Finite Group Databasebeta

← #10896355

Subgroups — #10896355

Order 768.

800 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#936C211yesmaximal, characteristic
2#75967C211yesmaximal, characteristic, complemented
3C3×#1864C211yesmaximal, characteristic
4#71607C211yesmaximal, characteristic
5#70590C211yesmaximal, characteristic, complemented
6#70564C211yesmaximal, characteristic, complemented
7#70183C211yesmaximal, characteristic
8#70163C211yesmaximal, characteristic
9#70162C211yesmaximal, characteristic
10#70162C211yesmaximal, characteristic
11#69631C211yesmaximal, characteristic, complemented
12#68910C211yesmaximal, characteristic, complemented
13#68798C211yesmaximal, characteristic, complemented
14#68683C211yesmaximal, characteristic, complemented
15#68605C211yesmaximal, characteristic, complemented
16#29262—31nomaximal, Sylow
17C2×C3×C4×D4C2211yescharacteristic
18C2×#936C2211yescharacteristic
19C2×#888C2211yescharacteristic
20C2×#887—21no—
21C2×C4×Dic6C2211yescharacteristic
22C2×C3×#106C2211yescharacteristic
23C2×C3×#102C2211yescharacteristic
24C2×C3×#102C2211yescharacteristic
25C3×#389C2211yescharacteristic
26#5221C2211yescharacteristic
27C2×#832C2211yescharacteristic
28C2×#832C2211yescharacteristic
29C4×#217—21no—
30C4×#217—21no—
31C2×#818C2211yescharacteristic
32C2×#816C2211yescharacteristic
33C2×#816C2211yescharacteristic
34C2×Dic3⋊C8C2211yescharacteristic
35C2×Dic3⋊C8C2211yescharacteristic
36#4977C2211yescharacteristic
37#4976—21no—
38#4975C2211yescharacteristic
39Dic3×C4⋊C4C2211yescharacteristic
40Dic3×C4⋊C4C2211yescharacteristic
41#4973C2211yescharacteristic
42C2×#810C2211yescharacteristic
43C2×#810C2211yescharacteristic
44C2×#808C2211yescharacteristic
45C2×#807C2211yescharacteristic
46#4945—21no—
47#4945—21no—
48C42×Dic3C2211yescharacteristic
49#4928C2211yescharacteristic
50#4649—21no—
51#4646—21no—
52C3×#340C2211yescharacteristic
53C3×#336C2211yescharacteristic
54#4551C2211yescharacteristic
55#4550C2211yescharacteristic
56#4550C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4524C2211yescharacteristic
60C4×C4⋊Q8—31no—
61C4×#124—31no—
62#2096—31no—
63#2093—31no—
64#2024—31no—
65#2023—31no—
66#1983—31no—
67#1983—31no—
68#1949—31no—
69#1949—31no—
70#1864—31no—
71#1864D311yescharacteristic
72#1863—31no—
73#1863—31no—
74#1831—31no—
75C22×C3×D4C2311yescharacteristic
76C23×C3×C4—21no—
77C22×Dic6—21no—
78C22×Dic6C2311yescharacteristic
79C22×C3×C8C2311yescharacteristic
80C22×C3×C8C2311yescharacteristic
81C3×C4×D4—41no—
82C3×C4×D4—41no—
83C2×C3×C4⋊C4—21no—
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C22⋊C4—21no—
88C2×C3×C42C2311yescharacteristic
89C2×#217—21no—
90C2×#217—21no—
91C2×#217—21no—
92C2×#217—21no—
93#936—22no—
94#936Q814yescomplemented
95C2×C4⋊Dic3C2311yescharacteristic
96C2×C4⋊Dic3C2311yescharacteristic
97C2×C4⋊Dic3C2311yescharacteristic
98C2×Dic3⋊C4—21no—
99C2×Dic3⋊C4—21no—
100C2×Dic3⋊C4—21no—
101C2×Dic3⋊C4—21no—
102C2×Dic3⋊C4—21no—
103C2×C4×Dic3—21no—
104C2×C4×Dic3—21no—
105C2×C4×Dic3—21no—
106C2×C4×Dic3D411yescharacteristic
107C2×C4×Dic3D411yescharacteristic
108C2×C4×Dic3D411yescharacteristic
109C2×C4×Dic3D411yescharacteristic
110C2×C4×Dic3C2311yescharacteristic
111C2×C4×Dic3C2311yescharacteristic
112C22×C3⋊C8C2311yescharacteristic
113C22×C3⋊C8C2311yescharacteristic
114#888—22no—
115#888—22no—
116#887—44no—
117C4×Dic6—41no—
118C4×Dic6—41no—
119C3×#106—41no—
120C3×#102—22no—
121C3×#102—41no—
122C3×#95—21no—
123#832—22no—
124#832—41no—
125C3⋊#95—21no—
126C3⋊#95—21no—
127C3⋊#95—21no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131C3⋊#95—21no—
132#818—41no—
133#816—22no—
134#816—41no—
135Dic3⋊C8—22no—
136Dic3⋊C8—22no—
137#810—22no—
138#810—41no—
139#808—41no—
140#807—41no—
141C2×C4⋊Q8—31no—
142C2×#124—31no—
143C2×C4×Q8—31no—
144C2×C4×Q8—61no—
145C2×C4×D4C2×D311yescharacteristic
146C2×#107—31no—
147C2×#107—31no—
148C2×#106—31no—
149C2×#106C2×D311yescharacteristic
150C2×C4⋊C8—31no—
151C2×C4⋊C8—31no—
152C2×#103—31no—
153C2×#103—31no—
154C2×#103—31no—
155C2×#103—31no—
156C2×#102—31no—
157C2×#102—31no—
158C2×#102C2×D311yescharacteristic
159C2×#102C2×D311yescharacteristic
160#391—31no—
161#391—61no—
162#390—31no—
163#389—31no—
164#389—61no—
165#389C2×D311yescharacteristic
166#388—61no—
167#388—61no—
168#384—31no—
169#382—31no—
170C4×C4⋊C4—31no—
171C4×C4⋊C4—31no—
172C4×C4⋊C4—61no—
173C4×C22⋊C4—61no—
174C4×C22⋊C4—61no—
175C43—31no—
176#340—31no—
177#340—31no—
178#340—31no—
179#340C2×D311yescharacteristic
180#336—31no—
181#336—31no—
182#336—31no—
183#336C2×D311yescharacteristic
184C24×C3—21no—
185C2×C3×D4—21no—
186C2×C3×D4—21no—
187C2×C3×D4—21no—
188C2×C3×D4—21no—
189C2×C3×D4—21no—
190C2×C3×D4C2×Q812yes—
191C22×C3×C4—21no—
192C22×C3×C4—21no—
193C22×C3×C4—21no—
194C22×C3×C4—42no—
195C22×C3×C4—42no—
196C22×C3×C4—42no—
197C22×C3×C4C2×D411yescharacteristic
198C22×C3×C4C2×D411yescharacteristic
199C22×C3×C4C2411yescharacteristic
200C22×Dic3—21no—
201C22×Dic3—21no—
202C22×Dic3—21no—
203C22×Dic3—21no—
204C22×Dic3C2×D411yescharacteristic
205C22×Dic3C2×D411yescharacteristic
206C22×Dic3C2×D411yescharacteristic
207C22×Dic3C2×D411yescharacteristic
208C2×Dic6—21no—
209C2×Dic6—21no—
210C2×Dic6—21no—
211C2×Dic6—21no—
212C2×Dic6—24no—
213C2×Dic6—21no—
214C2×Dic6—42no—
215C2×Dic6—42no—
216C2×Dic6C2×Q812yes—
217C2×C3×C8—21no—
218C2×C3×C8—21no—
219C2×C3×C8—21no—
220C2×C3×C8—21no—
221C2×C3×C8—21no—
222C2×C3×C8—21no—
223C3×C4⋊C4—21no—
224C3×C4⋊C4—21no—
225C3×C4⋊C4—21no—
226C3×C4⋊C4—21no—
227C3×C4⋊C4—21no—
228C3×C4⋊C4—42no—
229C3×C4⋊C4#4112yes—
230C3×C22⋊C4—42no—
231C3×C42—21no—
232C3×C42—21no—
233#217—24no—
234#217—24no—
235#217—42no—
236#217—42no—
237C4⋊Dic3—21no—
238C4⋊Dic3—21no—
239C4⋊Dic3—21no—
240C4⋊Dic3—21no—
241C4⋊Dic3—21no—
242C4⋊Dic3#4112yes—
243Dic3⋊C4—24no—
244Dic3⋊C4—42no—
245Dic3⋊C4—42no—
246Dic3⋊C4—42no—
247Dic3⋊C4—42no—
248C4×Dic3—22no—
249C4×Dic3—21no—
250C4×Dic3—21no—
251C4×Dic3—22no—
252C4×Dic3—22no—
253C4×Dic3—41no—
254C4×Dic3—42no—
255C4×Dic3—42no—
256C4×Dic3—42no—
257C4×Dic3C2×Q812yes—
258C4×Dic3#4112yes—
259C2×C3⋊C8—21no—
260C2×C3⋊C8—21no—
261C2×C3⋊C8—21no—
262C2×C3⋊C8—21no—
263C2×C3⋊C8—21no—
264C2×C3⋊C8—21no—
265C22×Q8—31no—
266C22×Q8—61no—
267C22×D4C22×D311yescharacteristic
268C23×C4—21no—
269C22×C8—31no—
270C22×C8—31no—
271C22×C8C22×D311yescharacteristic
272C22×C8C22×D311yescharacteristic
273C4⋊Q8—62no—
274C4⋊Q8—62no—
275#124—34no—
276#124—62no—
277C4×Q8—121no—
278C4×Q8—121no—
279C4×Q8—124no—
280C4×D4—41no—
281C4×D4—41no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—31no—
284C2×C4⋊C4—31no—
285C2×C4⋊C4—31no—
286C2×C4⋊C4—61no—
287C2×C4⋊C4—61no—
288C2×C4⋊C4—61no—
289C2×C4⋊C4—61no—
290C2×C4⋊C4—61no—
291C2×C4⋊C4C22×D311yescharacteristic
292C2×C4⋊C4C22×D311yescharacteristic
293C2×C4⋊C4C22×D311yescharacteristic
294C2×C22⋊C4—21no—
295C2×C22⋊C4—61no—
296C2×C22⋊C4—61no—
297C2×C22⋊C4—61no—
298C2×C22⋊C4—61no—
299C2×C42—31no—
300C2×C42—31no—
301C2×C42—31no—
302C2×C42—31no—
303C2×C42—31no—
304C2×C42—31no—
305C2×C42—61no—
306C2×C42—61no—
307C2×C42—61no—
308C2×C42C22×D311yescharacteristic
309#107—121no—
310#107—121no—
311#106—41no—
312#106—121no—
313C4⋊C8—62no—
314C4⋊C8—62no—
315#103—62no—
316#103—62no—
317#103—121no—
318#103—121no—
319#102—22no—
320#102—41no—
321#102—62no—
322#102—121no—
323#95—21no—
324#95—61no—
325#95—61no—
326#95—61no—
327#95—61no—
328#95—61no—
329#95—61no—
330#95—61no—
331C23×C3—22no—
332C23×C3—41no—
333C23×C3—41no—
334C23×C3—41no—
335C23×C3—41no—
336C23×C3—41no—
337C23×C3—41no—
338C23×C3C22×D411yescharacteristic
339C3×D4—22no—
340C3×D4—22no—
341C3×D4—41no—
342C3×D4—41no—
343C2×C3×C4—21no—
344C2×C3×C4—21no—
345C2×C3×C4—22no—
346C2×C3×C4—21no—
347C2×C3×C4—21no—
348C2×C3×C4—21no—
349C2×C3×C4—41no—
350C2×C3×C4—41no—
351C2×C3×C4—41no—
352C2×C3×C4—41no—
353C2×C3×C4—41no—
354C2×C3×C4—41no—
355C2×C3×C4—41no—
356C2×C3×C4—41no—
357C2×C3×C4—84no—
358C2×C3×C4#13712yes—
359C2×C3×C4C22×Q811yescharacteristic
360C2×C3×C4C2×#4111yescharacteristic
361C2×C3×C4C2×#4111yescharacteristic
362C2×C3×C4C2×#4111yescharacteristic
363C2×C3×C4H(5,𝔽2)11yescharacteristic
364C2×C3×C4H(5,𝔽2)11yescharacteristic
365C2×Dic3—21no—
366C2×Dic3—21no—
367C2×Dic3—21no—
368C2×Dic3—21no—
369C2×Dic3—21no—
370C2×Dic3—21no—
371C2×Dic3—21no—
372C2×Dic3—21no—
373C2×Dic3—22no—
374C2×Dic3—22no—
375C2×Dic3—41no—
376C2×Dic3—41no—
377C2×Dic3—41no—
378C2×Dic3—41no—
379C2×Dic3—41no—
380C2×Dic3—41no—
381C2×Dic3—41no—
382C2×Dic3—41no—
383C2×Dic3—41no—
384C2×Dic3—41no—
385C2×Dic3C22⋊Q812yes—
386C2×Dic3C22⋊Q812yes—
387C2×Dic3#13512yes—
388C2×Dic3AGL(1,ℤ/8)12yes—
389Dic6—22no—
390Dic6—22no—
391Dic6—48no—
392Dic6—41no—
393Dic6—41no—
394C3×C8—41no—
395C3×C8—41no—
396C3⋊C8—41no—
397C3⋊C8—41no—
398C24—21no—
399C2×Q8—32no—
400C2×Q8—64no—
401C2×Q8—61no—
402C2×Q8—61no—
403C2×Q8—61no—
404C2×Q8—61no—
405C2×Q8—61no—
406C2×Q8—122no—
407C2×Q8—122no—
408C2×D4—21no—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4D3×Q812yes—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—31no—
418C22×C4—31no—
419C22×C4—31no—
420C22×C4—31no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—61no—
425C22×C4—61no—
426C22×C4—61no—
427C22×C4—61no—
428C22×C4D3×D411yescharacteristic
429C22×C4D3×D411yescharacteristic
430C22×C4C23×D311yescharacteristic
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—61no—
438C2×C8—61no—
439C2×C8—61no—
440C2×C8—61no—
441C2×C8—61no—
442C2×C8—61no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—32no—
449C4⋊C4—42no—
450C4⋊C4—64no—
451C4⋊C4—61no—
452C4⋊C4—61no—
453C4⋊C4—61no—
454C4⋊C4—61no—
455C4⋊C4—61no—
456C4⋊C4—122no—
457C4⋊C4—122no—
458C4⋊C4—122no—
459C4⋊C4—122no—
460C4⋊C4#23512yes—
461C22⋊C4—42no—
462C22⋊C4—64no—
463C22⋊C4—64no—
464C22⋊C4—122no—
465C22⋊C4—122no—
466C42—21no—
467C42—21no—
468C42—32no—
469C42—32no—
470C42—62no—
471C42—62no—
472C42—62no—
473C42—61no—
474C42—61no—
475C42—121no—
476C42—122no—
477C42—122no—
478C42—122no—
479C22×C3—42no—
480C22×C3—42no—
481C22×C3—42no—
482C22×C3—81no—
483C22×C3—81no—
484C22×C3—81no—
485C22×C3—81no—
486C22×C3C2×C22⋊Q811yescharacteristic
487C22×C3#52511yescharacteristic
488C22×C3#53511yescharacteristic
489C22×C3C2×#13511yescharacteristic
490C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
491C22×C3C2×#13711yescharacteristic
492C22×C3#57511yescharacteristic
493C3×C4—22no—
494C3×C4—42no—
495C3×C4—41no—
496C3×C4—81no—
497C3×C4—81no—
498C3×C4#55411yescharacteristic
499C3×C4#55411yescharacteristic
500C3×C4#55911yescharacteristic
501C3×C4#55911yescharacteristic
502Dic3—22no—
503Dic3—22no—
504Dic3—22no—
505Dic3—22no—
506Dic3—42no—
507Dic3—42no—
508Dic3—81no—
509Dic3—81no—
510C23—22no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23C2×D3×D411yescharacteristic
518Q8—62no—
519Q8—62no—
520Q8—128no—
521Q8—121no—
522Q8—121no—
523D4—22no—
524D4—22no—
525D4—41no—
526D4—41no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—22no—
532C2×C4—21no—
533C2×C4—32no—
534C2×C4—32no—
535C2×C4—32no—
536C2×C4—32no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—61no—
546C2×C4—61no—
547C2×C4—61no—
548C2×C4—61no—
549C2×C4—61no—
550C2×C4—61no—
551C2×C4—61no—
552C2×C4—61no—
553C2×C4—62no—
554C2×C4—62no—
555C2×C4—84no—
556C2×C4—121no—
557C2×C4—121no—
558C2×C4—121no—
559C2×C4—121no—
560C2×C4—121no—
561C2×C4—121no—
562C2×C4—121no—
563C2×C4—121no—
564C2×C4—121no—
565C2×C4—121no—
566C2×C4#91512yes—
567C2×C4C2×#23511yescharacteristic
568C2×C4#100411yescharacteristic
569C2×C4#100411yescharacteristic
570C2×C4C2×D3×Q811yescharacteristic
571C2×C4D3×#4111yescharacteristic
572C2×C4D3×#4111yescharacteristic
573C8—41no—
574C8—41no—
575C8—121no—
576C8—121no—
577C2×C3—82no—
578C2×C3#258011yescharacteristic
579C2×C3#307611yescharacteristic
580C2×C3#307911yescharacteristic
581C2×C3#310811yescharacteristic
582C2×C3#311411yescharacteristic
583C2×C3#320711yescharacteristic
584C2×C3#321511yescharacteristic
585C22—42no—
586C22—42no—
587C22—42no—
588C22—81no—
589C22—81no—
590C22—81no—
591C22—81no—
592C22#554711yescharacteristic
593C22#559211yescharacteristic
594C22D3×C22⋊Q811yescharacteristic
595C22C2×#91511yescharacteristic
596C22D3×#13511yescharacteristic
597C22D3×AGL(1,ℤ/8)11yescharacteristic
598C22#577311yescharacteristic
599C4—22no—
600C4—41no—
601C4—42no—
602C4—62no—
603C4—62no—
604C4—62no—
605C4—62no—
606C4—81no—
607C4—81no—
608C4—122no—
609C4—122no—
610C4—241no—
611C4—241no—
612C4#551111yescharacteristic
613C4#555911yescharacteristic
614C4#563611yescharacteristic
615C4#566411yescharacteristic
616C3#2926211yescharacteristic, complemented, Sylow
617C2—82no—
618C2#7448711yescharacteristic
619C2#7965611yescharacteristic
620C2#7987111yescharacteristic
621C2#8007611yescharacteristic
622C2#8014011yescharacteristic
623C2#8106111yescharacteristic
624C2#8116511yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1864C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
hypercenter#1864D3
hypocenterC3#29262
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1864

Complements

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

NH ≅ G/Ntype
2. #75967C2semidirect
5. #70590C2semidirect
6. #70564C2semidirect
11. #69631C2semidirect
12. #68910C2semidirect
13. #68798C2semidirect
14. #68683C2semidirect
15. #68605C2semidirect
94. #936Q8semidirect
616. C3#29262semidirect

Maximal subgroups

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

subgroupindex
C4×#9362
#759672
C3×#18642
#716072
#705902
#705642
#701832
#701632
#701622
#696312
#689102
#687982
#686832
#686052
#292623