Finite Group Databasebeta

← #10877477

Subgroups — #10877477

Order 768.

772 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#73519C211yesmaximal, characteristic
2#73495C211yesmaximal, characteristic, complemented
3C3×C8×C22⋊C4C211yesmaximal, characteristic, complemented
4#71973C211yesmaximal, characteristic
5C2×#4551C211yesmaximal, characteristic, complemented
6#69932C211yesmaximal, characteristic
7#69925C211yesmaximal, characteristic, complemented
8#69778C211yesmaximal, characteristic, complemented
9#69768C211yesmaximal, characteristic, complemented
10#69768C211yesmaximal, characteristic, complemented
11#69763C211yesmaximal, characteristic, complemented
12#68679C211yesmaximal, characteristic, complemented
13#68644C211yesmaximal, characteristic, complemented
14#68644C211yesmaximal, characteristic, complemented
15#68637C211yesmaximal, characteristic, complemented
16#27701—31nomaximal, Sylow
17C23×C3×C8C2211yescharacteristic
18C2×#930C2211yescharacteristic
19C22×C4⋊Dic3C2211yescharacteristic
20C22×C4⋊Dic3C2211yescharacteristic
21C2×#870C2211yescharacteristic
22C2×C3×C22⋊C8C2211yescharacteristic
23C2×C3×C4×C8C2211yescharacteristic
24C2×C3×C4×C8C2211yescharacteristic
25C3×C4×C22⋊C4C2211yescharacteristic
26#5212C2211yescharacteristic
27C2×#821C2211yescharacteristic
28C2×#821C2211yescharacteristic
29C2×#821C2211yescharacteristic
30C2×#821C2211yescharacteristic
31#4958—21no—
32#4958—21no—
33#4946—21no—
34#4942—21no—
35#4939C2211yescharacteristic
36#4939C2211yescharacteristic
37#4936C2211yescharacteristic
38#4936C2211yescharacteristic
39#4936C2211yescharacteristic
40#4928C2211yescharacteristic
41#4928C2211yescharacteristic
42#4674—21no—
43#4674—21no—
44#4660—21no—
45#4657—21no—
46C3×#336C2211yescharacteristic
47C3×#336C2211yescharacteristic
48#4551C2211yescharacteristic
49#4551C2211yescharacteristic
50#4551C2211yescharacteristic
51#4551C2211yescharacteristic
52#4551C2211yescharacteristic
53#4551C2211yescharacteristic
54#4551C2211yescharacteristic
55#4551C2211yescharacteristic
56#4551C2211yescharacteristic
57#4551C2211yescharacteristic
58#4551C2211yescharacteristic
59#4551C2211yescharacteristic
60#2504—31no—
61#2442—31no—
62#2101—31no—
63#2101—31no—
64#2073—31no—
65#2025—31no—
66#1991—31no—
67#1934—31no—
68#1934—31no—
69#1893—31no—
70#1874—31no—
71#1835—31no—
72#1824—31no—
73C8×C22⋊C4D311yescharacteristic, complemented
74C2×#340—31no—
75C23×C3×C4C2311yescharacteristic
76C22×#67—21no—
77C23×Dic3—21no—
78C23×Dic3—21no—
79C22×D12C2311yescharacteristic
80C22×C3×C8—22no—
81C22×C3×C8—22no—
82C22×C3×C8—41no—
83C22×C3×C8C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C22×C3×C8C2311yescharacteristic
86C22×C3×C8C2311yescharacteristic
87C2×C3×C22⋊C4D411yescharacteristic
88C2×C3×C22⋊C4D411yescharacteristic
89C2×C3×C42C2311yescharacteristic
90C2×C3×C42C2311yescharacteristic
91C2×#217—21no—
92#930—22no—
93#930—22no—
94C2×(C2×D3)⋊C4—21no—
95C2×(C2×D3)⋊C4—21no—
96C2×(C2×D3)⋊C4—21no—
97C2×(C2×D3)⋊C4—21no—
98C2×(C2×D3)⋊C4—21no—
99C2×C4⋊Dic3—22no—
100C2×C4⋊Dic3—22no—
101C2×C4⋊Dic3—22no—
102C2×C4⋊Dic3—21no—
103C2×C4⋊Dic3—21no—
104C2×C4⋊Dic3—41no—
105C2×C4⋊Dic3C2311yescharacteristic
106C2×C4⋊Dic3C2311yescharacteristic
107C2×C4⋊Dic3C2311yescharacteristic
108C2×C4⋊Dic3C2311yescharacteristic
109C2×C4⋊Dic3C2311yescharacteristic
110C2×C4⋊Dic3C2311yescharacteristic
111C2×C4⋊Dic3C2311yescharacteristic
112C2×Dic3⋊C4—21no—
113C2×Dic3⋊C4—21no—
114C2×Dic3⋊C4—21no—
115C2×Dic3⋊C4—21no—
116#870—22no—
117#870—41no—
118C3×C22⋊C8—22no—
119C3×C4×C8—22no—
120C3×C4×C8—41no—
121C3×#95D411yescharacteristic
122C3×#95D411yescharacteristic
123C3⋊#95—21no—
124C3⋊#95—21no—
125C3⋊#95—21no—
126C3⋊#95—21no—
127C3⋊#95—21no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131C3⋊#95—21no—
132#821—22no—
133#821—22no—
134#821—41no—
135#821—41no—
136C23×C8C2×D311yescharacteristic
137C2×#125—31no—
138C2×#121—31no—
139C22×C4⋊C4—31no—
140C22×C4⋊C4—31no—
141C2×#102—31no—
142C2×#102—31no—
143C2×#102—31no—
144C2×#102—31no—
145C2×C22⋊C8C2×D311yescharacteristic
146C2×C4×C8C2×D311yescharacteristic
147C2×C4×C8C2×D311yescharacteristic
148#400—61no—
149#398—61no—
150#397—61no—
151#397—61no—
152#390—31no—
153#390—31no—
154#389—31no—
155#389—31no—
156#386—61no—
157#386—61no—
158#385—61no—
159#382—31no—
160#382—31no—
161#382—31no—
162#380—31no—
163#380—61no—
164C4×C22⋊C4C2×D311yescharacteristic
165#340—31no—
166#340—31no—
167#340—31no—
168#340—31no—
169#340—31no—
170#340—31no—
171#340—31no—
172#340—31no—
173#340—31no—
174#340—31no—
175#340—31no—
176#340—31no—
177#336C2×D311yescharacteristic
178#336C2×D311yescharacteristic
179C24×C3C2×D411yescharacteristic
180C23×D3—21no—
181C22×C3×C4—21no—
182C22×C3×C4—21no—
183C22×C3×C4—21no—
184C22×C3×C4—21no—
185C22×C3×C4C2×D411yescharacteristic
186C22×C3×C4C2×D411yescharacteristic
187C22×C3×C4C2×D411yescharacteristic
188C22×C3×C4C2×D411yescharacteristic
189C22×C3×C4C2×D411yescharacteristic
190C22×C3×C4#4112yes—
191C22×C3×C4#4112yes—
192C22×C3×C4C2411yescharacteristic
193C2×#67—42no—
194C2×#67—42no—
195C2×#67—42no—
196C22×Dic3—21no—
197C22×Dic3—21no—
198C22×Dic3—21no—
199C22×Dic3—21no—
200C22×Dic3—21no—
201C22×Dic3—21no—
202C22×Dic3—21no—
203C22×Dic3—42no—
204C22×Dic3—42no—
205C22×Dic3—42no—
206C22×Dic3—81no—
207C22×Dic3—81no—
208C22×Dic3—81no—
209C2×D12—21no—
210C2×D12—21no—
211C2×D12—21no—
212C2×D12—21no—
213C2×D12—21no—
214C2×D12—21no—
215C2×C3×C8—21no—
216C2×C3×C8—21no—
217C2×C3×C8—21no—
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—44no—
227C2×C3×C8#4112yes—
228C3×C22⋊C4—22no—
229C3×C22⋊C4—22no—
230C3×C42—21no—
231C3×C42—21no—
232C3×C42—21no—
233C3×C42—21no—
234#217—42no—
235(C2×D3)⋊C4—42no—
236(C2×D3)⋊C4—81no—
237(C2×D3)⋊C4—81no—
238(C2×D3)⋊C4—81no—
239(C2×D3)⋊C4—81no—
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—21no—
251C4⋊Dic3—21no—
252C4⋊Dic3—21no—
253C4⋊Dic3—42no—
254C4⋊Dic3—42no—
255C4⋊Dic3#4112yes—
256Dic3⋊C4—42no—
257Dic3⋊C4—42no—
258Dic3⋊C4—81no—
259Dic3⋊C4—81no—
260C22×D4—31no—
261C22×D4—61no—
262C23×C4—61no—
263C23×C4—61no—
264C23×C4C22×D311yescharacteristic
265C22×C8—22no—
266C22×C8—22no—
267C22×C8—41no—
268C22×C8C22×D311yescharacteristic
269C22×C8C22×D311yescharacteristic
270C22×C8C22×D311yescharacteristic
271C22×C8C22×D311yescharacteristic
272#125—62no—
273#125—121no—
274#121—62no—
275#121—62no—
276C2×C4⋊C4—31no—
277C2×C4⋊C4—31no—
278C2×C4⋊C4—31no—
279C2×C4⋊C4—31no—
280C2×C4⋊C4—31no—
281C2×C4⋊C4—31no—
282C2×C4⋊C4—31no—
283C2×C4⋊C4—62no—
284C2×C4⋊C4—62no—
285C2×C4⋊C4—62no—
286C2×C4⋊C4—61no—
287C2×C4⋊C4—61no—
288C2×C4⋊C4—61no—
289C2×C4⋊C4—61no—
290C2×C4⋊C4—61no—
291C2×C4⋊C4—61no—
292C2×C4⋊C4—121no—
293C2×C22⋊C4—61no—
294C2×C22⋊C4—61no—
295C2×C22⋊C4—61no—
296C2×C22⋊C4—61no—
297C2×C22⋊C4—61no—
298C2×C22⋊C4—61no—
299C2×C22⋊C4D1211yescharacteristic
300C2×C22⋊C4D1211yescharacteristic
301C2×C42C22×D311yescharacteristic
302C2×C42C22×D311yescharacteristic
303#102—62no—
304#102—62no—
305#102—121no—
306#102—121no—
307C22⋊C8—22no—
308C4×C8—22no—
309C4×C8—41no—
310#95—61no—
311#95—61no—
312#95—61no—
313#95—61no—
314#95—61no—
315#95—61no—
316#95—61no—
317#95—61no—
318#95—61no—
319#95D1211yescharacteristic
320#95D1211yescharacteristic
321C23×C3—21no—
322C23×C3—21no—
323C23×C3—21no—
324C23×C3—21no—
325C23×C3#12412yes—
326C23×C3#13512yes—
327C23×C3#13512yes—
328C23×C3C22×D411yescharacteristic
329C22×D3—41no—
330C22×D3—41no—
331C22×D3—41no—
332C22×D3—41no—
333C22×D3—41no—
334C22×D3—41no—
335C22×D3—41no—
336C2×C3×C4—21no—
337C2×C3×C4—21no—
338C2×C3×C4—22no—
339C2×C3×C4—22no—
340C2×C3×C4—22no—
341C2×C3×C4—22no—
342C2×C3×C4—21no—
343C2×C3×C4—21no—
344C2×C3×C4—21no—
345C2×C3×C4—21no—
346C2×C3×C4—21no—
347C2×C3×C4—21no—
348C2×C3×C4—21no—
349C2×C3×C4—21no—
350C2×C3×C4—21no—
351C2×C3×C4—21no—
352C2×C3×C4—21no—
353C2×C3×C4—21no—
354C2×C3×C4#12412yes—
355C2×C3×C4C2×#4111yescharacteristic
356C2×C3×C4C2×#4111yescharacteristic
357C2×C3×C4C2×#4111yescharacteristic
358C2×C3×C4C2×#4111yescharacteristic
359C2×C3×C4H(5,𝔽2)11yescharacteristic
360C2×C3×C42−1+411yescharacteristic
361#67—84no—
362C2×Dic3—22no—
363C2×Dic3—41no—
364C2×Dic3—41no—
365C2×Dic3—41no—
366C2×Dic3—41no—
367C2×Dic3—41no—
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—84no—
384C2×Dic3—84no—
385D12—41no—
386D12—41no—
387D12—41no—
388D12—41no—
389C3×C8—22no—
390C3×C8—41no—
391C3×C8—41no—
392C3×C8—41no—
393C24—61no—
394C24C2×D1211yescharacteristic
395C2×D4—61no—
396C2×D4—61no—
397C2×D4—61no—
398C2×D4—61no—
399C2×D4—61no—
400C2×D4—61no—
401C2×D4—122no—
402C2×D4—122no—
403C2×D4—122no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—61no—
409C22×C4—61no—
410C22×C4—61no—
411C22×C4—61no—
412C22×C4—61no—
413C22×C4—61no—
414C22×C4—61no—
415C22×C4—122no—
416C22×C4—122no—
417C22×C4—122no—
418C22×C4—241no—
419C22×C4—241no—
420C22×C4—241no—
421C22×C4C2×D1211yescharacteristic
422C22×C4C2×D1211yescharacteristic
423C22×C4C2×D1211yescharacteristic
424C22×C4C2×D1211yescharacteristic
425C22×C4C2×D1211yescharacteristic
426C22×C4#23512yes—
427C22×C4#23512yes—
428C22×C4C23×D311yescharacteristic
429C2×C8—21no—
430C2×C8—21no—
431C2×C8—21no—
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—44no—
441C2×C8#23712yes—
442C4⋊C4—32no—
443C4⋊C4—61no—
444C4⋊C4—61no—
445C4⋊C4—61no—
446C4⋊C4—61no—
447C4⋊C4—61no—
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—122no—
457C4⋊C4—122no—
458C4⋊C4—122no—
459C4⋊C4—122no—
460C4⋊C4—241no—
461C4⋊C4—241no—
462C22⋊C4—22no—
463C22⋊C4—22no—
464C22⋊C4—122no—
465C22⋊C4—122no—
466C22⋊C4—241no—
467C22⋊C4—241no—
468C22⋊C4—241no—
469C22⋊C4—241no—
470C42—21no—
471C42—21no—
472C42—21no—
473C42—21no—
474C22×C3—22no—
475C22×C3—22no—
476C22×C3—22no—
477C22×C3—22no—
478C22×C3—22no—
479C22×C3—22no—
480C22×C3#48814yes—
481C22×C3C2×#12411yescharacteristic
482C22×C3#54611yescharacteristic
483C22×C3#54711yescharacteristic
484C22×C3C2×#13511yescharacteristic
485C22×C3C2×#13511yescharacteristic
486C22×C3#57611yescharacteristic
487C22×C3#57711yescharacteristic
488C2×D3—81no—
489C2×D3—81no—
490C2×D3—81no—
491C2×D3—81no—
492C2×D3—81no—
493C2×D3—81no—
494C2×D3—81no—
495C3×C4—22no—
496C3×C4—41no—
497C3×C4—41no—
498C3×C4—41no—
499C3×C4—41no—
500C3×C4#52911yescharacteristic
501C3×C4#53911yescharacteristic
502C3×C4#55111yescharacteristic
503C3×C4#55211yescharacteristic
504Dic3—42no—
505Dic3—81no—
506Dic3—81no—
507Dic3—81no—
508Dic3—81no—
509Dic3—81no—
510Dic3—81no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—121no—
516C23—121no—
517C23—121no—
518C23—121no—
519C23—121no—
520C23—121no—
521C23—121no—
522C23#87512yes—
523C23#90412yes—
524C23#90412yes—
525C23C22×D1211yescharacteristic
526D4—121no—
527D4—121no—
528D4—121no—
529D4—121no—
530D4—244no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—62no—
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—121no—
564C2×C4—121no—
565C2×C4—121no—
566C2×C4—121no—
567C2×C4—121no—
568C2×C4—121no—
569C2×C4—121no—
570C2×C4—244no—
571C2×C4—244no—
572C2×C4#87512yes—
573C2×C4C2×#23511yescharacteristic
574C2×C4C2×#23511yescharacteristic
575C2×C4C2×#23711yescharacteristic
576C2×C4#100411yescharacteristic
577C2×C4D3×#4111yescharacteristic
578C2×C4#101011yescharacteristic
579C8—22no—
580C8—41no—
581C8—41no—
582C8—41no—
583C2×C3—24no—
584C2×C3#248811yescharacteristic
585C2×C3C2×#48811yescharacteristic
586C2×C3#313611yescharacteristic
587C2×C3#329711yescharacteristic
588C2×C3#330111yescharacteristic
589C2×C3#333711yescharacteristic
590C2×C3#333911yescharacteristic
591D3—161no—
592C22—22no—
593C22—22no—
594C22—22no—
595C22—22no—
596C22—22no—
597C22—22no—
598C22—241no—
599C22—241no—
600C22—241no—
601C22—241no—
602C22—241no—
603C22—241no—
604C22—241no—
605C22#469714yes—
606C22C2×#87511yescharacteristic
607C22#555411yescharacteristic
608C22#555511yescharacteristic
609C22C2×#90411yescharacteristic
610C22C2×#90411yescharacteristic
611C22#575111yescharacteristic
612C22#575211yescharacteristic
613C4—22no—
614C4—41no—
615C4—41no—
616C4—41no—
617C4—41no—
618C4—122no—
619C4—241no—
620C4—241no—
621C4—241no—
622C4—241no—
623C4—241no—
624C4—241no—
625C4#549511yescharacteristic
626C4#553111yescharacteristic
627C4#553811yescharacteristic
628C4#556411yescharacteristic
629C3#2770111yescharacteristic, complemented, Sylow
630C2—24no—
631C2—481no—
632C2#7350411yescharacteristic
633C2C2×#469711yescharacteristic
634C2#7856311yescharacteristic
635C2#7941411yescharacteristic
636C2#7941611yescharacteristic
637C2#7944111yescharacteristic
638C2#7945011yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×C8×C22⋊C4C2
socleC23×C3C22×D4
prime radicalC2×#930C22
hypercenterC8×C22⋊C4D3
hypocenterC3#27701
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ C8×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. #73495C2semidirect
3. C3×C8×C22⋊C4C2semidirect
5. C2×#4551C2semidirect
7. #69925C2semidirect
8. #69778C2semidirect
9. #69768C2semidirect
10. #69768C2semidirect
11. #69763C2semidirect
12. #68679C2semidirect
13. #68644C2semidirect
14. #68644C2semidirect
15. #68637C2semidirect
73. C8×C22⋊C4D3semidirect
629. C3#27701semidirect

Maximal subgroups

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

subgroupindex
#735192
#734952
C3×C8×C22⋊C42
#719732
C2×#45512
#699322
#699252
#697782
#697682
#697632
#686792
#686442
#686372
#277013