—
635C2×C4—122no—
636C2×C4—122no—
637C2×C4—122no—
638C2×C4—122no—
639C2×C4—121no—
640C2×C4—121no—
641C2×C4—121no—
642C2×C4—121no—
643C2×C4—121no—
644C2×C4—121no—
645C2×C4—241no—
646C2×C4—241no—
647C2×C4—241no—
648C2×C4—241no—
649C2×C4#67⋊C411yescharacteristic
650C2×C4#88311yescharacteristic
651C2×C4C4×#6711yescharacteristic
652C2×C4#93711yescharacteristic
653C8—42no—
654C8—122no—
655C2×C3—21no—
656C2×C3—21no—
657C2×C3—21no—
658C2×C3—21no—
659C2×C3—161no—
660C2×C3C2×#32711yescharacteristic
661C2×C3#150111yescharacteristic
662C2×C3#151611yescharacteristic
663C2×C3#151911yescharacteristic
664C2×C3#187111yescharacteristic
665C2×C3#189311yescharacteristic
666C2×C3#189811yescharacteristic
667D3—24no—
668D3—42no—
669C22—21no—
670C22—21no—
671C22—21no—
672C22—21no—
673C22—21no—
674C22—21no—
675C22—21no—
676C22—21no—
677C22—21no—
678C22—21no—
679C22—21no—
680C22—21no—
681C22—34no—
682C22—62no—
683C22—62no—
684C22—62no—
685C22—62no—
686C22—62no—
687C22—62no—
688C22—62no—
689C22—62no—
690C22—62no—
691C22—81no—
692C22—81no—
693C22—81no—
694C22—81no—
695C22—81no—
696C22—81no—
697C22—81no—
698C22—124no—
699C22—124no—
700C22—124no—
701C22—124no—
702C22—121no—
703C22—121no—
704C22—121no—
705C22—121no—
706C22D3×#9911yescharacteristic
707C22D3×#9911yescharacteristic
708C22D3×#10211yescharacteristic
709C22#477211yescharacteristic
710C22#477511yescharacteristic
711C22#477811yescharacteristic
712C22D3×#10411yescharacteristic
713C22#482111yescharacteristic
714C22#495611yescharacteristic
715C22#515711yescharacteristic
716C22#517111yescharacteristic
717C4—21no—
718C4—21no—
719C4—21no—
720C4—21no—
721C4—62no—
722C4—82no—
723C4—81no—
724C4—121no—
725C4—242no—
726C4—241no—
727C4—241no—
728C3#911311yescharacteristic, complemented, Sylow
729C2—21no—
730C2—21no—
731C2—21no—
732C2—21no—
733C2—64no—
734C2—122no—
735C2—161no—
736C2D3×#32711yescharacteristic
737C2#6830111yescharacteristic
738C2#6830611yescharacteristic
739C2#6830711yescharacteristic
740C2#7003611yescharacteristic
741C2#7020211yescharacteristic
742C2#7038111yescharacteristic
subgroupG/H
centerC23D3×C22⋊C4
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC2×C3×#327C2
socleC23×C3C2×C22⋊C4
prime radical#82723C2
cosocleC23×C3×C4C23
hypercenterC2×#327D3
hypocenterC3#9113
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×D3×C22⋊C8C2semidirect
3. C2×C3×#327C2semidirect
4. C2×#4537C2semidirect
5. (C2×D3)⋊#95C2semidirect
6. #67675C2semidirect
7. #67612C2semidirect
11. C2×#938C4semidirect
12. C2×#938C4semidirect
24. #4987C4semidirect
25. #4987C4semidirect
35. C2×#327D3semidirect
84. C3×C22⋊C8D4semidirect
94. #830D4semidirect
268. C22⋊C8#67semidirect
534. C2×D3#327semidirect
728. C3#9113semidirect

Maximal subgroups

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

subgroupindex
#827232
C2×D3×C22⋊C82
C2×C3×#3272
C2×#45372
(C2×D3)⋊#952
#676752
#676122
#91133