Finite Group Databasebeta

← #10947939

Subgroups — #10947939

Order 768.

1410 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#87284C211yesmaximal, characteristic
2C3×#2888C211yesmaximal, characteristic, complemented
3#84107C211yesmaximal, characteristic, complemented
4#70146C212yesmaximal, complemented
5#70145C212yesmaximal, complemented
6#70094C214yesmaximal, complemented
7C2×#4475C214yesmaximal, complemented
8#33348—31nomaximal, Sylow
9C2×C3×H(5,𝔽2)C2211yescharacteristic
10C22×D3×D4C411yescharacteristic, complemented
11C3×#568—21no—
12#5830C411yescharacteristic, complemented
13#5830C2211yescharacteristic
14#5829—21no—
15#5829—22no—
16#5829—21no—
17#5828C411yescharacteristic, complemented
18#5828C2211yescharacteristic
19#5804C412yescomplemented
20#5804C2212yes—
21D4×#67—22no—
22D4×#67—21no—
23D4×#67—21no—
24C2×#937C412yescomplemented
25C2×#937C2212yes—
26#5798—21no—
27#5798C411yescharacteristic, complemented
28C3×#413—22no—
29C3×#413C2212yescomplemented
30C2×C3×#100—21no—
31C2×C3×#100C2214yescomplemented
32#5240—21no—
33#5240C2212yescomplemented
34C2×#834—22no—
35C2×#834C2214yescomplemented
36#4476C2214yescomplemented
37#4475C2214yescomplemented
38#4475C2218yescomplemented
39#3484—31no—
40#2888D311yescharacteristic, complemented
41#2883—31no—
42#2122—32no—
43#2121—34no—
44#2120—32no—
45C2×AΣL(1,𝔽16)—34no—
46C24×D3C2×C412yescomplemented
47C3×H(5,𝔽2)—44no—
48C2×C3×#41C2×C411yescharacteristic
49C2×C3×#41D411yescharacteristic
50C2×C3×#41D412yes—
51C2×C3×#41D411yescharacteristic
52C2×C3×#41C2311yescharacteristic
53C22×C3×D4C2×C412yes—
54C22×C3×D4C2×C411yescharacteristic
55C22×C3×D4D412yes—
56C22×C3×D4D411yescharacteristic
57C22×C3×D4D411yescharacteristic
58C22×C3×D4C2312yes—
59C22×#67—22no—
60C22×#67—22no—
61C22×#67C2×C412yescomplemented
62C2×D3×D4—22no—
63C2×D3×D4—21no—
64C2×D3×D4—21no—
65C2×D3×D4—44no—
66C22×D12C2×C411yescharacteristic, complemented
67C22×C4×D3C2×C411yescharacteristic, complemented
68C3×#131—44no—
69C2×C3×M16—22no—
70C2×C3×M16—24no—
71#953—22no—
72#953—22no—
73C2×#217—24no—
74C2×#217C2×C412yescomplemented
75C2×#217C2×C412yescomplemented
76#947—21no—
77#947—21no—
78#946—21no—
79#946—21no—
80#940—21no—
81#940—21no—
82#939—22no—
83#939—22no—
84#939—22no—
85#939—22no—
86#938—21no—
87#938—21no—
88#937—22no—
89#937—22no—
90#937—22no—
91#937—22no—
92#937C2×C411yescharacteristic, complemented
93#937C2×C412yescomplemented
94#937C2×C412yescomplemented
95#937C2×C412yescomplemented
96#937C2×C412yescomplemented
97#937C2311yescharacteristic
98#937C2312yes—
99#935C2×C411yescharacteristic, complemented
100#935C2×C412yescomplemented
101#935C2311yescharacteristic
102D4×Dic3—22no—
103D4×Dic3—21no—
104D4×Dic3—21no—
105D4×Dic3—44no—
106#930—22no—
107#930—21no—
108#930—21no—
109#929—22no—
110#929—22no—
111C4×#67—22no—
112C4×#67—21no—
113C4×#67—21no—
114C2×(C2×D3)⋊C4—22no—
115C2×(C2×D3)⋊C4—22no—
116#926—22no—
117#926C2×C411yescharacteristic, complemented
118#926C2×C411yescharacteristic, complemented
119C3×#101—22no—
120C3×#100—24no—
121C3×#100C2314yes—
122#834—22no—
123#834—24no—
124#834C2314yes—
125C2×H(5,𝔽2)C2×D311yescharacteristic
126C23×D4—31no—
127#568—21no—
128#546—61no—
129#546—62no—
130#546—61no—
131D42—62no—
132D42—61no—
133D42—61no—
134#535—31no—
135#535—31no—
136#534—32no—
137#534—32no—
138#525—31no—
139#525—31no—
140C2×#120—32no—
141C2×#120—32no—
142#518—31no—
143#518—61no—
144#413—22no—
145#413C2×D312yescomplemented
146C2×#100—21no—
147C2×#100C2×D314yescomplemented
148#410—32no—
149#410—61no—
150C2×#99—34no—
151C2×#99—62no—
152#352—34no—
153AΣL(1,𝔽16)—34no—
154AΣL(1,𝔽16)—38no—
155C24×C3C22⋊C412yes—
156C24×C3C22×C412yes—
157C24×C3C2×D412yes—
158C23×D3—22no—
159C23×D3—22no—
160C23×D3—44no—
161C23×D3—84no—
162C23×D3C22×C411yescharacteristic
163C23×D3C22×C412yes—
164C3×#41—24no—
165C3×#41—42no—
166C3×#41—44no—
167C3×#41—42no—
168C3×#41—42no—
169C2×C3×Q8C22⋊C411yescharacteristic
170C2×C3×Q8C2×D411yescharacteristic
171C2×C3×D4—24no—
172C2×C3×D4—24no—
173C2×C3×D4—42no—
174C2×C3×D4—44no—
175C2×C3×D4—44no—
176C2×C3×D4—44no—
177C2×C3×D4C22⋊C412yes—
178C2×C3×D4C22⋊C411yescharacteristic
179C2×C3×D4C22⋊C412yes—
180C2×C3×D4C22⋊C412yes—
181C2×C3×D4C22⋊C411yescharacteristic
182C2×C3×D4C22⋊C411yescharacteristic
183C2×C3×D4C22×C411yescharacteristic
184C2×C3×D4C22×C412yes—
185C2×C3×D4C2×D412yes—
186C2×C3×D4C2×D411yescharacteristic
187C2×C3×D4C2×D412yes—
188C2×C3×D4C2411yescharacteristic
189C22×C3×C4C22⋊C412yes—
190C22×C3×C4C22⋊C412yes—
191C22×C3×C4C22×C411yescharacteristic
192C22×C3×C4C2×D411yescharacteristic
193C22×C3×C4C2×D412yes—
194C22×C3×C4C2×D411yescharacteristic
195C2×#67—22no—
196C2×#67—21no—
197C2×#67—22no—
198C2×#67—22no—
199C2×#67—21no—
200C2×#67—22no—
201C2×#67—22no—
202C2×#67—22no—
203C2×#67—22no—
204C2×#67—44no—
205C2×#67—84no—
206C2×#67—84no—
207C22×Dic3—22no—
208C22×Dic3—22no—
209C22×Dic3—44no—
210C22×Dic3C22×C411yescharacteristic
211D3×D4—48no—
212D3×D4—44no—
213D3×D4—44no—
214C2×D12—21no—
215C2×D12—21no—
216C2×D12—42no—
217C2×C4×D3—21no—
218C2×C4×D3—21no—
219C2×C4×D3—42no—
220C3×M16—24no—
221C3×M16—44no—
222C3×M16—42no—
223C2×C3×C8—44no—
224C2×C3×C8—42no—
225#217—22no—
226#217—21no—
227#217—22no—
228#217—22no—
229#217—21no—
230#217—24no—
231#217—44no—
232#217C22×C412yes—
233#217C22×C411yescharacteristic
234#217C22×C411yescharacteristic
235(C2×D3)⋊C4—21no—
236(C2×D3)⋊C4—22no—
237(C2×D3)⋊C4—22no—
238(C2×D3)⋊C4—22no—
239(C2×D3)⋊C4—22no—
240(C2×D3)⋊C4—21no—
241(C2×D3)⋊C4—22no—
242C4⋊Dic3—21no—
243C4⋊Dic3—21no—
244C4⋊Dic3—42no—
245Dic3⋊C4—21no—
246Dic3⋊C4—22no—
247Dic3⋊C4—21no—
248C4×Dic3—21no—
249C4×Dic3—21no—
250C4×Dic3—42no—
251C25—32no—
252H(5,𝔽2)—44no—
253C2×#41C4×D311yescharacteristic
254C2×#41D1211yescharacteristic
255C2×#41#6712yes—
256C2×#41#6711yescharacteristic
257C2×#41C22×D311yescharacteristic
258C22×D4—32no—
259C22×D4—31no—
260C22×D4—62no—
261C22×D4—62no—
262C22×D4—62no—
263C22×D4—61no—
264C22×D4—61no—
265C22×D4—124no—
266C22×D4C4×D312yes—
267C22×D4C4×D311yescharacteristic
268C22×D4D1212yes—
269C22×D4D1211yescharacteristic
270C22×D4#6711yescharacteristic
271C22×D4C22×D312yes—
272C23×C4—31no—
273#131—44no—
274C2×M16—22no—
275C2×M16—24no—
276#127—61no—
277#127—61no—
278#124—61no—
279#124—61no—
280#123—31no—
281#123—32no—
282#123—31no—
283#123—62no—
284#123—62no—
285C22⋊Q8—61no—
286C22⋊Q8—61no—
287#121—61no—
288#121—62no—
289#121—62no—
290#121—61no—
291#121—62no—
292#121—62no—
293#121—62no—
294#121—61no—
295#121—61no—
296#120—31no—
297#120—32no—
298#120—32no—
299#120—32no—
300#120—32no—
301#120—31no—
302#120—32no—
303#120—62no—
304#120—62no—
305#120—62no—
306#120—62no—
307#120—62no—
308#120—62no—
309C4×D4—62no—
310C4×D4—61no—
311C4×D4—61no—
312C4×D4—62no—
313C4×D4—61no—
314C4×D4—61no—
315C4×D4—124no—
316#117—31no—
317#117—31no—
318#117—62no—
319C2×C22⋊C4—32no—
320C2×C22⋊C4—32no—
321C2×C22⋊C4—62no—
322C2×C22⋊C4—62no—
323C2×C22⋊C4—64no—
324#101—22no—
325#100—24no—
326#100C22×D314yes—
327#99—34no—
328#99—64no—
329#99—62no—
330C23×C3—24no—
331C23×C3—24no—
332C23×C3—42no—
333C23×C3—82no—
334C23×C3—82no—
335C23×C3—82no—
336C23×C3C2×C22⋊C412yes—
337C23×C3C2×C22⋊C412yes—
338C23×C3C2×C22⋊C411yescharacteristic
339C23×C3C2×C22⋊C412yes—
340C23×C3C23×C411yescharacteristic
341C23×C3C22×D411yescharacteristic
342C22×D3—21no—
343C22×D3—22no—
344C22×D3—42no—
345C22×D3—44no—
346C22×D3—44no—
347C22×D3—41no—
348C22×D3—44no—
349C22×D3—44no—
350C22×D3—42no—
351C22×D3—82no—
352C22×D3—82no—
353C22×D3—82no—
354C22×D3—88no—
355C22×D3—88no—
356C3×Q8—22no—
357C3×Q8—41no—
358C3×D4—24no—
359C3×D4—24no—
360C3×D4—24no—
361C3×D4—22no—
362C3×D4—22no—
363C3×D4—41no—
364C3×D4—42no—
365C3×D4—41no—
366C3×D4—42no—
367C3×D4—42no—
368C3×D4—41no—
369C3×D4#9914yes—
370C2×C3×C4—24no—
371C2×C3×C4—22no—
372C2×C3×C4—22no—
373C2×C3×C4—22no—
374C2×C3×C4—42no—
375C2×C3×C4—42no—
376C2×C3×C4C2×C22⋊C411yescharacteristic
377C2×C3×C4C2×C22⋊C411yescharacteristic
378C2×C3×C4C2×C22⋊C412yes—
379C2×C3×C4C2×C22⋊C411yescharacteristic
380C2×C3×C4C22×D411yescharacteristic
381#67—44no—
382#67—42no—
383#67—42no—
384#67—44no—
385#67—44no—
386#67—82no—
387#67—82no—
388#67—82no—
389#67—82no—
390C2×Dic3—22no—
391C2×Dic3—21no—
392C2×Dic3—21no—
393C2×Dic3—21no—
394C2×Dic3—41no—
395C2×Dic3—44no—
396C2×Dic3—82no—
397C2×Dic3—82no—
398C2×Dic3—164no—
399D12—42no—
400D12—42no—
401C4×D3—42no—
402C4×D3—42no—
403C3×C8—44no—
404C24—31no—
405C24—32no—
406C24—62no—
407C24—62no—
408C24—124no—
409C24—244no—
410C24(C2×D3)⋊C412yes—
411C24C2×C4×D312yes—
412C24C2×D1212yes—
413#41—24no—
414#41—42no—
415#41—44no—
416#41—42no—
417#41—42no—
418C2×Q8(C2×D3)⋊C411yescharacteristic
419C2×Q8C2×#6711yescharacteristic
420C2×D4—24no—
421C2×D4—24no—
422C2×D4—42no—
423C2×D4—44no—
424C2×D4—44no—
425C2×D4—44no—
426C2×D4—62no—
427C2×D4—61no—
428C2×D4—62no—
429C2×D4—62no—
430C2×D4—61no—
431C2×D4—62no—
432C2×D4—62no—
433C2×D4—62no—
434C2×D4—62no—
435C2×D4—61no—
436C2×D4—61no—
437C2×D4—122no—
438C2×D4—128no—
439C2×D4—124no—
440C2×D4—124no—
441C2×D4—124no—
442C2×D4—244no—
443C2×D4—244no—
444C2×D4(C2×D3)⋊C412yes—
445C2×D4(C2×D3)⋊C411yescharacteristic
446C2×D4(C2×D3)⋊C412yes—
447C2×D4(C2×D3)⋊C412yes—
448C2×D4(C2×D3)⋊C411yescharacteristic
449C2×D4(C2×D3)⋊C411yescharacteristic
450C2×D4C2×C4×D311yescharacteristic
451C2×D4C2×C4×D312yes—
452C2×D4C2×D1212yes—
453C2×D4C2×D1211yescharacteristic
454C2×D4C2×#6712yes—
455C2×D4C23×D311yescharacteristic
456C22×C4—31no—
457C22×C4—62no—
458C22×C4—62no—
459C22×C4—61no—
460C22×C4—61no—
461C22×C4—122no—
462C22×C4—124no—
463C22×C4(C2×D3)⋊C412yes—
464C22×C4(C2×D3)⋊C412yes—
465C22×C4C2×C4×D311yescharacteristic
466C22×C4C2×D1211yescharacteristic
467C22×C4C2×#6712yes—
468C22×C4C2×#6711yescharacteristic
469M16—24no—
470M16—44no—
471M16—42no—
472C2×C8—44no—
473C2×C8—42no—
474C4⋊C4—61no—
475C4⋊C4—61no—
476C4⋊C4—61no—
477C4⋊C4—62no—
478C4⋊C4—61no—
479C4⋊C4—122no—
480C22⋊C4—31no—
481C22⋊C4—32no—
482C22⋊C4—31no—
483C22⋊C4—62no—
484C22⋊C4—62no—
485C22⋊C4—62no—
486C22⋊C4—61no—
487C22⋊C4—61no—
488C22⋊C4—61no—
489C22⋊C4—62no—
490C22⋊C4—62no—
491C22⋊C4—62no—
492C22⋊C4—62no—
493C22⋊C4—61no—
494C22⋊C4—62no—
495C22⋊C4—64no—
496C22⋊C4—124no—
497C42—61no—
498C42—61no—
499C42—122no—
500C22×C3—21no—
501C22×C3—21no—
502C22×C3—22no—
503C22×C3—21no—
504C22×C3—22no—
505C22×C3—42no—
506C22×C3—42no—
507C22×C3—42no—
508C22×C3—41no—
509C22×C3—41no—
510C22×C3—41no—
511C22×C3—84no—
512C22×C3—84no—
513C22×C3—162no—
514C22×C3C2×#9914yes—
515C22×C3C22×C22⋊C411yescharacteristic
516C2×D3—22no—
517C2×D3—41no—
518C2×D3—42no—
519C2×D3—41no—
520C2×D3—42no—
521C2×D3—42no—
522C2×D3—42no—
523C2×D3—42no—
524C2×D3—88no—
525C2×D3—88no—
526C2×D3—84no—
527C3×C4—21no—
528C3×C4—21no—
529C3×C4—21no—
530C3×C4—22no—
531C3×C4C2×#9912yes—
532Dic3—42no—
533Dic3—41no—
534Dic3—41no—
535Dic3—41no—
536Dic3—84no—
537C23—24no—
538C23—24no—
539C23—42no—
540C23—61no—
541C23—62no—
542C23—82no—
543C23—82no—
544C23—82no—
545C23—121no—
546C23—124no—
547C23—122no—
548C23—122no—
549C23—124no—
550C23—124no—
551C23—124no—
552C23—242no—
553C23—242no—
554C23—242no—
555C23—248no—
556C23—248no—
557C23C2×(C2×D3)⋊C412yes—
558C23C2×(C2×D3)⋊C412yes—
559C23C2×(C2×D3)⋊C411yescharacteristic
560C23C2×(C2×D3)⋊C412yes—
561C23C22×C4×D311yescharacteristic
562C23C22×D1211yescharacteristic
563Q8—22no—
564Q8—41no—
565D4—24no—
566D4—24no—
567D4—24no—
568D4—22no—
569D4—22no—
570D4—41no—
571D4—42no—
572D4—41no—
573D4—42no—
574D4—42no—
575D4—41no—
576D4—122no—
577D4—124no—
578D4—124no—
579D4—122no—
580D4—122no—
581D4—124no—
582D4—122no—
583D4—242no—
584D4—242no—
585D4—242no—
586D4—242no—
587D4#80614yes—
588C2×C4—24no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—62no—
595C2×C4—61no—
596C2×C4—61no—
597C2×C4—61no—
598C2×C4—122no—
599C2×C4—122no—
600C2×C4—121no—
601C2×C4—124no—
602C2×C4—242no—
603C2×C4—242no—
604C2×C4—484no—
605C2×C4C2×(C2×D3)⋊C411yescharacteristic
606C2×C4C2×(C2×D3)⋊C411yescharacteristic
607C2×C4C2×(C2×D3)⋊C412yes—
608C2×C4C2×(C2×D3)⋊C411yescharacteristic
609C2×C4C22×#6711yescharacteristic
610C8—44no—
611C2×C3—21no—
612C2×C3—24no—
613C2×C3—41no—
614C2×C3—41no—
615C2×C3—42no—
616C2×C3—42no—
617C2×C3—41no—
618C2×C3C22×#9911yescharacteristic
619D3—42no—
620D3—82no—
621C22—21no—
622C22—21no—
623C22—21no—
624C22—22no—
625C22—22no—
626C22—42no—
627C22—42no—
628C22—42no—
629C22—41no—
630C22—41no—
631C22—41no—
632C22—62no—
633C22—84no—
634C22—84no—
635C22—121no—
636C22—122no—
637C22—122no—
638C22—122no—
639C22—122no—
640C22—121no—
641C22—122no—
642C22—162no—
643C22—248no—
644C22—248no—
645C22—244no—
646C22C2×#80614yes—
647C22C22×(C2×D3)⋊C411yescharacteristic
648C4—21no—
649C4—21no—
650C4—21no—
651C4—22no—
652C4—122no—
653C4—121no—
654C4—121no—
655C4—121no—
656C4—244no—
657C4C2×#80612yes—
658C3#3334811yescharacteristic, complemented, Sylow
659C2—21no—
660C2—24no—
661C2—41no—
662C2—41no—
663C2—42no—
664C2—42no—
665C2—41no—
666C2—122no—
667C2—242no—
668C2C22×#80611yescharacteristic
subgroupG/H
centerC2C22×#806
commutatorC23×C3C23×C4
FrattiniC2×D4C23×D3
FittingC3×#2888C2
socleC2×C3C22×#99
prime radical#87284C2
cosocleC2×C3×D4C24
hypercenter#2888D3
hypocenterC3#33348
LCSG ▷ C23×C3 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C2×D4 ◁ C2×H(5,𝔽2) ◁ #2888

Complements

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

NH ≅ G/Ntype
2. C3×#2888C2semidirect
3. #84107C2semidirect
4. #70146C2semidirect
5. #70145C2semidirect
6. #70094C2semidirect
7. C2×#4475C2semidirect
10. C22×D3×D4C4semidirect
12. #5830C4semidirect
17. #5828C4semidirect
19. #5804C4semidirect
24. C2×#937C4semidirect
27. #5798C4semidirect
29. C3×#413C22semidirect
31. C2×C3×#100C22semidirect
33. #5240C22semidirect
35. C2×#834C22semidirect
36. #4476C22semidirect
37. #4475C22semidirect
38. #4475C22semidirect
40. #2888D3semidirect
46. C24×D3C2×C4semidirect
61. C22×#67C2×C4semidirect
66. C22×D12C2×C4semidirect
67. C22×C4×D3C2×C4semidirect
74. C2×#217C2×C4semidirect
75. C2×#217C2×C4semidirect
92. #937C2×C4semidirect
93. #937C2×C4semidirect
94. #937C2×C4semidirect
95. #937C2×C4semidirect
96. #937C2×C4semidirect
99. #935C2×C4semidirect
100. #935C2×C4semidirect
117. #926C2×C4semidirect
118. #926C2×C4semidirect
145. #413C2×D3semidirect
147. C2×#100C2×D3semidirect
658. C3#33348semidirect

Maximal subgroups

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

subgroupindex
#872842
C3×#28882
#841072
#701462
#701452
#700942
C2×#44752
#333483