Finite Group Databasebeta

← #10690769

Subgroups — #10690769

Order 768.

1030 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×#4726C211yesmaximal, 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#9117—31nomaximal, Sylow
9C23×C4×D3C2211yescharacteristic
10C2×C3×#121C2211yescharacteristic
11C2×#938C411yescharacteristic, complemented
12C2×#938C411yescharacteristic, complemented
13C2×#937—21no—
14C22×(C2×D3)⋊C4—21no—
15C22×ΓL(1,𝔽25)—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—
28#4726—22no—
29#4726—22no—
30C3×#327—22no—
31#4537—22no—
32C2×#407—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×ΓL(1,𝔽25)—24no—
81C2×ΓL(1,𝔽25)—42no—
82C2×ΓL(1,𝔽25)—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—
97C22×M16—61no—
98C2×#121—31no—
99C2×#121—31no—
100C2×#121C2×D311yescharacteristic
101C2×#120—61no—
102C22×C22⋊C4—61no—
103#407—62no—
104#407—62no—
105C2×C22⋊C8—31no—
106C2×C22⋊C8—61no—
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—22no—
124C23×D3—22no—
125C23×D3—21no—
126C23×D3—21no—
127C23×D3—21no—
128C23×D3—21no—
129C23×D3—41no—
130C23×D3—41no—
131C23×D3C22⋊C411yescharacteristic
132C23×D3C22⋊C411yescharacteristic
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—
219ΓL(1,𝔽25)—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
239C2×M16—64no—
240C2×M16—122no—
241C2×M16—122no—
242C22×C8—21no—
243C22×C8—61no—
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×C3AGL(1,ℤ/8)11yescharacteristic
292C23×C3AGL(1,ℤ/8)11yescharacteristic
293C23×C3AGL(1,ℤ/8)11yescharacteristic
294C23×C3#13711yescharacteristic
295C22×D3—24no—
296C22×D3—24no—
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—21no—
306C22×D3—21no—
307C22×D3—21no—
308C22×D3—21no—
309C22×D3—42no—
310C22×D3—42no—
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—41no—
320C22×D3—41no—
321C22×D3—41no—
322C22×D3—41no—
323C22×D3#9912yes—
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—62no—
390C24—62no—
391C24—61no—
392C24—61no—
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
467M16—128no—
468C2×C8—22no—
469C2×C8—41no—
470C2×C8—41no—
471C2×C8—62no—
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—128no—
482C22⋊C4—122no—
483C22⋊C4—122no—
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×C3#41711yescharacteristic
510C22×C3#41811yescharacteristic
511C22×C3C2×#10411yescharacteristic
512C22×C3#42111yescharacteristic
513C22×C3#46811yescharacteristic
514C22×C3#46811yescharacteristic
515C22×C3#46911yescharacteristic
516C22×C3#46911yescharacteristic
517C2×D3—22no—
518C2×D3—22no—
519C2×D3—22no—
520C2×D3—22no—
521C2×D3—22no—
522C2×D3—22no—
523C2×D3—44no—
524C2×D3—44no—
525C2×D3—44no—
526C2×D3—44no—
527C2×D3—41no—
528C2×D3—41no—
529C2×D3—41no—
530C2×D3—41no—
531C2×D3—41no—
532C2×D3—41no—
533C2×D3—41no—
534C2×D3—41no—
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—41no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23—41no—
558C23—41no—
559C23—64no—
560C23—64no—
561C23—61no—
562C23—61no—
563C23—61no—
564C23—61no—
565C23—61no—
566C23—61no—
567C23—61no—
568C23—61no—
569C23—61no—
570C23—61no—
571C23—61no—
572C23—61no—
573C23—122no—
574C23—122no—
575C23—122no—
576C23—122no—
577C23—122no—
578C23—122no—
579C23—122no—
580C23—122no—
581C23—122no—
582C23—122no—
583C23—121no—
584C23—121no—
585C23—121no—
586C23—121no—
587C23D3×C22⋊C411yescharacteristic
588C23#88411yescharacteristic
589C23#91111yescharacteristic
590C23#91111yescharacteristic
591C23#91411yescharacteristic
592C23#91511yescharacteristic
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×C3#150411yescharacteristic
661C2×C3#150711yescharacteristic
662C2×C3#151011yescharacteristic
663C2×C3#151311yescharacteristic
664C2×C3#187111yescharacteristic
665C2×C3#189511yescharacteristic
666C2×C3#189811yescharacteristic
667D3—42no—
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—62no—
682C22—62no—
683C22—62no—
684C22—62no—
685C22—62no—
686C22—62no—
687C22—81no—
688C22—81no—
689C22—81no—
690C22—81no—
691C22—81no—
692C22—81no—
693C22—81no—
694C22—124no—
695C22—124no—
696C22—124no—
697C22—124no—
698C22—121no—
699C22—121no—
700C22—121no—
701C22—121no—
702C22—121no—
703C22—121no—
704C22—121no—
705C22—121no—
706C22D3×#9911yescharacteristic
707C22D3×#9911yescharacteristic
708C22#477011yescharacteristic
709C22#477111yescharacteristic
710C22#478011yescharacteristic
711C22#478211yescharacteristic
712C22D3×#10411yescharacteristic
713C22#482111yescharacteristic
714C22#495611yescharacteristic
715C22#515911yescharacteristic
716C22#517411yescharacteristic
717C4—21no—
718C4—21no—
719C4—21no—
720C4—21no—
721C4—62no—
722C4—82no—
723C4—81no—
724C4—121no—
725C4—242no—
726C4—241no—
727C4—241no—
728C3#911711yescharacteristic, complemented, Sylow
729C2—21no—
730C2—21no—
731C2—21no—
732C2—21no—
733C2—122no—
734C2—122no—
735C2—161no—
736C2#6830211yescharacteristic
737C2#6830311yescharacteristic
738C2#6830411yescharacteristic
739C2#6830511yescharacteristic
740C2#7003611yescharacteristic
741C2#7020311yescharacteristic
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#9117
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×#4726C2semidirect
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
728. C3#9117semidirect

Maximal subgroups

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

subgroupindex
#827232
C2×#47262
C2×C3×#3272
C2×#45372
(C2×D3)⋊#952
#676752
#676122
#91173