Finite Group Databasebeta

← #10858766

Subgroups — #10858766

Order 768.

902 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#73493C211yesmaximal, characteristic, complemented
2#73467C211yesmaximal, characteristic
3C3×#1850C211yesmaximal, characteristic, complemented
4#71973C211yesmaximal, characteristic
5C2×#4551C211yesmaximal, characteristic, complemented
6#70483C211yesmaximal, characteristic, complemented
7#70479C211yesmaximal, characteristic, complemented
8#70464C211yesmaximal, characteristic, complemented
9#70459C211yesmaximal, characteristic, complemented
10#69932C211yesmaximal, characteristic
11#69928C211yesmaximal, characteristic, complemented
12#68678C211yesmaximal, characteristic, complemented
13#68671C211yesmaximal, characteristic, complemented
14#68643C211yesmaximal, characteristic, complemented
15#68623C211yesmaximal, characteristic, complemented
16#26410—31nomaximal, Sylow
17C23×C3×C8C2211yescharacteristic
18C2×#930C2211yescharacteristic
19C22×C4⋊Dic3C2211yescharacteristic
20C22×C4⋊Dic3C2211yescharacteristic
21C2×#67⋊C4—21no—
22C2×#878—21no—
23C2×C4×D12C2211yescharacteristic
24C2×#869C2211yescharacteristic
25C2×C3×C4⋊C8C2211yescharacteristic
26C2×C3×C4⋊C8C2211yescharacteristic
27C2×C3×C22⋊C8C2211yescharacteristic
28C3×C4×C22⋊C4C2211yescharacteristic
29#5212C2211yescharacteristic
30C2×#821C2211yescharacteristic
31C2×#821C2211yescharacteristic
32C2×#821C2211yescharacteristic
33C2×#821C2211yescharacteristic
34C2×#817C2211yescharacteristic
35C2×#817C2211yescharacteristic
36#4958—21no—
37#4950—21no—
38#4939C2211yescharacteristic
39#4935C2211yescharacteristic
40C4×C4⋊Dic3C2211yescharacteristic
41#4928C2211yescharacteristic
42#4928C2211yescharacteristic
43#4928C2211yescharacteristic
44#4670—21no—
45#4668—21no—
46#4661—21no—
47#4656—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#2361—31no—
62#2100—31no—
63#2088—31no—
64#2065—31no—
65#2023—31no—
66#1987—31no—
67#1920—31no—
68#1893—31no—
69#1874—31no—
70#1864—31no—
71#1853—31no—
72#1850D311yescharacteristic, complemented
73#1802—31no—
74C2×#340—31no—
75C23×C3×C4C2311yescharacteristic
76C22×#67—21no—
77C23×Dic3—21no—
78C23×Dic3—21no—
79C22×D12C2311yescharacteristic
80C22×C4×D3—21no—
81C22×Dic6—21no—
82C22×C3×C8—22no—
83C22×C3×C8—41no—
84C22×C3×C8—41no—
85C22×C3×C8C2311yescharacteristic
86C22×C3×C8C2311yescharacteristic
87C22×C3×C8C2311yescharacteristic
88C22×C3×C8C2311yescharacteristic
89C2×C3×C22⋊C4D411yescharacteristic
90C2×C3×C22⋊C4D411yescharacteristic
91C2×C3×C42C2311yescharacteristic
92C2×C3×C42C2311yescharacteristic
93C2×#217—21no—
94#930—22no—
95#930Q814yescomplemented
96C2×(C2×D3)⋊C4—21no—
97C2×(C2×D3)⋊C4—21no—
98C2×(C2×D3)⋊C4—21no—
99C2×(C2×D3)⋊C4—21no—
100C2×C4⋊Dic3—22no—
101C2×C4⋊Dic3—22no—
102C2×C4⋊Dic3—21no—
103C2×C4⋊Dic3—21no—
104C2×C4⋊Dic3—21no—
105C2×C4⋊Dic3—21no—
106C2×C4⋊Dic3—21no—
107C2×C4⋊Dic3—41no—
108C2×C4⋊Dic3—41no—
109C2×C4⋊Dic3C2311yescharacteristic
110C2×C4⋊Dic3C2311yescharacteristic
111C2×C4⋊Dic3C2311yescharacteristic
112C2×C4⋊Dic3C2311yescharacteristic
113C2×C4⋊Dic3C2311yescharacteristic
114C2×C4⋊Dic3C2311yescharacteristic
115C2×C4⋊Dic3C2311yescharacteristic
116C2×Dic3⋊C4—21no—
117C2×Dic3⋊C4—21no—
118C2×Dic3⋊C4—21no—
119C2×C4×Dic3—21no—
120#67⋊C4—44no—
121#878—44no—
122C4×D12—22no—
123C4×D12—41no—
124#869—22no—
125#869—41no—
126C3×C4⋊C8—41no—
127C3×C4⋊C8—41no—
128C3×C22⋊C8—22no—
129C3×#95D411yescharacteristic
130C3×#95D411yescharacteristic
131C3⋊#95—21no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136#821—22no—
137#821—41no—
138#821—41no—
139#821—41no—
140#817—22no—
141#817—41no—
142C23×C8C2×D311yescharacteristic
143C2×C4⋊Q8—31no—
144C2×C22⋊Q8—61no—
145C2×#121—31no—
146C2×C4×D4—31no—
147C2×C4×D4—61no—
148C22×C4⋊C4—31no—
149C22×C4⋊C4—31no—
150C2×#106—31no—
151C2×#106—31no—
152C2×C4⋊C8C2×D311yescharacteristic
153C2×C4⋊C8C2×D311yescharacteristic
154C2×#102—31no—
155C2×#102—31no—
156C2×#102—31no—
157C2×#102—31no—
158C2×C22⋊C8C2×D311yescharacteristic
159#398—61no—
160#395—61no—
161#390—31no—
162#389—31no—
163#389—31no—
164#389—31no—
165#388—61no—
166#386—61no—
167#385—61no—
168#385—61no—
169#384—31no—
170#380—31no—
171C4×C4⋊C4—31no—
172C4×C22⋊C4C2×D311yescharacteristic
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#340—31no—
183#336C2×D311yescharacteristic
184#336C2×D311yescharacteristic
185C24×C3C2×D411yescharacteristic
186C23×D3—21no—
187C22×C3×C4—21no—
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4—21no—
191C22×C3×C4C2×D411yescharacteristic
192C22×C3×C4C2×D411yescharacteristic
193C22×C3×C4C2×D411yescharacteristic
194C22×C3×C4C2×D411yescharacteristic
195C22×C3×C4C2×D411yescharacteristic
196C22×C3×C4C2×Q812yes—
197C22×C3×C4#4112yes—
198C22×C3×C4C2411yescharacteristic
199C2×#67—24no—
200C2×#67—42no—
201C2×#67—42no—
202C22×Dic3—21no—
203C22×Dic3—21no—
204C22×Dic3—21no—
205C22×Dic3—21no—
206C22×Dic3—21no—
207C22×Dic3—21no—
208C22×Dic3—21no—
209C22×Dic3—42no—
210C22×Dic3—42no—
211C22×Dic3—42no—
212C22×Dic3—42no—
213C22×Dic3—81no—
214C22×Dic3—81no—
215C2×D12—21no—
216C2×D12—21no—
217C2×D12—21no—
218C2×D12—21no—
219C2×D12—21no—
220C2×D12C2×Q812yes—
221C2×C4×D3—42no—
222C2×C4×D3—42no—
223C2×C4×D3—81no—
224C2×Dic6—42no—
225C2×Dic6—42no—
226C2×Dic6—81no—
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—21no—
234C2×C3×C8—21no—
235C2×C3×C8—21no—
236C2×C3×C8—21no—
237C2×C3×C8—21no—
238C2×C3×C8—21no—
239C2×C3×C8—44no—
240C3×C22⋊C4—22no—
241C3×C22⋊C4SD1614yes—
242C3×C42—21no—
243C3×C42—21no—
244C3×C42—21no—
245C3×C42—21no—
246#217—42no—
247(C2×D3)⋊C4—24no—
248(C2×D3)⋊C4—42no—
249(C2×D3)⋊C4—42no—
250(C2×D3)⋊C4—81no—
251C4⋊Dic3—21no—
252C4⋊Dic3—21no—
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—42no—
264C4⋊Dic3—42no—
265C4⋊Dic3—42no—
266C4⋊Dic3—81no—
267C4⋊Dic3C2×Q812yes—
268C4⋊Dic3#4112yes—
269C4⋊Dic3#4112yes—
270Dic3⋊C4—42no—
271Dic3⋊C4—42no—
272Dic3⋊C4—42no—
273C4×Dic3—42no—
274C22×Q8—61no—
275C22×D4—31no—
276C22×D4—61no—
277C23×C4—61no—
278C23×C4—61no—
279C23×C4—61no—
280C23×C4C22×D311yescharacteristic
281C22×C8—22no—
282C22×C8—41no—
283C22×C8—41no—
284C22×C8C22×D311yescharacteristic
285C22×C8C22×D311yescharacteristic
286C22×C8C22×D311yescharacteristic
287C22×C8C22×D311yescharacteristic
288C4⋊Q8—62no—
289C4⋊Q8—121no—
290C22⋊Q8—124no—
291#121—34no—
292#121—62no—
293C4×D4—62no—
294C4×D4—121no—
295C4×D4—124no—
296C2×C4⋊C4—31no—
297C2×C4⋊C4—31no—
298C2×C4⋊C4—31no—
299C2×C4⋊C4—31no—
300C2×C4⋊C4—31no—
301C2×C4⋊C4—31no—
302C2×C4⋊C4—31no—
303C2×C4⋊C4—62no—
304C2×C4⋊C4—62no—
305C2×C4⋊C4—61no—
306C2×C4⋊C4—61no—
307C2×C4⋊C4—61no—
308C2×C4⋊C4—61no—
309C2×C4⋊C4—61no—
310C2×C4⋊C4—61no—
311C2×C4⋊C4—61no—
312C2×C4⋊C4—61no—
313C2×C4⋊C4—121no—
314C2×C4⋊C4—121no—
315C2×C22⋊C4—61no—
316C2×C22⋊C4—61no—
317C2×C22⋊C4—61no—
318C2×C22⋊C4—61no—
319C2×C22⋊C4—61no—
320C2×C22⋊C4D1211yescharacteristic
321C2×C22⋊C4D1211yescharacteristic
322C2×C42—61no—
323C2×C42C22×D311yescharacteristic
324C2×C42C22×D311yescharacteristic
325#106—62no—
326#106—121no—
327C4⋊C8—41no—
328C4⋊C8—41no—
329#102—62no—
330#102—121no—
331#102—121no—
332#102—121no—
333C22⋊C8—22no—
334#95—61no—
335#95—61no—
336#95—61no—
337#95—61no—
338#95—61no—
339#95D1211yescharacteristic
340#95D1211yescharacteristic
341C23×C3—21no—
342C23×C3—21no—
343C23×C3—21no—
344C23×C3—21no—
345C23×C3C22⋊Q812yes—
346C23×C3C2×SD1612yes—
347C23×C3AGL(1,ℤ/8)12yes—
348C23×C3C22×D411yescharacteristic
349C22×D3—22no—
350C22×D3—41no—
351C22×D3—41no—
352C22×D3—41no—
353C22×D3—41no—
354C22×D3—41no—
355C22×D3—41no—
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—22no—
363C2×C3×C4—22no—
364C2×C3×C4—22no—
365C2×C3×C4—22no—
366C2×C3×C4—21no—
367C2×C3×C4—21no—
368C2×C3×C4—21no—
369C2×C3×C4—21no—
370C2×C3×C4—21no—
371C2×C3×C4—21no—
372C2×C3×C4C22⋊Q812yes—
373C2×C3×C4C2×SD1612yes—
374C2×C3×C4C2×SD1612yes—
375C2×C3×C4C22×Q811yescharacteristic
376C2×C3×C4C2×#4111yescharacteristic
377C2×C3×C4C2×#4111yescharacteristic
378C2×C3×C4C2×#4111yescharacteristic
379C2×C3×C4H(5,𝔽2)11yescharacteristic
380C2×C3×C4H(5,𝔽2)11yescharacteristic
381#67—48no—
382C2×Dic3—22no—
383C2×Dic3—22no—
384C2×Dic3—22no—
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—
397C2×Dic3—41no—
398C2×Dic3—41no—
399C2×Dic3—41no—
400C2×Dic3—41no—
401C2×Dic3—41no—
402C2×Dic3—41no—
403C2×Dic3—84no—
404C2×Dic3—84no—
405D12—22no—
406D12—22no—
407D12—41no—
408D12—41no—
409C4×D3—84no—
410Dic6—84no—
411C3×C8—41no—
412C3×C8—41no—
413C3×C8—41no—
414C3×C8—41no—
415C24—61no—
416C24C2×D1211yescharacteristic
417C2×Q8—122no—
418C2×Q8—122no—
419C2×Q8—241no—
420C2×D4—32no—
421C2×D4—64no—
422C2×D4—61no—
423C2×D4—61no—
424C2×D4—61no—
425C2×D4—61no—
426C2×D4—61no—
427C2×D4—122no—
428C2×D4—122no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—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—122no—
443C22×C4—122no—
444C22×C4—122no—
445C22×C4—122no—
446C22×C4—241no—
447C22×C4—241no—
448C22×C4—241no—
449C22×C4C2×D1211yescharacteristic
450C22×C4C2×D1211yescharacteristic
451C22×C4C2×D1211yescharacteristic
452C22×C4C2×D1211yescharacteristic
453C22×C4C2×D1211yescharacteristic
454C22×C4#23712yes—
455C22×C4D3×Q812yes—
456C22×C4C23×D311yescharacteristic
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—44no—
470C4⋊C4—32no—
471C4⋊C4—32no—
472C4⋊C4—32no—
473C4⋊C4—61no—
474C4⋊C4—61no—
475C4⋊C4—61no—
476C4⋊C4—61no—
477C4⋊C4—61no—
478C4⋊C4—61no—
479C4⋊C4—61no—
480C4⋊C4—61no—
481C4⋊C4—61no—
482C4⋊C4—61no—
483C4⋊C4—61no—
484C4⋊C4—122no—
485C4⋊C4—122no—
486C4⋊C4—122no—
487C4⋊C4—122no—
488C4⋊C4—122no—
489C4⋊C4—122no—
490C4⋊C4—122no—
491C4⋊C4—241no—
492C22⋊C4—22no—
493C22⋊C4—64no—
494C22⋊C4—122no—
495C22⋊C4—122no—
496C22⋊C4—122no—
497C22⋊C4—241no—
498C22⋊C4#20414yes—
499C42—21no—
500C42—21no—
501C42—21no—
502C42—21no—
503C42—122no—
504C22×C3—22no—
505C22×C3—22no—
506C22×C3—22no—
507C22×C3—22no—
508C22×C3—22no—
509C22×C3—22no—
510C22×C3#47614yes—
511C22×C3C2×C22⋊Q811yescharacteristic
512C22×C3#54611yescharacteristic
513C22×C3#54611yescharacteristic
514C22×C3C22×SD1611yescharacteristic
515C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
516C22×C3#57611yescharacteristic
517C22×C3#57811yescharacteristic
518C2×D3—42no—
519C2×D3—42no—
520C2×D3—42no—
521C2×D3—81no—
522C2×D3—81no—
523C2×D3—81no—
524C2×D3—81no—
525C3×C4—22no—
526C3×C4—22no—
527C3×C4—22no—
528C3×C4—41no—
529C3×C4—41no—
530C3×C4#52911yescharacteristic
531C3×C4#53811yescharacteristic
532C3×C4#55411yescharacteristic
533C3×C4#55411yescharacteristic
534Dic3—42no—
535Dic3—42no—
536Dic3—42no—
537Dic3—81no—
538Dic3—81no—
539Dic3—81no—
540Dic3—81no—
541C23—21no—
542C23—21no—
543C23—21no—
544C23—21no—
545C23—62no—
546C23—121no—
547C23—121no—
548C23—121no—
549C23—121no—
550C23—121no—
551C23—121no—
552C23#89712yes—
553C23C2×#20412yes—
554C23#90812yes—
555C23C22×D1211yescharacteristic
556Q8—244no—
557D4—62no—
558D4—62no—
559D4—128no—
560D4—121no—
561D4—121no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—62no—
579C2×C4—62no—
580C2×C4—62no—
581C2×C4—121no—
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—121no—
599C2×C4—244no—
600C2×C4—244no—
601C2×C4—244no—
602C2×C4#89712yes—
603C2×C4C2×#20412yes—
604C2×C4C2×#20412yes—
605C2×C4C2×#23711yescharacteristic
606C2×C4#100411yescharacteristic
607C2×C4#100411yescharacteristic
608C2×C4C2×D3×Q811yescharacteristic
609C2×C4D3×#4111yescharacteristic
610C2×C4D3×#4111yescharacteristic
611C8—41no—
612C8—41no—
613C8—41no—
614C8—41no—
615C2×C3—24no—
616C2×C3#245011yescharacteristic
617C2×C3C2×#47611yescharacteristic
618C2×C3#308311yescharacteristic
619C2×C3#329511yescharacteristic
620C2×C3#329811yescharacteristic
621C2×C3#330811yescharacteristic
622C2×C3#331811yescharacteristic
623D3—82no—
624C22—22no—
625C22—22no—
626C22—22no—
627C22—22no—
628C22—22no—
629C22—22no—
630C22—122no—
631C22—122no—
632C22—122no—
633C22—241no—
634C22—241no—
635C22—241no—
636C22—241no—
637C22#484214yes—
638C22C2×#89711yescharacteristic
639C22#555411yescharacteristic
640C22#555411yescharacteristic
641C22C22×#20411yescharacteristic
642C22C2×#90811yescharacteristic
643C22#575211yescharacteristic
644C22#575311yescharacteristic
645C4—22no—
646C4—22no—
647C4—22no—
648C4—41no—
649C4—41no—
650C4—122no—
651C4—122no—
652C4—122no—
653C4—241no—
654C4—241no—
655C4—241no—
656C4—241no—
657C4#559311yescharacteristic
658C4#563611yescharacteristic
659C4#563611yescharacteristic
660C4#564911yescharacteristic
661C3#2641011yescharacteristic, complemented, Sylow
662C2—24no—
663C2—242no—
664C2#7442411yescharacteristic
665C2C2×#484211yescharacteristic
666C2#7909111yescharacteristic
667C2#7943911yescharacteristic
668C2#7944711yescharacteristic
669C2#7950511yescharacteristic
670C2#7950911yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1850C2
socleC23×C3C22×D4
prime radicalC2×#930C22
hypercenter#1850D3
hypocenterC3#26410
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #1850

Complements

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

NH ≅ G/Ntype
1. #73493C2semidirect
3. C3×#1850C2semidirect
5. C2×#4551C2semidirect
6. #70483C2semidirect
7. #70479C2semidirect
8. #70464C2semidirect
9. #70459C2semidirect
11. #69928C2semidirect
12. #68678C2semidirect
13. #68671C2semidirect
14. #68643C2semidirect
15. #68623C2semidirect
72. #1850D3semidirect
95. #930Q8semidirect
661. C3#26410semidirect

Maximal subgroups

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

subgroupindex
#734932
#734672
C3×#18502
#719732
C2×#45512
#704832
#704792
#704642
#704592
#699322
#699282
#686782
#686712
#686432
#686232
#264103