Finite Group Databasebeta

← #10868453

Subgroups — #10868453

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
1C3×#2367C211yesmaximal, characteristic, complemented
2#73692C211yesmaximal, characteristic, complemented
3#71728C211yesmaximal, characteristic
4#71722C211yesmaximal, characteristic, complemented
5#71633C211yesmaximal, characteristic, complemented
6#71626C211yesmaximal, characteristic, complemented
7#71613C211yesmaximal, characteristic
8C4×#832C211yesmaximal, characteristic, complemented
9#69676C211yesmaximal, characteristic, complemented
10#69666C211yesmaximal, characteristic, complemented
11#69656C211yesmaximal, characteristic, complemented
12#69650C211yesmaximal, characteristic
13#69482C211yesmaximal, characteristic, complemented
14#69463C211yesmaximal, characteristic, complemented
15#69460C211yesmaximal, characteristic, complemented
16#27071—31nomaximal, Sylow
17C2×C3×#124C2211yescharacteristic
18C2×C3×C4×Q8C2211yescharacteristic
19C2×C3×C4×D4C2211yescharacteristic
20C22×#215C2211yescharacteristic
21C22×#213C2211yescharacteristic
22C2×#898—21no—
23C3×#391C2211yescharacteristic
24C3×#390C2211yescharacteristic
25C3×#388—21no—
26C3×#388—21no—
27C3×#385—21no—
28C3×C4×C4⋊C4C2211yescharacteristic
29C3×C4×C4⋊C4C2211yescharacteristic
30C3×C4×C22⋊C4—21no—
31C2×#835C2211yescharacteristic
32C2×#835C2211yescharacteristic
33C2×#835C2211yescharacteristic
34C2×#832C2211yescharacteristic
35C2×#832C2211yescharacteristic
36C2×#832C2211yescharacteristic
37#4994—21no—
38C2×#809C2211yescharacteristic
39C2×#809C2211yescharacteristic
40C2×#808C2211yescharacteristic
41C2×#808C2211yescharacteristic
42#4939C2211yescharacteristic
43#4939C2211yescharacteristic
44#4939C2211yescharacteristic
45#4935C2211yescharacteristic
46C4×C4⋊Dic3C2211yescharacteristic
47#4928C2211yescharacteristic
48C2×C4⋊(C3⋊C8)C2211yescharacteristic
49C2×C4×C3⋊C8C2211yescharacteristic
50#4674—21no—
51#4670—21no—
52#4529C2211yescharacteristic
53#4529C2211yescharacteristic
54#4529C2211yescharacteristic
55#4529C2211yescharacteristic
56#4524C2211yescharacteristic
57#4524C2211yescharacteristic
58#4524C2211yescharacteristic
59#4524C2211yescharacteristic
60#2512—31no—
61#2367D311yescharacteristic, complemented
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#1916—31no—
71#1880—31no—
72#1879—31no—
73#1803—31no—
74C4×#102—31no—
75C22×C3×Q8C2311yescharacteristic
76C22×C3×D4C2311yescharacteristic
77C23×C3×C4—21no—
78C22×D12C2311yescharacteristic
79C3×#124—22no—
80C3×#124—22no—
81C3×C4×Q8—22no—
82C3×C4×Q8—41no—
83C3×C4×D4—22no—
84C3×C4×D4—41no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4D411yescharacteristic
87C2×C3×C4⋊C4D411yescharacteristic
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C22⋊C4—21no—
91C2×C3×C22⋊C4—21no—
92C2×C3×C22⋊C4—21no—
93C2×C3×C22⋊C4—21no—
94C2×C3×C22⋊C4—21no—
95C2×C3×C42—21no—
96C2×C3×C42—21no—
97C2×C3×C42C2311yescharacteristic
98C2×C3×C42C2311yescharacteristic
99C2×C3×C42C2311yescharacteristic
100C2×#215—22no—
101C2×#215—22no—
102C2×#215—41no—
103C2×#213—22no—
104C2×#213—41no—
105C2×#213—41no—
106C2×(C2×D3)⋊C4—21no—
107C2×(C2×D3)⋊C4—21no—
108C2×(C2×D3)⋊C4—21no—
109C2×(C2×D3)⋊C4—21no—
110C2×(C2×D3)⋊C4—21no—
111C2×(C2×D3)⋊C4—21no—
112C2×C4⋊Dic3—21no—
113C2×C4⋊Dic3—21no—
114C2×C4⋊Dic3C2311yescharacteristic
115C2×C4⋊Dic3C2311yescharacteristic
116C2×C4⋊Dic3C2311yescharacteristic
117C2×Dic3⋊C4—21no—
118C2×C4×Dic3—21no—
119C22×C3⋊C8C2311yescharacteristic
120C22×C3⋊C8C2311yescharacteristic
121C22×C3⋊C8C2311yescharacteristic
122C22×C3⋊C8C2311yescharacteristic
123#898—44no—
124C3×#95—21no—
125C3×#95—21no—
126C3×#95—21no—
127C3×#95D411yescharacteristic
128C3×#95D411yescharacteristic
129#835—22no—
130#835—41no—
131#835—41no—
132#832—22no—
133#832—22no—
134#832—41no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137#809—22no—
138#809—41no—
139#808—22no—
140#808—41no—
141C4⋊(C3⋊C8)—41no—
142C4×C3⋊C8—41no—
143C22×SD16—31no—
144C22×D8—31no—
145C2×#126—61no—
146C2×#124C2×D311yescharacteristic
147C2×C4×Q8C2×D311yescharacteristic
148C2×C4×D4C2×D311yescharacteristic
149C2×#106—31no—
150C2×#106—31no—
151C2×C4⋊C8—31no—
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×C8—31no—
161#397—61no—
162#396—61no—
163#391C2×D311yescharacteristic
164#390—31no—
165#390—31no—
166#390—31no—
167#390C2×D311yescharacteristic
168#389—31no—
169#388—21no—
170#388—21no—
171#388—61no—
172#385—21no—
173#384—31no—
174C4×C4⋊C4—31no—
175C4×C4⋊C4C2×D311yescharacteristic
176C4×C4⋊C4C2×D311yescharacteristic
177C4×C22⋊C4—21no—
178#340—31no—
179#340—31no—
180#340—31no—
181#340—31no—
182#336—31no—
183#336—31no—
184#336—31no—
185#336—31no—
186C24×C3—21no—
187C23×D3—21no—
188C2×C3×Q8—21no—
189C2×C3×Q8—21no—
190C2×C3×Q8—21no—
191C2×C3×Q8—21no—
192C2×C3×Q8—21no—
193C2×C3×Q8#4112yes—
194C2×C3×D4—21no—
195C2×C3×D4—21no—
196C2×C3×D4—21no—
197C2×C3×D4—21no—
198C2×C3×D4—21no—
199C2×C3×D4#4112yes—
200C22×C3×C4—21no—
201C22×C3×C4—21no—
202C22×C3×C4—21no—
203C22×C3×C4—42no—
204C22×C3×C4—42no—
205C22×C3×C4—81no—
206C22×C3×C4C2×D411yescharacteristic
207C22×C3×C4C2×D411yescharacteristic
208C22×C3×C4C2×D411yescharacteristic
209C22×C3×C4C2×D411yescharacteristic
210C22×C3×C4C2×D411yescharacteristic
211C22×C3×C4C2×D411yescharacteristic
212C22×C3×C4C2411yescharacteristic
213C22×Dic3—21no—
214C22×Dic3—21no—
215C22×Dic3—21no—
216C2×D12—21no—
217C2×D12—21no—
218C2×D12—21no—
219C2×D12—21no—
220C2×D12—21no—
221C2×D12—21no—
222C3×C4⋊C4—22no—
223C3×C4⋊C4—22no—
224C3×C4⋊C4—21no—
225C3×C4⋊C4—21no—
226C3×C4⋊C4—21no—
227C3×C4⋊C4—42no—
228C3×C4⋊C4#4112yes—
229C3×C22⋊C4—24no—
230C3×C22⋊C4—24no—
231C3×C22⋊C4—42no—
232C3×C22⋊C4—42no—
233C3×C22⋊C4—81no—
234C3×C42—21no—
235C3×C42—21no—
236C3×C42—21no—
237C3×C42—21no—
238C3×C42—21no—
239C3×C42—21no—
240C3×C42—42no—
241C3×C42—81no—
242#215—44no—
243#213—44no—
244(C2×D3)⋊C4—42no—
245(C2×D3)⋊C4—42no—
246(C2×D3)⋊C4—42no—
247(C2×D3)⋊C4—42no—
248(C2×D3)⋊C4—81no—
249(C2×D3)⋊C4—81no—
250C4⋊Dic3—21no—
251C4⋊Dic3—21no—
252C4⋊Dic3—21no—
253C4⋊Dic3—21no—
254C4⋊Dic3—21no—
255C4⋊Dic3—21no—
256C4⋊Dic3—42no—
257C4⋊Dic3—81no—
258Dic3⋊C4—42no—
259C4×Dic3—42no—
260C2×C3⋊C8—21no—
261C2×C3⋊C8—21no—
262C2×C3⋊C8—21no—
263C2×C3⋊C8—21no—
264C2×C3⋊C8—21no—
265C2×C3⋊C8—21no—
266C2×C3⋊C8—21no—
267C2×C3⋊C8—21no—
268C2×C3⋊C8—21no—
269C2×C3⋊C8—21no—
270C2×C3⋊C8—21no—
271C2×C3⋊C8#4112yes—
272C22×Q8C22×D311yescharacteristic
273C22×D4—31no—
274C22×D4C22×D311yescharacteristic
275C23×C4—21no—
276C2×SD16—62no—
277C2×SD16—62no—
278C2×SD16—121no—
279C2×D8—62no—
280C2×D8—121no—
281C2×D8—121no—
282C22×C8—31no—
283C22×C8—31no—
284C22×C8—31no—
285C22×C8—31no—
286#126—124no—
287#124—22no—
288#124—22no—
289C4×Q8—22no—
290C4×Q8—41no—
291C4×D4—22no—
292C4×D4—41no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—31no—
295C2×C4⋊C4—31no—
296C2×C4⋊C4—31no—
297C2×C4⋊C4—61no—
298C2×C4⋊C4—61no—
299C2×C4⋊C4—61no—
300C2×C4⋊C4#6711yescharacteristic
301C2×C4⋊C4#6711yescharacteristic
302C2×C4⋊C4C22×D311yescharacteristic
303C2×C4⋊C4C22×D311yescharacteristic
304C2×C22⋊C4—21no—
305C2×C22⋊C4—21no—
306C2×C22⋊C4—21no—
307C2×C22⋊C4—21no—
308C2×C22⋊C4—21no—
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—21no—
316C2×C42—21no—
317C2×C42—61no—
318C2×C42C22×D311yescharacteristic
319C2×C42C22×D311yescharacteristic
320C2×C42C22×D311yescharacteristic
321#106—62no—
322#106—121no—
323C4⋊C8—121no—
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—121no—
333#95—21no—
334#95—21no—
335#95—21no—
336#95—61no—
337#95—61no—
338#95#6711yescharacteristic
339#95#6711yescharacteristic
340C23×C3—22no—
341C23×C3—41no—
342C23×C3—41no—
343C23×C3—41no—
344C23×C3—41no—
345C23×C3—41no—
346C23×C3—41no—
347C23×C3C22×D411yescharacteristic
348C22×D3—41no—
349C22×D3—41no—
350C22×D3—41no—
351C22×D3—41no—
352C22×D3—41no—
353C22×D3—41no—
354C22×D3—41no—
355C3×Q8—22no—
356C3×Q8—22no—
357C3×Q8—41no—
358C3×Q8—41no—
359C3×D4—22no—
360C3×D4—22no—
361C3×D4—41no—
362C3×D4—41no—
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—22no—
370C2×C3×C4—21no—
371C2×C3×C4—21no—
372C2×C3×C4—22no—
373C2×C3×C4—21no—
374C2×C3×C4—21no—
375C2×C3×C4—21no—
376C2×C3×C4—21no—
377C2×C3×C4—21no—
378C2×C3×C4—21no—
379C2×C3×C4—21no—
380C2×C3×C4—21no—
381C2×C3×C4—41no—
382C2×C3×C4—41no—
383C2×C3×C4—41no—
384C2×C3×C4—41no—
385C2×C3×C4—41no—
386C2×C3×C4—41no—
387C2×C3×C4—41no—
388C2×C3×C4—84no—
389C2×C3×C4#13512yes—
390C2×C3×C4AGL(1,ℤ/8)12yes—
391C2×C3×C4C2×#4111yescharacteristic
392C2×C3×C4C2×#4111yescharacteristic
393C2×C3×C4C2×#4111yescharacteristic
394C2×C3×C4C2×#4111yescharacteristic
395C2×C3×C4H(5,𝔽2)11yescharacteristic
396C2×C3×C4H(5,𝔽2)11yescharacteristic
397C2×Dic3—41no—
398C2×Dic3—41no—
399C2×Dic3—41no—
400C2×Dic3—41no—
401C2×Dic3—41no—
402C2×Dic3—41no—
403C2×Dic3—41no—
404C2×Dic3—41no—
405C2×Dic3—41no—
406D12—41no—
407D12—41no—
408D12—41no—
409D12—41no—
410C3⋊C8—22no—
411C3⋊C8—41no—
412C3⋊C8—41no—
413C3⋊C8—41no—
414C24—21no—
415C24—61no—
416C2×Q8—21no—
417C2×Q8—21no—
418C2×Q8—21no—
419C2×Q8—21no—
420C2×Q8—21no—
421C2×Q8#23512yes—
422C2×D4—21no—
423C2×D4—21no—
424C2×D4—21no—
425C2×D4—21no—
426C2×D4—21no—
427C2×D4—61no—
428C2×D4—61no—
429C2×D4—61no—
430C2×D4—61no—
431C2×D4—61no—
432C2×D4—61no—
433C2×D4#23512yes—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—61no—
440C22×C4—61no—
441C22×C4—61no—
442C22×C4—81no—
443C22×C4C2×#6711yescharacteristic
444C22×C4C2×#6711yescharacteristic
445C22×C4C2×#6711yescharacteristic
446C22×C4C2×#6711yescharacteristic
447C22×C4C2×#6711yescharacteristic
448C22×C4C2×#6711yescharacteristic
449C22×C4C23×D311yescharacteristic
450SD16—124no—
451D8—124no—
452C2×C8—32no—
453C2×C8—61no—
454C2×C8—61no—
455C2×C8—61no—
456C2×C8—61no—
457C2×C8—61no—
458C2×C8—61no—
459C2×C8—61no—
460C2×C8—61no—
461C2×C8—61no—
462C2×C8—61no—
463C2×C8—61no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—42no—
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—241no—
479C4⋊C4#23712yes—
480C22⋊C4—24no—
481C22⋊C4—24no—
482C22⋊C4—42no—
483C22⋊C4—42no—
484C22⋊C4—81no—
485C22⋊C4—122no—
486C22⋊C4—122no—
487C22⋊C4—122no—
488C22⋊C4—122no—
489C22⋊C4—241no—
490C22⋊C4—241no—
491C42—21no—
492C42—21no—
493C42—21no—
494C42—21no—
495C42—21no—
496C42—21no—
497C42—42no—
498C42—81no—
499C42—122no—
500C22×C3—42no—
501C22×C3—42no—
502C22×C3—42no—
503C22×C3—81no—
504C22×C3—81no—
505C22×C3—81no—
506C22×C3—81no—
507C22×C3#52511yescharacteristic
508C22×C3#54611yescharacteristic
509C22×C3#55011yescharacteristic
510C22×C3C2×#13511yescharacteristic
511C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
512C22×C3#57711yescharacteristic
513C22×C3#57711yescharacteristic
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—42no—
526C3×C4—41no—
527C3×C4—41no—
528C3×C4—42no—
529C3×C4—81no—
530C3×C4#52911yescharacteristic
531C3×C4#53811yescharacteristic
532C3×C4#55111yescharacteristic
533C3×C4#55311yescharacteristic
534Dic3—81no—
535Dic3—81no—
536Dic3—81no—
537C23—22no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—121no—
545C23—121no—
546C23—121no—
547C23—121no—
548C23—121no—
549C23—121no—
550C23—121no—
551C23C22×#6711yescharacteristic
552Q8—22no—
553Q8—22no—
554Q8—41no—
555Q8—41no—
556D4—22no—
557D4—22no—
558D4—41no—
559D4—41no—
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—22no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—84no—
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—121no—
599C2×C4#94912yes—
600C2×C4#95012yes—
601C2×C4C2×#23511yescharacteristic
602C2×C4C2×#23511yescharacteristic
603C2×C4C2×#23711yescharacteristic
604C2×C4#100411yescharacteristic
605C2×C4D3×#4111yescharacteristic
606C2×C4#100911yescharacteristic
607C8—62no—
608C8—121no—
609C8—121no—
610C8—121no—
611C2×C3—82no—
612C2×C3#246711yescharacteristic
613C2×C3#310611yescharacteristic
614C2×C3#312511yescharacteristic
615C2×C3#330111yescharacteristic
616C2×C3#330611yescharacteristic
617C2×C3#339411yescharacteristic
618C2×C3#339711yescharacteristic
619D3—161no—
620C22—42no—
621C22—42no—
622C22—42no—
623C22—81no—
624C22—81no—
625C22—81no—
626C22—81no—
627C22—241no—
628C22—241no—
629C22—241no—
630C22—241no—
631C22—241no—
632C22—241no—
633C22—241no—
634C22#579111yescharacteristic
635C22#580611yescharacteristic
636C22C2×#94911yescharacteristic
637C22C2×#95011yescharacteristic
638C22#583411yescharacteristic
639C22#583611yescharacteristic
640C22#583711yescharacteristic
641C4—22no—
642C4—22no—
643C4—41no—
644C4—42no—
645C4—41no—
646C4—41no—
647C4—41no—
648C4—42no—
649C4—81no—
650C4—241no—
651C4—241no—
652C4—241no—
653C4#553811yescharacteristic
654C4#555611yescharacteristic
655C4#556011yescharacteristic
656C4#556511yescharacteristic
657C3#2707111yescharacteristic, complemented, Sylow
658C2—82no—
659C2—481no—
660C2#8237011yescharacteristic
661C2#8252911yescharacteristic
662C2#8257611yescharacteristic
663C2#8349611yescharacteristic
664C2#8353511yescharacteristic
665C2#8367111yescharacteristic
666C2#8371011yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2367C2
socleC23×C3C22×D4
prime radicalC22×#213C22
hypercenter#2367D3
hypocenterC3#27071
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2367

Complements

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

NH ≅ G/Ntype
1. C3×#2367C2semidirect
2. #73692C2semidirect
4. #71722C2semidirect
5. #71633C2semidirect
6. #71626C2semidirect
8. C4×#832C2semidirect
9. #69676C2semidirect
10. #69666C2semidirect
11. #69656C2semidirect
13. #69482C2semidirect
14. #69463C2semidirect
15. #69460C2semidirect
61. #2367D3semidirect
657. C3#27071semidirect

Maximal subgroups

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

subgroupindex
C3×#23672
#736922
#717282
#717222
#716332
#716262
#716132
C4×#8322
#696762
#696662
#696562
#696502
#694822
#694632
#694602
#270713