Finite Group Databasebeta

← #10868457

Subgroups — #10868457

Order 768.

802 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#73692C211yesmaximal, characteristic, complemented
2#73655C211yesmaximal, characteristic, complemented
3C3×#1916C211yesmaximal, characteristic, complemented
4#70485C211yesmaximal, characteristic, complemented
5#70481C211yesmaximal, characteristic
6#69774C211yesmaximal, characteristic, complemented
7C4×#821C211yesmaximal, characteristic, complemented
8#68686C211yesmaximal, characteristic, complemented
9#68682C211yesmaximal, characteristic, complemented
10#68675C211yesmaximal, characteristic
11#68673C211yesmaximal, characteristic
12#68630C211yesmaximal, characteristic, complemented
13#68627C211yesmaximal, characteristic, complemented
14#68626C211yesmaximal, characteristic, complemented
15#68623C211yesmaximal, characteristic, complemented
16#27071—31nomaximal, Sylow
17C22×D24C2211yescharacteristic
18C22×#204C2211yescharacteristic
19C2×#898—21no—
20C2×#875C2211yescharacteristic
21C2×C4×D12C2211yescharacteristic
22C2×C4×Dic6C2211yescharacteristic
23C2×C3×C4⋊C8C2211yescharacteristic
24C2×C3×C4×C8C2211yescharacteristic
25C3×C4×C4⋊C4C2211yescharacteristic
26C2×#821C2211yescharacteristic
27C2×#821C2211yescharacteristic
28C2×#821C2211yescharacteristic
29C2×#821C2211yescharacteristic
30C2×#821C2211yescharacteristic
31C2×#817C2211yescharacteristic
32C2×#817C2211yescharacteristic
33C2×#816C2211yescharacteristic
34C2×#816C2211yescharacteristic
35C2×#816C2211yescharacteristic
36#4994—21no—
37#4990—21no—
38#4989—21no—
39#4939C2211yescharacteristic
40#4939C2211yescharacteristic
41#4939C2211yescharacteristic
42#4939C2211yescharacteristic
43#4935C2211yescharacteristic
44C4×C4⋊Dic3C2211yescharacteristic
45C4×C4⋊Dic3C2211yescharacteristic
46#4929C2211yescharacteristic
47#4928C2211yescharacteristic
48#4674—21no—
49#4670—21no—
50#4669—21no—
51#4666—21no—
52C3×#336C2211yescharacteristic
53C3×#336C2211yescharacteristic
54C3×#336C2211yescharacteristic
55C3×#336C2211yescharacteristic
56#4551C2211yescharacteristic
57#4551C2211yescharacteristic
58#4551C2211yescharacteristic
59#4551C2211yescharacteristic
60#2512—31no—
61#2367—31no—
62#2094—31no—
63#2054—31no—
64#2042—31no—
65#2012—31no—
66#1968—31no—
67#1927—31no—
68#1926—31no—
69#1920—31no—
70#1916D311yescharacteristic, complemented
71#1880—31no—
72#1879—31no—
73#1803—31no—
74C4×#102—31no—
75C22×D12C2311yescharacteristic
76C22×D12C2311yescharacteristic
77C22×C4×D3—21no—
78C22×Dic6C2311yescharacteristic
79C22×C3×C8C2311yescharacteristic
80C22×C3×C8C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C2×C3×C4⋊C4D411yescharacteristic
84C2×C3×C4⋊C4D411yescharacteristic
85C2×C3×C42C2311yescharacteristic
86C2×C3×C42C2311yescharacteristic
87C2×C3×C42C2311yescharacteristic
88C2×(C2×D3)⋊C4—21no—
89C2×(C2×D3)⋊C4—21no—
90C2×(C2×D3)⋊C4—21no—
91C2×(C2×D3)⋊C4—21no—
92C2×(C2×D3)⋊C4—21no—
93C2×(C2×D3)⋊C4—21no—
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—21no—
100C2×C4⋊Dic3—21no—
101C2×C4⋊Dic3C2311yescharacteristic
102C2×C4⋊Dic3C2311yescharacteristic
103C2×C4⋊Dic3C2311yescharacteristic
104C2×C4⋊Dic3C2311yescharacteristic
105C2×C4⋊Dic3C2311yescharacteristic
106C2×Dic3⋊C4—21no—
107C2×Dic3⋊C4—21no—
108C2×C4×Dic3—21no—
109C2×C4×Dic3—21no—
110C2×C4×Dic3—21no—
111C2×D24—22no—
112C2×D24—41no—
113C2×D24—41no—
114C2×#204—22no—
115C2×#204—22no—
116C2×#204—41no—
117#898—44no—
118#875—22no—
119#875—22no—
120C4×D12—22no—
121C4×D12—41no—
122C4×Dic6—22no—
123C4×Dic6—41no—
124C3×C4⋊C8—41no—
125C3×C4×C8—41no—
126C3×#95D411yescharacteristic
127C3×#95D411yescharacteristic
128C3⋊#95—21no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133#821—22no—
134#821—22no—
135#821—22no—
136#821—41no—
137#821—41no—
138#817—22no—
139#817—41no—
140#816—22no—
141#816—41no—
142#816—41no—
143C22×SD16—31no—
144C22×D8—31no—
145C2×#126—61no—
146C2×#124—31no—
147C2×C4×Q8—31no—
148C2×C4×D4—31no—
149C2×#106—31no—
150C2×#106—31no—
151C2×C4⋊C8C2×D311yescharacteristic
152C2×#103—31no—
153C2×#103—31no—
154C2×#103—31no—
155C2×#102—31no—
156C2×#102—31no—
157C2×#102—31no—
158C2×#102—31no—
159C2×#102—31no—
160C2×C4×C8C2×D311yescharacteristic
161#397—61no—
162#396—61no—
163#391—31no—
164#390—31no—
165#390—31no—
166#390—31no—
167#390—31no—
168#389—31no—
169#388—61no—
170#388—61no—
171#388—61no—
172#385—61no—
173#384—31no—
174C4×C4⋊C4—31no—
175C4×C4⋊C4—31no—
176C4×C4⋊C4C2×D311yescharacteristic
177C4×C22⋊C4—61no—
178#340—31no—
179#340—31no—
180#340—31no—
181#340—31no—
182#336C2×D311yescharacteristic
183#336C2×D311yescharacteristic
184#336C2×D311yescharacteristic
185#336C2×D311yescharacteristic
186C23×D3—21no—
187C23×D3—21no—
188C22×C3×C4C2×D411yescharacteristic
189C22×C3×C4C2×D411yescharacteristic
190C22×C3×C4C2×D411yescharacteristic
191C22×C3×C4C2×D411yescharacteristic
192C22×C3×C4C2×D411yescharacteristic
193C22×C3×C4C2×D411yescharacteristic
194C22×C3×C4C2411yescharacteristic
195C22×Dic3—21no—
196C22×Dic3—21no—
197C22×Dic3—21no—
198C22×Dic3—21no—
199C22×Dic3—21no—
200C22×Dic3—21no—
201C2×D12—21no—
202C2×D12—21no—
203C2×D12—21no—
204C2×D12—21no—
205C2×D12—21no—
206C2×D12—21no—
207C2×D12—21no—
208C2×D12—21no—
209C2×D12—21no—
210C2×D12—21no—
211C2×D12—21no—
212C2×D12#4112yes—
213C2×C4×D3—42no—
214C2×C4×D3—42no—
215C2×C4×D3—81no—
216C2×Dic6—21no—
217C2×Dic6—21no—
218C2×Dic6—21no—
219C2×Dic6—21no—
220C2×Dic6—21no—
221C2×Dic6#4112yes—
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—
230C2×C3×C8—21no—
231C2×C3×C8—21no—
232C2×C3×C8—21no—
233C2×C3×C8#4112yes—
234C3×C4⋊C4—22no—
235C3×C4⋊C4—22no—
236C3×C42—21no—
237C3×C42—21no—
238C3×C42—21no—
239C3×C42—21no—
240C3×C42—21no—
241C3×C42—21no—
242(C2×D3)⋊C4—24no—
243(C2×D3)⋊C4—24no—
244(C2×D3)⋊C4—42no—
245(C2×D3)⋊C4—42no—
246(C2×D3)⋊C4—42no—
247(C2×D3)⋊C4—42no—
248(C2×D3)⋊C4—42no—
249(C2×D3)⋊C4—42no—
250(C2×D3)⋊C4—81no—
251(C2×D3)⋊C4—81no—
252(C2×D3)⋊C4—81no—
253C4⋊Dic3—21no—
254C4⋊Dic3—21no—
255C4⋊Dic3—21no—
256C4⋊Dic3—21no—
257C4⋊Dic3—21no—
258C4⋊Dic3—21no—
259C4⋊Dic3—21no—
260C4⋊Dic3—21no—
261C4⋊Dic3—21no—
262C4⋊Dic3—42no—
263C4⋊Dic3—81no—
264C4⋊Dic3#4112yes—
265Dic3⋊C4—42no—
266Dic3⋊C4—42no—
267C4×Dic3—42no—
268C4×Dic3—42no—
269C4×Dic3—81no—
270D24—44no—
271#204—44no—
272C22×Q8—31no—
273C22×D4—31no—
274C22×D4—31no—
275C23×C4—61no—
276C2×SD16—62no—
277C2×SD16—62no—
278C2×SD16—121no—
279C2×D8—62no—
280C2×D8—121no—
281C2×D8—121no—
282C22×C8C22×D311yescharacteristic
283C22×C8C22×D311yescharacteristic
284C22×C8C22×D311yescharacteristic
285C22×C8C22×D311yescharacteristic
286#126—124no—
287#124—62no—
288#124—62no—
289C4×Q8—62no—
290C4×Q8—121no—
291C4×D4—62no—
292C4×D4—121no—
293C2×C4⋊C4—31no—
294C2×C4⋊C4—31no—
295C2×C4⋊C4—31no—
296C2×C4⋊C4—31no—
297C2×C4⋊C4—31no—
298C2×C4⋊C4—61no—
299C2×C4⋊C4—61no—
300C2×C4⋊C4—61no—
301C2×C4⋊C4—61no—
302C2×C4⋊C4D1211yescharacteristic
303C2×C4⋊C4D1211yescharacteristic
304C2×C22⋊C4—61no—
305C2×C22⋊C4—61no—
306C2×C22⋊C4—61no—
307C2×C22⋊C4—61no—
308C2×C22⋊C4—61no—
309C2×C22⋊C4—61no—
310C2×C22⋊C4—61no—
311C2×C22⋊C4—61no—
312C2×C22⋊C4—61no—
313C2×C22⋊C4—61no—
314C2×C22⋊C4—61no—
315C2×C42—61no—
316C2×C42—61no—
317C2×C42—61no—
318C2×C42C22×D311yescharacteristic
319C2×C42C22×D311yescharacteristic
320C2×C42C22×D311yescharacteristic
321#106—62no—
322#106—121no—
323C4⋊C8—41no—
324#103—62no—
325#103—121no—
326#103—121no—
327#102—62no—
328#102—62no—
329#102—62no—
330#102—121no—
331#102—121no—
332C4×C8—41no—
333#95—61no—
334#95—61no—
335#95—61no—
336#95—61no—
337#95—61no—
338#95D1211yescharacteristic
339#95D1211yescharacteristic
340C23×C3C22×D411yescharacteristic
341C22×D3—22no—
342C22×D3—41no—
343C22×D3—41no—
344C22×D3—41no—
345C22×D3—41no—
346C22×D3—41no—
347C22×D3—41no—
348C22×D3—41no—
349C22×D3—41no—
350C22×D3—41no—
351C22×D3—41no—
352C22×D3—41no—
353C22×D3—41no—
354C22×D3—41no—
355C2×C3×C4—21no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—21no—
361C2×C3×C4—21no—
362C2×C3×C4—21no—
363C2×C3×C4—21no—
364C2×C3×C4—21no—
365C2×C3×C4—21no—
366C2×C3×C4—21no—
367C2×C3×C4—21no—
368C2×C3×C4—21no—
369C2×C3×C4—21no—
370C2×C3×C4—21no—
371C2×C3×C4#13512yes—
372C2×C3×C4AGL(1,ℤ/8)12yes—
373C2×C3×C4C2×#4111yescharacteristic
374C2×C3×C4C2×#4111yescharacteristic
375C2×C3×C4C2×#4111yescharacteristic
376C2×C3×C4C2×#4111yescharacteristic
377C2×C3×C4H(5,𝔽2)11yescharacteristic
378C2×C3×C4H(5,𝔽2)11yescharacteristic
379C2×Dic3—22no—
380C2×Dic3—22no—
381C2×Dic3—41no—
382C2×Dic3—41no—
383C2×Dic3—41no—
384C2×Dic3—41no—
385C2×Dic3—41no—
386C2×Dic3—41no—
387C2×Dic3—41no—
388C2×Dic3—41no—
389C2×Dic3—41no—
390C2×Dic3—41no—
391C2×Dic3—41no—
392C2×Dic3—41no—
393C2×Dic3—41no—
394C2×Dic3—41no—
395C2×Dic3—41no—
396C2×Dic3—41no—
397D12—22no—
398D12—22no—
399D12—41no—
400D12—41no—
401D12—41no—
402D12—41no—
403D12—41no—
404D12—41no—
405C4×D3—84no—
406Dic6—22no—
407Dic6—22no—
408Dic6—41no—
409Dic6—41no—
410C3×C8—22no—
411C3×C8—41no—
412C3×C8—41no—
413C3×C8—41no—
414C24—61no—
415C24—61no—
416C2×Q8—32no—
417C2×Q8—61no—
418C2×Q8—61no—
419C2×Q8—61no—
420C2×Q8—61no—
421C2×Q8—61no—
422C2×D4—32no—
423C2×D4—61no—
424C2×D4—61no—
425C2×D4—61no—
426C2×D4—61no—
427C2×D4—61no—
428C2×D4—61no—
429C2×D4—61no—
430C2×D4—61no—
431C2×D4—61no—
432C2×D4—61no—
433C2×D4—61no—
434C22×C4—61no—
435C22×C4—61no—
436C22×C4—61no—
437C22×C4—61no—
438C22×C4—61no—
439C22×C4—61no—
440C22×C4—122no—
441C22×C4—122no—
442C22×C4—241no—
443C22×C4C2×D1211yescharacteristic
444C22×C4C2×D1211yescharacteristic
445C22×C4C2×D1211yescharacteristic
446C22×C4C2×D1211yescharacteristic
447C22×C4C2×D1211yescharacteristic
448C22×C4C2×D1211yescharacteristic
449C22×C4C23×D311yescharacteristic
450SD16—124no—
451D8—124no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8#23912yes—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—32no—
467C4⋊C4—61no—
468C4⋊C4—61no—
469C4⋊C4—61no—
470C4⋊C4—61no—
471C4⋊C4—61no—
472C4⋊C4—61no—
473C4⋊C4—61no—
474C4⋊C4—61no—
475C4⋊C4—61no—
476C4⋊C4—122no—
477C4⋊C4—122no—
478C4⋊C4—122no—
479C4⋊C4—241no—
480C22⋊C4—64no—
481C22⋊C4—64no—
482C22⋊C4—122no—
483C22⋊C4—122no—
484C22⋊C4—122no—
485C22⋊C4—122no—
486C22⋊C4—122no—
487C22⋊C4—122no—
488C22⋊C4—241no—
489C22⋊C4—241no—
490C22⋊C4—241no—
491C42—21no—
492C42—21no—
493C42—21no—
494C42—21no—
495C42—21no—
496C42—21no—
497C42—122no—
498C42—122no—
499C42—241no—
500C22×C3#52511yescharacteristic
501C22×C3#54611yescharacteristic
502C22×C3#55011yescharacteristic
503C22×C3C2×#13511yescharacteristic
504C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
505C22×C3#57711yescharacteristic
506C22×C3#57711yescharacteristic
507C2×D3—42no—
508C2×D3—42no—
509C2×D3—42no—
510C2×D3—81no—
511C2×D3—81no—
512C2×D3—81no—
513C2×D3—81no—
514C2×D3—81no—
515C2×D3—81no—
516C2×D3—81no—
517C2×D3—81no—
518C2×D3—81no—
519C2×D3—81no—
520C2×D3—81no—
521C3×C4—22no—
522C3×C4—22no—
523C3×C4—41no—
524C3×C4—41no—
525C3×C4—41no—
526C3×C4—41no—
527C3×C4#52911yescharacteristic
528C3×C4#53811yescharacteristic
529C3×C4#55111yescharacteristic
530C3×C4#55311yescharacteristic
531Dic3—42no—
532Dic3—42no—
533Dic3—81no—
534Dic3—81no—
535Dic3—81no—
536Dic3—81no—
537C23—62no—
538C23—121no—
539C23—121no—
540C23—121no—
541C23—121no—
542C23—121no—
543C23—121no—
544C23—121no—
545C23—121no—
546C23—121no—
547C23—121no—
548C23—121no—
549C23—121no—
550C23—121no—
551C23C22×D1211yescharacteristic
552Q8—62no—
553Q8—62no—
554Q8—121no—
555Q8—121no—
556D4—62no—
557D4—62no—
558D4—121no—
559D4—121no—
560D4—121no—
561D4—121no—
562D4—121no—
563D4—121no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—62no—
581C2×C4—62no—
582C2×C4—121no—
583C2×C4—121no—
584C2×C4—121no—
585C2×C4—121no—
586C2×C4—121no—
587C2×C4—121no—
588C2×C4—121no—
589C2×C4—121no—
590C2×C4—121no—
591C2×C4—121no—
592C2×C4—121no—
593C2×C4—121no—
594C2×C4—121no—
595C2×C4—121no—
596C2×C4—121no—
597C2×C4—121no—
598C2×C4—244no—
599C2×C4#90412yes—
600C2×C4#90812yes—
601C2×C4#100411yescharacteristic
602C2×C4C2×#23911yescharacteristic
603C2×C4D3×#4111yescharacteristic
604C2×C4D3×#4111yescharacteristic
605C2×C4D3×#4111yescharacteristic
606C2×C4#100911yescharacteristic
607C8—22no—
608C8—41no—
609C8—41no—
610C8—41no—
611C2×C3#246711yescharacteristic
612C2×C3#310611yescharacteristic
613C2×C3#312511yescharacteristic
614C2×C3#330111yescharacteristic
615C2×C3#330611yescharacteristic
616C2×C3#339411yescharacteristic
617C2×C3#339711yescharacteristic
618D3—82no—
619D3—161no—
620C22—122no—
621C22—122no—
622C22—122no—
623C22—241no—
624C22—241no—
625C22—241no—
626C22—241no—
627C22—241no—
628C22—241no—
629C22—241no—
630C22—241no—
631C22—241no—
632C22—241no—
633C22—241no—
634C22#552411yescharacteristic
635C22#555411yescharacteristic
636C22#557711yescharacteristic
637C22C2×#90411yescharacteristic
638C22C2×#90811yescharacteristic
639C22#575111yescharacteristic
640C22#575111yescharacteristic
641C4—22no—
642C4—22no—
643C4—41no—
644C4—41no—
645C4—41no—
646C4—41no—
647C4—122no—
648C4—122no—
649C4—241no—
650C4—241no—
651C4—241no—
652C4—241no—
653C4#563711yescharacteristic
654C4#565711yescharacteristic
655C4#569111yescharacteristic
656C4#570911yescharacteristic
657C3#2707111yescharacteristic, complemented, Sylow
658C2—242no—
659C2—481no—
660C2#7500111yescharacteristic
661C2#7930111yescharacteristic
662C2#7938911yescharacteristic
663C2#7945011yescharacteristic
664C2#7949911yescharacteristic
665C2#7991311yescharacteristic
666C2#7995011yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1916C2
socleC23×C3C22×D4
prime radicalC22×D24C22
hypercenter#1916D3
hypocenterC3#27071
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #1916

Complements

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

NH ≅ G/Ntype
1. #73692C2semidirect
2. #73655C2semidirect
3. C3×#1916C2semidirect
4. #70485C2semidirect
6. #69774C2semidirect
7. C4×#821C2semidirect
8. #68686C2semidirect
9. #68682C2semidirect
12. #68630C2semidirect
13. #68627C2semidirect
14. #68626C2semidirect
15. #68623C2semidirect
70. #1916D3semidirect
657. C3#27071semidirect

Maximal subgroups

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

subgroupindex
#736922
#736552
C3×#19162
#704852
#704812
#697742
C4×#8212
#686862
#686822
#686752
#686732
#686302
#686272
#686262
#686232
#270713