Finite Group Databasebeta

← #10763253

Subgroups — #10763253

Order 768.

1306 conjugacy classes of subgroups.

The subgroup lattice has 716 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D3×#124C211yesmaximal, characteristic
2#79331C211yesmaximal, characteristic, complemented
3#78517C211yesmaximal, characteristic, complemented
4C3×#1527C211yesmaximal, characteristic, complemented
5#71718C211yesmaximal, characteristic, complemented
6#70804C211yesmaximal, characteristic, complemented
7#69860C211yesmaximal, characteristic, complemented
8#68347C214yesmaximal, complemented
9#68343C214yesmaximal, complemented
10#17666—31nomaximal, Sylow
11C22×D3×Q8C411yescharacteristic
12C22×D3×D4C411yescharacteristic
13C2×C3×#124C2211yescharacteristic
14C2×#947C411yescharacteristic
15C2×#936C411yescharacteristic
16D3×#124—21no—
17D3×#124—21no—
18D3×#124—21no—
19D3×#124—21no—
20D3×#124C412yes—
21D3×#124C412yes—
22D3×#124C2212yes—
23D3×#124C2212yes—
24C2×#885—21no—
25C2×#885—21no—
26C2×D3×C22⋊C4—21no—
27C2×D3×C22⋊C4—21no—
28C2×#875C2211yescharacteristic
29C2×C42×D3C2211yescharacteristic
30C3×#432C2211yescharacteristic, complemented
31C3×C4×M16C2211yescharacteristic, complemented
32#5126—21no—
33#5118—21no—
34#4926C2211yescharacteristic, complemented
35C4×#208C2211yescharacteristic, complemented
36#4837—21no—
37#4833—21no—
38#4726—21no—
39#4718—21no—
40#4707C2212yescomplemented
41#4705C2212yescomplemented
42#4705C2212yescomplemented
43#4705C2212yescomplemented
44#4685—21no—
45#4685—21no—
46C3×#329C2214yescomplemented
47#4540C2214yescomplemented
48#4464C2214yescomplemented
49#4464C2214yescomplemented
50C22×#124—31no—
51#2920—31no—
52#2920—31no—
53#1533—34no—
54#1527—31no—
55#1527—31no—
56#1527—34no—
57#1527—31no—
58#1527D311yescharacteristic, complemented
59C24×D3—21no—
60C22×C3×Q8C2×C411yescharacteristic
61C22×C3×D4C2×C411yescharacteristic
62C22×#67—21no—
63C2×D3×Q8—21no—
64C2×D3×Q8—21no—
65C2×D3×Q8—44no—
66C2×D3×Q8C2×C412yes—
67C2×D3×Q8C2×C412yes—
68C2×D3×D4—21no—
69C2×D3×D4—21no—
70C2×D3×D4—44no—
71C2×D3×D4C2×C412yes—
72C2×D3×D4C2×C412yes—
73C22×D12C2×C411yescharacteristic
74C22×C4×D3—21no—
75C22×C4×D3C2×C411yescharacteristic
76C22×C4×D3D411yescharacteristic
77C22×C4×D3D411yescharacteristic
78C22×Dic6—21no—
79C22×Dic6C2×C411yescharacteristic
80C2×C3×M16—21no—
81C3×#124—21no—
82C3×#124—21no—
83C3×#124C2×C412yes—
84C3×#124C2312yes—
85C2×C3×C22⋊C4—21no—
86C2×C3×C22⋊C4—21no—
87C2×C3×C42C2311yescharacteristic
88C2×#217—21no—
89C2×#217—21no—
90#947—21no—
91#947—21no—
92#947C2×C412yes—
93#947C2×C412yes—
94#936—21no—
95#936—21no—
96#936C2×C412yes—
97#936C2×C412yes—
98C2×(C2×D3)⋊C4—21no—
99C2×(C2×D3)⋊C4—21no—
100C2×(C2×D3)⋊C4—21no—
101C2×(C2×D3)⋊C4—21no—
102C2×C4×Dic3C2×C411yescharacteristic
103C2×C4×Dic3D411yescharacteristic
104C2×C4×Dic3D411yescharacteristic
105C2×#208—21no—
106C2×ΓL(1,𝔽25)—22no—
107#885—22no—
108#885—22no—
109#885—22no—
110#885—22no—
111#885—22no—
112#885—22no—
113#885—22no—
114#885—22no—
115D3×C22⋊C4—22no—
116D3×C22⋊C4—22no—
117D3×C22⋊C4—22no—
118D3×C22⋊C4—22no—
119D3×C22⋊C4—22no—
120D3×C22⋊C4—22no—
121D3×C22⋊C4—22no—
122D3×C22⋊C4—22no—
123#875—21no—
124#875—21no—
125#875C2×C412yes—
126#875C2312yes—
127C42×D3C2×C411yescharacteristic
128C42×D3C2×C411yescharacteristic
129C42×D3D411yescharacteristic
130C42×D3D411yescharacteristic
131C42×D3D411yescharacteristic
132C42×D3D411yescharacteristic
133C42×D3C2311yescharacteristic
134C42×D3C2311yescharacteristic
135C3×C4⋊C8—21no—
136C3×C22⋊C8—21no—
137C3×#97C2312yescomplemented
138C3×#97C2312yescomplemented
139C3×C4×C8—21no—
140#830—21no—
141(C2×D3)⋊C8—22no—
142(C2×D3)⋊C8—22no—
143(C2×D3)⋊C8—22no—
144Dic3⋊C8—22no—
145Dic3⋊C8—22no—
146Dic3⋊C8—22no—
147C8×Dic3—22no—
148C4⋊(C3⋊C8)—21no—
149#803C2312yescomplemented
150#803C2312yescomplemented
151C4×C3⋊C8—21no—
152C23×Q8—31no—
153C23×D4—31no—
154C2×#124—31no—
155C2×#124—31no—
156C2×#124—32no—
157C2×#124—32no—
158C2×#124—32no—
159C2×#124—32no—
160C2×#124—31no—
161C2×#124—61no—
162C2×#124—61no—
163C2×#124—61no—
164C2×#124—61no—
165C2×#124—61no—
166C2×#124—61no—
167C2×#124C2×D311yescharacteristic
168C22×C22⋊C4—61no—
169C22×C22⋊C4—61no—
170C22×C42—31no—
171#432—31no—
172#432—32no—
173#432—32no—
174#432—32no—
175#432C2×D311yescharacteristic, complemented
176#431—61no—
177#431—61no—
178#431—61no—
179#431—61no—
180#423—61no—
181#423—61no—
182#407—61no—
183#407—61no—
184C4×M16—31no—
185C4×M16—32no—
186C4×M16C2×D311yescharacteristic, complemented
187#329—34no—
188#329—34no—
189#329—34no—
190#329C2×D314yescomplemented
191C24×C3—21no—
192C23×D3—21no—
193C23×D3—22no—
194C23×D3—22no—
195C23×D3—44no—
196C23×D3—44no—
197C23×D3—44no—
198C23×D3C22⋊C411yescharacteristic
199C2×C3×Q8—21no—
200C2×C3×Q8—42no—
201C2×C3×Q8C22×C412yes—
202C2×C3×D4—21no—
203C2×C3×D4—42no—
204C2×C3×D4C22×C412yes—
205C22×C3×C4—21no—
206C22×C3×C4C22×C411yescharacteristic
207C22×C3×C4C2×D411yescharacteristic
208C22×C3×C4C2×D411yescharacteristic
209C2×#67—22no—
210C2×#67—22no—
211C2×#67—44no—
212C22×Dic3—21no—
213C22×Dic3C22⋊C411yescharacteristic
214C22×Dic3C22⋊C411yescharacteristic
215C22×Dic3C22⋊C411yescharacteristic
216D3×Q8—44no—
217D3×Q8—44no—
218D3×Q8—44no—
219D3×Q8—44no—
220D3×D4—44no—
221D3×D4—44no—
222D3×D4—44no—
223D3×D4—44no—
224C2×D12—21no—
225C2×D12—42no—
226C2×D12C22×C412yes—
227C2×C4×D3—24no—
228C2×C4×D3—22no—
229C2×C4×D3—22no—
230C2×C4×D3—42no—
231C2×C4×D3—42no—
232C2×C4×D3—44no—
233C2×C4×D3C22⋊C411yescharacteristic
234C2×C4×D3C22⋊C411yescharacteristic
235C2×C4×D3C22⋊C411yescharacteristic
236C2×C4×D3C22⋊C411yescharacteristic
237C2×C4×D3C22⋊C411yescharacteristic
238C2×C4×D3C22⋊C411yescharacteristic
239C2×C4×D3C22×C411yescharacteristic
240C2×C4×D3C22×C411yescharacteristic
241C2×C4×D3C2×D411yescharacteristic
242C2×C4×D3C2×D411yescharacteristic
243C2×C4×D3C2×D411yescharacteristic
244C2×C4×D3C2×D411yescharacteristic
245C2×Dic6—21no—
246C2×Dic6—22no—
247C2×Dic6—22no—
248C2×Dic6—44no—
249C2×Dic6—42no—
250C2×Dic6C22×C412yes—
251C3×M16—42no—
252C2×C3×C8—22no—
253C2×C3×C8—22no—
254C3×C22⋊C4—22no—
255C3×C22⋊C4—22no—
256C3×C22⋊C4—22no—
257C3×C22⋊C4—22no—
258C3×C42C22×C411yescharacteristic
259C3×C42C2×D411yescharacteristic
260C3×C42C2×D411yescharacteristic
261C3×C42C2411yescharacteristic
262#217—22no—
263#217—22no—
264#217—22no—
265#217—22no—
266(C2×D3)⋊C4—22no—
267(C2×D3)⋊C4—22no—
268(C2×D3)⋊C4—22no—
269(C2×D3)⋊C4—22no—
270(C2×D3)⋊C4—22no—
271(C2×D3)⋊C4—22no—
272(C2×D3)⋊C4—22no—
273(C2×D3)⋊C4—22no—
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×Dic3C2×D411yescharacteristic
283C4×Dic3C2×D411yescharacteristic
284C4×Dic3C2×D411yescharacteristic
285C4×Dic3C2×D411yescharacteristic
286#208—42no—
287C2×C3⋊C8—22no—
288C2×C3⋊C8—22no—
289ΓL(1,𝔽25)—44no—
290C25—61no—
291C22×Q8—32no—
292C22×Q8—32no—
293C22×Q8—31no—
294C22×Q8—61no—
295C22×Q8—61no—
296C22×Q8—61no—
297C22×Q8—124no—
298C22×Q8C4×D311yescharacteristic
299C22×D4—32no—
300C22×D4—32no—
301C22×D4—31no—
302C22×D4—61no—
303C22×D4—61no—
304C22×D4—61no—
305C22×D4—124no—
306C22×D4C4×D311yescharacteristic
307C23×C4—31no—
308C23×C4—31no—
309C23×C4—31no—
310C23×C4—61no—
311C2×M16—21no—
312C2×M16—62no—
313C2×M16—61no—
314#124—21no—
315#124—21no—
316#124—32no—
317#124—32no—
318#124—32no—
319#124—32no—
320#124—32no—
321#124—32no—
322#124—62no—
323#124—62no—
324#124—62no—
325#124—62no—
326#124—62no—
327#124—62no—
328#124—62no—
329#124—62no—
330#124—61no—
331#124—61no—
332#124—61no—
333#124—61no—
334#124—61no—
335#124—61no—
336#124C4×D312yes—
337#124C22×D312yes—
338C2×C22⋊C4—21no—
339C2×C22⋊C4—21no—
340C2×C22⋊C4—62no—
341C2×C22⋊C4—62no—
342C2×C22⋊C4—62no—
343C2×C22⋊C4—62no—
344C2×C22⋊C4—62no—
345C2×C22⋊C4—62no—
346C2×C22⋊C4—62no—
347C2×C22⋊C4—62no—
348C2×C22⋊C4—61no—
349C2×C22⋊C4—61no—
350C2×C22⋊C4—61no—
351C2×C22⋊C4—61no—
352C2×C22⋊C4—61no—
353C2×C22⋊C4—61no—
354C2×C42—31no—
355C2×C42—31no—
356C2×C42—31no—
357C2×C42—31no—
358C2×C42—31no—
359C2×C42—31no—
360C2×C42—31no—
361C2×C42—31no—
362C2×C42—31no—
363C2×C42—31no—
364C2×C42—31no—
365C2×C42C22×D311yescharacteristic
366C4⋊C8—21no—
367C4⋊C8—62no—
368C4⋊C8—62no—
369C4⋊C8—62no—
370C4⋊C8—61no—
371C22⋊C8—21no—
372C22⋊C8—62no—
373C22⋊C8—62no—
374C22⋊C8—62no—
375C22⋊C8—61no—
376#97—32no—
377#97—32no—
378#97C22×D312yescomplemented
379#97C22×D312yescomplemented
380C4×C8—21no—
381C4×C8—62no—
382C4×C8—61no—
383C23×C3—22no—
384C23×C3—42no—
385C23×C3—42no—
386C23×C3—42no—
387C23×C3C2×C22⋊C411yescharacteristic
388C22×D3—22no—
389C22×D3—22no—
390C22×D3—22no—
391C22×D3—22no—
392C22×D3—44no—
393C22×D3—44no—
394C22×D3—44no—
395C22×D3—44no—
396C22×D3—42no—
397C22×D3—42no—
398C22×D3—42no—
399C22×D3—44no—
400C22×D3—44no—
401C22×D3—816no—
402C22×D3C2×C22⋊C411yescharacteristic
403C22×D3C2×C22⋊C411yescharacteristic
404C3×Q8—42no—
405C3×Q8—42no—
406C3×D4—42no—
407C3×D4—42no—
408C2×C3×C4—22no—
409C2×C3×C4—22no—
410C2×C3×C4—22no—
411C2×C3×C4—41no—
412C2×C3×C4—42no—
413C2×C3×C4C2×C22⋊C411yescharacteristic
414C2×C3×C4C2×C22⋊C411yescharacteristic
415C2×C3×C4C2×C22⋊C411yescharacteristic
416C2×C3×C4C23×C411yescharacteristic
417C2×C3×C4C22×D411yescharacteristic
418C2×C3×C4C22×D411yescharacteristic
419#67—44no—
420#67—44no—
421C2×Dic3—22no—
422C2×Dic3—22no—
423C2×Dic3—22no—
424C2×Dic3—41no—
425C2×Dic3—42no—
426C2×Dic3C2×C22⋊C411yescharacteristic
427C2×Dic3C2×C22⋊C411yescharacteristic
428C2×Dic3C2×C22⋊C411yescharacteristic
429C2×Dic3C2×C22⋊C411yescharacteristic
430C2×Dic3C2×C22⋊C411yescharacteristic
431C2×Dic3C2×C22⋊C411yescharacteristic
432D12—42no—
433D12—42no—
434C4×D3—22no—
435C4×D3—22no—
436C4×D3—22no—
437C4×D3—22no—
438C4×D3—22no—
439C4×D3—41no—
440C4×D3—41no—
441C4×D3—41no—
442C4×D3—41no—
443C4×D3—41no—
444C4×D3—41no—
445C4×D3—44no—
446C4×D3—44no—
447C4×D3#10014yes—
448Dic6—44no—
449Dic6—42no—
450Dic6—44no—
451Dic6—42no—
452C3×C8—42no—
453C3×C8—42no—
454C3⋊C8—42no—
455C3⋊C8—42no—
456C24—21no—
457C24—31no—
458C24—62no—
459C24—62no—
460C24—61no—
461C24—124no—
462C24—124no—
463C24—124no—
464C2×Q8—21no—
465C2×Q8—32no—
466C2×Q8—42no—
467C2×Q8—62no—
468C2×Q8—62no—
469C2×Q8—61no—
470C2×Q8—122no—
471C2×Q8—124no—
472C2×Q8—124no—
473C2×Q8—124no—
474C2×Q8—124no—
475C2×Q8—124no—
476C2×Q8C2×C4×D312yes—
477C2×D4—21no—
478C2×D4—32no—
479C2×D4—42no—
480C2×D4—62no—
481C2×D4—62no—
482C2×D4—61no—
483C2×D4—122no—
484C2×D4—124no—
485C2×D4—124no—
486C2×D4—124no—
487C2×D4—124no—
488C2×D4—124no—
489C2×D4C2×C4×D312yes—
490C22×C4—21no—
491C22×C4—31no—
492C22×C4—31no—
493C22×C4—31no—
494C22×C4—31no—
495C22×C4—31no—
496C22×C4—31no—
497C22×C4—31no—
498C22×C4—31no—
499C22×C4—31no—
500C22×C4—31no—
501C22×C4—31no—
502C22×C4—31no—
503C22×C4—31no—
504C22×C4—31no—
505C22×C4—31no—
506C22×C4—64no—
507C22×C4—62no—
508C22×C4—62no—
509C22×C4—61no—
510C22×C4—122no—
511C22×C4—122no—
512C22×C4—124no—
513C22×C4C2×C4×D311yescharacteristic
514C22×C4D3×D411yescharacteristic
515C22×C4D3×D411yescharacteristic
516M16—42no—
517M16—124no—
518M16—122no—
519C2×C8—22no—
520C2×C8—22no—
521C2×C8—62no—
522C2×C8—62no—
523C22⋊C4—22no—
524C22⋊C4—22no—
525C22⋊C4—22no—
526C22⋊C4—22no—
527C22⋊C4—62no—
528C22⋊C4—62no—
529C22⋊C4—62no—
530C22⋊C4—62no—
531C22⋊C4—62no—
532C22⋊C4—62no—
533C22⋊C4—62no—
534C22⋊C4—62no—
535C22⋊C4—62no—
536C22⋊C4—62no—
537C22⋊C4—62no—
538C22⋊C4—62no—
539C42—31no—
540C42—31no—
541C42—31no—
542C42—31no—
543C42—31no—
544C42—31no—
545C42—31no—
546C42—31no—
547C42—31no—
548C42—31no—
549C42—31no—
550C42—31no—
551C42C2×C4×D311yescharacteristic
552C42D3×D411yescharacteristic
553C42D3×D411yescharacteristic
554C42C23×D311yescharacteristic
555C22×C3—21no—
556C22×C3—21no—
557C22×C3—42no—
558C22×C3—42no—
559C22×C3—42no—
560C22×C3—84no—
561C22×C3#42112yes—
562C22×C3C22×C22⋊C411yescharacteristic
563C2×D3—28no—
564C2×D3—21no—
565C2×D3—21no—
566C2×D3—21no—
567C2×D3—21no—
568C2×D3—42no—
569C2×D3—42no—
570C2×D3—42no—
571C2×D3—88no—
572C2×D3—88no—
573C2×D3—84no—
574C2×D3C2×#10012yes—
575C2×D3#42112yes—
576C3×C4—21no—
577C3×C4—21no—
578C3×C4—21no—
579C3×C4—21no—
580C3×C4—21no—
581C3×C4—42no—
582C3×C4C2×#10012yes—
583Dic3—21no—
584Dic3—21no—
585Dic3—21no—
586Dic3—21no—
587Dic3—21no—
588Dic3—42no—
589Dic3C2×#10012yes—
590C23—22no—
591C23—31no—
592C23—31no—
593C23—42no—
594C23—42no—
595C23—42no—
596C23—62no—
597C23—62no—
598C23—62no—
599C23—62no—
600C23—122no—
601C23—122no—
602C23—124no—
603C23—124no—
604C23—124no—
605C23—124no—
606C23—124no—
607C23—124no—
608C23—122no—
609C23—2416no—
610C23D3×C22⋊C411yescharacteristic
611Q8—42no—
612Q8—42no—
613Q8—122no—
614Q8—124no—
615Q8—124no—
616Q8—122no—
617D4—42no—
618D4—42no—
619D4—122no—
620D4—124no—
621D4—124no—
622D4—122no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—31no—
627C2×C4—34no—
628C2×C4—31no—
629C2×C4—31no—
630C2×C4—31no—
631C2×C4—31no—
632C2×C4—31no—
633C2×C4—41no—
634C2×C4—42no—
635C2×C4—62no—
636C2×C4—62no—
637C2×C4—62no—
638C2×C4—62no—
639C2×C4—62no—
640C2×C4—62no—
641C2×C4—62no—
642C2×C4—62no—
643C2×C4—121no—
644C2×C4—121no—
645C2×C4—121no—
646C2×C4—121no—
647C2×C4—121no—
648C2×C4—121no—
649C2×C4—124no—
650C2×C4—124no—
651C2×C4—121no—
652C2×C4—122no—
653C2×C4D3×C22⋊C411yescharacteristic
654C2×C4D3×C22⋊C411yescharacteristic
655C2×C4D3×C22⋊C411yescharacteristic
656C2×C4C22×C4×D311yescharacteristic
657C2×C4C2×D3×D411yescharacteristic
658C2×C4C2×D3×D411yescharacteristic
659C8—42no—
660C8—42no—
661C8—122no—
662C8—122no—
663C2×C3—22no—
664C2×C3—82no—
665C2×C3C22×#10011yescharacteristic
666C2×C3C2×#42111yescharacteristic
667C2×C3C2×#42111yescharacteristic
668D3—22no—
669D3—22no—
670D3—82no—
671C22—21no—
672C22—21no—
673C22—32no—
674C22—32no—
675C22—42no—
676C22—42no—
677C22—42no—
678C22—68no—
679C22—61no—
680C22—61no—
681C22—61no—
682C22—61no—
683C22—84no—
684C22—122no—
685C22—122no—
686C22—122no—
687C22—244no—
688C22—248no—
689C22—248no—
690C22#482112yes—
691C22C2×D3×C22⋊C411yescharacteristic
692C4—21no—
693C4—21no—
694C4—21no—
695C4—21no—
696C4—21no—
697C4—32no—
698C4—42no—
699C4—61no—
700C4—61no—
701C4—61no—
702C4—61no—
703C4—61no—
704C4—122no—
705C4D3×#10012yes—
706C3#1766611yescharacteristic, complemented, Sylow
707C2—22no—
708C2—62no—
709C2—62no—
710C2—82no—
711C2—242no—
712C2C2×D3×#10011yescharacteristic
713C2C2×#482111yescharacteristic
714C2D3×#42111yescharacteristic
subgroupG/H
centerC22C2×D3×C22⋊C4
commutatorC2×C3×C4C23×C4
FrattiniC42C23×D3
FittingC3×#1527C2
socleC22×C3C22×C22⋊C4
prime radicalC22×D3×D4C4
cosocleC3×C42C24
hypercenter#1527D3
hypocenterC3#17666
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1527

Complements

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

NH ≅ G/Ntype
2. #79331C2semidirect
3. #78517C2semidirect
4. C3×#1527C2semidirect
5. #71718C2semidirect
6. #70804C2semidirect
7. #69860C2semidirect
8. #68347C2semidirect
9. #68343C2semidirect
30. C3×#432C22semidirect
31. C3×C4×M16C22semidirect
34. #4926C22semidirect
35. C4×#208C22semidirect
40. #4707C22semidirect
41. #4705C22semidirect
42. #4705C22semidirect
43. #4705C22semidirect
46. C3×#329C22semidirect
47. #4540C22semidirect
48. #4464C22semidirect
49. #4464C22semidirect
58. #1527D3semidirect
137. C3×#97C23semidirect
138. C3×#97C23semidirect
149. #803C23semidirect
150. #803C23semidirect
175. #432C2×D3semidirect
186. C4×M16C2×D3semidirect
190. #329C2×D3semidirect
378. #97C22×D3semidirect
379. #97C22×D3semidirect
706. C3#17666semidirect

Maximal subgroups

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

subgroupindex
C2×D3×#1242
#793312
#785172
C3×#15272
#717182
#708042
#698602
#683472
#683432
#176663