Finite Group Databasebeta

← #10690585

Subgroups — #10690585

Order 768.

928 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4989C211yesmaximal, characteristic
2C2×#4726C211yesmaximal, characteristic
3C2×C3×#324C211yesmaximal, characteristic, complemented
4C2×#4528C211yesmaximal, characteristic, complemented
5#69955C211yesmaximal, characteristic
6#67674C211yesmaximal, characteristic, complemented
7#67674C211yesmaximal, characteristic, complemented
8#9085—31nomaximal, Sylow
9C23×C4×D3C2211yescharacteristic
10C22×C3×C4⋊C4C2211yescharacteristic
11C22×(C2×D3)⋊C4C411yescharacteristic
12C22×(C2×D3)⋊C4C411yescharacteristic
13C22×Dic3⋊C4C411yescharacteristic
14C22×ΓL(1,𝔽25)—21no—
15C2×C3×C22⋊C8C2211yescharacteristic
16C2×C3×C22⋊C8C2211yescharacteristic
17C2×C3⋊#95C411yescharacteristic
18C2×C3⋊#95C2211yescharacteristic
19C2×#830C2211yescharacteristic
20C2×#830C2211yescharacteristic
21C2×(C2×D3)⋊C8—21no—
22C2×(C2×D3)⋊C8—21no—
23C2×(C2×D3)⋊C8—21no—
24#4989—21no—
25#4989—21no—
26#4989—21no—
27#4989—21no—
28#4726—22no—
29#4726—22no—
30C3×#324—22no—
31#4528—22no—
32C2×#407—31no—
33C2×#385—31no—
34#1779—31no—
35C2×#324—31no—
36C2×#324D311yescharacteristic, complemented
37#1280—31no—
38#1280—31no—
39C24×D3C2×C411yescharacteristic
40C23×C3×C4C2×C411yescharacteristic
41C23×C3×C4C2×C411yescharacteristic
42C23×C3×C4C2311yescharacteristic
43C23×Dic3C2×C411yescharacteristic
44C23×Dic3C2×C411yescharacteristic
45C23×Dic3C2×C411yescharacteristic
46C22×C4×D3—21no—
47C22×C4×D3—21no—
48C22×C4×D3—21no—
49C22×C4×D3—21no—
50C22×C4×D3—42no—
51C22×C4×D3—42no—
52C22×C4×D3D411yescharacteristic
53C22×C4×D3D411yescharacteristic
54C22×C4×D3D411yescharacteristic
55C22×C4×D3D411yescharacteristic
56C22×C3×C8—21no—
57C22×C3×C8—21no—
58C2×C3×C4⋊C4—21no—
59C2×C3×C4⋊C4—21no—
60C2×C3×C4⋊C4—21no—
61C2×C3×C4⋊C4—21no—
62C2×C3×C4⋊C4—21no—
63C2×C3×C4⋊C4—21no—
64C2×(C2×D3)⋊C4—21no—
65C2×(C2×D3)⋊C4—21no—
66C2×(C2×D3)⋊C4—22no—
67C2×(C2×D3)⋊C4—22no—
68C2×(C2×D3)⋊C4—21no—
69C2×(C2×D3)⋊C4—21no—
70C2×(C2×D3)⋊C4—42no—
71C2×Dic3⋊C4—21no—
72C2×Dic3⋊C4—21no—
73C2×Dic3⋊C4—42no—
74C22×C3⋊C8—21no—
75C22×C3⋊C8—21no—
76C2×ΓL(1,𝔽25)—24no—
77C2×ΓL(1,𝔽25)—42no—
78C2×ΓL(1,𝔽25)—42no—
79C3×C22⋊C8—21no—
80C3×C22⋊C8—21no—
81C3×C22⋊C8—21no—
82C3×C22⋊C8D412yes—
83C3⋊#95—21no—
84C3⋊#95—21no—
85C3⋊#95—21no—
86C3⋊#95—21no—
87#830—21no—
88#830—21no—
89#830—21no—
90#830D412yes—
91(C2×D3)⋊C8—24no—
92(C2×D3)⋊C8—42no—
93(C2×D3)⋊C8—42no—
94C24×C4—31no—
95C22×M16—61no—
96C22×C4⋊C4—31no—
97C22×C4⋊C4C2×D311yescharacteristic
98C22×C22⋊C4—31no—
99C22×C22⋊C4—31no—
100#407—62no—
101#407—62no—
102C2×C22⋊C8—31no—
103C2×C22⋊C8—31no—
104C2×C22⋊C8—61no—
105C2×C22⋊C8—61no—
106C2×C22⋊C8—61no—
107C2×C22⋊C8C2×D311yescharacteristic
108C2×C22⋊C8C2×D311yescharacteristic
109#385—61no—
110#385—61no—
111#385—61no—
112#385—61no—
113C2×#95—31no—
114C2×#95—31no—
115#324—22no—
116#324—62no—
117C24×C3C22×C411yescharacteristic
118C23×D3—21no—
119C23×D3—21no—
120C23×D3—22no—
121C23×D3—22no—
122C23×D3—21no—
123C23×D3—21no—
124C23×D3—21no—
125C23×D3—21no—
126C23×D3—41no—
127C23×D3—41no—
128C23×D3C22⋊C411yescharacteristic
129C23×D3C22⋊C411yescharacteristic
130C22×C3×C4—21no—
131C22×C3×C4—21no—
132C22×C3×C4—21no—
133C22×C3×C4—21no—
134C22×C3×C4—21no—
135C22×C3×C4—21no—
136C22×C3×C4—21no—
137C22×C3×C4—21no—
138C22×C3×C4—21no—
139C22×C3×C4—41no—
140C22×C3×C4—41no—
141C22×C3×C4—41no—
142C22×C3×C4—41no—
143C22×C3×C4M1612yescomplemented
144C22×C3×C4C2×D411yescharacteristic
145C22×C3×C4C2×D411yescharacteristic
146C22×C3×C4C2×D411yescharacteristic
147C22×C3×C4C2×D411yescharacteristic
148C22×C3×C4#4111yescharacteristic
149C22×C3×C4#4111yescharacteristic
150C22×Dic3—21no—
151C22×Dic3—21no—
152C22×Dic3—21no—
153C22×Dic3—21no—
154C22×Dic3—22no—
155C22×Dic3—21no—
156C22×Dic3—21no—
157C22×Dic3—21no—
158C22×Dic3—41no—
159C22×Dic3—41no—
160C22×Dic3—41no—
161C22×Dic3—41no—
162C22×Dic3—41no—
163C22×Dic3C22⋊C411yescharacteristic
164C22×Dic3C22⋊C411yescharacteristic
165C22×Dic3M1612yescomplemented
166C2×C4×D3—22no—
167C2×C4×D3—21no—
168C2×C4×D3—21no—
169C2×C4×D3—22no—
170C2×C4×D3—22no—
171C2×C4×D3—21no—
172C2×C4×D3—21no—
173C2×C4×D3—21no—
174C2×C4×D3—21no—
175C2×C4×D3—21no—
176C2×C4×D3—21no—
177C2×C4×D3—22no—
178C2×C4×D3—44no—
179C2×C4×D3—44no—
180C2×C4×D3—44no—
181C2×C4×D3—44no—
182C2×C4×D3—41no—
183C2×C4×D3—41no—
184C2×C4×D3—41no—
185C2×C4×D3—41no—
186C2×C3×C8—22no—
187C2×C3×C8—41no—
188C2×C3×C8—41no—
189C2×C3×C8—41no—
190C2×C3×C8—41no—
191C2×C3×C8—41no—
192C3×C4⋊C4—41no—
193C3×C4⋊C4—41no—
194C3×C4⋊C4—41no—
195C3×C4⋊C4—41no—
196(C2×D3)⋊C4—24no—
197(C2×D3)⋊C4—24no—
198(C2×D3)⋊C4—42no—
199(C2×D3)⋊C4—42no—
200Dic3⋊C4—42no—
201Dic3⋊C4—42no—
202C2×C3⋊C8—22no—
203C2×C3⋊C8—41no—
204C2×C3⋊C8—41no—
205C2×C3⋊C8—41no—
206C2×C3⋊C8—41no—
207C2×C3⋊C8—41no—
208ΓL(1,𝔽25)—48no—
209C25—31no—
210C23×C4—31no—
211C23×C4—31no—
212C23×C4—31no—
213C23×C4—31no—
214C23×C4—31no—
215C23×C4—31no—
216C23×C4—31no—
217C23×C4—61no—
218C23×C4—61no—
219C23×C4—61no—
220C23×C4—61no—
221C23×C4—122no—
222C23×C4—122no—
223C23×C4C4×D311yescharacteristic
224C23×C4C4×D311yescharacteristic
225C23×C4C22×D311yescharacteristic
226C2×M16—64no—
227C2×M16—122no—
228C2×M16—122no—
229C22×C8—21no—
230C22×C8—21no—
231C22×C8—61no—
232C22×C8—61no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—61no—
240C2×C4⋊C4—61no—
241C2×C4⋊C4—122no—
242C2×C22⋊C4—62no—
243C2×C22⋊C4—61no—
244C2×C22⋊C4—61no—
245C2×C22⋊C4—62no—
246C2×C22⋊C4—61no—
247C2×C22⋊C4—61no—
248C2×C22⋊C4—122no—
249C22⋊C8—21no—
250C22⋊C8—21no—
251C22⋊C8—21no—
252C22⋊C8—32no—
253C22⋊C8—64no—
254C22⋊C8—61no—
255C22⋊C8—61no—
256C22⋊C8—61no—
257C22⋊C8—122no—
258C22⋊C8—122no—
259C22⋊C8#6712yes—
260#95—61no—
261#95—61no—
262#95—61no—
263#95—61no—
264C23×C3—21no—
265C23×C3—21no—
266C23×C3—21no—
267C23×C3—21no—
268C23×C3C2×C22⋊C411yescharacteristic
269C23×C3C4×D411yescharacteristic
270C23×C3C4×D411yescharacteristic
271C23×C3C2×M1611yescharacteristic
272C23×C3C2×M1611yescharacteristic
273C23×C3#13111yescharacteristic
274C23×C3#13111yescharacteristic
275C22×D3—24no—
276C22×D3—24no—
277C22×D3—21no—
278C22×D3—21no—
279C22×D3—21no—
280C22×D3—21no—
281C22×D3—21no—
282C22×D3—21no—
283C22×D3—21no—
284C22×D3—21no—
285C22×D3—21no—
286C22×D3—21no—
287C22×D3—21no—
288C22×D3—21no—
289C22×D3—42no—
290C22×D3—42no—
291C22×D3—42no—
292C22×D3—42no—
293C22×D3—42no—
294C22×D3—42no—
295C22×D3—42no—
296C22×D3—42no—
297C22×D3—42no—
298C22×D3—42no—
299C22×D3—41no—
300C22×D3—41no—
301C22×D3—41no—
302C22×D3—41no—
303C22×D3#9912yes—
304C22×D3#10112yes—
305C2×C3×C4—21no—
306C2×C3×C4—21no—
307C2×C3×C4—21no—
308C2×C3×C4—21no—
309C2×C3×C4—22no—
310C2×C3×C4—22no—
311C2×C3×C4—22no—
312C2×C3×C4—41no—
313C2×C3×C4—41no—
314C2×C3×C4—41no—
315C2×C3×C4—41no—
316C2×C3×C4—41no—
317C2×C3×C4—41no—
318C2×C3×C4—41no—
319C2×C3×C4—41no—
320C2×C3×C4—41no—
321C2×C3×C4—41no—
322C2×C3×C4—41no—
323C2×C3×C4—81no—
324C2×C3×C4—81no—
325C2×C3×C4#12011yescharacteristic
326C2×C3×C4#12111yescharacteristic
327C2×C3×C4#12111yescharacteristic
328C2×C3×C4#12311yescharacteristic
329C2×Dic3—21no—
330C2×Dic3—21no—
331C2×Dic3—21no—
332C2×Dic3—22no—
333C2×Dic3—21no—
334C2×Dic3—21no—
335C2×Dic3—42no—
336C2×Dic3—42no—
337C2×Dic3—42no—
338C2×Dic3—41no—
339C2×Dic3—41no—
340C2×Dic3—42no—
341C2×Dic3—41no—
342C2×Dic3—41no—
343C2×Dic3—41no—
344C2×Dic3—81no—
345C2×Dic3—81no—
346C2×Dic3#9912yes—
347C4×D3—24no—
348C4×D3—24no—
349C4×D3—42no—
350C4×D3—42no—
351C4×D3—42no—
352C4×D3—42no—
353C4×D3—42no—
354C4×D3—42no—
355C3×C8—42no—
356C3×C8—81no—
357C3⋊C8—42no—
358C3⋊C8—81no—
359C24—31no—
360C24—31no—
361C24—62no—
362C24—62no—
363C24—61no—
364C24—61no—
365C24—61no—
366C24—61no—
367C24—61no—
368C24—61no—
369C24—121no—
370C24—121no—
371C24C2×C4×D311yescharacteristic
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—31no—
382C22×C4—31no—
383C22×C4—32no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—62no—
389C22×C4—62no—
390C22×C4—62no—
391C22×C4—62no—
392C22×C4—62no—
393C22×C4—61no—
394C22×C4—61no—
395C22×C4—61no—
396C22×C4—61no—
397C22×C4—61no—
398C22×C4—61no—
399C22×C4—61no—
400C22×C4—61no—
401C22×C4—61no—
402C22×C4—61no—
403C22×C4—61no—
404C22×C4—61no—
405C22×C4—61no—
406C22×C4—61no—
407C22×C4—61no—
408C22×C4—121no—
409C22×C4—121no—
410C22×C4—121no—
411C22×C4—121no—
412C22×C4—124no—
413C22×C4—124no—
414C22×C4—124no—
415C22×C4—124no—
416C22×C4—121no—
417C22×C4—121no—
418C22×C4—121no—
419C22×C4—121no—
420C22×C4—121no—
421C22×C4ΓL(1,𝔽25)12yescomplemented
422C22×C4#23511yescharacteristic
423C22×C4D3×D411yescharacteristic
424C22×C4D3×D411yescharacteristic
425C22×C4D3×D411yescharacteristic
426C22×C4#23711yescharacteristic
427C22×C4C2×#6711yescharacteristic
428M16—128no—
429C2×C8—22no—
430C2×C8—41no—
431C2×C8—41no—
432C2×C8—41no—
433C2×C8—41no—
434C2×C8—41no—
435C2×C8—62no—
436C2×C8—121no—
437C2×C8—121no—
438C2×C8—121no—
439C2×C8—121no—
440C2×C8—121no—
441C4⋊C4—41no—
442C4⋊C4—41no—
443C4⋊C4—41no—
444C4⋊C4—41no—
445C4⋊C4—122no—
446C4⋊C4—122no—
447C22⋊C4—64no—
448C22⋊C4—64no—
449C22⋊C4—122no—
450C22⋊C4—122no—
451C22×C3—21no—
452C22×C3—21no—
453C22×C3—21no—
454C22×C3—21no—
455C22×C3—21no—
456C22×C3—21no—
457C22×C3—21no—
458C22×C3—21no—
459C22×C3—21no—
460C22×C3—21no—
461C22×C3—21no—
462C22×C3—21no—
463C22×C3#38611yescharacteristic
464C22×C3#40711yescharacteristic
465C22×C3#40811yescharacteristic
466C22×C3C2×#9911yescharacteristic
467C22×C3C2×#9911yescharacteristic
468C22×C3C2×#10111yescharacteristic
469C22×C3#41311yescharacteristic
470C22×C3#43511yescharacteristic
471C22×C3#43511yescharacteristic
472C22×C3#43611yescharacteristic
473C22×C3#43611yescharacteristic
474C2×D3—22no—
475C2×D3—22no—
476C2×D3—22no—
477C2×D3—22no—
478C2×D3—22no—
479C2×D3—22no—
480C2×D3—44no—
481C2×D3—44no—
482C2×D3—44no—
483C2×D3—44no—
484C2×D3—41no—
485C2×D3—41no—
486C2×D3—41no—
487C2×D3—41no—
488C2×D3—41no—
489C2×D3—41no—
490C2×D3—41no—
491C2×D3—41no—
492C3×C4—21no—
493C3×C4—21no—
494C3×C4—21no—
495C3×C4—21no—
496C3×C4—42no—
497C3×C4—81no—
498Dic3—22no—
499Dic3—41no—
500Dic3—42no—
501Dic3—81no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—32no—
507C23—32no—
508C23—64no—
509C23—64no—
510C23—61no—
511C23—61no—
512C23—61no—
513C23—61no—
514C23—61no—
515C23—61no—
516C23—61no—
517C23—61no—
518C23—61no—
519C23—61no—
520C23—61no—
521C23—61no—
522C23—122no—
523C23—122no—
524C23—122no—
525C23—122no—
526C23—122no—
527C23—122no—
528C23—122no—
529C23—122no—
530C23—122no—
531C23—122no—
532C23—121no—
533C23—121no—
534C23—121no—
535C23—121no—
536C23D3×C22⋊C411yescharacteristic
537C23#67⋊C411yescharacteristic
538C23C2×ΓL(1,𝔽25)11yescharacteristic
539C23#90111yescharacteristic
540C23D3×M1611yescharacteristic
541C23#90711yescharacteristic
542C23C4×#6711yescharacteristic
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—32no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—64no—
563C2×C4—64no—
564C2×C4—61no—
565C2×C4—61no—
566C2×C4—61no—
567C2×C4—62no—
568C2×C4—61no—
569C2×C4—61no—
570C2×C4—81no—
571C2×C4—81no—
572C2×C4—122no—
573C2×C4—122no—
574C2×C4—122no—
575C2×C4—122no—
576C2×C4—122no—
577C2×C4—122no—
578C2×C4—122no—
579C2×C4—122no—
580C2×C4—122no—
581C2×C4—121no—
582C2×C4—121no—
583C2×C4—122no—
584C2×C4—121no—
585C2×C4—121no—
586C2×C4—121no—
587C2×C4—241no—
588C2×C4—241no—
589C2×C4#88311yescharacteristic
590C2×C4#88411yescharacteristic
591C2×C4#93711yescharacteristic
592C2×C4#93811yescharacteristic
593C8—42no—
594C8—81no—
595C8—122no—
596C8—241no—
597C2×C3—21no—
598C2×C3—21no—
599C2×C3—21no—
600C2×C3—21no—
601C2×C3#146311yescharacteristic
602C2×C3#146511yescharacteristic
603C2×C3#146811yescharacteristic
604C2×C3#147211yescharacteristic
605C2×C3#187011yescharacteristic
606C2×C3#189811yescharacteristic
607C2×C3#190111yescharacteristic
608D3—42no—
609D3—42no—
610C22—21no—
611C22—21no—
612C22—21no—
613C22—21no—
614C22—21no—
615C22—21no—
616C22—21no—
617C22—21no—
618C22—21no—
619C22—21no—
620C22—21no—
621C22—21no—
622C22—62no—
623C22—62no—
624C22—62no—
625C22—62no—
626C22—62no—
627C22—62no—
628C22—124no—
629C22—124no—
630C22—124no—
631C22—124no—
632C22—121no—
633C22—121no—
634C22—121no—
635C22—121no—
636C22—121no—
637C22—121no—
638C22—121no—
639C22—121no—
640C22#472611yescharacteristic
641C22#472711yescharacteristic
642C22#472911yescharacteristic
643C22#473011yescharacteristic
644C22D3×#9911yescharacteristic
645C22D3×#9911yescharacteristic
646C22D3×#10111yescharacteristic
647C22#475111yescharacteristic
648C22#495611yescharacteristic
649C22#511111yescharacteristic
650C22#512811yescharacteristic
651C4—21no—
652C4—21no—
653C4—21no—
654C4—21no—
655C4—42no—
656C4—62no—
657C4—81no—
658C4—122no—
659C4—121no—
660C4—241no—
661C3#908511yescharacteristic, complemented, Sylow
662C2—21no—
663C2—21no—
664C2—21no—
665C2—21no—
666C2—122no—
667C2—122no—
668C2#6814411yescharacteristic
669C2#6814511yescharacteristic
670C2#6814611yescharacteristic
671C2#6814711yescharacteristic
672C2#6995811yescharacteristic
673C2#7003611yescharacteristic
674C2#7011511yescharacteristic
subgroupG/H
centerC23D3×C22⋊C4
commutatorC24×C3C22×C4
FrattiniC23×C4C22×D3
FittingC2×C3×#324C2
socleC23×C3C2×C22⋊C4
prime radicalC24×D3C2×C4
cosocleC23×C3×C4C23
hypercenterC2×#324D3
hypocenterC3#9085
LCSG ▷ C24×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#324

Complements

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

NH ≅ G/Ntype
3. C2×C3×#324C2semidirect
4. C2×#4528C2semidirect
6. #67674C2semidirect
7. #67674C2semidirect
36. C2×#324D3semidirect
143. C22×C3×C4M16semidirect
165. C22×Dic3M16semidirect
421. C22×C4ΓL(1,𝔽25)semidirect
661. C3#9085semidirect

Maximal subgroups

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

subgroupindex
C2×#49892
C2×#47262
C2×C3×#3242
C2×#45282
#699552
#676742
#90853