Finite Group Databasebeta

← #10690577

Subgroups — #10690577

Order 768.

1100 conjugacy classes of subgroups.

The subgroup lattice has 758 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4726C211yesmaximal, characteristic, complemented
2C2×#4665C211yesmaximal, characteristic
3#69955C211yesmaximal, characteristic, complemented
4C2×#4468C211yesmaximal, characteristic, complemented
5C2×#4468C211yesmaximal, characteristic, complemented
6C3×#1280C211yesmaximal, characteristic, complemented
7#67613C211yesmaximal, characteristic, complemented
8#9083—31nomaximal, Sylow
9C23×D12C411yescharacteristic
10C23×C4×D3C2211yescharacteristic
11C22×(C2×D3)⋊C4C411yescharacteristic
12C22×(C2×D3)⋊C4C411yescharacteristic
13C22×(C2×D3)⋊C4C2211yescharacteristic
14C22×ΓL(1,𝔽25)—21no—
15C2×C3×C22⋊C8C2211yescharacteristic
16C2×C3×C22⋊C8C2211yescharacteristic
17C2×C3×#95C2211yescharacteristic
18C2×C3⋊#95C411yescharacteristic
19C2×#830C2211yescharacteristic
20C2×#830C2211yescharacteristic
21C2×(C2×D3)⋊C8—21no—
22C2×(C2×D3)⋊C8—21no—
23C2×(C2×D3)⋊C8—21no—
24#4726—22no—
25#4726—22no—
26#4665—21no—
27#4665—21no—
28#4665—21no—
29#4665—21no—
30#4468—22no—
31#4468—22no—
32C2×#407—31no—
33C2×#386—31no—
34#1779—31no—
35C2×#323—31no—
36C2×#323—31no—
37#1280—31no—
38#1280D311yescharacteristic, complemented
39C24×D3C2×C411yescharacteristic
40C24×D3C2×C411yescharacteristic
41C23×C3×C4C2×C411yescharacteristic
42C23×C3×C4C2×C411yescharacteristic
43C23×C3×C4C2311yescharacteristic
44C23×Dic3C2×C411yescharacteristic
45C23×Dic3C2×C411yescharacteristic
46C22×D12—21no—
47C22×D12—21no—
48C22×D12—22no—
49C22×D12—22no—
50C22×D12—42no—
51C22×D12—42no—
52C22×C4×D3—21no—
53C22×C4×D3—21no—
54C22×C4×D3—21no—
55C22×C4×D3—21no—
56C22×C4×D3—42no—
57C22×C4×D3—42no—
58C22×C4×D3D411yescharacteristic
59C22×C4×D3D411yescharacteristic
60C22×C4×D3D411yescharacteristic
61C22×C4×D3D411yescharacteristic
62C22×C3×C8—21no—
63C22×C3×C8—21no—
64C2×(C2×D3)⋊C4—21no—
65C2×(C2×D3)⋊C4—21no—
66C2×(C2×D3)⋊C4—21no—
67C2×(C2×D3)⋊C4—21no—
68C2×(C2×D3)⋊C4—21no—
69C2×(C2×D3)⋊C4—21no—
70C2×(C2×D3)⋊C4—21no—
71C2×(C2×D3)⋊C4—21no—
72C2×(C2×D3)⋊C4—21no—
73C2×(C2×D3)⋊C4—21no—
74C2×(C2×D3)⋊C4—42no—
75C2×(C2×D3)⋊C4—42no—
76C22×C3⋊C8—21no—
77C22×C3⋊C8—21no—
78C2×ΓL(1,𝔽25)—24no—
79C2×ΓL(1,𝔽25)—42no—
80C2×ΓL(1,𝔽25)—42no—
81C3×C22⋊C8—21no—
82C3×C22⋊C8—21no—
83C3×C22⋊C8—21no—
84C3×C22⋊C8D412yescomplemented
85C3×#95—21no—
86C3×#95—21no—
87C3⋊#95—21no—
88C3⋊#95—21no—
89#830—21no—
90#830—21no—
91#830—21no—
92#830D412yescomplemented
93(C2×D3)⋊C8—24no—
94(C2×D3)⋊C8—42no—
95(C2×D3)⋊C8—42no—
96C23×D4—31no—
97C24×C4—31no—
98C22×M16—61no—
99C22×C22⋊C4—31no—
100C22×C22⋊C4—31no—
101C22×C22⋊C4—31no—
102#407—62no—
103#407—62no—
104C2×C22⋊C8—31no—
105C2×C22⋊C8—31no—
106C2×C22⋊C8—61no—
107C2×C22⋊C8—61no—
108C2×C22⋊C8—61no—
109C2×C22⋊C8C2×D311yescharacteristic
110C2×C22⋊C8C2×D311yescharacteristic
111#386—61no—
112#386—61no—
113#386—61no—
114#386—61no—
115C2×#95—31no—
116C2×#95C2×D311yescharacteristic
117#323—62no—
118#323—62no—
119C24×C3C22×C411yescharacteristic
120C23×D3—21no—
121C23×D3—21no—
122C23×D3—22no—
123C23×D3—22no—
124C23×D3—21no—
125C23×D3—21no—
126C23×D3—21no—
127C23×D3—21no—
128C23×D3—21no—
129C23×D3—21no—
130C23×D3—21no—
131C23×D3—21no—
132C23×D3—21no—
133C23×D3—21no—
134C23×D3—41no—
135C23×D3—41no—
136C23×D3—41no—
137C23×D3—41no—
138C23×D3—41no—
139C23×D3—41no—
140C23×D3C22⋊C411yescharacteristic
141C23×D3C22⋊C411yescharacteristic
142C23×D3M1612yescomplemented
143C22×C3×C4—21no—
144C22×C3×C4—21no—
145C22×C3×C4—21no—
146C22×C3×C4—21no—
147C22×C3×C4—21no—
148C22×C3×C4—41no—
149C22×C3×C4—41no—
150C22×C3×C4—41no—
151C22×C3×C4—41no—
152C22×C3×C4—41no—
153C22×C3×C4—41no—
154C22×C3×C4M1612yescomplemented
155C22×C3×C4C2×D411yescharacteristic
156C22×C3×C4C2×D411yescharacteristic
157C22×C3×C4C2×D411yescharacteristic
158C22×C3×C4C2×D411yescharacteristic
159C22×C3×C4#4111yescharacteristic
160C22×C3×C4#4111yescharacteristic
161C22×Dic3—21no—
162C22×Dic3—21no—
163C22×Dic3—22no—
164C22×Dic3—21no—
165C22×Dic3—21no—
166C22×Dic3—21no—
167C22×Dic3—41no—
168C22×Dic3—41no—
169C22×Dic3—41no—
170C22×Dic3C22⋊C411yescharacteristic
171C22×Dic3C22⋊C411yescharacteristic
172C2×D12—24no—
173C2×D12—24no—
174C2×D12—48no—
175C2×D12—42no—
176C2×D12—42no—
177C2×D12—48no—
178C2×D12—42no—
179C2×D12—42no—
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—
203C2×C3×C8—41no—
204C2×C3×C8—41no—
205C2×C3×C8—41no—
206(C2×D3)⋊C4—42no—
207(C2×D3)⋊C4—42no—
208(C2×D3)⋊C4—42no—
209(C2×D3)⋊C4—41no—
210(C2×D3)⋊C4—41no—
211(C2×D3)⋊C4—41no—
212(C2×D3)⋊C4—41no—
213(C2×D3)⋊C4—42no—
214C2×C3⋊C8—22no—
215C2×C3⋊C8—41no—
216C2×C3⋊C8—41no—
217C2×C3⋊C8—41no—
218C2×C3⋊C8—41no—
219C2×C3⋊C8—41no—
220ΓL(1,𝔽25)—48no—
221C25—31no—
222C25—31no—
223C22×D4—61no—
224C22×D4—61no—
225C22×D4—62no—
226C22×D4—62no—
227C22×D4—122no—
228C22×D4—122no—
229C23×C4—31no—
230C23×C4—31no—
231C23×C4—31no—
232C23×C4—31no—
233C23×C4—31no—
234C23×C4—31no—
235C23×C4—61no—
236C23×C4—61no—
237C23×C4—61no—
238C23×C4—61no—
239C23×C4—122no—
240C23×C4—122no—
241C23×C4C4×D311yescharacteristic
242C23×C4C4×D311yescharacteristic
243C23×C4C22×D311yescharacteristic
244C2×M16—64no—
245C2×M16—122no—
246C2×M16—122no—
247C22×C8—21no—
248C22×C8—21no—
249C22×C8—61no—
250C22×C8—61no—
251C2×C22⋊C4—61no—
252C2×C22⋊C4—61no—
253C2×C22⋊C4—61no—
254C2×C22⋊C4—61no—
255C2×C22⋊C4—61no—
256C2×C22⋊C4—61no—
257C2×C22⋊C4—61no—
258C2×C22⋊C4—61no—
259C2×C22⋊C4—61no—
260C2×C22⋊C4—61no—
261C2×C22⋊C4—122no—
262C2×C22⋊C4—122no—
263C22⋊C8—21no—
264C22⋊C8—21no—
265C22⋊C8—21no—
266C22⋊C8—32no—
267C22⋊C8—64no—
268C22⋊C8—61no—
269C22⋊C8—61no—
270C22⋊C8—61no—
271C22⋊C8—122no—
272C22⋊C8—122no—
273C22⋊C8D1212yescomplemented
274#95—21no—
275#95—21no—
276#95—61no—
277#95—61no—
278C23×C3—21no—
279C23×C3—21no—
280C23×C3—21no—
281C23×C3—21no—
282C23×C3C2×C22⋊C411yescharacteristic
283C23×C3C4×D411yescharacteristic
284C23×C3C4×D411yescharacteristic
285C23×C3C2×M1611yescharacteristic
286C23×C3C2×M1611yescharacteristic
287C23×C3#13111yescharacteristic
288C23×C3#13111yescharacteristic
289C22×D3—24no—
290C22×D3—24no—
291C22×D3—21no—
292C22×D3—21no—
293C22×D3—21no—
294C22×D3—21no—
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—22no—
304C22×D3—22no—
305C22×D3—22no—
306C22×D3—42no—
307C22×D3—42no—
308C22×D3—42no—
309C22×D3—42no—
310C22×D3—42no—
311C22×D3—42no—
312C22×D3—42no—
313C22×D3—42no—
314C22×D3—42no—
315C22×D3—42no—
316C22×D3—41no—
317C22×D3—41no—
318C22×D3—41no—
319C22×D3—41no—
320C22×D3—42no—
321C22×D3—42no—
322C22×D3—41no—
323C22×D3—41no—
324C22×D3—41no—
325C22×D3—41no—
326C22×D3—41no—
327C22×D3—41no—
328C22×D3—41no—
329C22×D3—41no—
330C22×D3—42no—
331C22×D3—42no—
332C22×D3—81no—
333C22×D3—81no—
334C22×D3—81no—
335C22×D3—81no—
336C22×D3—81no—
337C22×D3—81no—
338C22×D3—81no—
339C22×D3—81no—
340C22×D3#9912yes—
341C22×D3#10012yes—
342C2×C3×C4—21no—
343C2×C3×C4—21no—
344C2×C3×C4—21no—
345C2×C3×C4—21no—
346C2×C3×C4—22no—
347C2×C3×C4—41no—
348C2×C3×C4—41no—
349C2×C3×C4—41no—
350C2×C3×C4—41no—
351C2×C3×C4—42no—
352C2×C3×C4—41no—
353C2×C3×C4—41no—
354C2×C3×C4—42no—
355C2×C3×C4—41no—
356C2×C3×C4—41no—
357C2×C3×C4—41no—
358C2×C3×C4—81no—
359C2×C3×C4—81no—
360C2×C3×C4#12011yescharacteristic
361C2×C3×C4#12111yescharacteristic
362C2×C3×C4#12111yescharacteristic
363C2×C3×C4#12311yescharacteristic
364C2×Dic3—21no—
365C2×Dic3—21no—
366C2×Dic3—21no—
367C2×Dic3—21no—
368C2×Dic3—21no—
369C2×Dic3—42no—
370C2×Dic3—42no—
371C2×Dic3—41no—
372C2×Dic3—41no—
373C2×Dic3—41no—
374C2×Dic3—81no—
375C2×Dic3—81no—
376C2×Dic3#9912yes—
377D12—48no—
378D12—48no—
379C4×D3—24no—
380C4×D3—24no—
381C4×D3—42no—
382C4×D3—42no—
383C4×D3—42no—
384C4×D3—42no—
385C4×D3—42no—
386C4×D3—42no—
387C3×C8—42no—
388C3×C8—81no—
389C3⋊C8—42no—
390C3⋊C8—81no—
391C24—31no—
392C24—31no—
393C24—32no—
394C24—62no—
395C24—62no—
396C24—61no—
397C24—61no—
398C24—61no—
399C24—61no—
400C24—61no—
401C24—61no—
402C24—61no—
403C24—61no—
404C24—61no—
405C24—61no—
406C24—61no—
407C24—61no—
408C24—121no—
409C24—121no—
410C24—121no—
411C24—121no—
412C24—121no—
413C24—121no—
414C24C2×C4×D311yescharacteristic
415C2×D4—64no—
416C2×D4—64no—
417C2×D4—122no—
418C2×D4—122no—
419C2×D4—128no—
420C2×D4—122no—
421C2×D4—122no—
422C2×D4—128no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—31no—
429C22×C4—31no—
430C22×C4—41no—
431C22×C4—41no—
432C22×C4—41no—
433C22×C4—41no—
434C22×C4—41no—
435C22×C4—41no—
436C22×C4—62no—
437C22×C4—62no—
438C22×C4—62no—
439C22×C4—62no—
440C22×C4—62no—
441C22×C4—61no—
442C22×C4—61no—
443C22×C4—61no—
444C22×C4—61no—
445C22×C4—61no—
446C22×C4—61no—
447C22×C4—61no—
448C22×C4—61no—
449C22×C4—61no—
450C22×C4—61no—
451C22×C4—61no—
452C22×C4—61no—
453C22×C4—61no—
454C22×C4—121no—
455C22×C4—121no—
456C22×C4—121no—
457C22×C4—121no—
458C22×C4—124no—
459C22×C4—124no—
460C22×C4—124no—
461C22×C4—124no—
462C22×C4—121no—
463C22×C4—121no—
464C22×C4—121no—
465C22×C4ΓL(1,𝔽25)12yescomplemented
466C22×C4C2×D1211yescharacteristic
467C22×C4#23511yescharacteristic
468C22×C4D3×D411yescharacteristic
469C22×C4D3×D411yescharacteristic
470C22×C4D3×D411yescharacteristic
471C22×C4#23911yescharacteristic
472M16—128no—
473C2×C8—22no—
474C2×C8—41no—
475C2×C8—41no—
476C2×C8—41no—
477C2×C8—41no—
478C2×C8—41no—
479C2×C8—62no—
480C2×C8—121no—
481C2×C8—121no—
482C2×C8—121no—
483C2×C8—121no—
484C2×C8—121no—
485C22⋊C4—122no—
486C22⋊C4—122no—
487C22⋊C4—122no—
488C22⋊C4—122no—
489C22⋊C4—121no—
490C22⋊C4—121no—
491C22⋊C4—121no—
492C22⋊C4—121no—
493C22×C3—21no—
494C22×C3—21no—
495C22×C3—21no—
496C22×C3—21no—
497C22×C3—21no—
498C22×C3—21no—
499C22×C3—21no—
500C22×C3—21no—
501C22×C3—21no—
502C22×C3—21no—
503C22×C3—21no—
504C22×C3—21no—
505C22×C3#38611yescharacteristic
506C22×C3#40711yescharacteristic
507C22×C3#40811yescharacteristic
508C22×C3C2×#9911yescharacteristic
509C22×C3C2×#9911yescharacteristic
510C22×C3C2×#10011yescharacteristic
511C22×C3#41311yescharacteristic
512C22×C3#43511yescharacteristic
513C22×C3#43511yescharacteristic
514C22×C3#43611yescharacteristic
515C22×C3#43611yescharacteristic
516C2×D3—22no—
517C2×D3—22no—
518C2×D3—22no—
519C2×D3—22no—
520C2×D3—22no—
521C2×D3—22no—
522C2×D3—44no—
523C2×D3—44no—
524C2×D3—44no—
525C2×D3—44no—
526C2×D3—41no—
527C2×D3—41no—
528C2×D3—41no—
529C2×D3—41no—
530C2×D3—41no—
531C2×D3—41no—
532C2×D3—41no—
533C2×D3—41no—
534C2×D3—42no—
535C2×D3—42no—
536C2×D3—42no—
537C2×D3—82no—
538C2×D3—82no—
539C2×D3—81no—
540C2×D3—81no—
541C2×D3—81no—
542C2×D3—81no—
543C2×D3—82no—
544C2×D3—82no—
545C3×C4—21no—
546C3×C4—21no—
547C3×C4—21no—
548C3×C4—21no—
549C3×C4—42no—
550C3×C4—81no—
551Dic3—22no—
552Dic3—41no—
553Dic3—81no—
554C23—21no—
555C23—21no—
556C23—21no—
557C23—21no—
558C23—32no—
559C23—32no—
560C23—64no—
561C23—64no—
562C23—61no—
563C23—61no—
564C23—61no—
565C23—61no—
566C23—61no—
567C23—61no—
568C23—61no—
569C23—61no—
570C23—61no—
571C23—61no—
572C23—61no—
573C23—61no—
574C23—62no—
575C23—62no—
576C23—62no—
577C23—122no—
578C23—122no—
579C23—122no—
580C23—122no—
581C23—122no—
582C23—122no—
583C23—122no—
584C23—122no—
585C23—122no—
586C23—122no—
587C23—122no—
588C23—122no—
589C23—122no—
590C23—122no—
591C23—121no—
592C23—121no—
593C23—121no—
594C23—121no—
595C23—121no—
596C23—121no—
597C23—121no—
598C23—121no—
599C23—121no—
600C23—121no—
601C23—121no—
602C23—121no—
603C23—241no—
604C23—241no—
605C23—241no—
606C23—241no—
607C23—241no—
608C23—241no—
609C23—241no—
610C23—241no—
611C23C4×D1211yescharacteristic
612C23D3×C22⋊C411yescharacteristic
613C23#89311yescharacteristic
614C23C2×ΓL(1,𝔽25)11yescharacteristic
615C23#90111yescharacteristic
616C23D3×M1611yescharacteristic
617C23#90711yescharacteristic
618D4—128no—
619D4—128no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—22no—
625C2×C4—32no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—42no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—42no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—41no—
637C2×C4—64no—
638C2×C4—64no—
639C2×C4—61no—
640C2×C4—61no—
641C2×C4—61no—
642C2×C4—61no—
643C2×C4—61no—
644C2×C4—81no—
645C2×C4—81no—
646C2×C4—122no—
647C2×C4—122no—
648C2×C4—122no—
649C2×C4—122no—
650C2×C4—122no—
651C2×C4—122no—
652C2×C4—122no—
653C2×C4—122no—
654C2×C4—121no—
655C2×C4—121no—
656C2×C4—121no—
657C2×C4—241no—
658C2×C4—241no—
659C2×C4#88211yescharacteristic
660C2×C4#88411yescharacteristic
661C2×C4#89411yescharacteristic
662C2×C4#89511yescharacteristic
663C8—42no—
664C8—81no—
665C8—122no—
666C8—241no—
667C2×C3—21no—
668C2×C3—21no—
669C2×C3—21no—
670C2×C3—21no—
671C2×C3#146211yescharacteristic
672C2×C3#146511yescharacteristic
673C2×C3#146811yescharacteristic
674C2×C3#147111yescharacteristic
675C2×C3#187011yescharacteristic
676C2×C3#189811yescharacteristic
677C2×C3#190011yescharacteristic
678D3—42no—
679D3—42no—
680D3—82no—
681C22—21no—
682C22—21no—
683C22—21no—
684C22—21no—
685C22—21no—
686C22—21no—
687C22—21no—
688C22—21no—
689C22—21no—
690C22—21no—
691C22—21no—
692C22—21no—
693C22—62no—
694C22—62no—
695C22—62no—
696C22—62no—
697C22—62no—
698C22—62no—
699C22—124no—
700C22—124no—
701C22—124no—
702C22—124no—
703C22—121no—
704C22—121no—
705C22—121no—
706C22—121no—
707C22—121no—
708C22—121no—
709C22—121no—
710C22—121no—
711C22—122no—
712C22—122no—
713C22—122no—
714C22—242no—
715C22—242no—
716C22—242no—
717C22—242no—
718C22—241no—
719C22—241no—
720C22—241no—
721C22—241no—
722C22#466511yescharacteristic
723C22#468811yescharacteristic
724C22#470611yescharacteristic
725C22#472611yescharacteristic
726C22#472711yescharacteristic
727C22D3×#9911yescharacteristic
728C22D3×#9911yescharacteristic
729C22D3×#10011yescharacteristic
730C22#475111yescharacteristic
731C22#483611yescharacteristic
732C22#483811yescharacteristic
733C4—21no—
734C4—21no—
735C4—21no—
736C4—21no—
737C4—42no—
738C4—62no—
739C4—81no—
740C4—121no—
741C4—241no—
742C3#908311yescharacteristic, complemented, Sylow
743C2—21no—
744C2—21no—
745C2—21no—
746C2—21no—
747C2—122no—
748C2—122no—
749C2—242no—
750C2#6812011yescharacteristic
751C2#6812311yescharacteristic
752C2#6814511yescharacteristic
753C2#6814611yescharacteristic
754C2#6866911yescharacteristic
755C2#6897011yescharacteristic
756C2#6899211yescharacteristic
subgroupG/H
centerC23D3×C22⋊C4
commutatorC24×C3C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1280C2
socleC23×C3C2×C22⋊C4
prime radicalC23×D12C4
cosocleC23×C3×C4C23
hypercenter#1280D3
hypocenterC3#9083
LCSG ▷ C24×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1280

Complements

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

NH ≅ G/Ntype
1. C2×#4726C2semidirect
3. #69955C2semidirect
4. C2×#4468C2semidirect
5. C2×#4468C2semidirect
6. C3×#1280C2semidirect
7. #67613C2semidirect
38. #1280D3semidirect
84. C3×C22⋊C8D4semidirect
92. #830D4semidirect
142. C23×D3M16semidirect
154. C22×C3×C4M16semidirect
273. C22⋊C8D12semidirect
465. C22×C4ΓL(1,𝔽25)semidirect
742. C3#9083semidirect

Maximal subgroups

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

subgroupindex
C2×#47262
C2×#46652
#699552
C2×#44682
C3×#12802
#676132
#90833