Finite Group Databasebeta

← #10816992

Subgroups — #10816992

Order 768.

898 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D4×Dic3⋊C4C211yesmaximal, characteristic
2C2×#4761C211yesmaximal, characteristic
3#75987C211yesmaximal, characteristic, complemented
4C3×#1864C211yesmaximal, characteristic
5C4×#832C211yesmaximal, characteristic
6#70563C211yesmaximal, characteristic, complemented
7#70559C211yesmaximal, characteristic, complemented
8#70169C211yesmaximal, characteristic
9Dic3×#102C211yesmaximal, characteristic
10Dic3×#102C211yesmaximal, characteristic
11#69618C211yesmaximal, characteristic, complemented
12#68803C211yesmaximal, characteristic, complemented
13#68788C211yesmaximal, characteristic, complemented
14#68651C211yesmaximal, characteristic, complemented
15#68622C211yesmaximal, characteristic, complemented
16#22963—31nomaximal, Sylow
17C2×C3×C4×D4C2211yescharacteristic
18C2×D4×Dic3C2211yescharacteristic
19C2×D4×Dic3C2211yescharacteristic
20C22×Dic3⋊C4—21no—
21C2×#888C2211yescharacteristic
22C2×C3×#106C2211yescharacteristic
23C2×C3×#102C2211yescharacteristic
24C2×C3×#102C2211yescharacteristic
25C3×#389C2211yescharacteristic
26C2×#832C2211yescharacteristic
27C2×#832C2211yescharacteristic
28#5210—21no—
29C2×#817C2211yescharacteristic
30C2×Dic3⋊C8C2211yescharacteristic
31C2×C8×Dic3C2211yescharacteristic
32C2×C8×Dic3C2211yescharacteristic
33#4976—21no—
34#4975C2211yescharacteristic
35Dic3×C4⋊C4C2211yescharacteristic
36Dic3×C4⋊C4C2211yescharacteristic
37#4973—21no—
38#4972C2211yescharacteristic
39#4972C2211yescharacteristic
40C2×#808C2211yescharacteristic
41C2×#808C2211yescharacteristic
42#4946—21no—
43#4942—21no—
44C4×C4⋊Dic3C2211yescharacteristic
45C4×Dic3⋊C4C2211yescharacteristic
46#4928C2211yescharacteristic
47C2×C4×C3⋊C8C2211yescharacteristic
48#4761C418yescomplemented
49#4649—21no—
50#4646—21no—
51C3×#340C2211yescharacteristic
52C3×#336C2211yescharacteristic
53#4551C2211yescharacteristic
54#4551C2211yescharacteristic
55#4551C2211yescharacteristic
56#4550C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61C2×#476—31no—
62#2369—31no—
63D4×C4⋊C4—31no—
64#1990—31no—
65#1948—31no—
66#1941—31no—
67#1941—31no—
68#1919—31no—
69#1919—31no—
70#1864—31no—
71#1864D311yescharacteristic
72C8×C4⋊C4—31no—
73C4×#102—31no—
74C4×#102—31no—
75C4×#102—31no—
76C22×C3×D4C2311yescharacteristic
77C23×C3×C4—21no—
78C23×Dic3—21no—
79C23×Dic3—21no—
80C22×Dic6—21no—
81C22×C3×C8C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C3×C4×D4—22no—
84C3×C4×D4—41no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C22⋊C4—21no—
90C2×C3×C42C2311yescharacteristic
91C2×#217—21no—
92C2×#217—21no—
93D4×Dic3—22no—
94D4×Dic3—22no—
95D4×Dic3—41no—
96D4×Dic3C2×C414yes—
97C2×C4⋊Dic3C2311yescharacteristic
98C2×C4⋊Dic3C2311yescharacteristic
99C2×C4⋊Dic3C2311yescharacteristic
100C2×C4⋊Dic3C2311yescharacteristic
101C2×Dic3⋊C4—21no—
102C2×Dic3⋊C4—21no—
103C2×Dic3⋊C4—21no—
104C2×Dic3⋊C4—21no—
105C2×Dic3⋊C4—21no—
106C2×Dic3⋊C4—21no—
107C2×Dic3⋊C4—44no—
108C2×Dic3⋊C4D411yescharacteristic
109C2×Dic3⋊C4D411yescharacteristic
110C2×C4×Dic3—21no—
111C2×C4×Dic3—21no—
112C2×C4×Dic3—21no—
113C2×C4×Dic3C2311yescharacteristic
114C2×C4×Dic3C2311yescharacteristic
115C22×C3⋊C8C2311yescharacteristic
116C22×C3⋊C8C2311yescharacteristic
117#888—22no—
118#888C2×C414yescomplemented
119C3×#106—41no—
120C3×#102—22no—
121C3×#102C2×C414yes—
122C3×#95—21no—
123#832C2×C414yes—
124#832Q814yescomplemented
125C3⋊#95—21no—
126C3⋊#95—21no—
127C3⋊#95—21no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131C3⋊#95D411yescharacteristic
132C3⋊#95D411yescharacteristic
133#817C2×C414yescomplemented
134Dic3⋊C8C2×C414yescomplemented
135C8×Dic3—22no—
136C8×Dic3—41no—
137#808—41no—
138#808C2×C414yescomplemented
139C4×C3⋊C8—41no—
140C2×C4⋊Q8—31no—
141C2×C4×D4—31no—
142C2×C4×D4—31no—
143C2×C4×D4C2×D311yescharacteristic
144C22×C4⋊C4—61no—
145#476—38no—
146C2×#106—31no—
147C2×#106—31no—
148C2×#106—31no—
149C2×#106C2×D311yescharacteristic
150C2×C4⋊C8—31no—
151C2×#102—31no—
152C2×#102—31no—
153C2×#102C2×D311yescharacteristic
154C2×#102C2×D311yescharacteristic
155C2×C4×C8—31no—
156C2×C4×C8—31no—
157C2×C4×C8—31no—
158#391—61no—
159#391—61no—
160#389—31no—
161#389—31no—
162#389—31no—
163#389—61no—
164#389C2×D311yescharacteristic
165#385—61no—
166#385—61no—
167#384—31no—
168#380—61no—
169C4×C4⋊C4—31no—
170C4×C4⋊C4—31no—
171C4×C4⋊C4—31no—
172C4×C4⋊C4—31no—
173C4×C4⋊C4—61no—
174#340—31no—
175#340—31no—
176#340—31no—
177#340—31no—
178#340—31no—
179#340—31no—
180#340—31no—
181#340C2×D311yescharacteristic
182#336—31no—
183#336C2×D311yescharacteristic
184C24×C3—21no—
185C2×C3×D4—21no—
186C2×C3×D4—21no—
187C2×C3×D4—21no—
188C2×C3×D4C22×C412yes—
189C2×C3×D4C2×Q812yes—
190C2×C3×D4#4112yes—
191C22×C3×C4—21no—
192C22×C3×C4—21no—
193C22×C3×C4—21no—
194C22×C3×C4—42no—
195C22×C3×C4—42no—
196C22×C3×C4—81no—
197C22×C3×C4C2×D411yescharacteristic
198C22×C3×C4C2×D411yescharacteristic
199C22×C3×C4C2411yescharacteristic
200C22×Dic3—21no—
201C22×Dic3—21no—
202C22×Dic3—21no—
203C22×Dic3—24no—
204C22×Dic3—21no—
205C22×Dic3—42no—
206C22×Dic3—42no—
207C22×Dic3—42no—
208C22×Dic3—42no—
209C22×Dic3—81no—
210C22×Dic3C2×D411yescharacteristic
211C22×Dic3C2×D411yescharacteristic
212C22×Dic3C2×D411yescharacteristic
213C22×Dic3C2×D411yescharacteristic
214C2×Dic6—24no—
215C2×Dic6—81no—
216C2×Dic6—81no—
217C2×C3×C8—21no—
218C2×C3×C8—21no—
219C2×C3×C8—21no—
220C2×C3×C8—21no—
221C2×C3×C8—21no—
222C2×C3×C8C22×C412yes—
223C3×C4⋊C4—21no—
224C3×C4⋊C4—21no—
225C3×C4⋊C4—21no—
226C3×C4⋊C4—21no—
227C3×C4⋊C4—21no—
228C3×C4⋊C4—81no—
229C3×C4⋊C4C22×C412yes—
230C3×C22⋊C4—42no—
231C3×C42—21no—
232C3×C42—21no—
233#217—24no—
234#217—42no—
235C4⋊Dic3—21no—
236C4⋊Dic3—21no—
237C4⋊Dic3—21no—
238C4⋊Dic3—21no—
239C4⋊Dic3—21no—
240C4⋊Dic3C22×C412yes—
241C4⋊Dic3C22×C412yes—
242C4⋊Dic3C2×Q812yes—
243Dic3⋊C4—24no—
244Dic3⋊C4—22no—
245Dic3⋊C4—42no—
246Dic3⋊C4—42no—
247Dic3⋊C4—42no—
248Dic3⋊C4—81no—
249Dic3⋊C4—81no—
250Dic3⋊C4SD1614yes—
251C4×Dic3—21no—
252C4×Dic3—21no—
253C4×Dic3—42no—
254C4×Dic3—81no—
255C4×Dic3—81no—
256C4×Dic3C22×C412yes—
257C4×Dic3#4112yes—
258C2×C3⋊C8—21no—
259C2×C3⋊C8—21no—
260C2×C3⋊C8—21no—
261C2×C3⋊C8—21no—
262C2×C3⋊C8C22×C412yes—
263C2×C3⋊C8C2×Q812yes—
264C22×Q8—61no—
265C22×D4C22×D311yescharacteristic
266C23×C4—21no—
267C23×C4—61no—
268C23×C4—61no—
269C22×C8—31no—
270C22×C8—31no—
271C22×C8C22×D311yescharacteristic
272C22×C8C22×D311yescharacteristic
273C4⋊Q8—34no—
274C4⋊Q8—62no—
275C4×D4—22no—
276C4×D4—34no—
277C4×D4—41no—
278C4×D4—62no—
279C4×D4—62no—
280C4×D4—121no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—31no—
283C2×C4⋊C4—31no—
284C2×C4⋊C4—31no—
285C2×C4⋊C4—31no—
286C2×C4⋊C4—31no—
287C2×C4⋊C4—31no—
288C2×C4⋊C4—61no—
289C2×C4⋊C4—61no—
290C2×C4⋊C4—61no—
291C2×C4⋊C4—61no—
292C2×C4⋊C4—61no—
293C2×C4⋊C4—61no—
294C2×C4⋊C4—124no—
295C2×C4⋊C4C22×D311yescharacteristic
296C2×C4⋊C4C22×D311yescharacteristic
297C2×C4⋊C4C22×D311yescharacteristic
298C2×C22⋊C4—21no—
299C2×C22⋊C4—61no—
300C2×C22⋊C4—61no—
301C2×C42—31no—
302C2×C42—31no—
303C2×C42—61no—
304C2×C42—61no—
305C2×C42—61no—
306C2×C42C22×D311yescharacteristic
307#106—34no—
308#106—34no—
309#106—41no—
310#106—121no—
311C4⋊C8—34no—
312#102—22no—
313#102—34no—
314#102—34no—
315#102C4×D314yes—
316C4×C8—62no—
317C4×C8—121no—
318C4×C8—121no—
319#95—21no—
320#95—31no—
321#95—31no—
322#95—61no—
323#95—61no—
324#95—61no—
325#95—61no—
326#95—61no—
327#95—61no—
328C23×C3—22no—
329C23×C3—22no—
330C23×C3—22no—
331C23×C3—41no—
332C23×C3—41no—
333C23×C3—41no—
334C23×C3—41no—
335C23×C3C22×D411yescharacteristic
336C3×D4—22no—
337C3×D4—22no—
338C3×D4—22no—
339C3×D4C4×Q814yes—
340C2×C3×C4—21no—
341C2×C3×C4—21no—
342C2×C3×C4—22no—
343C2×C3×C4—21no—
344C2×C3×C4—21no—
345C2×C3×C4—21no—
346C2×C3×C4—41no—
347C2×C3×C4—41no—
348C2×C3×C4—41no—
349C2×C3×C4—41no—
350C2×C3×C4—41no—
351C2×C3×C4—41no—
352C2×C3×C4—41no—
353C2×C3×C4—41no—
354C2×C3×C4—84no—
355C2×C3×C4C2×SD1612yes—
356C2×C3×C4C23×C411yescharacteristic
357C2×C3×C4C22×Q811yescharacteristic
358C2×C3×C4C2×#4111yescharacteristic
359C2×C3×C4C2×#4111yescharacteristic
360C2×C3×C4H(5,𝔽2)11yescharacteristic
361C2×C3×C42−1+411yescharacteristic
362C2×Dic3—21no—
363C2×Dic3—21no—
364C2×Dic3—21no—
365C2×Dic3—21no—
366C2×Dic3—22no—
367C2×Dic3—22no—
368C2×Dic3—21no—
369C2×Dic3—22no—
370C2×Dic3—21no—
371C2×Dic3—21no—
372C2×Dic3—41no—
373C2×Dic3—41no—
374C2×Dic3—41no—
375C2×Dic3—41no—
376C2×Dic3—41no—
377C2×Dic3—41no—
378C2×Dic3—48no—
379C2×Dic3—41no—
380C2×Dic3—41no—
381C2×Dic3—41no—
382C2×Dic3—84no—
383C2×Dic3C4×D412yes—
384C2×Dic3C4×D412yes—
385C2×Dic3C2×SD1612yes—
386C2×Dic3C2×SD1612yes—
387C2×Dic3#13512yes—
388Dic6—84no—
389C3×C8—22no—
390C3×C8—41no—
391C3⋊C8—22no—
392C3⋊C8—22no—
393C24—21no—
394C2×Q8—64no—
395C2×Q8—241no—
396C2×Q8—241no—
397C2×D4—21no—
398C2×D4—21no—
399C2×D4—21no—
400C2×D4C2×C4×D312yes—
401C2×D4#23712yes—
402C2×D4D3×Q812yes—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—31no—
407C22×C4—31no—
408C22×C4—31no—
409C22×C4—31no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—64no—
413C22×C4—61no—
414C22×C4—61no—
415C22×C4—61no—
416C22×C4—61no—
417C22×C4—81no—
418C22×C4—122no—
419C22×C4—122no—
420C22×C4—122no—
421C22×C4—122no—
422C22×C4—241no—
423C22×C4D3×D411yescharacteristic
424C22×C4D3×D411yescharacteristic
425C22×C4C23×D311yescharacteristic
426C2×C8—21no—
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—21no—
430C2×C8—21no—
431C2×C8—32no—
432C2×C8—32no—
433C2×C8—61no—
434C2×C8—61no—
435C2×C8—61no—
436C2×C8—61no—
437C2×C8C2×C4×D312yes—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—34no—
444C4⋊C4—32no—
445C4⋊C4—32no—
446C4⋊C4—32no—
447C4⋊C4—62no—
448C4⋊C4—64no—
449C4⋊C4—61no—
450C4⋊C4—61no—
451C4⋊C4—61no—
452C4⋊C4—61no—
453C4⋊C4—61no—
454C4⋊C4—81no—
455C4⋊C4—122no—
456C4⋊C4—122no—
457C4⋊C4—122no—
458C4⋊C4—241no—
459C4⋊C4—241no—
460C4⋊C4C2×C4×D312yes—
461C22⋊C4—42no—
462C22⋊C4—64no—
463C22⋊C4—122no—
464C42—21no—
465C42—21no—
466C42—32no—
467C42—32no—
468C42—61no—
469C42—61no—
470C42—122no—
471C42—241no—
472C42—241no—
473C22×C3—24no—
474C22×C3—42no—
475C22×C3—42no—
476C22×C3—42no—
477C22×C3—42no—
478C22×C3—42no—
479C22×C3—42no—
480C22×C3C2×C4×D411yescharacteristic
481C22×C3#54611yescharacteristic
482C22×C3D4×Q811yescharacteristic
483C22×C3C22×SD1611yescharacteristic
484C22×C3C2×#13511yescharacteristic
485C22×C3#57611yescharacteristic
486C22×C3#57711yescharacteristic
487C3×C4—22no—
488C3×C4—42no—
489C3×C4—41no—
490C3×C4—81no—
491C3×C4—81no—
492C3×C4C2×C4×Q811yescharacteristic
493C3×C4#52011yescharacteristic
494C3×C4#55211yescharacteristic
495C3×C4#55411yescharacteristic
496Dic3—22no—
497Dic3—22no—
498Dic3—42no—
499Dic3—42no—
500Dic3—42no—
501Dic3—41no—
502Dic3—81no—
503Dic3C4×SD1614yes—
504C23—22no—
505C23—22no—
506C23—22no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23C2×D3×D411yescharacteristic
512Q8—244no—
513D4—22no—
514D4—22no—
515D4—22no—
516D4#88714yes—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—22no—
522C2×C4—21no—
523C2×C4—32no—
524C2×C4—32no—
525C2×C4—32no—
526C2×C4—32no—
527C2×C4—32no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—61no—
537C2×C4—61no—
538C2×C4—61no—
539C2×C4—61no—
540C2×C4—62no—
541C2×C4—61no—
542C2×C4—61no—
543C2×C4—62no—
544C2×C4—62no—
545C2×C4—61no—
546C2×C4—84no—
547C2×C4—128no—
548C2×C4—121no—
549C2×C4—121no—
550C2×C4—121no—
551C2×C4—121no—
552C2×C4—121no—
553C2×C4—121no—
554C2×C4—121no—
555C2×C4—121no—
556C2×C4—121no—
557C2×C4—244no—
558C2×C4D3×SD1612yes—
559C2×C4C22×C4×D311yescharacteristic
560C2×C4C2×#23711yescharacteristic
561C2×C4#100411yescharacteristic
562C2×C4C2×D3×Q811yescharacteristic
563C2×C4D3×#4111yescharacteristic
564C2×C4#101011yescharacteristic
565C8—22no—
566C8—41no—
567C8—62no—
568C8—62no—
569C2×C3—44no—
570C2×C3#236011yescharacteristic
571C2×C3C2×C4×SD1611yescharacteristic
572C2×C3#294611yescharacteristic
573C2×C3#329511yescharacteristic
574C2×C3#330311yescharacteristic
575C2×C3Q8×SD1611yescharacteristic
576C2×C3#338011yescharacteristic
577C22—24no—
578C22—42no—
579C22—42no—
580C22—42no—
581C22—42no—
582C22—42no—
583C22—42no—
584C22C4×D3×D411yescharacteristic
585C22#559111yescharacteristic
586C22#563211yescharacteristic
587C22C2×D3×SD1611yescharacteristic
588C22D3×#13511yescharacteristic
589C22#577411yescharacteristic
590C22#577511yescharacteristic
591C4—22no—
592C4—34no—
593C4—41no—
594C4—42no—
595C4—62no—
596C4—62no—
597C4—81no—
598C4—81no—
599C4—122no—
600C4—122no—
601C4—121no—
602C4—122no—
603C4—241no—
604C4C2×#88711yescharacteristic
605C4#554611yescharacteristic
606C4#563611yescharacteristic
607C4#565011yescharacteristic
608C3#2296311yescharacteristic, complemented, Sylow
609C2—44no—
610C2#7439611yescharacteristic
611C2C4×D3×SD1611yescharacteristic
612C2#7986611yescharacteristic
613C2#8010411yescharacteristic
614C2#8015811yescharacteristic
615C2#8105311yescharacteristic
616C2#8115611yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1864C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
cosocleC22×C3×C4C24
hypercenter#1864D3
hypocenterC3#22963
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1864

Complements

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

NH ≅ G/Ntype
3. #75987C2semidirect
6. #70563C2semidirect
7. #70559C2semidirect
11. #69618C2semidirect
12. #68803C2semidirect
13. #68788C2semidirect
14. #68651C2semidirect
15. #68622C2semidirect
48. #4761C4semidirect
118. #888C2×C4semidirect
124. #832Q8semidirect
133. #817C2×C4semidirect
134. Dic3⋊C8C2×C4semidirect
138. #808C2×C4semidirect
608. C3#22963semidirect

Maximal subgroups

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

subgroupindex
D4×Dic3⋊C42
C2×#47612
#759872
C3×#18642
C4×#8322
#705632
#705592
#701692
Dic3×#1022
#696182
#688032
#687882
#686512
#686222
#229633