Finite Group Databasebeta

← #10796326

Subgroups — #10796326

Order 768.

1938 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5828C211yesmaximal, characteristic
2C22×D3×M16C211yesmaximal, characteristic, complemented
3C2×C3×#408C211yesmaximal, characteristic, complemented
4C2×#5234C211yesmaximal, characteristic, complemented
5C22×(C2×D3)⋊C8C211yesmaximal, characteristic, complemented
6#71980C211yesmaximal, characteristic, complemented
7#71965C211yesmaximal, characteristic, complemented
8#69958C214yesmaximal, complemented
9#69957C214yesmaximal, complemented
10#20897—31nomaximal, Sylow
11C22×C3×#41C2211yescharacteristic
12C23×C4×D3C2211yescharacteristic
13C22×C3×M16C2211yescharacteristic
14C23×C3×C8C2211yescharacteristic
15#5828—24no—
16#5828—24no—
17C2×#946C411yescharacteristic
18C2×#946C411yescharacteristic
19C2×#938C411yescharacteristic
20C2×#938C411yescharacteristic
21C2×#937C412yes—
22C2×#935C412yes—
23C2×C4×#67C2212yes—
24C2×C4×#67C2212yes—
25C2×#926C2211yescharacteristic
26C22×#208C2211yescharacteristic
27C23×C3⋊C8C2211yescharacteristic
28C2×D3×M16—24no—
29C2×D3×M16—24no—
30C2×D3×M16—24no—
31C22×ΓL(1,𝔽25)C2212yes—
32C22×D3×C8C2212yes—
33C3×#408—24no—
34C2×C3×C22⋊C8C2214yescomplemented
35#5234—24no—
36C2×#830C2214yescomplemented
37C2×(C2×D3)⋊C8—24no—
38C2×(C2×D3)⋊C8C2212yes—
39C2×(C2×D3)⋊C8C2212yes—
40#4550C2214yescomplemented
41#4550C2214yescomplemented
42C23×M16—31no—
43C2×#525—31no—
44C2×#408—31no—
45C2×#408D311yescharacteristic, complemented
46C22×C22⋊C8—31no—
47#1870—34no—
48#1869—34no—
49#1729—31no—
50#1729—31no—
51C24×D3C2×C411yescharacteristic
52C2×C3×#41—22no—
53C2×C3×#41—22no—
54C2×C3×#41—22no—
55C2×C3×#41—22no—
56C2×C3×#41—22no—
57C2×C3×#41—22no—
58C22×C3×Q8C2×C411yescharacteristic
59C22×C3×D4C2×C412yes—
60C22×C3×D4C2×C411yescharacteristic
61C23×C3×C4C2312yes—
62C23×C3×C4C2311yescharacteristic
63C22×#67C2×C412yes—
64C22×#67C2×C412yes—
65C23×Dic3C2×C411yescharacteristic
66C22×C4×D3—22no—
67C22×C4×D3—22no—
68C22×C4×D3—22no—
69C22×C4×D3—22no—
70C22×C4×D3C2×C411yescharacteristic
71C22×C4×D3C2×C411yescharacteristic
72C22×C4×D3D414yes—
73C22×C4×D3D414yes—
74C22×C4×D3C2311yescharacteristic
75C22×C4×D3C2311yescharacteristic
76C2×C3×M16—22no—
77C2×C3×M16—22no—
78C2×C3×M16D414yescomplemented
79C22×C3×C8—22no—
80C22×C3×C8—22no—
81C22×C3×C8—22no—
82C22×C3×C8C2312yes—
83C22×C3×C8C2312yes—
84C2×#217C2×C412yes—
85C2×#217C2×C412yes—
86C2×#217C2×C412yes—
87#946—24no—
88#946—24no—
89#938—24no—
90#938—24no—
91#937—28no—
92#935—28no—
93C4×#67—24no—
94C4×#67—24no—
95C4×#67—24no—
96C4×#67—24no—
97C2×(C2×D3)⋊C4C2×C412yes—
98C2×(C2×D3)⋊C4C2×C412yes—
99#926—22no—
100#926—22no—
101C2×C4⋊Dic3C2×C411yescharacteristic
102C2×C4⋊Dic3C2×C411yescharacteristic
103C2×Dic3⋊C4C2×C412yes—
104C2×Dic3⋊C4C2×C412yes—
105C2×C4×Dic3C2312yes—
106C2×#208—22no—
107C2×#208—22no—
108C2×#208D414yescomplemented
109C22×C3⋊C8—22no—
110C22×C3⋊C8—22no—
111C22×C3⋊C8—22no—
112C22×C3⋊C8C2312yes—
113C22×C3⋊C8C2312yes—
114D3×M16—232no—
115C2×ΓL(1,𝔽25)—24no—
116C2×ΓL(1,𝔽25)—24no—
117C2×ΓL(1,𝔽25)—24no—
118C2×D3×C8—24no—
119C2×D3×C8—24no—
120C2×D3×C8—24no—
121C3×C22⋊C8—28no—
122#830—28no—
123(C2×D3)⋊C8—24no—
124(C2×D3)⋊C8—24no—
125C22×#41C2×D311yescharacteristic
126C24×C4—31no—
127C22×M16—31no—
128C22×M16—32no—
129C22×M16—64no—
130C22×M16—64no—
131C22×M16—64no—
132C22×M16C2×D311yescharacteristic
133C23×C8—31no—
134C23×C8—32no—
135C23×C8C2×D311yescharacteristic
136#525—64no—
137#525—64no—
138C2×#123—32no—
139C2×C22⋊Q8—31no—
140C2×C22⋊Q8—31no—
141C2×#121—31no—
142C2×#121—31no—
143C2×#120—32no—
144C2×C4×D4—32no—
145C2×C4×D4—32no—
146C2×#117—31no—
147#408—24no—
148#408—64no—
149C2×C22⋊C8—34no—
150C2×C22⋊C8—32no—
151C2×C22⋊C8—32no—
152C2×C22⋊C8—64no—
153C2×C22⋊C8C2×D314yescomplemented
154#336—34no—
155#336—34no—
156C24×C3C22×C412yes—
157C24×C3C22×C411yescharacteristic
158C23×D3—22no—
159C23×D3—22no—
160C23×D3—24no—
161C23×D3—42no—
162C23×D3C22⋊C414yes—
163C23×D3C22×C411yescharacteristic
164C23×D3C22×C411yescharacteristic
165C3×#41—44no—
166C3×#41—44no—
167C3×#41—44no—
168C3×#41—44no—
169C2×C3×Q8—22no—
170C2×C3×Q8—22no—
171C2×C3×Q8—22no—
172C2×C3×D4—24no—
173C2×C3×D4—24no—
174C2×C3×D4—24no—
175C2×C3×D4—22no—
176C2×C3×D4—22no—
177C2×C3×D4—22no—
178C22×C3×C4—22no—
179C22×C3×C4—22no—
180C22×C3×C4—22no—
181C22×C3×C4—22no—
182C22×C3×C4—22no—
183C22×C3×C4—22no—
184C22×C3×C4C22×C412yes—
185C22×C3×C4C22×C411yescharacteristic
186C22×C3×C4C2×D412yes—
187C22×C3×C4C2×D412yes—
188C22×C3×C4C2×D412yes—
189C22×C3×C4C2×D412yes—
190C22×C3×C4#4112yes—
191C22×C3×C4#4112yes—
192C22×C3×C4C2411yescharacteristic
193C2×#67—24no—
194C2×#67—24no—
195C2×#67—28no—
196C2×#67—44no—
197C22×Dic3—22no—
198C22×Dic3—22no—
199C22×Dic3C22⋊C414yes—
200C22×Dic3C22×C412yes—
201C22×Dic3C22×C412yes—
202C22×Dic3C22×C411yescharacteristic
203C22×Dic3C22×C411yescharacteristic
204C2×C4×D3—22no—
205C2×C4×D3—22no—
206C2×C4×D3—22no—
207C2×C4×D3—22no—
208C2×C4×D3—28no—
209C2×C4×D3—28no—
210C2×C4×D3—28no—
211C2×C4×D3—28no—
212C2×C4×D3—21no—
213C2×C4×D3—21no—
214C2×C4×D3—21no—
215C2×C4×D3—21no—
216C2×C4×D3—21no—
217C2×C4×D3—21no—
218C2×C4×D3—21no—
219C2×C4×D3—21no—
220C2×C4×D3C22⋊C414yes—
221C2×C4×D3C22⋊C414yes—
222C2×C4×D3C2×D412yes—
223C2×C4×D3C2×D412yes—
224C2×C4×D3C2×D412yes—
225C2×C4×D3C2×D412yes—
226C3×M16—28no—
227C2×C3×C8—22no—
228C2×C3×C8—22no—
229C2×C3×C8—22no—
230C2×C3×C8—22no—
231C2×C3×C8—22no—
232C2×C3×C8—44no—
233C2×C3×C8C2×D414yes—
234#217—24no—
235#217—24no—
236#217—24no—
237(C2×D3)⋊C4—24no—
238(C2×D3)⋊C4—24no—
239C4⋊Dic3—22no—
240C4⋊Dic3—22no—
241Dic3⋊C4—24no—
242Dic3⋊C4—24no—
243C4×Dic3—22no—
244C4×Dic3—22no—
245#208—28no—
246C2×C3⋊C8—22no—
247C2×C3⋊C8—22no—
248C2×C3⋊C8—22no—
249C2×C3⋊C8—22no—
250C2×C3⋊C8—22no—
251C2×C3⋊C8—44no—
252C2×C3⋊C8C2×D414yes—
253ΓL(1,𝔽25)—216no—
254D3×C8—216no—
255C25—31no—
256C2×#41—22no—
257C2×#41—22no—
258C2×#41—22no—
259C2×#41—22no—
260C2×#41—22no—
261C2×#41—22no—
262C22×Q8C4×D311yescharacteristic
263C22×D4—32no—
264C22×D4—32no—
265C22×D4C4×D312yes—
266C22×D4C4×D311yescharacteristic
267C23×C4—31no—
268C23×C4—31no—
269C23×C4—31no—
270C23×C4—34no—
271C23×C4—34no—
272C23×C4—31no—
273C23×C4—31no—
274C23×C4—62no—
275C23×C4—62no—
276C23×C4—62no—
277C23×C4—62no—
278C23×C4C22×D312yes—
279C23×C4C22×D311yescharacteristic
280C2×M16—22no—
281C2×M16—22no—
282C2×M16—34no—
283C2×M16—64no—
284C2×M16—64no—
285C2×M16—632no—
286C2×M16—62no—
287C2×M16—62no—
288C2×M16—64no—
289C2×M16#6714yescomplemented
290C22×C8—22no—
291C22×C8—22no—
292C22×C8—22no—
293C22×C8—32no—
294C22×C8—32no—
295C22×C8—64no—
296C22×C8—64no—
297C22×C8—64no—
298C22×C8—62no—
299C22×C8—62no—
300C22×C8—62no—
301C22×C8C22×D312yes—
302C22×C8C22×D312yes—
303#123—68no—
304C22⋊Q8—64no—
305C22⋊Q8—64no—
306#121—64no—
307#121—64no—
308#120—68no—
309C4×D4—64no—
310C4×D4—64no—
311C4×D4—64no—
312C4×D4—64no—
313#117—62no—
314#117—62no—
315C2×C4⋊C4—31no—
316C2×C4⋊C4—31no—
317C2×C4⋊C4—32no—
318C2×C4⋊C4—32no—
319C2×C22⋊C4—32no—
320C2×C22⋊C4—32no—
321C2×C22⋊C4—32no—
322C2×C22⋊C4—32no—
323C2×C22⋊C4—32no—
324C2×C42—32no—
325C22⋊C8—28no—
326C22⋊C8—64no—
327C22⋊C8—64no—
328C22⋊C8—68no—
329C23×C3—22no—
330C23×C3—22no—
331C23×C3—22no—
332C23×C3—22no—
333C23×C3—22no—
334C23×C3—24no—
335C23×C3—24no—
336C23×C3C2×C22⋊C412yes—
337C23×C3C4×D412yes—
338C23×C3C4×D412yes—
339C23×C3C23×C411yescharacteristic
340C22×D3—21no—
341C22×D3—21no—
342C22×D3—22no—
343C22×D3—21no—
344C22×D3—21no—
345C22×D3—22no—
346C22×D3—22no—
347C22×D3—28no—
348C22×D3—28no—
349C22×D3—44no—
350C22×D3—48no—
351C22×D3—48no—
352C22×D3C2×C22⋊C412yes—
353C22×D3C2×C22⋊C412yes—
354C22×D3#13114yes—
355C3×Q8—44no—
356C3×D4—48no—
357C3×D4—44no—
358C2×C3×C4—22no—
359C2×C3×C4—22no—
360C2×C3×C4—22no—
361C2×C3×C4—42no—
362C2×C3×C4—42no—
363C2×C3×C4—42no—
364C2×C3×C4—42no—
365C2×C3×C4C2×C22⋊C412yes—
366C2×C3×C4C4×D412yes—
367C2×C3×C4C4×D412yes—
368C2×C3×C4#12014yes—
369C2×C3×C4#12114yes—
370C2×C3×C4#12114yes—
371C2×C3×C4#12314yes—
372C2×C3×C4C22×D411yescharacteristic
373C2×C3×C4C22×D411yescharacteristic
374C2×C3×C4C22×D411yescharacteristic
375C2×C3×C4C22×D411yescharacteristic
376C2×C3×C4C2×#4111yescharacteristic
377C2×C3×C4C2×#4111yescharacteristic
378#67—48no—
379#67—48no—
380C2×Dic3—21no—
381C2×Dic3—21no—
382C2×Dic3—21no—
383C2×Dic3—21no—
384C2×Dic3—28no—
385C2×Dic3—22no—
386C2×Dic3—22no—
387C2×Dic3—24no—
388C2×Dic3—24no—
389C2×Dic3C2×C22⋊C412yes—
390C2×Dic3C2×C22⋊C412yes—
391C4×D3—22no—
392C4×D3—22no—
393C4×D3—22no—
394C4×D3—22no—
395C4×D3—22no—
396C4×D3—22no—
397C4×D3—22no—
398C4×D3—22no—
399C4×D3—28no—
400C4×D3—28no—
401C3×C8—24no—
402C3×C8—42no—
403C3⋊C8—24no—
404C3⋊C8—42no—
405C24—31no—
406C24—31no—
407C24—34no—
408C24—64no—
409C24—62no—
410C24—62no—
411C24—122no—
412C24C2×C4×D312yes—
413C24C2×C4×D311yescharacteristic
414#41—44no—
415#41—44no—
416#41—44no—
417#41—44no—
418C2×Q8—22no—
419C2×Q8—22no—
420C2×Q8—22no—
421C2×D4—24no—
422C2×D4—24no—
423C2×D4—24no—
424C2×D4—22no—
425C2×D4—22no—
426C2×D4—22no—
427C2×D4—64no—
428C2×D4—64no—
429C2×D4—68no—
430C2×D4—124no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—34no—
438C22×C4—31no—
439C22×C4—31no—
440C22×C4—34no—
441C22×C4—34no—
442C22×C4—32no—
443C22×C4—32no—
444C22×C4—32no—
445C22×C4—32no—
446C22×C4—32no—
447C22×C4—32no—
448C22×C4—62no—
449C22×C4—62no—
450C22×C4—62no—
451C22×C4—62no—
452C22×C4—68no—
453C22×C4—68no—
454C22×C4—62no—
455C22×C4—62no—
456C22×C4—68no—
457C22×C4—68no—
458C22×C4—61no—
459C22×C4—61no—
460C22×C4—61no—
461C22×C4—61no—
462C22×C4—61no—
463C22×C4—61no—
464C22×C4—61no—
465C22×C4—61no—
466C22×C4C2×C4×D312yes—
467C22×C4C2×C4×D311yescharacteristic
468C22×C4#23512yes—
469C22×C4D3×D412yes—
470C22×C4D3×D412yes—
471C22×C4D3×D412yes—
472C22×C4#23712yes—
473C22×C4C2×#6712yes—
474C22×C4C23×D311yescharacteristic
475M16—28no—
476M16—616no—
477M16—68no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—34no—
484C2×C8—44no—
485C2×C8—616no—
486C2×C8—62no—
487C2×C8—62no—
488C2×C8—62no—
489C2×C8—62no—
490C2×C8—62no—
491C2×C8—124no—
492C2×C8C2×#6714yes—
493C4⋊C4—64no—
494C4⋊C4—64no—
495C4⋊C4—62no—
496C4⋊C4—62no—
497C22⋊C4—64no—
498C22⋊C4—64no—
499C22⋊C4—64no—
500C22⋊C4—64no—
501C22⋊C4—64no—
502C42—62no—
503C42—62no—
504C22×C3—22no—
505C22×C3—22no—
506C22×C3—24no—
507C22×C3—24no—
508C22×C3—42no—
509C22×C3—42no—
510C22×C3—44no—
511C22×C3—44no—
512C22×C3—42no—
513C22×C3#38614yes—
514C22×C3C22×C22⋊C411yescharacteristic
515C22×C3C2×C4×D411yescharacteristic
516C22×C3C2×C4×D411yescharacteristic
517C22×C3C2×#13112yes—
518C22×C3#56812yes—
519C2×D3—22no—
520C2×D3—24no—
521C2×D3—24no—
522C2×D3—22no—
523C2×D3—22no—
524C2×D3—22no—
525C2×D3—22no—
526C2×D3—416no—
527C2×D3—42no—
528C2×D3—42no—
529C2×D3#40814yescomplemented
530C3×C4—42no—
531C3×C4#38614yes—
532C3×C4C2×#12011yescharacteristic
533C3×C4C2×#12111yescharacteristic
534C3×C4C2×#12111yescharacteristic
535C3×C4C2×#12311yescharacteristic
536Dic3—22no—
537Dic3—22no—
538Dic3—42no—
539Dic3—42no—
540C23—22no—
541C23—22no—
542C23—22no—
543C23—22no—
544C23—24no—
545C23—24no—
546C23—22no—
547C23—32no—
548C23—34no—
549C23—32no—
550C23—68no—
551C23—61no—
552C23—62no—
553C23—61no—
554C23—68no—
555C23—61no—
556C23—62no—
557C23—62no—
558C23—61no—
559C23—124no—
560C23—128no—
561C23—128no—
562C23D3×C22⋊C412yes—
563C23#67⋊C412yes—
564C23C4×#6712yes—
565C23C22×C4×D311yescharacteristic
566Q8—44no—
567D4—48no—
568D4—44no—
569D4—128no—
570D4—128no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—32no—
575C2×C4—32no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—68no—
581C2×C4—61no—
582C2×C4—61no—
583C2×C4—61no—
584C2×C4—61no—
585C2×C4—68no—
586C2×C4—68no—
587C2×C4—62no—
588C2×C4—62no—
589C2×C4—62no—
590C2×C4—62no—
591C2×C4—62no—
592C2×C4—62no—
593C2×C4—62no—
594C2×C4—62no—
595C2×C4—62no—
596C2×C4—62no—
597C2×C4—64no—
598C2×C4—64no—
599C2×C4D3×C22⋊C412yes—
600C2×C4#67⋊C412yes—
601C2×C4#88314yes—
602C2×C4#88414yes—
603C2×C4C4×#6712yes—
604C2×C4#93714yes—
605C2×C4#93814yes—
606C2×C4C2×#23511yescharacteristic
607C2×C4C2×D3×D411yescharacteristic
608C2×C4C2×D3×D411yescharacteristic
609C2×C4C2×D3×D411yescharacteristic
610C2×C4C2×#23711yescharacteristic
611C2×C4C22×#6711yescharacteristic
612C8—24no—
613C8—42no—
614C8—64no—
615C8—122no—
616C2×C3—24no—
617C2×C3—82no—
618C2×C3C2×#38611yescharacteristic
619C2×C3C2×#40811yescharacteristic
620C2×C3#287911yescharacteristic
621C2×C3#293312yes—
622C2×C3#293312yes—
623D3—24no—
624D3—42no—
625C22—24no—
626C22—22no—
627C22—22no—
628C22—24no—
629C22—34no—
630C22—44no—
631C22—42no—
632C22—42no—
633C22—44no—
634C22—42no—
635C22—64no—
636C22—62no—
637C22—64no—
638C22—62no—
639C22—62no—
640C22—62no—
641C22—62no—
642C22—1216no—
643C22—122no—
644C22—122no—
645C22#495614yes—
646C22C2×D3×C22⋊C411yescharacteristic
647C22C2×#67⋊C411yescharacteristic
648C22D3×#13112yes—
649C22#575012yes—
650C22C2×C4×#6711yescharacteristic
651C4—42no—
652C4—62no—
653C4—62no—
654C4—122no—
655C4—122no—
656C4#495614yes—
657C4C2×#88311yescharacteristic
658C4C2×#88411yescharacteristic
659C4C2×#93711yescharacteristic
660C4C2×#93811yescharacteristic
661C3#2089711yescharacteristic, complemented, Sylow
662C2—24no—
663C2—64no—
664C2—82no—
665C2—122no—
666C2#7865912yes—
667C2D3×#40811yescharacteristic
668C2#7866211yescharacteristic
669C2C2×#495611yescharacteristic
670C2#8379612yes—
subgroupG/H
center, FrattiniC22×C4C23×D3
commutator, socleC23×C3C23×C4
FittingC2×C3×#408C2
prime radicalC2×#5828C2
cosocleC22×C3×C4C24
hypercenterC2×#408D3
hypocenterC3#20897
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ C2×#408

Complements

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

NH ≅ G/Ntype
2. C22×D3×M16C2semidirect
3. C2×C3×#408C2semidirect
4. C2×#5234C2semidirect
5. C22×(C2×D3)⋊C8C2semidirect
6. #71980C2semidirect
7. #71965C2semidirect
8. #69958C2semidirect
9. #69957C2semidirect
34. C2×C3×C22⋊C8C22semidirect
36. C2×#830C22semidirect
40. #4550C22semidirect
41. #4550C22semidirect
45. C2×#408D3semidirect
78. C2×C3×M16D4semidirect
108. C2×#208D4semidirect
153. C2×C22⋊C8C2×D3semidirect
289. C2×M16#67semidirect
529. C2×D3#408semidirect
661. C3#20897semidirect

Maximal subgroups

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

subgroupindex
C2×#58282
C22×D3×M162
C2×C3×#4082
C2×#52342
C22×(C2×D3)⋊C82
#719802
#719652
#699582
#699572
#208973