Finite Group Databasebeta

← #10858921

Subgroups — #10858921

Order 768.

858 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#76206C211yesmaximal, characteristic
2#74163C211yesmaximal, characteristic, complemented
3C3×#2101C211yesmaximal, characteristic, complemented
4#69623C211yesmaximal, characteristic, complemented
5#69556C211yesmaximal, characteristic, complemented
6#68901C211yesmaximal, characteristic, complemented
7#68899C211yesmaximal, characteristic, complemented
8#68883C211yesmaximal, characteristic
9#68880C211yesmaximal, characteristic
10#68874C211yesmaximal, characteristic, complemented
11D3×#340C211yesmaximal, characteristic, complemented
12#68820C211yesmaximal, characteristic, complemented
13#68801C211yesmaximal, characteristic, complemented
14#68670C211yesmaximal, characteristic, complemented
15#68666C211yesmaximal, characteristic, complemented
16#26420—31nomaximal, Sylow
17C22×D3×C8C2211yescharacteristic
18C2×#895C2211yescharacteristic
19C2×#893—21no—
20C2×D3×C4⋊C4C2211yescharacteristic
21C2×D3×C4⋊C4C2211yescharacteristic
22C2×#890C2211yescharacteristic
23C2×C4×D12C2211yescharacteristic
24C3×#389C2211yescharacteristic
25C3×#382C2211yescharacteristic
26C2×#821C2211yescharacteristic
27C2×#821C2211yescharacteristic
28C2×(C2×D3)⋊C8C2211yescharacteristic
29C2×#818C2211yescharacteristic
30C2×Dic3⋊C8C2211yescharacteristic
31#4989—21no—
32#4988C2211yescharacteristic
33#4987C2211yescharacteristic
34Dic3×C4⋊C4C2211yescharacteristic
35#4972C2211yescharacteristic
36C2×#809C2211yescharacteristic
37C2×#809C2211yescharacteristic
38C2×#807C2211yescharacteristic
39C4×(C2×D3)⋊C4C2211yescharacteristic
40#4928C2211yescharacteristic
41C2×C4⋊(C3⋊C8)C2211yescharacteristic
42#4669—21no—
43#4668—21no—
44#4667—21no—
45#4665—21no—
46#4658—21no—
47#4657—21no—
48C3×#340C2211yescharacteristic
49C3×#340C2211yescharacteristic
50C3×#340C2211yescharacteristic
51C3×#340C2211yescharacteristic
52C3×#336C2211yescharacteristic
53#4551C2211yescharacteristic
54#4550C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#2442—31no—
61#2361—31no—
62#2101D311yescharacteristic, complemented
63#2087—31no—
64#2067—31no—
65#2025—31no—
66#1989—31no—
67#1921—31no—
68#1893—31no—
69#1874—31no—
70#1862—31no—
71#1852—31no—
72#1850—31no—
73#1804—31no—
74C2×#340—31no—
75C22×D12—21no—
76C22×D12C2311yescharacteristic
77C22×C4×D3—21no—
78C22×C4×D3—21no—
79C22×C4×D3—21no—
80C22×C4×D3C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C2×C3×C4⋊C4—21no—
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C42C2311yescharacteristic
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)⋊C4D411yescharacteristic
95C2×(C2×D3)⋊C4D411yescharacteristic
96C2×C4⋊Dic3—21no—
97C2×C4⋊Dic3C2311yescharacteristic
98C2×C4⋊Dic3C2311yescharacteristic
99C2×C4⋊Dic3C2311yescharacteristic
100C2×Dic3⋊C4—21no—
101C2×Dic3⋊C4—21no—
102C2×Dic3⋊C4—21no—
103C2×Dic3⋊C4—21no—
104C2×Dic3⋊C4—21no—
105C2×C4×Dic3—21no—
106C2×C4×Dic3C2311yescharacteristic
107C22×C3⋊C8C2311yescharacteristic
108C22×C3⋊C8C2311yescharacteristic
109C2×D3×C8—22no—
110C2×D3×C8—41no—
111C2×D3×C8—41no—
112#895—22no—
113#895Q814yes—
114#893—44no—
115D3×C4⋊C4—22no—
116D3×C4⋊C4—22no—
117D3×C4⋊C4—41no—
118D3×C4⋊C4—41no—
119#890—22no—
120#890—41no—
121C4×D12—22no—
122C4×D12—41no—
123C3×#95—21no—
124C3×#95—21no—
125C3×#95—21no—
126C3⋊#95—21no—
127C3⋊#95—21no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130C3⋊#95D411yescharacteristic
131C3⋊#95D411yescharacteristic
132#821—22no—
133#821—41no—
134(C2×D3)⋊C8—22no—
135#818—22no—
136Dic3⋊C8—41no—
137#809—41no—
138#809—41no—
139#807—41no—
140C4⋊(C3⋊C8)—41no—
141C23×C8—31no—
142C2×#125—31no—
143C2×#121—31no—
144C2×C4×D4—31no—
145C2×C4×D4—61no—
146C22×C4⋊C4—31no—
147C22×C4⋊C4—31no—
148C2×#107—31no—
149C2×#107—31no—
150C2×C4⋊C8—31no—
151C2×C4⋊C8—31no—
152C2×#102—31no—
153C2×#102—31no—
154C2×#102—31no—
155C2×#102—31no—
156C2×C22⋊C8—31no—
157#400—61no—
158#398—61no—
159#390—31no—
160#389—31no—
161#389—31no—
162#389C2×D311yescharacteristic
163#388—61no—
164#386—61no—
165#385—61no—
166#385—61no—
167#382C2×D311yescharacteristic
168#380—31no—
169#380—61no—
170C4×C4⋊C4—31no—
171C4×C22⋊C4—31no—
172#340—31no—
173#340—31no—
174#340—31no—
175#340—31no—
176#340—31no—
177#340—31no—
178#340C2×D311yescharacteristic
179#340C2×D311yescharacteristic
180#340C2×D311yescharacteristic
181#340C2×D311yescharacteristic
182#336—31no—
183#336C2×D311yescharacteristic
184C23×D3—21no—
185C23×D3C2×D411yescharacteristic
186C22×C3×C4—21no—
187C22×C3×C4—21no—
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4C2×D411yescharacteristic
191C22×C3×C4C2×D411yescharacteristic
192C22×C3×C4C2411yescharacteristic
193C22×Dic3—21no—
194C22×Dic3—21no—
195C22×Dic3—21no—
196C22×Dic3C2×D411yescharacteristic
197C22×Dic3C2×D411yescharacteristic
198C22×Dic3C2×D411yescharacteristic
199C2×D12—21no—
200C2×D12—21no—
201C2×D12—21no—
202C2×D12—21no—
203C2×D12—24no—
204C2×D12—21no—
205C2×D12—42no—
206C2×D12—42no—
207C2×D12C2×Q812yes—
208C2×C4×D3—21no—
209C2×C4×D3—21no—
210C2×C4×D3—21no—
211C2×C4×D3—21no—
212C2×C4×D3—42no—
213C2×C4×D3—42no—
214C2×C4×D3—42no—
215C2×C4×D3—42no—
216C2×C4×D3—42no—
217C2×C4×D3—42no—
218C2×C4×D3—81no—
219C2×C4×D3—81no—
220C2×C4×D3—81no—
221C2×C4×D3C2×Q812yes—
222C2×C4×D3#4112yes—
223C2×C3×C8—21no—
224C2×C3×C8—21no—
225C2×C3×C8—21no—
226C2×C3×C8—21no—
227C2×C3×C8—21no—
228C2×C3×C8—21no—
229C3×C4⋊C4—21no—
230C3×C4⋊C4—21no—
231C3×C4⋊C4—21no—
232C3×C4⋊C4—21no—
233C3×C4⋊C4—21no—
234C3×C4⋊C4—21no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—42no—
237C3×C4⋊C4C2×Q812yes—
238C3×C42—21no—
239C3×C42—21no—
240(C2×D3)⋊C4—22no—
241(C2×D3)⋊C4—22no—
242(C2×D3)⋊C4—24no—
243(C2×D3)⋊C4—42no—
244(C2×D3)⋊C4—42no—
245(C2×D3)⋊C4—42no—
246(C2×D3)⋊C4—81no—
247C4⋊Dic3—21no—
248C4⋊Dic3—21no—
249C4⋊Dic3—21no—
250C4⋊Dic3—21no—
251C4⋊Dic3—42no—
252C4⋊Dic3#4112yes—
253C4⋊Dic3#4112yes—
254Dic3⋊C4—42no—
255Dic3⋊C4—42no—
256Dic3⋊C4—42no—
257Dic3⋊C4—81no—
258Dic3⋊C4—81no—
259C4×Dic3—21no—
260C4×Dic3—21no—
261C4×Dic3—42no—
262C2×C3⋊C8—21no—
263C2×C3⋊C8—21no—
264C2×C3⋊C8—21no—
265C2×C3⋊C8—21no—
266C2×C3⋊C8—21no—
267C2×C3⋊C8—21no—
268D3×C8—44no—
269C22×D4—31no—
270C22×D4—61no—
271C23×C4—31no—
272C23×C4—61no—
273C23×C4—61no—
274C23×C4—61no—
275C22×C8—31no—
276C22×C8—31no—
277C22×C8—62no—
278C22×C8—121no—
279C22×C8—121no—
280C22×C8C22×D311yescharacteristic
281C22×C8C22×D311yescharacteristic
282#125—62no—
283#125—121no—
284#121—34no—
285#121—62no—
286C4×D4—62no—
287C4×D4—121no—
288C4×D4—124no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—31no—
291C2×C4⋊C4—31no—
292C2×C4⋊C4—31no—
293C2×C4⋊C4—62no—
294C2×C4⋊C4—62no—
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—121no—
302C2×C4⋊C4—121no—
303C2×C4⋊C4C22×D311yescharacteristic
304C2×C4⋊C4C22×D311yescharacteristic
305C2×C4⋊C4C22×D311yescharacteristic
306C2×C4⋊C4C22×D311yescharacteristic
307C2×C22⋊C4—31no—
308C2×C22⋊C4—31no—
309C2×C22⋊C4—61no—
310C2×C22⋊C4—61no—
311C2×C22⋊C4—61no—
312C2×C22⋊C4—61no—
313C2×C22⋊C4—61no—
314C2×C42—31no—
315C2×C42—61no—
316C2×C42C22×D311yescharacteristic
317#107—62no—
318#107—121no—
319C4⋊C8—121no—
320C4⋊C8—121no—
321#102—62no—
322#102—121no—
323#102—121no—
324#102—121no—
325C22⋊C8—62no—
326#95—21no—
327#95—21no—
328#95—21no—
329#95—31no—
330#95—31no—
331#95—61no—
332#95—61no—
333#95—61no—
334#95—61no—
335C23×C3C22×D411yescharacteristic
336C22×D3—21no—
337C22×D3—21no—
338C22×D3—21no—
339C22×D3—21no—
340C22×D3—22no—
341C22×D3—41no—
342C22×D3—41no—
343C22×D3—41no—
344C22×D3—41no—
345C22×D3—41no—
346C22×D3—41no—
347C22×D3C22⋊Q812yes—
348C22×D3#13512yes—
349C22×D3AGL(1,ℤ/8)12yes—
350C2×C3×C4—22no—
351C2×C3×C4—21no—
352C2×C3×C4—21no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4—21no—
356C2×C3×C4—21no—
357C2×C3×C4—41no—
358C2×C3×C4—41no—
359C2×C3×C4—41no—
360C2×C3×C4—41no—
361C2×C3×C4—41no—
362C2×C3×C4—41no—
363C2×C3×C4—41no—
364C2×C3×C4—41no—
365C2×C3×C4—41no—
366C2×C3×C4—41no—
367C2×C3×C4—41no—
368C2×C3×C4C22×Q811yescharacteristic
369C2×C3×C4C2×#4111yescharacteristic
370C2×C3×C4C2×#4111yescharacteristic
371C2×C3×C4C2×#4111yescharacteristic
372C2×C3×C4H(5,𝔽2)11yescharacteristic
373C2×C3×C4H(5,𝔽2)11yescharacteristic
374C2×Dic3—21no—
375C2×Dic3—21no—
376C2×Dic3—21no—
377C2×Dic3—21no—
378C2×Dic3—22no—
379C2×Dic3—22no—
380C2×Dic3—21no—
381C2×Dic3—21no—
382C2×Dic3—21no—
383C2×Dic3—21no—
384C2×Dic3—41no—
385C2×Dic3—41no—
386C2×Dic3—41no—
387C2×Dic3—41no—
388C2×Dic3—41no—
389C2×Dic3—41no—
390C2×Dic3—41no—
391C2×Dic3C22⋊Q812yes—
392D12—22no—
393D12—22no—
394D12—48no—
395D12—41no—
396D12—41no—
397C4×D3—22no—
398C4×D3—22no—
399C4×D3—22no—
400C4×D3—22no—
401C4×D3—84no—
402C4×D3—84no—
403C4×D3—84no—
404C3×C8—41no—
405C3×C8—41no—
406C3⋊C8—41no—
407C3⋊C8—41no—
408C24—31no—
409C24—61no—
410C2×D4—32no—
411C2×D4—64no—
412C2×D4—61no—
413C2×D4—61no—
414C2×D4—61no—
415C2×D4—61no—
416C2×D4—61no—
417C2×D4—122no—
418C2×D4—122no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—31no—
424C22×C4—32no—
425C22×C4—32no—
426C22×C4—31no—
427C22×C4—31no—
428C22×C4—61no—
429C22×C4—61no—
430C22×C4—61no—
431C22×C4—61no—
432C22×C4—61no—
433C22×C4—61no—
434C22×C4—61no—
435C22×C4—122no—
436C22×C4—122no—
437C22×C4—122no—
438C22×C4—122no—
439C22×C4—122no—
440C22×C4—122no—
441C22×C4—241no—
442C22×C4—241no—
443C22×C4—241no—
444C22×C4D3×D411yescharacteristic
445C22×C4D3×D411yescharacteristic
446C22×C4C23×D311yescharacteristic
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—61no—
454C2×C8—61no—
455C2×C8—61no—
456C2×C8—61no—
457C2×C8—61no—
458C2×C8—61no—
459C2×C8—124no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—32no—
468C4⋊C4—32no—
469C4⋊C4—42no—
470C4⋊C4—61no—
471C4⋊C4—61no—
472C4⋊C4—61no—
473C4⋊C4—61no—
474C4⋊C4—122no—
475C4⋊C4—122no—
476C4⋊C4—122no—
477C4⋊C4—122no—
478C4⋊C4—241no—
479C4⋊C4—241no—
480C4⋊C4D3×Q812yes—
481C22⋊C4—64no—
482C22⋊C4—62no—
483C22⋊C4—62no—
484C22⋊C4—122no—
485C22⋊C4—122no—
486C22⋊C4—122no—
487C22⋊C4—241no—
488C42—21no—
489C42—21no—
490C42—61no—
491C42—61no—
492C42—122no—
493C22×C3C2×C22⋊Q811yescharacteristic
494C22×C3#54611yescharacteristic
495C22×C3#54611yescharacteristic
496C22×C3C2×#13511yescharacteristic
497C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
498C22×C3#57611yescharacteristic
499C22×C3#57811yescharacteristic
500C2×D3—22no—
501C2×D3—22no—
502C2×D3—22no—
503C2×D3—22no—
504C2×D3—22no—
505C2×D3—22no—
506C2×D3—42no—
507C2×D3—42no—
508C2×D3—42no—
509C2×D3—81no—
510C2×D3—81no—
511C2×D3—81no—
512C2×D3—81no—
513C2×D3#47814yes—
514C3×C4—42no—
515C3×C4—41no—
516C3×C4—41no—
517C3×C4—81no—
518C3×C4—81no—
519C3×C4—81no—
520C3×C4#52911yescharacteristic
521C3×C4#53811yescharacteristic
522C3×C4#55411yescharacteristic
523C3×C4#55411yescharacteristic
524Dic3—22no—
525Dic3—42no—
526Dic3—41no—
527Dic3—41no—
528Dic3—42no—
529Dic3—81no—
530C23—32no—
531C23—32no—
532C23—32no—
533C23—61no—
534C23—61no—
535C23—61no—
536C23—61no—
537C23—62no—
538C23—121no—
539C23—121no—
540C23—121no—
541C23—121no—
542C23—121no—
543C23—121no—
544C23C2×D3×D411yescharacteristic
545D4—62no—
546D4—62no—
547D4—128no—
548D4—121no—
549D4—121no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—22no—
557C2×C4—32no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—61no—
570C2×C4—61no—
571C2×C4—62no—
572C2×C4—62no—
573C2×C4—62no—
574C2×C4—62no—
575C2×C4—62no—
576C2×C4—61no—
577C2×C4—61no—
578C2×C4—62no—
579C2×C4—61no—
580C2×C4—61no—
581C2×C4—61no—
582C2×C4—61no—
583C2×C4—121no—
584C2×C4—121no—
585C2×C4—121no—
586C2×C4—121no—
587C2×C4—121no—
588C2×C4—121no—
589C2×C4—121no—
590C2×C4—244no—
591C2×C4—244no—
592C2×C4—244no—
593C2×C4C2×D3×Q811yescharacteristic
594C2×C4D3×#4111yescharacteristic
595C2×C4D3×#4111yescharacteristic
596C2×C4D3×#4111yescharacteristic
597C2×C4#100911yescharacteristic
598C2×C4#100911yescharacteristic
599C8—41no—
600C8—41no—
601C8—121no—
602C8—121no—
603C2×C3#245011yescharacteristic
604C2×C3C2×#47811yescharacteristic
605C2×C3#308311yescharacteristic
606C2×C3#329711yescharacteristic
607C2×C3#330011yescharacteristic
608C2×C3#330811yescharacteristic
609C2×C3#331811yescharacteristic
610D3—24no—
611D3—82no—
612C22—34no—
613C22—62no—
614C22—62no—
615C22—62no—
616C22—62no—
617C22—62no—
618C22—62no—
619C22—122no—
620C22—122no—
621C22—122no—
622C22—241no—
623C22—241no—
624C22—241no—
625C22—241no—
626C22D3×C22⋊Q811yescharacteristic
627C22#565411yescharacteristic
628C22#567611yescharacteristic
629C22D3×#13511yescharacteristic
630C22#576911yescharacteristic
631C22D3×AGL(1,ℤ/8)11yescharacteristic
632C22#577911yescharacteristic
633C4—42no—
634C4—41no—
635C4—41no—
636C4—62no—
637C4—81no—
638C4—81no—
639C4—81no—
640C4—121no—
641C4—122no—
642C4—121no—
643C4—122no—
644C4—241no—
645C4#569011yescharacteristic
646C4#569011yescharacteristic
647C4#570411yescharacteristic
648C4#570611yescharacteristic
649C3#2642011yescharacteristic, complemented, Sylow
650C2—64no—
651C2—242no—
652C2#7834411yescharacteristic
653C2D3×#47811yescharacteristic
654C2#8123411yescharacteristic
655C2#8138511yescharacteristic
656C2#8152411yescharacteristic
657C2#8165311yescharacteristic
658C2#8169711yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2101C2
socleC23×C3C22×D4
prime radicalC2×#895C22
hypercenter#2101D3
hypocenterC3#26420
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2101

Complements

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

NH ≅ G/Ntype
2. #74163C2semidirect
3. C3×#2101C2semidirect
4. #69623C2semidirect
5. #69556C2semidirect
6. #68901C2semidirect
7. #68899C2semidirect
10. #68874C2semidirect
11. D3×#340C2semidirect
12. #68820C2semidirect
13. #68801C2semidirect
14. #68670C2semidirect
15. #68666C2semidirect
62. #2101D3semidirect
649. C3#26420semidirect

Maximal subgroups

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

subgroupindex
#762062
#741632
C3×#21012
#696232
#695562
#689012
#688992
#688832
#688802
#688742
D3×#3402
#688202
#688012
#686702
#686662
#264203