Finite Group Databasebeta

← #10800214

Subgroups — #10800214

Order 768.

1256 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#86257C211yesmaximal, characteristic
2C4×D3×M16C211yesmaximal, characteristic, complemented
3C4×#892C211yesmaximal, characteristic, complemented
4C3×#1986C211yesmaximal, characteristic, complemented
5#71841C211yesmaximal, characteristic, complemented
6#69867C211yesmaximal, characteristic, complemented
7#69492C211yesmaximal, characteristic, complemented
8#68755C214yesmaximal, complemented
9#68728C214yesmaximal, complemented
10#21382—31nomaximal, Sylow
11C3×#533C2211yescharacteristic
12C2×D3×M16C2212yes—
13#5728C411yescharacteristic, complemented
14#5728C411yescharacteristic, complemented
15#5725C411yescharacteristic, complemented
16#5725C411yescharacteristic, complemented
17#5691C412yescomplemented
18#5691C412yescomplemented
19#5627—21no—
20#5627—21no—
21#5624—21no—
22#5624—21no—
23#5580—21no—
24#5580—21no—
25C4×#239—21no—
26C4×#239—21no—
27#5524C2212yes—
28#5522C2212yes—
29C2×#892—21no—
30#5477C2211yescharacteristic
31C2×C42×D3C2211yescharacteristic
32C3×C4×M16C2211yescharacteristic
33C3×C4×C4⋊C4C2211yescharacteristic
34C2×#825C2214yes—
35Dic3×M16C2212yes—
36Dic3×C4⋊C4—21no—
37C4×(C2×D3)⋊C4C2212yes—
38C4×C4⋊Dic3C2211yescharacteristic
39C42×Dic3C2212yes—
40C2×#805C2214yes—
41C4×#208C2211yescharacteristic
42#4707—21no—
43D3×#97—21no—
44C4×ΓL(1,𝔽25)—21no—
45C4×D3×C8—21no—
46#4666—21no—
47#4648—21no—
48C3×#339C2214yes—
49#4532C2214yes—
50#3444—31no—
51C2×C4×M16—31no—
52C4×#117—31no—
53#1986—31no—
54#1986D311yescharacteristic, complemented
55#1980—31no—
56#1980—31no—
57#1871—34no—
58C4×#104—34no—
59C2×#239—21no—
60C2×#239—21no—
61C2×#235C2312yes—
62C22×C4×D3D411yescharacteristic
63C22×C4×D3D411yescharacteristic
64C22×C4×D3C2311yescharacteristic
65C2×C3×M16C2312yes—
66C3×C4⋊Q8C2×C411yescharacteristic
67C3×C4⋊Q8C2×C411yescharacteristic
68C3×#125C2×C412yes—
69C3×C22⋊Q8—21no—
70C3×C22⋊Q8—21no—
71C3×C4×Q8—21no—
72C3×C4×Q8—21no—
73C3×#117C2312yes—
74C2×C3×C4⋊C4—21no—
75C2×C3×C42C2312yes—
76C2×C3×C42C2311yescharacteristic
77#947—21no—
78#947—21no—
79#946—21no—
80#946—21no—
81Q8×Dic3—21no—
82Q8×Dic3—21no—
83#930—21no—
84#930—21no—
85C2×(C2×D3)⋊C4—22no—
86C2×C4⋊Dic3—21no—
87C2×C4×Dic3—22no—
88C2×C4×Dic3—22no—
89C2×C4×Dic3D411yescharacteristic
90C2×C4×Dic3D411yescharacteristic
91C2×C4×Dic3C2312yes—
92C2×C4×Dic3C2311yescharacteristic
93C2×#208C2312yes—
94D3×M16—24no—
95D3×M16—24no—
96C2×ΓL(1,𝔽25)—22no—
97C2×D3×C8—22no—
98#898—22no—
99#898—22no—
100#897C2×C412yes—
101#897C2×C412yes—
102#895—21no—
103#895—21no—
104#895C2×C412yes—
105#895C2×C412yes—
106#894—22no—
107#894—22no—
108#893—22no—
109#893—22no—
110#893—21no—
111#893—21no—
112#893C2×C412yes—
113#893C2×C412yes—
114#892—28no—
115#892—21no—
116#892—21no—
117#892—22no—
118#892—22no—
119#890—21no—
120#890—21no—
121#887C2×C412yes—
122#887C2×C412yes—
123#886—22no—
124#882—22no—
125#67⋊C4—22no—
126#67⋊C4—22no—
127#875C2×C412yes—
128#874C2×C411yescharacteristic
129C4×D12—22no—
130C4×D12—22no—
131C4×D12—21no—
132C4×D12—21no—
133#872—22no—
134C42×D3—22no—
135C42×D3C2×C411yescharacteristic
136C42×D3C2×C411yescharacteristic
137C42×D3C2×C411yescharacteristic
138C42×D3C2×C411yescharacteristic
139C42×D3D411yescharacteristic
140C42×D3D411yescharacteristic
141C42×D3D411yescharacteristic
142C42×D3D411yescharacteristic
143#869C2×C411yescharacteristic
144C3×#97—21no—
145C3×C4×C8—21no—
146C3×#95—21no—
147C3⋊#95—21no—
148C3⋊#95—22no—
149#825—28no—
150#815—22no—
151C8×Dic3—22no—
152#805—28no—
153#803—21no—
154C4×C3⋊C8—21no—
155C22×M16—32no—
156#533—31no—
157#533—31no—
158#533C2×D311yescharacteristic
159#532—31no—
160#532—31no—
161#532—31no—
162#529—32no—
163#529—32no—
164#529—61no—
165#529—61no—
166#529—61no—
167#529—61no—
168#525—32no—
169#525—61no—
170#525—61no—
171C4×#41—32no—
172C4×#41—61no—
173C4×#41—61no—
174C2×#117—61no—
175C22×C42—31no—
176C2×#104—34no—
177C2×#104—34no—
178C4×M16—32no—
179C4×M16—31no—
180C4×M16—61no—
181C4×M16—61no—
182C4×M16C2×D311yescharacteristic
183C2×#97—61no—
184C2×C4×C8—61no—
185C4×C4⋊C4—31no—
186C4×C4⋊C4—61no—
187C4×C4⋊C4C2×D311yescharacteristic
188C4×C22⋊C4—32no—
189C4×C22⋊C4—61no—
190#376—61no—
191C43—32no—
192#339—34no—
193#339C2×D314yes—
194C23×D3C2×D411yescharacteristic
195C2×C3×Q8—21no—
196C2×C3×Q8—21no—
197C22×C3×C4—22no—
198C22×C3×C4C2×D411yescharacteristic
199C22×C3×C4C2×D411yescharacteristic
200C22×C3×C4C2411yescharacteristic
201C2×#67—22no—
202C22×Dic3—22no—
203C22×Dic3C2×D411yescharacteristic
204C22×Dic3C2×D411yescharacteristic
205C22×Dic3C2×D411yescharacteristic
206#239—44no—
207#239—44no—
208#235—24no—
209#235—24no—
210C2×D12—22no—
211C2×D12—22no—
212C2×D12—21no—
213C2×D12—21no—
214C2×D12C22×C412yes—
215C2×C4×D3—22no—
216C2×C4×D3—22no—
217C2×C4×D3—24no—
218C2×C4×D3—22no—
219C2×C4×D3—22no—
220C2×C4×D3—22no—
221C2×C4×D3—42no—
222C2×C4×D3C22⋊C411yescharacteristic
223C2×C4×D3C22⋊C411yescharacteristic
224C2×C4×D3C22⋊C411yescharacteristic
225C2×C4×D3C22⋊C411yescharacteristic
226C2×C4×D3C22⋊C411yescharacteristic
227C2×C4×D3C22⋊C411yescharacteristic
228C2×C4×D3C22⋊C411yescharacteristic
229C2×C4×D3C22⋊C411yescharacteristic
230C2×C4×D3C22×C411yescharacteristic
231C2×C4×D3C22×C411yescharacteristic
232C2×C4×D3C2×D411yescharacteristic
233C2×C4×D3C2×D411yescharacteristic
234C2×Dic6C22×C412yes—
235C3×M16—22no—
236C3×M16#4114yescomplemented
237C2×C3×C8—22no—
238C3×C4⋊C4—24no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—22no—
242C3×C4⋊C4—22no—
243C3×C4⋊C4C22×C412yes—
244C3×C4⋊C4C22×C412yes—
245C3×C22⋊C4—22no—
246C3×C42—22no—
247C3×C42—24no—
248C3×C42C22×C411yescharacteristic
249C3×C42C22×C411yescharacteristic
250C3×C42C2×D411yescharacteristic
251C3×C42C2×D411yescharacteristic
252(C2×D3)⋊C4—28no—
253(C2×D3)⋊C4—22no—
254(C2×D3)⋊C4—22no—
255(C2×D3)⋊C4—22no—
256(C2×D3)⋊C4—22no—
257(C2×D3)⋊C4—22no—
258(C2×D3)⋊C4—22no—
259(C2×D3)⋊C4—22no—
260(C2×D3)⋊C4—22no—
261C4⋊Dic3—24no—
262C4⋊Dic3—21no—
263C4⋊Dic3—21no—
264C4⋊Dic3—22no—
265C4⋊Dic3—22no—
266Dic3⋊C4—22no—
267Dic3⋊C4—22no—
268C4×Dic3—28no—
269C4×Dic3—22no—
270C4×Dic3—22no—
271C4×Dic3—24no—
272C4×Dic3—44no—
273C4×Dic3C22⋊C411yescharacteristic
274C4×Dic3C22⋊C411yescharacteristic
275C4×Dic3C22⋊C411yescharacteristic
276C4×Dic3C22⋊C411yescharacteristic
277C4×Dic3C22⋊C411yescharacteristic
278C4×Dic3C22⋊C411yescharacteristic
279C4×Dic3C22⋊C411yescharacteristic
280C4×Dic3C22⋊C411yescharacteristic
281C4×Dic3C22×C411yescharacteristic
282C4×Dic3C22×C411yescharacteristic
283C4×Dic3C2×D411yescharacteristic
284C4×Dic3C2×D411yescharacteristic
285#208—22no—
286#208#4114yescomplemented
287C2×C3⋊C8—22no—
288ΓL(1,𝔽25)—28no—
289D3×C8—28no—
290C2×#41—32no—
291C2×#41—61no—
292C2×#41—61no—
293C23×C4—31no—
294C23×C4—31no—
295C23×C4—31no—
296C2×M16—32no—
297C2×M16—62no—
298C2×M16—64no—
299C2×M16—64no—
300C2×M16C22×D312yes—
301C22×C8—62no—
302C4⋊Q8—31no—
303C4⋊Q8C4×D311yescharacteristic
304C4⋊Q8C4×D311yescharacteristic
305#127—31no—
306#126—62no—
307#126—62no—
308#125—61no—
309#125—61no—
310#125C4×D312yes—
311#124—32no—
312#124—61no—
313#124—61no—
314#123—62no—
315#123—62no—
316#123—62no—
317C22⋊Q8—21no—
318C22⋊Q8—21no—
319C22⋊Q8—32no—
320C22⋊Q8—32no—
321C22⋊Q8—61no—
322C22⋊Q8—61no—
323#121—32no—
324#121—32no—
325#121—61no—
326#121—61no—
327#121—61no—
328#121—61no—
329#120—62no—
330C4×Q8—21no—
331C4×Q8—21no—
332C4×Q8—32no—
333C4×Q8—32no—
334C4×Q8—61no—
335C4×Q8—61no—
336C4×D4—32no—
337C4×D4—32no—
338C4×D4—62no—
339C4×D4—62no—
340C4×D4—61no—
341C4×D4—61no—
342C4×D4—62no—
343C4×D4—62no—
344C4×D4—62no—
345C4×D4—62no—
346C4×D4—61no—
347C4×D4—61no—
348#117—62no—
349#117—62no—
350#117—62no—
351#117—61no—
352#117—61no—
353#117—68no—
354#117C22×D312yes—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—61no—
357C2×C22⋊C4—62no—
358C2×C42—31no—
359C2×C42—31no—
360C2×C42—31no—
361C2×C42—31no—
362C2×C42—31no—
363C2×C42—31no—
364C2×C42—31no—
365C2×C42—31no—
366C2×C42—31no—
367C2×C42—31no—
368C2×C42—31no—
369C2×C42—32no—
370C2×C42—62no—
371C2×C42—62no—
372C2×C42—62no—
373C2×C42C22×D312yes—
374C2×C42C22×D311yescharacteristic
375#104—68no—
376#104—68no—
377#97—21no—
378#97—62no—
379#97—61no—
380C4×C8—21no—
381C4×C8—62no—
382C4×C8—61no—
383#95—21no—
384#95—62no—
385#95—61no—
386C23×C3C22×D411yescharacteristic
387C22×D3—21no—
388C22×D3—21no—
389C22×D3—22no—
390C22×D3—22no—
391C22×D3—41no—
392C22×D3C2×C22⋊C411yescharacteristic
393C22×D3C2×C22⋊C411yescharacteristic
394C3×Q8—42no—
395C3×Q8—42no—
396C2×C3×C4—22no—
397C2×C3×C4—24no—
398C2×C3×C4—22no—
399C2×C3×C4—22no—
400C2×C3×C4—41no—
401C2×C3×C4—42no—
402C2×C3×C4—42no—
403C2×C3×C4C2×C22⋊C411yescharacteristic
404C2×C3×C4C2×C22⋊C411yescharacteristic
405C2×C3×C4C2×C22⋊C411yescharacteristic
406C2×C3×C4C2×C22⋊C411yescharacteristic
407C2×C3×C4C23×C411yescharacteristic
408C2×C3×C4C22×D411yescharacteristic
409C2×C3×C4C2×#4112yes—
410C2×C3×C4C2×#4112yes—
411#67—42no—
412#67—42no—
413C2×Dic3—21no—
414C2×Dic3—21no—
415C2×Dic3—22no—
416C2×Dic3—22no—
417C2×Dic3—24no—
418C2×Dic3—22no—
419C2×Dic3—42no—
420C2×Dic3—42no—
421C2×Dic3C2×C22⋊C411yescharacteristic
422C2×Dic3C2×C22⋊C411yescharacteristic
423C2×Dic3C2×C22⋊C411yescharacteristic
424C2×Dic3C2×C22⋊C411yescharacteristic
425C2×Dic3C2×C22⋊C411yescharacteristic
426C2×Dic3C2×C22⋊C411yescharacteristic
427D12—24no—
428D12—42no—
429D12—44no—
430D12—44no—
431D12—42no—
432C4×D3—22no—
433C4×D3—22no—
434C4×D3—22no—
435C4×D3—22no—
436C4×D3—22no—
437C4×D3—22no—
438C4×D3—22no—
439C4×D3—22no—
440C4×D3—22no—
441C4×D3—41no—
442C4×D3—41no—
443C4×D3—44no—
444C4×D3—44no—
445C4×D3—44no—
446C4×D3#10414yescomplemented
447Dic6—24no—
448C3×C8—24no—
449C3⋊C8—24no—
450C24—31no—
451#41—64no—
452#41—64no—
453#41—124no—
454#41—124no—
455C2×Q8—21no—
456C2×Q8—21no—
457C2×Q8—32no—
458C2×D4—32no—
459C2×D4—62no—
460C2×D4—62no—
461C2×D4—62no—
462C2×D4—61no—
463C2×D4—61no—
464C22×C4—22no—
465C22×C4—31no—
466C22×C4—31no—
467C22×C4—31no—
468C22×C4—31no—
469C22×C4—31no—
470C22×C4—31no—
471C22×C4—31no—
472C22×C4—31no—
473C22×C4—31no—
474C22×C4—31no—
475C22×C4—31no—
476C22×C4—31no—
477C22×C4—31no—
478C22×C4—31no—
479C22×C4—31no—
480C22×C4—64no—
481C22×C4—62no—
482C22×C4—62no—
483C22×C4—62no—
484C22×C4—62no—
485C22×C4—62no—
486C22×C4—62no—
487C22×C4—122no—
488C22×C4D3×D411yescharacteristic
489C22×C4D3×D411yescharacteristic
490C22×C4C23×D311yescharacteristic
491M16—22no—
492M16—34no—
493M16—68no—
494M16—62no—
495M16#23914yescomplemented
496C2×C8—22no—
497C2×C8—68no—
498C2×C8—62no—
499C4⋊C4—24no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—62no—
505C4⋊C4—62no—
506C4⋊C4—62no—
507C4⋊C4—62no—
508C4⋊C4—64no—
509C4⋊C4—61no—
510C4⋊C4—61no—
511C4⋊C4C2×C4×D312yes—
512C4⋊C4C2×C4×D312yes—
513C22⋊C4—22no—
514C22⋊C4—62no—
515C22⋊C4—62no—
516C22⋊C4—62no—
517C22⋊C4—62no—
518C22⋊C4—62no—
519C22⋊C4—62no—
520C22⋊C4—68no—
521C22⋊C4—62no—
522C22⋊C4—62no—
523C42—22no—
524C42—24no—
525C42—31no—
526C42—31no—
527C42—31no—
528C42—31no—
529C42—31no—
530C42—31no—
531C42—31no—
532C42—31no—
533C42—31no—
534C42—31no—
535C42—31no—
536C42—31no—
537C42—62no—
538C42—62no—
539C42—68no—
540C42—64no—
541C42—124no—
542C42C2×C4×D311yescharacteristic
543C42C2×C4×D311yescharacteristic
544C42D3×D411yescharacteristic
545C42D3×D411yescharacteristic
546C22×C3—21no—
547C22×C3—21no—
548C22×C3C22×C22⋊C411yescharacteristic
549C22×C3#53211yescharacteristic
550C22×C3#53211yescharacteristic
551C2×D3—21no—
552C2×D3—21no—
553C2×D3—21no—
554C2×D3—21no—
555C2×D3—21no—
556C2×D3—42no—
557C2×D3—42no—
558C2×D3—42no—
559C2×D3—42no—
560C2×D3—42no—
561C2×D3C2×#10412yes—
562C3×C4—21no—
563C3×C4—21no—
564C3×C4—21no—
565C3×C4—44no—
566C3×C4—42no—
567C3×C4—42no—
568C3×C4C2×#10412yes—
569C3×C4C4×#4112yes—
570C3×C4#52512yes—
571Dic3—21no—
572Dic3—21no—
573Dic3—21no—
574Dic3—21no—
575Dic3—44no—
576Dic3—42no—
577Dic3C2×#10412yes—
578Dic3#42112yes—
579C23—31no—
580C23—31no—
581C23—62no—
582C23—61no—
583C23—61no—
584C23—62no—
585C23—121no—
586C23C2×D3×D411yescharacteristic
587Q8—42no—
588Q8—42no—
589Q8—64no—
590D4—64no—
591D4—122no—
592D4—124no—
593D4—124no—
594D4—122no—
595D4—122no—
596D4—122no—
597C2×C4—22no—
598C2×C4—24no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—31no—
602C2×C4—31no—
603C2×C4—34no—
604C2×C4—31no—
605C2×C4—31no—
606C2×C4—31no—
607C2×C4—31no—
608C2×C4—41no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—62no—
612C2×C4—62no—
613C2×C4—62no—
614C2×C4—62no—
615C2×C4—62no—
616C2×C4—62no—
617C2×C4—61no—
618C2×C4—61no—
619C2×C4—62no—
620C2×C4—62no—
621C2×C4—62no—
622C2×C4—62no—
623C2×C4—62no—
624C2×C4—64no—
625C2×C4—62no—
626C2×C4—121no—
627C2×C4—121no—
628C2×C4—124no—
629C2×C4—124no—
630C2×C4—122no—
631C2×C4—122no—
632C2×C4—124no—
633C2×C4D3×C22⋊C411yescharacteristic
634C2×C4D3×C22⋊C411yescharacteristic
635C2×C4D3×C22⋊C411yescharacteristic
636C2×C4D3×C22⋊C411yescharacteristic
637C2×C4C22×C4×D311yescharacteristic
638C2×C4C2×D3×D411yescharacteristic
639C2×C4C2×#23912yes—
640C2×C4D3×#4112yes—
641C8—24no—
642C8—64no—
643C2×C3—21no—
644C2×C3—21no—
645C2×C3#229711yescharacteristic
646C2×C3C22×#10411yescharacteristic
647C2×C3C2×#42111yescharacteristic
648D3—22no—
649D3—41no—
650D3—82no—
651C22—21no—
652C22—21no—
653C22—32no—
654C22—61no—
655C22—61no—
656C22—61no—
657C22—61no—
658C22—61no—
659C22—122no—
660C22—122no—
661C22—122no—
662C22—122no—
663C22—122no—
664C22C2×D3×C22⋊C411yescharacteristic
665C22#566811yescharacteristic
666C22#572511yescharacteristic
667C4—21no—
668C4—21no—
669C4—21no—
670C4—32no—
671C4—32no—
672C4—44no—
673C4—42no—
674C4—42no—
675C4—61no—
676C4—61no—
677C4—61no—
678C4—61no—
679C4—124no—
680C4—122no—
681C4D3×#10412yes—
682C4C4×#23912yes—
683C4#562712yes—
684C3#2138211yescharacteristic, complemented, Sylow
685C2—21no—
686C2—21no—
687C2—62no—
688C2—121no—
689C2—242no—
690C2#7647811yescharacteristic
691C2C2×D3×#10411yescharacteristic
692C2D3×#42111yescharacteristic
subgroupG/H
centerC2×C4C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1986C2
socleC22×C3C22×C22⋊C4
prime radical#86257C2
cosocleC22×C3×C4C24
hypercenter#1986D3
hypocenterC3#21382
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1986

Complements

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

NH ≅ G/Ntype
2. C4×D3×M16C2semidirect
3. C4×#892C2semidirect
4. C3×#1986C2semidirect
5. #71841C2semidirect
6. #69867C2semidirect
7. #69492C2semidirect
8. #68755C2semidirect
9. #68728C2semidirect
13. #5728C4semidirect
14. #5728C4semidirect
15. #5725C4semidirect
16. #5725C4semidirect
17. #5691C4semidirect
18. #5691C4semidirect
54. #1986D3semidirect
236. C3×M16#41semidirect
286. #208#41semidirect
446. C4×D3#104semidirect
495. M16#239semidirect
684. C3#21382semidirect

Maximal subgroups

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

subgroupindex
#862572
C4×D3×M162
C4×#8922
C3×#19862
#718412
#698672
#694922
#687552
#687282
#213823