Finite Group Databasebeta

← #10859051

Subgroups — #10859051

Order 768.

784 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
1#73493C211yesmaximal, characteristic
2#73468C211yesmaximal, characteristic
3C3×#1851C211yesmaximal, characteristic
4#71989C211yesmaximal, characteristic
5#71985C211yesmaximal, characteristic
6#70488C211yesmaximal, characteristic, complemented
7#70485C211yesmaximal, characteristic, complemented
8#70464C211yesmaximal, characteristic, complemented
9#70460C211yesmaximal, characteristic, complemented
10#69930C211yesmaximal, characteristic
11#69929C211yesmaximal, characteristic
12#68689C211yesmaximal, characteristic, complemented
13#68683C211yesmaximal, characteristic, complemented
14#68643C211yesmaximal, characteristic, complemented
15#68621C211yesmaximal, characteristic, complemented
16#26429—31nomaximal, Sylow
17C22×C3×M16C2211yescharacteristic
18C2×#926C2211yescharacteristic
19C22×C4⋊Dic3C2211yescharacteristic
20C2×#924C2211yescharacteristic
21C2×#878—21no—
22C2×#869C2211yescharacteristic
23C2×C4×Dic6C2211yescharacteristic
24C2×C3×C4⋊C8C2211yescharacteristic
25C2×C3×C4⋊C8C2211yescharacteristic
26C2×C3×C22⋊C8C2211yescharacteristic
27C3×C4×C22⋊C4C2211yescharacteristic
28#5212C2211yescharacteristic
29C2×#818C2211yescharacteristic
30C2×#817C2211yescharacteristic
31C2×#816C2211yescharacteristic
32C2×#816C2211yescharacteristic
33C2×#816C2211yescharacteristic
34C2×#816C2211yescharacteristic
35#4950—21no—
36#4945—21no—
37#4944—21no—
38#4935C2211yescharacteristic
39C4×C4⋊Dic3C2211yescharacteristic
40#4929C2211yescharacteristic
41#4928C2211yescharacteristic
42#4928C2211yescharacteristic
43#4928C2211yescharacteristic
44#4661—21no—
45#4656—21no—
46#4652—21no—
47#4649—21no—
48C3×#336C2211yescharacteristic
49C3×#336C2211yescharacteristic
50#4551C2211yescharacteristic
51#4551C2211yescharacteristic
52#4551C2211yescharacteristic
53#4551C2211yescharacteristic
54#4551C2211yescharacteristic
55#4551C2211yescharacteristic
56#4551C2211yescharacteristic
57#4551C2211yescharacteristic
58#4551C2211yescharacteristic
59#4551C2211yescharacteristic
60#2440—31no—
61#2362—31no—
62#2100—31no—
63#2088—31no—
64#2066—31no—
65#2024—31no—
66#1988—31no—
67#1918—31no—
68#1894—31no—
69#1872—31no—
70#1863—31no—
71#1854—31no—
72#1851D311yescharacteristic
73#1803—31no—
74#1736—31no—
75C23×C3×C4C2311yescharacteristic
76C23×Dic3—21no—
77C22×Dic6—21no—
78C22×Dic6C2311yescharacteristic
79C2×C3×M16—22no—
80C2×C3×M16—41no—
81C2×C3×M16—41no—
82C22×C3×C8C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C22×C3×C8C2311yescharacteristic
86C2×C3×C22⋊C4D411yescharacteristic
87C2×C3×C22⋊C4D411yescharacteristic
88C2×C3×C42C2311yescharacteristic
89C2×C3×C42C2311yescharacteristic
90C2×#217—21no—
91C2×#217—21no—
92C2×#217—21no—
93#926—22no—
94#926—41no—
95C2×C4⋊Dic3—22no—
96C2×C4⋊Dic3—21no—
97C2×C4⋊Dic3—21no—
98C2×C4⋊Dic3—21no—
99C2×C4⋊Dic3—21no—
100C2×C4⋊Dic3—41no—
101C2×C4⋊Dic3C2311yescharacteristic
102C2×C4⋊Dic3C2311yescharacteristic
103C2×C4⋊Dic3C2311yescharacteristic
104C2×C4⋊Dic3C2311yescharacteristic
105C2×C4⋊Dic3C2311yescharacteristic
106C2×C4⋊Dic3C2311yescharacteristic
107C2×C4⋊Dic3C2311yescharacteristic
108#924—22no—
109#924Q814yescomplemented
110C2×Dic3⋊C4—21no—
111C2×Dic3⋊C4—21no—
112C2×Dic3⋊C4—21no—
113C2×Dic3⋊C4—21no—
114C2×Dic3⋊C4—21no—
115C2×C4×Dic3—21no—
116C2×C4×Dic3—21no—
117C2×C4×Dic3—21no—
118#878—44no—
119#869—22no—
120#869—41no—
121C4×Dic6—22no—
122C4×Dic6—41no—
123C3×C4⋊C8—41no—
124C3×C4⋊C8—41no—
125C3×C22⋊C8—22no—
126C3×#95D411yescharacteristic
127C3×#95D411yescharacteristic
128C3⋊#95—21no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135#818—22no—
136#817—41no—
137#816—22no—
138#816—41no—
139#816—41no—
140#816—41no—
141C22×M16C2×D311yescharacteristic
142C2×C4⋊Q8—31no—
143C2×C22⋊Q8—31no—
144C2×C22⋊Q8—61no—
145C2×C4×Q8—31no—
146C2×#117—31no—
147C22×C4⋊C4—31no—
148C2×#107—31no—
149C2×#106—31no—
150C2×C4⋊C8C2×D311yescharacteristic
151C2×C4⋊C8C2×D311yescharacteristic
152C2×#103—31no—
153C2×#103—31no—
154C2×#103—31no—
155C2×#103—31no—
156C2×C22⋊C8C2×D311yescharacteristic
157#398—61no—
158#395—61no—
159#391—31no—
160#389—31no—
161#389—31no—
162#389—31no—
163#388—61no—
164#388—61no—
165#387—61no—
166#385—61no—
167#384—31no—
168#380—31no—
169C4×C4⋊C4—31no—
170C4×C4⋊C4—61no—
171C4×C22⋊C4C2×D311yescharacteristic
172#340—31no—
173#340—31no—
174#340—31no—
175#340—31no—
176#340—31no—
177#340—31no—
178#340—31no—
179#340—31no—
180#340—31no—
181#340—31no—
182#336C2×D311yescharacteristic
183#336C2×D311yescharacteristic
184C24×C3C2×D411yescharacteristic
185C22×C3×C4—21no—
186C22×C3×C4—21no—
187C22×C3×C4—21no—
188C22×C3×C4—21no—
189C22×C3×C4C2×D411yescharacteristic
190C22×C3×C4C2×D411yescharacteristic
191C22×C3×C4C2×D411yescharacteristic
192C22×C3×C4C2×D411yescharacteristic
193C22×C3×C4C2×D411yescharacteristic
194C22×C3×C4C2×Q812yes—
195C22×C3×C4#4112yes—
196C22×C3×C4C2411yescharacteristic
197C22×Dic3—21no—
198C22×Dic3—21no—
199C22×Dic3—21no—
200C22×Dic3—21no—
201C22×Dic3—21no—
202C22×Dic3—21no—
203C22×Dic3—21no—
204C22×Dic3—21no—
205C22×Dic3—42no—
206C22×Dic3—42no—
207C22×Dic3—81no—
208C2×Dic6—21no—
209C2×Dic6—21no—
210C2×Dic6—21no—
211C2×Dic6—21no—
212C2×Dic6—21no—
213C2×Dic6—42no—
214C2×Dic6—42no—
215C2×Dic6—81no—
216C2×Dic6C2×Q812yes—
217C3×M16—44no—
218C2×C3×C8—21no—
219C2×C3×C8—21no—
220C2×C3×C8—21no—
221C2×C3×C8—21no—
222C2×C3×C8—21no—
223C2×C3×C8—21no—
224C2×C3×C8—21no—
225C2×C3×C8—21no—
226C2×C3×C8—21no—
227C2×C3×C8—21no—
228C2×C3×C8—21no—
229C2×C3×C8—21no—
230C3×C22⋊C4—22no—
231C3×C22⋊C4—22no—
232C3×C42—21no—
233C3×C42—21no—
234C3×C42—21no—
235C3×C42—21no—
236#217—24no—
237#217—42no—
238#217—42no—
239C4⋊Dic3—21no—
240C4⋊Dic3—21no—
241C4⋊Dic3—21no—
242C4⋊Dic3—21no—
243C4⋊Dic3—21no—
244C4⋊Dic3—21no—
245C4⋊Dic3—21no—
246C4⋊Dic3—21no—
247C4⋊Dic3—21no—
248C4⋊Dic3—21no—
249C4⋊Dic3—21no—
250C4⋊Dic3—42no—
251C4⋊Dic3—42no—
252C4⋊Dic3—42no—
253C4⋊Dic3—81no—
254C4⋊Dic3C2×Q812yes—
255C4⋊Dic3#4112yes—
256C4⋊Dic3#4112yes—
257Dic3⋊C4—24no—
258Dic3⋊C4—42no—
259Dic3⋊C4—42no—
260Dic3⋊C4—42no—
261Dic3⋊C4—42no—
262C4×Dic3—42no—
263C4×Dic3—42no—
264C4×Dic3—42no—
265C22×Q8—31no—
266C22×Q8—61no—
267C23×C4—61no—
268C23×C4C22×D311yescharacteristic
269C2×M16—22no—
270C2×M16—41no—
271C2×M16—41no—
272C22×C8C22×D311yescharacteristic
273C22×C8C22×D311yescharacteristic
274C22×C8C22×D311yescharacteristic
275C22×C8C22×D311yescharacteristic
276C4⋊Q8—62no—
277C4⋊Q8—121no—
278C22⋊Q8—34no—
279C22⋊Q8—62no—
280C22⋊Q8—124no—
281C4×Q8—62no—
282C4×Q8—121no—
283#117—62no—
284#117—121no—
285C2×C4⋊C4—31no—
286C2×C4⋊C4—31no—
287C2×C4⋊C4—31no—
288C2×C4⋊C4—31no—
289C2×C4⋊C4—31no—
290C2×C4⋊C4—31no—
291C2×C4⋊C4—31no—
292C2×C4⋊C4—62no—
293C2×C4⋊C4—61no—
294C2×C4⋊C4—61no—
295C2×C4⋊C4—61no—
296C2×C4⋊C4—61no—
297C2×C4⋊C4—61no—
298C2×C4⋊C4—61no—
299C2×C4⋊C4—61no—
300C2×C4⋊C4—61no—
301C2×C4⋊C4—61no—
302C2×C4⋊C4—121no—
303C2×C22⋊C4—61no—
304C2×C22⋊C4—61no—
305C2×C22⋊C4—61no—
306C2×C22⋊C4D1211yescharacteristic
307C2×C22⋊C4D1211yescharacteristic
308C2×C42—61no—
309C2×C42—61no—
310C2×C42—61no—
311C2×C42C22×D311yescharacteristic
312C2×C42C22×D311yescharacteristic
313#107—62no—
314#106—121no—
315C4⋊C8—41no—
316C4⋊C8—41no—
317#103—62no—
318#103—121no—
319#103—121no—
320#103—121no—
321C22⋊C8—22no—
322#95—61no—
323#95—61no—
324#95—61no—
325#95—61no—
326#95—61no—
327#95—61no—
328#95—61no—
329#95D1211yescharacteristic
330#95D1211yescharacteristic
331C23×C3—21no—
332C23×C3—21no—
333C23×C3—21no—
334C23×C3—21no—
335C23×C3—21no—
336C23×C3—21no—
337C23×C3C22⋊Q812yes—
338C23×C3C22×D411yescharacteristic
339C2×C3×C4—21no—
340C2×C3×C4—21no—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—21no—
344C2×C3×C4—21no—
345C2×C3×C4—22no—
346C2×C3×C4—22no—
347C2×C3×C4—22no—
348C2×C3×C4—22no—
349C2×C3×C4—21no—
350C2×C3×C4—21no—
351C2×C3×C4—21no—
352C2×C3×C4—21no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4C22⋊Q812yes—
356C2×C3×C4AGL(1,ℤ/8)12yes—
357C2×C3×C4#13712yes—
358C2×C3×C4C22×Q811yescharacteristic
359C2×C3×C4C2×#4111yescharacteristic
360C2×C3×C4C2×#4111yescharacteristic
361C2×C3×C4C2×#4111yescharacteristic
362C2×C3×C4H(5,𝔽2)11yescharacteristic
363C2×C3×C4H(5,𝔽2)11yescharacteristic
364C2×Dic3—22no—
365C2×Dic3—22no—
366C2×Dic3—22no—
367C2×Dic3—22no—
368C2×Dic3—41no—
369C2×Dic3—41no—
370C2×Dic3—41no—
371C2×Dic3—41no—
372C2×Dic3—41no—
373C2×Dic3—41no—
374C2×Dic3—41no—
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×Dic3—41no—
386C2×Dic3—41no—
387C2×Dic3—41no—
388C2×Dic3—84no—
389Dic6—22no—
390Dic6—22no—
391Dic6—41no—
392Dic6—41no—
393Dic6—84no—
394C3×C8—41no—
395C3×C8—41no—
396C3×C8—41no—
397C3×C8—41no—
398C24C2×D1211yescharacteristic
399C2×Q8—32no—
400C2×Q8—61no—
401C2×Q8—61no—
402C2×Q8—61no—
403C2×Q8—61no—
404C2×Q8—61no—
405C2×Q8—122no—
406C2×Q8—122no—
407C2×Q8—241no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—61no—
413C22×C4—61no—
414C22×C4—61no—
415C22×C4—61no—
416C22×C4—61no—
417C22×C4—61no—
418C22×C4—61no—
419C22×C4—61no—
420C22×C4—122no—
421C22×C4—122no—
422C22×C4—241no—
423C22×C4C2×D1211yescharacteristic
424C22×C4C2×D1211yescharacteristic
425C22×C4C2×D1211yescharacteristic
426C22×C4C2×D1211yescharacteristic
427C22×C4C2×D1211yescharacteristic
428C22×C4#23712yes—
429C22×C4D3×Q812yes—
430C22×C4C23×D311yescharacteristic
431M16—44no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C4⋊C4—32no—
445C4⋊C4—32no—
446C4⋊C4—32no—
447C4⋊C4—64no—
448C4⋊C4—61no—
449C4⋊C4—61no—
450C4⋊C4—61no—
451C4⋊C4—61no—
452C4⋊C4—61no—
453C4⋊C4—61no—
454C4⋊C4—61no—
455C4⋊C4—61no—
456C4⋊C4—61no—
457C4⋊C4—61no—
458C4⋊C4—61no—
459C4⋊C4—122no—
460C4⋊C4—122no—
461C4⋊C4—122no—
462C4⋊C4—122no—
463C4⋊C4—122no—
464C4⋊C4—122no—
465C4⋊C4—122no—
466C4⋊C4—241no—
467C22⋊C4—22no—
468C22⋊C4—22no—
469C22⋊C4—64no—
470C22⋊C4—122no—
471C22⋊C4—122no—
472C42—21no—
473C42—21no—
474C42—21no—
475C42—21no—
476C42—122no—
477C42—122no—
478C42—122no—
479C22×C3—22no—
480C22×C3—22no—
481C22×C3—22no—
482C22×C3—41no—
483C22×C3—41no—
484C22×C3—41no—
485C22×C3—41no—
486C22×C3C2×C22⋊Q811yescharacteristic
487C22×C3#54611yescharacteristic
488C22×C3#54611yescharacteristic
489C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
490C22×C3C2×#13711yescharacteristic
491C22×C3#57711yescharacteristic
492C22×C3#57811yescharacteristic
493C3×C4—22no—
494C3×C4—22no—
495C3×C4—22no—
496C3×C4—41no—
497C3×C4—41no—
498C3×C4#52911yescharacteristic
499C3×C4#53811yescharacteristic
500C3×C4#55411yescharacteristic
501C3×C4#55411yescharacteristic
502Dic3—42no—
503Dic3—42no—
504Dic3—42no—
505Dic3—42no—
506Dic3—81no—
507Dic3—81no—
508Dic3—81no—
509Dic3—81no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23#89712yes—
517C23C22×D1211yescharacteristic
518Q8—62no—
519Q8—62no—
520Q8—121no—
521Q8—121no—
522Q8—244no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—62no—
540C2×C4—62no—
541C2×C4—62no—
542C2×C4—62no—
543C2×C4—121no—
544C2×C4—121no—
545C2×C4—121no—
546C2×C4—121no—
547C2×C4—121no—
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—121no—
558C2×C4—121no—
559C2×C4—121no—
560C2×C4—121no—
561C2×C4—121no—
562C2×C4—121no—
563C2×C4—244no—
564C2×C4#89712yes—
565C2×C4#90812yes—
566C2×C4#90912yes—
567C2×C4C2×#23711yescharacteristic
568C2×C4#100411yescharacteristic
569C2×C4#100411yescharacteristic
570C2×C4C2×D3×Q811yescharacteristic
571C2×C4D3×#4111yescharacteristic
572C2×C4D3×#4111yescharacteristic
573C8—41no—
574C8—41no—
575C8—41no—
576C8—41no—
577C2×C3—42no—
578C2×C3#245011yescharacteristic
579C2×C3#307911yescharacteristic
580C2×C3#308211yescharacteristic
581C2×C3#331011yescharacteristic
582C2×C3#331611yescharacteristic
583C2×C3#332811yescharacteristic
584C2×C3#333011yescharacteristic
585C22—22no—
586C22—22no—
587C22—22no—
588C22—41no—
589C22—41no—
590C22—41no—
591C22—41no—
592C22C2×#89711yescharacteristic
593C22#555411yescharacteristic
594C22#555411yescharacteristic
595C22C2×#90811yescharacteristic
596C22C2×#90911yescharacteristic
597C22#575111yescharacteristic
598C22#575311yescharacteristic
599C4—22no—
600C4—22no—
601C4—22no—
602C4—41no—
603C4—41no—
604C4—122no—
605C4—122no—
606C4—122no—
607C4—122no—
608C4—241no—
609C4—241no—
610C4—241no—
611C4—241no—
612C4#559311yescharacteristic
613C4#563611yescharacteristic
614C4#563611yescharacteristic
615C4#564911yescharacteristic
616C3#2642911yescharacteristic, complemented, Sylow
617C2—42no—
618C2#7442411yescharacteristic
619C2#7906011yescharacteristic
620C2#7910111yescharacteristic
621C2#7950111yescharacteristic
622C2#7951311yescharacteristic
623C2#7955711yescharacteristic
624C2#7956311yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1851C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenter#1851D3
hypocenterC3#26429
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #1851

Complements

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

NH ≅ G/Ntype
6. #70488C2semidirect
7. #70485C2semidirect
8. #70464C2semidirect
9. #70460C2semidirect
12. #68689C2semidirect
13. #68683C2semidirect
14. #68643C2semidirect
15. #68621C2semidirect
109. #924Q8semidirect
616. C3#26429semidirect

Maximal subgroups

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

subgroupindex
#734932
#734682
C3×#18512
#719892
#719852
#704882
#704852
#704642
#704602
#699302
#699292
#686892
#686832
#686432
#686212
#264293