Finite Group Databasebeta

← #10800222

Subgroups — #10800222

Order 768.

1164 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#85680C211yesmaximal, characteristic
2C2×C3×C4×M16C211yesmaximal, characteristic, complemented
3#82060C211yesmaximal, characteristic, complemented
4#71984C214yesmaximal, complemented
5#69868C212yesmaximal, complemented
6#69867C212yesmaximal, complemented
7#69866C214yesmaximal, complemented
8#21383—31nomaximal, Sylow
9C22×C3×M16C2212yes—
10C22×C3×C42C2211yescharacteristic
11#5791—21no—
12#5791—21no—
13#5791C2212yes—
14#5479—21no—
15#5479—21no—
16#5479—21no—
17#5479—21no—
18#5479C412yescomplemented
19#5479C412yescomplemented
20#5477C411yescharacteristic, complemented
21#5477C411yescharacteristic, complemented
22#5477C2211yescharacteristic
23C4×#235—21no—
24C4×#235—21no—
25C4×#235C2212yes—
26#5470C411yescharacteristic, complemented
27#5470C411yescharacteristic, complemented
28#5470C2211yescharacteristic
29C3×C4×M16—21no—
30C3×C4×M16—21no—
31C3×C4×M16C2212yes—
32C3×C4×M16C2212yes—
33C2×C3×#97—21no—
34C2×C3×C4×C8—21no—
35#5212—21no—
36#5211—21no—
37C4×#217C2212yes—
38C2×#825C2218yes—
39#4937—21no—
40#4936—21no—
41C4×C4⋊Dic3C2212yes—
42#4932C2212yes—
43#4555C2218yes—
44#3444—31no—
45C2×C4×M16D311yescharacteristic, complemented
46#2283—31no—
47#1986—32no—
48#1980—32no—
49#1871—34no—
50#1759—34no—
51C23×C3×C4D411yescharacteristic
52C23×C3×C4D411yescharacteristic
53C23×C3×C4C2311yescharacteristic
54C2×#235—21no—
55C2×#235—21no—
56C2×#235C2312yes—
57C2×C3×M16—22no—
58C2×C3×M16—28no—
59C2×C3×M16C2314yes—
60C22×C3×C8—22no—
61C2×C3×C42C2×C411yescharacteristic
62C2×C3×C42C2×C411yescharacteristic
63C2×C3×C42C2×C411yescharacteristic
64C2×C3×C42C2×C411yescharacteristic
65C2×C3×C42D411yescharacteristic
66C2×C3×C42D411yescharacteristic
67C2×C3×C42D411yescharacteristic
68C2×C3×C42D411yescharacteristic
69C2×C3×C42D411yescharacteristic
70C2×C3×C42D411yescharacteristic
71C2×C3×C42C2311yescharacteristic
72C2×C3×C42C2311yescharacteristic
73#953—22no—
74C2×#217—22no—
75#930—21no—
76#930—21no—
77#930—21no—
78#930—21no—
79#930C2×C412yes—
80#930C2×C412yes—
81#929—22no—
82#929—22no—
83#929—22no—
84C4×#67—22no—
85C4×#67—22no—
86C4×#67—22no—
87C4×#67—22no—
88C4×#67—22no—
89C4×#67—22no—
90#926—21no—
91#926—21no—
92#926C2312yes—
93C2×C4⋊Dic3—22no—
94#924—21no—
95#924—21no—
96#924—21no—
97#924—21no—
98#924C2×C412yes—
99#924C2×C412yes—
100C2×C4×Dic3C2314yes—
101#876—22no—
102#876—22no—
103#875—21no—
104#875—21no—
105#875C2×C412yes—
106#874C2×C411yescharacteristic
107C4×D12—21no—
108C4×D12—21no—
109C4×D12—21no—
110C4×D12—21no—
111C4×D12C2×C412yes—
112C4×D12C2×C412yes—
113#872—22no—
114#872—22no—
115#872—22no—
116C42×D3—22no—
117#870—21no—
118#870—21no—
119#870C2×C412yes—
120#869C2×C411yescharacteristic
121#869C2×C411yescharacteristic
122#869C2×C411yescharacteristic
123C4×Dic6—21no—
124C4×Dic6—21no—
125C4×Dic6—21no—
126C4×Dic6—21no—
127C4×Dic6C2×C412yes—
128C4×Dic6C2×C412yes—
129C3×#97—22no—
130C3×#97—22no—
131C3×C4×C8—22no—
132C3×C4×C8—22no—
133C3⋊#95—22no—
134C3⋊#95—22no—
135C3⋊#95—22no—
136C3⋊#95—22no—
137#825—216no—
138C22×M16C2×D312yes—
139#533—31no—
140#533—31no—
141#533—31no—
142#532—31no—
143#532—31no—
144#532—31no—
145#529—32no—
146#529—32no—
147#529—61no—
148#529—61no—
149#529—61no—
150#529—61no—
151#525—32no—
152#525—61no—
153#525—61no—
154C4×#41—32no—
155C4×#41—61no—
156C4×#41—61no—
157C22×C42C2×D311yescharacteristic
158C2×#104—38no—
159C4×M16—21no—
160C4×M16—21no—
161C4×M16C2×D312yes—
162C4×M16C2×D312yes—
163C2×#97—21no—
164C2×C4×C8—21no—
165#383—61no—
166#382—61no—
167#381—61no—
168#380—61no—
169C4×C4⋊C4—32no—
170C4×C22⋊C4—32no—
171#376—32no—
172#339—38no—
173C24×C3C2×D411yescharacteristic
174C22×C3×C4—24no—
175C22×C3×C4—42no—
176C22×C3×C4—42no—
177C22×C3×C4C22⋊C411yescharacteristic
178C22×C3×C4C22⋊C411yescharacteristic
179C22×C3×C4C22⋊C411yescharacteristic
180C22×C3×C4C22⋊C411yescharacteristic
181C22×C3×C4C22⋊C411yescharacteristic
182C22×C3×C4C22⋊C411yescharacteristic
183C22×C3×C4C22⋊C411yescharacteristic
184C22×C3×C4C22⋊C411yescharacteristic
185C22×C3×C4C22×C411yescharacteristic
186C22×C3×C4C22×C411yescharacteristic
187C22×C3×C4C2×D411yescharacteristic
188C22×C3×C4C2×D411yescharacteristic
189C22×C3×C4C2×D411yescharacteristic
190C22×C3×C4C2×D411yescharacteristic
191C22×C3×C4C2×D411yescharacteristic
192C22×C3×C4C2×D411yescharacteristic
193C22×C3×C4C2×D411yescharacteristic
194C22×C3×C4C2411yescharacteristic
195C2×#67—22no—
196C2×#67—22no—
197C2×#67—22no—
198C22×Dic3—24no—
199#235—28no—
200#235—44no—
201#235—44no—
202C2×D12—21no—
203C2×D12—21no—
204C2×D12C22×C412yes—
205C2×C4×D3—22no—
206C2×C4×D3—22no—
207C2×C4×D3—22no—
208C2×Dic6—21no—
209C2×Dic6—21no—
210C2×Dic6C22×C412yes—
211C3×M16—28no—
212C3×M16—24no—
213C3×M16#4118yescomplemented
214C2×C3×C8—28no—
215C2×C3×C8—24no—
216C3×C42C22⋊C411yescharacteristic
217C3×C42C22⋊C411yescharacteristic
218C3×C42C22⋊C411yescharacteristic
219C3×C42C22⋊C411yescharacteristic
220C3×C42C22⋊C411yescharacteristic
221C3×C42C22⋊C411yescharacteristic
222C3×C42C22⋊C411yescharacteristic
223C3×C42C22⋊C411yescharacteristic
224C3×C42C22×C411yescharacteristic
225C3×C42C22×C411yescharacteristic
226C3×C42C22×C411yescharacteristic
227C3×C42C22×C411yescharacteristic
228C3×C42C2×D411yescharacteristic
229C3×C42C2×D411yescharacteristic
230C3×C42C2×D411yescharacteristic
231C3×C42C2×D411yescharacteristic
232#217—22no—
233#217—22no—
234#217—22no—
235#217—44no—
236(C2×D3)⋊C4—22no—
237(C2×D3)⋊C4—22no—
238(C2×D3)⋊C4—22no—
239(C2×D3)⋊C4—22no—
240(C2×D3)⋊C4—22no—
241(C2×D3)⋊C4—22no—
242C4⋊Dic3—21no—
243C4⋊Dic3—21no—
244C4⋊Dic3—21no—
245C4⋊Dic3—21no—
246C4⋊Dic3—44no—
247C4⋊Dic3C22×C412yes—
248C4⋊Dic3C22×C412yes—
249Dic3⋊C4—22no—
250Dic3⋊C4—22no—
251Dic3⋊C4—22no—
252Dic3⋊C4—22no—
253Dic3⋊C4—22no—
254Dic3⋊C4—22no—
255C4×Dic3—22no—
256C4×Dic3—22no—
257C4×Dic3—22no—
258C4×Dic3—28no—
259C2×#41—32no—
260C2×#41—61no—
261C2×#41—61no—
262C23×C4D1211yescharacteristic
263C23×C4D1211yescharacteristic
264C23×C4C22×D311yescharacteristic
265C2×M16—22no—
266C2×M16—28no—
267C2×M16C22×D314yes—
268C22×C8—22no—
269C4⋊Q8—31no—
270C4⋊Q8—31no—
271C4⋊Q8—31no—
272#127—31no—
273#126—62no—
274#126—62no—
275#125—32no—
276#125—61no—
277#125—61no—
278#124—32no—
279#124—61no—
280#124—61no—
281#123—62no—
282#123—62no—
283#123—62no—
284C22⋊Q8—32no—
285C22⋊Q8—32no—
286C22⋊Q8—61no—
287C22⋊Q8—61no—
288C22⋊Q8—61no—
289C22⋊Q8—61no—
290#121—32no—
291#121—32no—
292#121—61no—
293#121—61no—
294#121—61no—
295#121—61no—
296#120—62no—
297C4×Q8—32no—
298C4×Q8—32no—
299C4×Q8—61no—
300C4×Q8—61no—
301C4×Q8—61no—
302C4×Q8—61no—
303C4×D4—32no—
304C4×D4—32no—
305C4×D4—62no—
306C4×D4—62no—
307C4×D4—62no—
308C4×D4—62no—
309C4×D4—62no—
310C4×D4—62no—
311C4×D4—61no—
312C4×D4—61no—
313C4×D4—61no—
314C4×D4—61no—
315#117—32no—
316#117—61no—
317#117—61no—
318#117—62no—
319#117—62no—
320#117—62no—
321C2×C4⋊C4—62no—
322C2×C22⋊C4—62no—
323C2×C42—34no—
324C2×C42—62no—
325C2×C42C4×D311yescharacteristic
326C2×C42C4×D311yescharacteristic
327C2×C42C4×D311yescharacteristic
328C2×C42C4×D311yescharacteristic
329C2×C42D1211yescharacteristic
330C2×C42D1211yescharacteristic
331C2×C42#6711yescharacteristic
332C2×C42#6711yescharacteristic
333C2×C42#6711yescharacteristic
334C2×C42#6711yescharacteristic
335C2×C42C22×D311yescharacteristic
336C2×C42C22×D311yescharacteristic
337#104—616no—
338#97—22no—
339#97—22no—
340C4×C8—22no—
341C4×C8—22no—
342#95—62no—
343#95—62no—
344#95—62no—
345#95—62no—
346C23×C3—22no—
347C23×C3—22no—
348C23×C3—41no—
349C23×C3C2×C22⋊C411yescharacteristic
350C23×C3C2×C22⋊C411yescharacteristic
351C23×C3C22×D411yescharacteristic
352C22×D3—22no—
353C2×C3×C4—22no—
354C2×C3×C4—22no—
355C2×C3×C4—22no—
356C2×C3×C4—22no—
357C2×C3×C4—22no—
358C2×C3×C4—22no—
359C2×C3×C4—22no—
360C2×C3×C4—22no—
361C2×C3×C4—22no—
362C2×C3×C4—22no—
363C2×C3×C4—41no—
364C2×C3×C4—41no—
365C2×C3×C4—41no—
366C2×C3×C4—41no—
367C2×C3×C4—41no—
368C2×C3×C4—41no—
369C2×C3×C4—41no—
370C2×C3×C4C2×C22⋊C411yescharacteristic
371C2×C3×C4C2×C22⋊C411yescharacteristic
372C2×C3×C4C2×C22⋊C411yescharacteristic
373C2×C3×C4C2×C22⋊C411yescharacteristic
374C2×C3×C4C2×C22⋊C411yescharacteristic
375C2×C3×C4C2×C22⋊C411yescharacteristic
376C2×C3×C4C2×C22⋊C411yescharacteristic
377C2×C3×C4C2×C22⋊C411yescharacteristic
378C2×C3×C4C2×C22⋊C411yescharacteristic
379C2×C3×C4C2×C22⋊C411yescharacteristic
380C2×C3×C4C23×C411yescharacteristic
381C2×C3×C4C22×D411yescharacteristic
382C2×C3×C4C2×#4114yes—
383#67—44no—
384#67—44no—
385#67—44no—
386C2×Dic3—22no—
387C2×Dic3—22no—
388C2×Dic3—22no—
389C2×Dic3—44no—
390C2×Dic3—44no—
391C2×Dic3—44no—
392D12—24no—
393D12—42no—
394D12—42no—
395C4×D3—44no—
396C4×D3—44no—
397C4×D3—44no—
398Dic6—24no—
399Dic6—42no—
400Dic6—42no—
401C3×C8—28no—
402C24C2×D1211yescharacteristic
403#41—68no—
404#41—124no—
405#41—124no—
406C2×Q8—32no—
407C2×Q8—61no—
408C2×Q8—61no—
409C2×D4—32no—
410C2×D4—62no—
411C2×D4—62no—
412C2×D4—62no—
413C2×D4—61no—
414C2×D4—61no—
415C22×C4—24no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—62no—
419C22×C4—62no—
420C22×C4—62no—
421C22×C4—64no—
422C22×C4(C2×D3)⋊C411yescharacteristic
423C22×C4(C2×D3)⋊C411yescharacteristic
424C22×C4(C2×D3)⋊C411yescharacteristic
425C22×C4(C2×D3)⋊C411yescharacteristic
426C22×C4(C2×D3)⋊C411yescharacteristic
427C22×C4(C2×D3)⋊C411yescharacteristic
428C22×C4(C2×D3)⋊C411yescharacteristic
429C22×C4(C2×D3)⋊C411yescharacteristic
430C22×C4C2×C4×D311yescharacteristic
431C22×C4C2×C4×D311yescharacteristic
432C22×C4C2×D1211yescharacteristic
433C22×C4C2×D1211yescharacteristic
434C22×C4C2×D1211yescharacteristic
435C22×C4C2×D1211yescharacteristic
436C22×C4C2×D1211yescharacteristic
437C22×C4C2×#6711yescharacteristic
438C22×C4C2×#6711yescharacteristic
439C22×C4C23×D311yescharacteristic
440M16—28no—
441M16—24no—
442M16#23518yescomplemented
443C2×C8—28no—
444C2×C8—24no—
445C4⋊C4—32no—
446C4⋊C4—32no—
447C4⋊C4—62no—
448C4⋊C4—62no—
449C4⋊C4—61no—
450C4⋊C4—61no—
451C4⋊C4—61no—
452C4⋊C4—61no—
453C4⋊C4—62no—
454C4⋊C4—62no—
455C4⋊C4—62no—
456C4⋊C4—62no—
457C4⋊C4—124no—
458C22⋊C4—62no—
459C22⋊C4—62no—
460C22⋊C4—62no—
461C22⋊C4—62no—
462C22⋊C4—62no—
463C22⋊C4—62no—
464C22⋊C4—62no—
465C22⋊C4—62no—
466C22⋊C4—62no—
467C22⋊C4—124no—
468C42—62no—
469C42—62no—
470C42—62no—
471C42—68no—
472C42(C2×D3)⋊C411yescharacteristic
473C42(C2×D3)⋊C411yescharacteristic
474C42(C2×D3)⋊C411yescharacteristic
475C42(C2×D3)⋊C411yescharacteristic
476C42(C2×D3)⋊C411yescharacteristic
477C42(C2×D3)⋊C411yescharacteristic
478C42(C2×D3)⋊C411yescharacteristic
479C42(C2×D3)⋊C411yescharacteristic
480C42C2×C4×D311yescharacteristic
481C42C2×C4×D311yescharacteristic
482C42C2×C4×D311yescharacteristic
483C42C2×C4×D311yescharacteristic
484C42C2×#6711yescharacteristic
485C42C2×#6711yescharacteristic
486C42C2×#6711yescharacteristic
487C42C2×#6711yescharacteristic
488C22×C3—21no—
489C22×C3—21no—
490C22×C3—21no—
491C22×C3—21no—
492C22×C3—21no—
493C22×C3—21no—
494C22×C3—21no—
495C22×C3—44no—
496C22×C3#42112yes—
497C22×C3C22×C22⋊C411yescharacteristic
498C22×C3#53212yes—
499C2×D3—42no—
500C2×D3—42no—
501C2×D3—42no—
502C3×C4—21no—
503C3×C4—21no—
504C3×C4—21no—
505C3×C4—21no—
506C3×C4—21no—
507C3×C4—21no—
508C3×C4—21no—
509C3×C4—21no—
510C3×C4—21no—
511C3×C4#42112yes—
512C3×C4C4×#4112yes—
513C3×C4#52512yes—
514Dic3—42no—
515Dic3—42no—
516Dic3—42no—
517Dic3—84no—
518C23—22no—
519C23—22no—
520C23—41no—
521C23—62no—
522C23C2×(C2×D3)⋊C411yescharacteristic
523C23C2×(C2×D3)⋊C411yescharacteristic
524C23C22×D1211yescharacteristic
525Q8—64no—
526Q8—122no—
527Q8—122no—
528D4—64no—
529D4—122no—
530D4—124no—
531D4—124no—
532D4—124no—
533D4—122no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—62no—
552C2×C4—62no—
553C2×C4—62no—
554C2×C4—124no—
555C2×C4—124no—
556C2×C4—124no—
557C2×C4—124no—
558C2×C4—124no—
559C2×C4—124no—
560C2×C4C2×(C2×D3)⋊C411yescharacteristic
561C2×C4C2×(C2×D3)⋊C411yescharacteristic
562C2×C4C2×(C2×D3)⋊C411yescharacteristic
563C2×C4C2×(C2×D3)⋊C411yescharacteristic
564C2×C4C2×(C2×D3)⋊C411yescharacteristic
565C2×C4C2×(C2×D3)⋊C411yescharacteristic
566C2×C4C2×(C2×D3)⋊C411yescharacteristic
567C2×C4C2×(C2×D3)⋊C411yescharacteristic
568C2×C4C2×(C2×D3)⋊C411yescharacteristic
569C2×C4C2×(C2×D3)⋊C411yescharacteristic
570C2×C4C22×C4×D311yescharacteristic
571C2×C4C2×#23514yes—
572C2×C4C22×#6711yescharacteristic
573C8—28no—
574C2×C3—22no—
575C2×C3—22no—
576C2×C3—41no—
577C2×C3#229711yescharacteristic
578C2×C3C2×#42111yescharacteristic
579C2×C3C2×#42111yescharacteristic
580D3—82no—
581C22—21no—
582C22—21no—
583C22—21no—
584C22—21no—
585C22—21no—
586C22—21no—
587C22—21no—
588C22—44no—
589C22—122no—
590C22—122no—
591C22—122no—
592C22#513912yes—
593C22#547712yes—
594C22C22×(C2×D3)⋊C411yescharacteristic
595C4—21no—
596C4—21no—
597C4—21no—
598C4—21no—
599C4—21no—
600C4—21no—
601C4—21no—
602C4—21no—
603C4—21no—
604C4—122no—
605C4—122no—
606C4—122no—
607C4—244no—
608C4#513912yes—
609C4C4×#23512yes—
610C4#579112yes—
611C3#2138311yescharacteristic, complemented, Sylow
612C2—22no—
613C2—22no—
614C2—41no—
615C2—242no—
616C2#8207011yescharacteristic
617C2C2×#513911yescharacteristic
618C2C2×#513911yescharacteristic
subgroupG/H
centerC2×C4C22×#67
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC2×C3×C4×M16C2
socleC22×C3C22×C22⋊C4
prime radical#85680C2
cosocleC22×C3×C4C24
hypercenterC2×C4×M16D3
hypocenterC3#21383
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ C2×C4×M16

Complements

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

NH ≅ G/Ntype
2. C2×C3×C4×M16C2semidirect
3. #82060C2semidirect
4. #71984C2semidirect
5. #69868C2semidirect
6. #69867C2semidirect
7. #69866C2semidirect
18. #5479C4semidirect
19. #5479C4semidirect
20. #5477C4semidirect
21. #5477C4semidirect
26. #5470C4semidirect
27. #5470C4semidirect
45. C2×C4×M16D3semidirect
213. C3×M16#41semidirect
442. M16#235semidirect
611. C3#21383semidirect

Maximal subgroups

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

subgroupindex
#856802
C2×C3×C4×M162
#820602
#719842
#698682
#698672
#698662
#213833