Finite Group Databasebeta

← #10798790

Subgroups — #10798790

Order 768.

2010 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×#67C211yesmaximal, characteristic
2C2×C3×#407C211yesmaximal, characteristic, complemented
3C2×#5207C211yesmaximal, characteristic, complemented
4C2×#5126C211yesmaximal, characteristic, complemented
5C2×#5126C211yesmaximal, characteristic, complemented
6C2×#5118C211yesmaximal, characteristic, complemented
7C2×Dic3×M16C211yesmaximal, characteristic, complemented
8#69917C214yesmaximal, complemented
9#69916C214yesmaximal, complemented
10#21190—31nomaximal, Sylow
11C24×C3×C4C2211yescharacteristic
12C23×#67C411yescharacteristic
13C23×C4×D3C2211yescharacteristic
14C22×C3×M16C2211yescharacteristic
15C22×C3×M16C2211yescharacteristic
16C22×#217C411yescharacteristic
17C2×C4×#67—24no—
18C2×C4×#67—24no—
19C2×C4×#67C412yes—
20C2×C4×#67C412yes—
21C2×C4×#67C2212yes—
22C2×C4×#67C2212yes—
23C22×(C2×D3)⋊C4C411yescharacteristic
24C22×Dic3⋊C4C411yescharacteristic
25C22×C4×Dic3C2211yescharacteristic
26C22×#208C2211yescharacteristic
27C22×#208C2211yescharacteristic
28C3×#407—24no—
29C2×C3×C22⋊C8C2214yescomplemented
30#5207—24no—
31C2×#830C2214yescomplemented
32#5126—24no—
33#5126—24no—
34#5118—24no—
35Dic3×M16—24no—
36C2×(C2×D3)⋊C8C2214yescomplemented
37C2×(C2×D3)⋊C8C2214yescomplemented
38C2×#815C2212yescomplemented
39C2×Dic3⋊C8C2212yescomplemented
40C2×Dic3⋊C8C2212yescomplemented
41C2×C8×Dic3C2212yescomplemented
42C22×C4×D4—31no—
43C2×#423—31no—
44C2×#407—31no—
45C2×#407—31no—
46C2×#407—31no—
47C2×#407D311yescharacteristic, complemented
48C2×C4×M16—31no—
49#1955—34no—
50#1954—34no—
51C25×C3C2×C411yescharacteristic
52C24×D3C2×C411yescharacteristic
53C23×C3×C4—22no—
54C23×C3×C4—22no—
55C23×C3×C4—22no—
56C23×C3×C4—22no—
57C23×C3×C4—22no—
58C23×C3×C4—22no—
59C23×C3×C4C2×C411yescharacteristic
60C23×C3×C4C2×C412yes—
61C23×C3×C4C2312yes—
62C23×C3×C4C2311yescharacteristic
63C22×#67—28no—
64C22×#67—24no—
65C22×#67C2×C412yes—
66C22×#67C2×C412yes—
67C23×Dic3C2×C411yescharacteristic
68C23×Dic3C2×C411yescharacteristic
69C23×Dic3C2×C411yescharacteristic
70C22×C4×D3—22no—
71C22×C4×D3—22no—
72C22×C4×D3—22no—
73C22×C4×D3—22no—
74C22×C4×D3—22no—
75C22×C4×D3—22no—
76C22×C4×D3C2×C412yes—
77C22×C4×D3C2312yes—
78C2×C3×M16—22no—
79C2×C3×M16—22no—
80C2×C3×M16—22no—
81C2×C3×M16—22no—
82C2×C3×M16—22no—
83C2×C3×M16D414yescomplemented
84C22×C3×C8C2312yes—
85C22×C3×C8C2312yes—
86C2×#217—24no—
87C2×#217C2×C412yes—
88C2×#217C2×C412yes—
89C4×#67—28no—
90C4×#67—28no—
91C4×#67—24no—
92C4×#67—24no—
93C4×#67—24no—
94C4×#67—24no—
95C2×(C2×D3)⋊C4—24no—
96C2×(C2×D3)⋊C4C2×C412yes—
97C2×(C2×D3)⋊C4C2×C412yes—
98C2×Dic3⋊C4—24no—
99C2×Dic3⋊C4C2×C412yes—
100C2×Dic3⋊C4C2×C412yes—
101C2×C4×Dic3C2×C411yescharacteristic
102C2×C4×Dic3C2×C411yescharacteristic
103C2×C4×Dic3D414yes—
104C2×C4×Dic3D414yes—
105C2×C4×Dic3C2311yescharacteristic
106C2×C4×Dic3C2311yescharacteristic
107C2×#208—22no—
108C2×#208—22no—
109C2×#208—22no—
110C2×#208—22no—
111C2×#208—22no—
112C2×#208D414yescomplemented
113C22×C3⋊C8C2312yes—
114C22×C3⋊C8C2312yes—
115C3×C22⋊C8—28no—
116#830—28no—
117(C2×D3)⋊C8—28no—
118(C2×D3)⋊C8—28no—
119#815—24no—
120Dic3⋊C8—24no—
121Dic3⋊C8—24no—
122C8×Dic3—24no—
123C23×D4—31no—
124C24×C4—31no—
125C24×C4C2×D311yescharacteristic
126C22×M16—31no—
127C22×M16—31no—
128C22×M16C2×D311yescharacteristic
129C22×M16C2×D311yescharacteristic
130C2×C4×D4—32no—
131C2×C4×D4—32no—
132C2×C4×D4—32no—
133C2×C4×D4—32no—
134C2×C4×D4—64no—
135C2×C4×D4—64no—
136C22×C4⋊C4—31no—
137C22×C22⋊C4—31no—
138C22×C22⋊C4—31no—
139C22×C42—31no—
140#423—64no—
141C2×C4⋊C8—32no—
142C2×C4⋊C8—32no—
143#407—24no—
144#407—64no—
145#407—64no—
146#407—64no—
147C2×C22⋊C8—34no—
148C2×C22⋊C8—34no—
149C2×C22⋊C8—34no—
150C2×C22⋊C8C2×D314yescomplemented
151C4×M16—64no—
152C2×#97—32no—
153C2×C4×C8—32no—
154C24×C3—22no—
155C24×C3—24no—
156C24×C3—22no—
157C24×C3—22no—
158C24×C3—42no—
159C24×C3C22×C412yes—
160C24×C3C22×C411yescharacteristic
161C23×D3—22no—
162C23×D3—24no—
163C23×D3—22no—
164C23×D3—22no—
165C23×D3—42no—
166C23×D3C22×C412yes—
167C22×C3×C4—22no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—24no—
171C22×C3×C4—24no—
172C22×C3×C4—24no—
173C22×C3×C4—22no—
174C22×C3×C4—22no—
175C22×C3×C4—22no—
176C22×C3×C4—22no—
177C22×C3×C4—22no—
178C22×C3×C4—22no—
179C22×C3×C4—44no—
180C22×C3×C4—44no—
181C22×C3×C4—44no—
182C22×C3×C4—44no—
183C22×C3×C4C22×C412yes—
184C22×C3×C4C22×C411yescharacteristic
185C22×C3×C4C2×D412yes—
186C22×C3×C4C2×D412yes—
187C22×C3×C4C2×D412yes—
188C22×C3×C4C2×D412yes—
189C22×C3×C4#4112yes—
190C22×C3×C4#4112yes—
191C22×C3×C4C2411yescharacteristic
192C2×#67—232no—
193C2×#67—24no—
194C2×#67—24no—
195C2×#67—24no—
196C2×#67—24no—
197C2×#67—24no—
198C2×#67M1618yescomplemented
199C22×Dic3—24no—
200C22×Dic3—22no—
201C22×Dic3—22no—
202C22×Dic3—22no—
203C22×Dic3—42no—
204C22×Dic3C22⋊C414yes—
205C22×Dic3C22⋊C414yes—
206C22×Dic3C22×C412yes—
207C22×Dic3C22×C411yescharacteristic
208C22×Dic3C22×C411yescharacteristic
209C22×Dic3C22×C411yescharacteristic
210C22×Dic3C22×C411yescharacteristic
211C2×C4×D3—24no—
212C2×C4×D3—24no—
213C2×C4×D3—24no—
214C2×C4×D3—22no—
215C2×C4×D3—22no—
216C2×C4×D3—22no—
217C2×C4×D3—22no—
218C2×C4×D3—22no—
219C2×C4×D3—22no—
220C2×C4×D3—44no—
221C2×C4×D3—44no—
222C2×C4×D3—44no—
223C2×C4×D3—44no—
224C3×M16—28no—
225C3×M16—44no—
226C2×C3×C8—22no—
227C2×C3×C8—22no—
228C2×C3×C8—22no—
229C2×C3×C8—22no—
230C2×C3×C8—22no—
231C2×C3×C8C2×D414yescomplemented
232#217—24no—
233#217—24no—
234(C2×D3)⋊C4—24no—
235(C2×D3)⋊C4—24no—
236Dic3⋊C4—24no—
237Dic3⋊C4—24no—
238C4×Dic3C22⋊C414yes—
239C4×Dic3C22⋊C414yes—
240C4×Dic3C2×D412yes—
241C4×Dic3C2×D412yes—
242C4×Dic3C2×D412yes—
243C4×Dic3C2×D412yes—
244#208—28no—
245#208—44no—
246C2×C3⋊C8—22no—
247C2×C3⋊C8—22no—
248C2×C3⋊C8—22no—
249C2×C3⋊C8—22no—
250C2×C3⋊C8—22no—
251C2×C3⋊C8C2×D414yescomplemented
252C25—31no—
253C25C4×D311yescharacteristic
254C22×D4—32no—
255C22×D4—32no—
256C22×D4—64no—
257C22×D4—68no—
258C23×C4—22no—
259C23×C4—22no—
260C23×C4—22no—
261C23×C4—22no—
262C23×C4—22no—
263C23×C4—22no—
264C23×C4—31no—
265C23×C4—31no—
266C23×C4—31no—
267C23×C4—32no—
268C23×C4—32no—
269C23×C4—62no—
270C23×C4—62no—
271C23×C4—62no—
272C23×C4—62no—
273C23×C4—62no—
274C23×C4—62no—
275C23×C4C4×D311yescharacteristic
276C23×C4C4×D312yes—
277C23×C4C22×D312yes—
278C23×C4C22×D311yescharacteristic
279C2×M16—22no—
280C2×M16—22no—
281C2×M16—22no—
282C2×M16—22no—
283C2×M16—22no—
284C2×M16—34no—
285C2×M16—62no—
286C2×M16—62no—
287C2×M16—62no—
288C2×M16—62no—
289C2×M16—62no—
290C2×M16#6714yescomplemented
291C22×C8—32no—
292C22×C8—32no—
293C22×C8C22×D312yes—
294C22×C8C22×D312yes—
295C4×D4—68no—
296C4×D4—68no—
297C4×D4—64no—
298C4×D4—64no—
299C4×D4—64no—
300C4×D4—64no—
301C2×C4⋊C4—32no—
302C2×C4⋊C4—32no—
303C2×C4⋊C4—64no—
304C2×C22⋊C4—32no—
305C2×C22⋊C4—32no—
306C2×C22⋊C4—32no—
307C2×C22⋊C4—32no—
308C2×C22⋊C4—64no—
309C2×C22⋊C4—64no—
310C2×C42—31no—
311C2×C42—31no—
312C2×C42—34no—
313C2×C42—34no—
314C2×C42—31no—
315C2×C42—31no—
316C4⋊C8—64no—
317C4⋊C8—64no—
318C22⋊C8—28no—
319C22⋊C8—68no—
320C22⋊C8—68no—
321C22⋊C8—68no—
322#97—64no—
323C4×C8—64no—
324C23×C3—22no—
325C23×C3—22no—
326C23×C3—28no—
327C23×C3—22no—
328C23×C3—22no—
329C23×C3—22no—
330C23×C3—22no—
331C23×C3—22no—
332C23×C3—22no—
333C23×C3—48no—
334C23×C3—48no—
335C23×C3—44no—
336C23×C3—44no—
337C23×C3C2×C22⋊C412yes—
338C23×C3C4×D412yes—
339C23×C3C4×D412yes—
340C23×C3C2×M1614yes—
341C23×C3C23×C411yescharacteristic
342C22×D3—28no—
343C22×D3—22no—
344C22×D3—22no—
345C22×D3—22no—
346C22×D3—22no—
347C22×D3—22no—
348C22×D3—22no—
349C22×D3—48no—
350C22×D3—48no—
351C22×D3—44no—
352C22×D3—44no—
353C22×D3C2×M1614yes—
354C2×C3×C4—22no—
355C2×C3×C4—22no—
356C2×C3×C4—22no—
357C2×C3×C4—44no—
358C2×C3×C4—42no—
359C2×C3×C4—42no—
360C2×C3×C4—42no—
361C2×C3×C4—42no—
362C2×C3×C4—48no—
363C2×C3×C4C2×C22⋊C412yes—
364C2×C3×C4C4×D412yes—
365C2×C3×C4C4×D412yes—
366C2×C3×C4#12114yes—
367C2×C3×C4#12114yes—
368C2×C3×C4#12414yes—
369C2×C3×C4#12714yes—
370C2×C3×C4C22×D411yescharacteristic
371C2×C3×C4C22×D411yescharacteristic
372C2×C3×C4C22×D411yescharacteristic
373C2×C3×C4C22×D411yescharacteristic
374C2×C3×C4C2×#4111yescharacteristic
375C2×C3×C4C2×#4111yescharacteristic
376#67—216no—
377#67—216no—
378C2×Dic3—22no—
379C2×Dic3—22no—
380C2×Dic3—22no—
381C2×Dic3—28no—
382C2×Dic3—22no—
383C2×Dic3—22no—
384C2×Dic3—22no—
385C2×Dic3—44no—
386C2×Dic3—44no—
387C2×Dic3C2×C22⋊C412yes—
388C2×Dic3C2×C22⋊C412yes—
389C2×Dic3C2×C22⋊C412yes—
390C2×Dic3C2×C22⋊C412yes—
391C2×Dic3C2×M1614yes—
392C4×D3—42no—
393C4×D3—42no—
394C4×D3—42no—
395C4×D3—42no—
396C4×D3—48no—
397C3×C8—24no—
398C3×C8—42no—
399C3⋊C8—24no—
400C3⋊C8—42no—
401C24—22no—
402C24—24no—
403C24—22no—
404C24—22no—
405C24—32no—
406C24—42no—
407C24—64no—
408C24—62no—
409C24—62no—
410C24—62no—
411C24—122no—
412C24C2×C4×D312yes—
413C24C2×C4×D311yescharacteristic
414C2×D4—38no—
415C2×D4—64no—
416C2×D4—64no—
417C2×D4—64no—
418C2×D4—64no—
419C2×D4—632no—
420C2×D4—64no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—31no—
434C22×C4—34no—
435C22×C4—31no—
436C22×C4—31no—
437C22×C4—31no—
438C22×C4—34no—
439C22×C4—32no—
440C22×C4—44no—
441C22×C4—44no—
442C22×C4—44no—
443C22×C4—44no—
444C22×C4—64no—
445C22×C4—62no—
446C22×C4—62no—
447C22×C4—62no—
448C22×C4—64no—
449C22×C4—64no—
450C22×C4—64no—
451C22×C4—62no—
452C22×C4—62no—
453C22×C4—62no—
454C22×C4—62no—
455C22×C4—62no—
456C22×C4—62no—
457C22×C4—122no—
458C22×C4—124no—
459C22×C4—124no—
460C22×C4—124no—
461C22×C4—124no—
462C22×C4C2×C4×D312yes—
463C22×C4C2×C4×D311yescharacteristic
464C22×C4#23512yes—
465C22×C4D3×D412yes—
466C22×C4D3×D412yes—
467C22×C4D3×D412yes—
468C22×C4#23712yes—
469C22×C4C2×#6712yes—
470C22×C4C23×D311yescharacteristic
471M16—28no—
472M16—44no—
473M16—68no—
474M16—124no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—34no—
481C2×C8—62no—
482C2×C8—62no—
483C2×C8—62no—
484C2×C8—62no—
485C2×C8—62no—
486C2×C8C2×#6714yescomplemented
487C4⋊C4—64no—
488C4⋊C4—64no—
489C22⋊C4—64no—
490C22⋊C4—64no—
491C22⋊C4—64no—
492C22⋊C4—64no—
493C42—34no—
494C42—34no—
495C42—32no—
496C42—32no—
497C42—32no—
498C42—32no—
499C22×C3—22no—
500C22×C3—22no—
501C22×C3—24no—
502C22×C3—24no—
503C22×C3—24no—
504C22×C3—24no—
505C22×C3—24no—
506C22×C3—416no—
507C22×C3—42no—
508C22×C3—42no—
509C22×C3—42no—
510C22×C3—42no—
511C22×C3#39014yes—
512C22×C3C22×C22⋊C411yescharacteristic
513C22×C3C2×C4×D411yescharacteristic
514C22×C3C2×C4×D411yescharacteristic
515C22×C3C22×M1612yes—
516C22×C3#56812yes—
517C2×D3—24no—
518C2×D3—24no—
519C2×D3—24no—
520C2×D3—416no—
521C2×D3—42no—
522C2×D3—42no—
523C2×D3—42no—
524C2×D3—42no—
525C3×C4—42no—
526C3×C4#39014yes—
527C3×C4C2×#12111yescharacteristic
528C3×C4C2×#12111yescharacteristic
529C3×C4C2×#12411yescharacteristic
530C3×C4C2×#12711yescharacteristic
531Dic3—22no—
532Dic3—22no—
533Dic3—22no—
534Dic3—42no—
535Dic3#40714yescomplemented
536C23—22no—
537C23—22no—
538C23—28no—
539C23—22no—
540C23—22no—
541C23—22no—
542C23—22no—
543C23—22no—
544C23—22no—
545C23—34no—
546C23—48no—
547C23—48no—
548C23—44no—
549C23—44no—
550C23—68no—
551C23—62no—
552C23—62no—
553C23—62no—
554C23—62no—
555C23—62no—
556C23—62no—
557C23—128no—
558C23—128no—
559C23—124no—
560C23—124no—
561C23D3×C22⋊C412yes—
562C23#67⋊C412yes—
563C23D3×M1614yes—
564C23C4×#6712yes—
565C23C22×C4×D311yescharacteristic
566D4—616no—
567D4—616no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—32no—
572C2×C4—34no—
573C2×C4—32no—
574C2×C4—32no—
575C2×C4—32no—
576C2×C4—44no—
577C2×C4—48no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—68no—
583C2×C4—62no—
584C2×C4—62no—
585C2×C4—62no—
586C2×C4—62no—
587C2×C4—62no—
588C2×C4—62no—
589C2×C4—124no—
590C2×C4—124no—
591C2×C4—128no—
592C2×C4—122no—
593C2×C4—122no—
594C2×C4—122no—
595C2×C4—122no—
596C2×C4D3×C22⋊C412yes—
597C2×C4#67⋊C412yes—
598C2×C4#88414yes—
599C2×C4#88514yes—
600C2×C4C4×#6712yes—
601C2×C4#93914yes—
602C2×C4#94014yes—
603C2×C4C2×#23511yescharacteristic
604C2×C4C2×D3×D411yescharacteristic
605C2×C4C2×D3×D411yescharacteristic
606C2×C4C2×D3×D411yescharacteristic
607C2×C4C2×#23711yescharacteristic
608C2×C4C22×#6711yescharacteristic
609C8—24no—
610C8—42no—
611C8—64no—
612C8—122no—
613C2×C3—24no—
614C2×C3—44no—
615C2×C3C2×#39011yescharacteristic
616C2×C3C2×#40711yescharacteristic
617C2×C3#287911yescharacteristic
618C2×C3D4×M1612yes—
619C2×C3D4×M1612yes—
620D3—44no—
621C22—24no—
622C22—22no—
623C22—22no—
624C22—24no—
625C22—24no—
626C22—24no—
627C22—24no—
628C22—416no—
629C22—42no—
630C22—42no—
631C22—42no—
632C22—42no—
633C22—64no—
634C22—64no—
635C22—64no—
636C22—1216no—
637C22—122no—
638C22—122no—
639C22—122no—
640C22—122no—
641C22#495714yes—
642C22C2×D3×C22⋊C411yescharacteristic
643C22C2×#67⋊C411yescharacteristic
644C22C2×D3×M1612yes—
645C22#574512yes—
646C22C2×C4×#6711yescharacteristic
647C4—34no—
648C4—42no—
649C4—62no—
650C4—62no—
651C4—62no—
652C4—122no—
653C4#495714yes—
654C4C2×#88411yescharacteristic
655C4C2×#88511yescharacteristic
656C4C2×#93911yescharacteristic
657C4C2×#94011yescharacteristic
658C3#2119011yescharacteristic, complemented, Sylow
659C2—24no—
660C2—44no—
661C2—124no—
662C2#7862812yes—
663C2D3×#40711yescharacteristic
664C2#7863011yescharacteristic
665C2C2×#495711yescharacteristic
666C2M16×#6712yes—
subgroupG/H
center, FrattiniC22×C4C23×D3
commutator, socleC23×C3C23×C4
FittingC2×C3×#407C2
prime radicalC23×#67C4
cosocleC22×C3×C4C24
hypercenterC2×#407D3
hypocenterC3#21190
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ C2×#407

Complements

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

NH ≅ G/Ntype
2. C2×C3×#407C2semidirect
3. C2×#5207C2semidirect
4. C2×#5126C2semidirect
5. C2×#5126C2semidirect
6. C2×#5118C2semidirect
7. C2×Dic3×M16C2semidirect
8. #69917C2semidirect
9. #69916C2semidirect
29. C2×C3×C22⋊C8C22semidirect
31. C2×#830C22semidirect
36. C2×(C2×D3)⋊C8C22semidirect
37. C2×(C2×D3)⋊C8C22semidirect
38. C2×#815C22semidirect
39. C2×Dic3⋊C8C22semidirect
40. C2×Dic3⋊C8C22semidirect
41. C2×C8×Dic3C22semidirect
47. C2×#407D3semidirect
83. C2×C3×M16D4semidirect
112. C2×#208D4semidirect
150. C2×C22⋊C8C2×D3semidirect
198. C2×#67M16semidirect
231. C2×C3×C8C2×D4semidirect
251. C2×C3⋊C8C2×D4semidirect
290. C2×M16#67semidirect
486. C2×C8C2×#67semidirect
535. Dic3#407semidirect
658. C3#21190semidirect

Maximal subgroups

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

subgroupindex
C22×C4×#672
C2×C3×#4072
C2×#52072
C2×#51262
C2×#51182
C2×Dic3×M162
#699172
#699162
#211903