Finite Group Databasebeta

← #10858740

Subgroups — #10858740

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#82163C211yesmaximal, characteristic
2#73899C211yesmaximal, characteristic, complemented
3C3×#1852C211yesmaximal, characteristic, complemented
4C2×#4529C211yesmaximal, characteristic, complemented
5#70639C211yesmaximal, characteristic, complemented
6#70638C211yesmaximal, characteristic, complemented
7#70621C211yesmaximal, characteristic, complemented
8#70609C211yesmaximal, characteristic, complemented
9#69950C211yesmaximal, characteristic, complemented
10#69932C211yesmaximal, characteristic
11#69515C211yesmaximal, characteristic
12#68898C211yesmaximal, characteristic, complemented
13#68892C211yesmaximal, characteristic, complemented
14#68810C211yesmaximal, characteristic, complemented
15#68800C211yesmaximal, characteristic, complemented
16#26408—31nomaximal, Sylow
17C22×C3×C4⋊C4C2211yescharacteristic
18C2×#930C2211yescharacteristic
19C2×C4×#67—21no—
20C22×C4⋊Dic3C2211yescharacteristic
21C23×C3⋊C8C2211yescharacteristic
22C2×#893C2211yescharacteristic
23C2×#888C2211yescharacteristic
24C2×#878—21no—
25C2×C3×#107C2211yescharacteristic
26C2×C3×#107C2211yescharacteristic
27C2×C3×C22⋊C8C2211yescharacteristic
28C3×#380C2211yescharacteristic
29#5213—21no—
30C4×#217C2211yescharacteristic
31C2×#821C2211yescharacteristic
32C2×#821C2211yescharacteristic
33C2×Dic3⋊C8C2211yescharacteristic
34C2×Dic3⋊C8C2211yescharacteristic
35#4990—21no—
36#4989—21no—
37#4988C2211yescharacteristic
38#4984C2211yescharacteristic
39#4975C2211yescharacteristic
40Dic3×C4⋊C4C2211yescharacteristic
41#4972C2211yescharacteristic
42#4972C2211yescharacteristic
43C2×#809C2211yescharacteristic
44C2×#809C2211yescharacteristic
45#4946—21no—
46#4657—21no—
47#4656—21no—
48C3×#340C2211yescharacteristic
49C3×#340C2211yescharacteristic
50#4551C2211yescharacteristic
51#4551C2211yescharacteristic
52#4550C2211yescharacteristic
53#4550C2211yescharacteristic
54#4529C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#2440—31no—
61#2361—31no—
62#2099—31no—
63#2086—31no—
64#2067—31no—
65#2023—31no—
66#1988—31no—
67#1918—31no—
68#1893—31no—
69#1874—31no—
70#1862—31no—
71#1852D311yescharacteristic, complemented
72#1850—31no—
73#1802—31no—
74C2×#340—31no—
75C23×C3×C4—21no—
76C23×C3×C4C2311yescharacteristic
77C22×#67—21no—
78C23×Dic3—21no—
79C22×D12C2311yescharacteristic
80C22×C4×D3—21no—
81C22×Dic6—21no—
82C22×C3×C8C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C2×C3×C4⋊C4—22no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4—41no—
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C22⋊C4—21no—
92C2×#217D411yescharacteristic
93C2×#217D411yescharacteristic
94#930—22no—
95#930Q814yescomplemented
96C4×#67—44no—
97C2×(C2×D3)⋊C4—21no—
98C2×(C2×D3)⋊C4—21no—
99C2×(C2×D3)⋊C4—21no—
100C2×(C2×D3)⋊C4—21no—
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×Dic3⋊C4—21no—
109C2×Dic3⋊C4—21no—
110C2×Dic3⋊C4—21no—
111C2×Dic3⋊C4—21no—
112C2×Dic3⋊C4—21no—
113C2×C4×Dic3—21no—
114C2×C4×Dic3C2311yescharacteristic
115C2×C4×Dic3C2311yescharacteristic
116C22×C3⋊C8—22no—
117C22×C3⋊C8—41no—
118C22×C3⋊C8—41no—
119C22×C3⋊C8C2311yescharacteristic
120C22×C3⋊C8C2311yescharacteristic
121#893—22no—
122#893—41no—
123#888—22no—
124#888—41no—
125#878—44no—
126C3×#107—22no—
127C3×#107—41no—
128C3×C22⋊C8—22no—
129C3×#95—21no—
130C3⋊#95—21no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95D411yescharacteristic
135C3⋊#95D411yescharacteristic
136#821—22no—
137#821—41no—
138Dic3⋊C8—41no—
139Dic3⋊C8—41no—
140#809—41no—
141#809—41no—
142C23×C8—31no—
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⋊C4C2×D311yescharacteristic
150C2×#107C2×D311yescharacteristic
151C2×#107C2×D311yescharacteristic
152C2×C4⋊C8—31no—
153C2×C4⋊C8—31no—
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#380C2×D311yescharacteristic
171C4×C4⋊C4—31no—
172C4×C22⋊C4—31no—
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#340C2×D311yescharacteristic
182#340C2×D311yescharacteristic
183#336—31no—
184#336—31no—
185C24×C3C2×D411yescharacteristic
186C23×D3—21no—
187C22×C3×C4—21no—
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4—21no—
191C22×C3×C4—21no—
192C22×C3×C4—21no—
193C22×C3×C4—21no—
194C22×C3×C4—21no—
195C22×C3×C4—42no—
196C22×C3×C4—42no—
197C22×C3×C4—81no—
198C22×C3×C4C2×D411yescharacteristic
199C22×C3×C4C2×Q812yes—
200C22×C3×C4#4112yes—
201C22×C3×C4C2411yescharacteristic
202C2×#67—24no—
203C2×#67—42no—
204C2×#67—42no—
205C22×Dic3—21no—
206C22×Dic3—21no—
207C22×Dic3—21no—
208C22×Dic3—42no—
209C22×Dic3—42no—
210C22×Dic3—81no—
211C22×Dic3C2×D411yescharacteristic
212C22×Dic3C2×D411yescharacteristic
213C22×Dic3C2×D411yescharacteristic
214C22×Dic3C2×D411yescharacteristic
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—
233C3×C4⋊C4—21no—
234C3×C4⋊C4—21no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—42no—
241C3×C4⋊C4#4112yes—
242C3×C22⋊C4—42no—
243#217—22no—
244#217D814yes—
245(C2×D3)⋊C4—24no—
246(C2×D3)⋊C4—42no—
247(C2×D3)⋊C4—42no—
248(C2×D3)⋊C4—81no—
249C4⋊Dic3—21no—
250C4⋊Dic3—21no—
251C4⋊Dic3—21no—
252C4⋊Dic3—21no—
253C4⋊Dic3—42no—
254C4⋊Dic3—42no—
255C4⋊Dic3C2×Q812yes—
256C4⋊Dic3#4112yes—
257Dic3⋊C4—42no—
258Dic3⋊C4—42no—
259Dic3⋊C4—42no—
260Dic3⋊C4—42no—
261Dic3⋊C4—81no—
262C4×Dic3—21no—
263C4×Dic3—21no—
264C4×Dic3—21no—
265C4×Dic3—21no—
266C4×Dic3—42no—
267C2×C3⋊C8—21no—
268C2×C3⋊C8—21no—
269C2×C3⋊C8—21no—
270C2×C3⋊C8—21no—
271C2×C3⋊C8—21no—
272C2×C3⋊C8—21no—
273C2×C3⋊C8—44no—
274C22×Q8—61no—
275C22×D4—31no—
276C22×D4—61no—
277C23×C4—21no—
278C23×C4—61no—
279C23×C4—61no—
280C23×C4C22×D311yescharacteristic
281C22×C8—31no—
282C22×C8—31no—
283C22×C8—62no—
284C22×C8—121no—
285C22×C8—121no—
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—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—31no—
299C2×C4⋊C4—31no—
300C2×C4⋊C4—31no—
301C2×C4⋊C4—41no—
302C2×C4⋊C4—62no—
303C2×C4⋊C4—61no—
304C2×C4⋊C4—61no—
305C2×C4⋊C4—61no—
306C2×C4⋊C4—61no—
307C2×C4⋊C4—61no—
308C2×C4⋊C4—61no—
309C2×C4⋊C4—61no—
310C2×C4⋊C4—121no—
311C2×C4⋊C4C22×D311yescharacteristic
312C2×C4⋊C4C22×D311yescharacteristic
313C2×C4⋊C4C22×D311yescharacteristic
314C2×C4⋊C4C22×D311yescharacteristic
315C2×C22⋊C4—21no—
316C2×C22⋊C4—31no—
317C2×C22⋊C4—31no—
318C2×C22⋊C4—61no—
319C2×C22⋊C4—61no—
320C2×C22⋊C4—61no—
321C2×C22⋊C4—61no—
322C2×C42—31no—
323C2×C42—31no—
324C2×C42—61no—
325#107—22no—
326#107—41no—
327C4⋊C8—121no—
328C4⋊C8—121no—
329#102—62no—
330#102—121no—
331#102—121no—
332#102—121no—
333C22⋊C8—22no—
334#95—21no—
335#95—31no—
336#95—31no—
337#95—61no—
338#95—61no—
339#95—61no—
340#95—61no—
341C23×C3—21no—
342C23×C3—21no—
343C23×C3—21no—
344C23×C3—21no—
345C23×C3C22⋊Q812yes—
346C23×C3C2×D812yes—
347C23×C3#13712yes—
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—22no—
359C2×C3×C4—22no—
360C2×C3×C4—22no—
361C2×C3×C4—22no—
362C2×C3×C4—22no—
363C2×C3×C4—41no—
364C2×C3×C4—41no—
365C2×C3×C4—41no—
366C2×C3×C4—41no—
367C2×C3×C4—41no—
368C2×C3×C4—41no—
369C2×C3×C4—41no—
370C2×C3×C4—41no—
371C2×C3×C4—41no—
372C2×C3×C4—41no—
373C2×C3×C4—41no—
374C2×C3×C4—84no—
375C2×C3×C4C22⋊Q812yes—
376C2×C3×C4C22×Q811yescharacteristic
377C2×C3×C4C2×#4111yescharacteristic
378C2×C3×C4C2×#4111yescharacteristic
379C2×C3×C4C2×#4111yescharacteristic
380C2×C3×C4H(5,𝔽2)11yescharacteristic
381C2×C3×C4H(5,𝔽2)11yescharacteristic
382#67—48no—
383C2×Dic3—21no—
384C2×Dic3—21no—
385C2×Dic3—21no—
386C2×Dic3—21no—
387C2×Dic3—22no—
388C2×Dic3—21no—
389C2×Dic3—21no—
390C2×Dic3—22no—
391C2×Dic3—21no—
392C2×Dic3—21no—
393C2×Dic3—21no—
394C2×Dic3—21no—
395C2×Dic3—41no—
396C2×Dic3—41no—
397C2×Dic3—41no—
398C2×Dic3—41no—
399C2×Dic3—41no—
400C2×Dic3—41no—
401C2×Dic3—41no—
402C2×Dic3—84no—
403C2×Dic3C2×D812yes—
404C2×Dic3C2×D812yes—
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—
416C24D3×D411yescharacteristic
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—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—31no—
438C22×C4—31no—
439C22×C4—31no—
440C22×C4—31no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—61no—
444C22×C4—61no—
445C22×C4—61no—
446C22×C4—81no—
447C22×C4—122no—
448C22×C4—122no—
449C22×C4—122no—
450C22×C4—122no—
451C22×C4—241no—
452C22×C4—241no—
453C22×C4#23512yes—
454C22×C4D3×D411yescharacteristic
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—61no—
464C2×C8—61no—
465C2×C8—61no—
466C2×C8—61no—
467C2×C8—61no—
468C2×C8—61no—
469C2×C8—124no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—32no—
478C4⋊C4—32no—
479C4⋊C4—42no—
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—241no—
491C4⋊C4#23712yes—
492C22⋊C4—34no—
493C22⋊C4—42no—
494C22⋊C4—62no—
495C22⋊C4—64no—
496C22⋊C4—122no—
497C22⋊C4—122no—
498C22⋊C4—241no—
499C42—61no—
500C42—61no—
501C42—61no—
502C42—61no—
503C42—122no—
504C22×C3—22no—
505C22×C3—22no—
506C22×C3—22no—
507C22×C3—22no—
508C22×C3—22no—
509C22×C3—22no—
510C22×C3#47414yes—
511C22×C3C2×C22⋊Q811yescharacteristic
512C22×C3#54611yescharacteristic
513C22×C3#54611yescharacteristic
514C22×C3C22×D811yescharacteristic
515C22×C3C2×#13711yescharacteristic
516C22×C3#57711yescharacteristic
517C22×C3#57711yescharacteristic
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—42no—
527C3×C4—81no—
528C3×C4—81no—
529C3×C4—81no—
530C3×C4#52911yescharacteristic
531C3×C4#53811yescharacteristic
532C3×C4#55411yescharacteristic
533C3×C4#55411yescharacteristic
534Dic3—22no—
535Dic3—22no—
536Dic3—42no—
537Dic3—41no—
538Dic3—41no—
539Dic3—42no—
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#89612yes—
553C23D3×D812yes—
554C23#91812yes—
555C23C2×D3×D411yescharacteristic
556Q8—244no—
557D4—62no—
558D4—62no—
559D4—128no—
560D4—121no—
561D4—121no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—32no—
570C2×C4—32no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—61no—
583C2×C4—61no—
584C2×C4—61no—
585C2×C4—61no—
586C2×C4—61no—
587C2×C4—62no—
588C2×C4—62no—
589C2×C4—61no—
590C2×C4—61no—
591C2×C4—61no—
592C2×C4—61no—
593C2×C4—61no—
594C2×C4—84no—
595C2×C4—121no—
596C2×C4—121no—
597C2×C4—121no—
598C2×C4—121no—
599C2×C4—121no—
600C2×C4—121no—
601C2×C4—121no—
602C2×C4—244no—
603C2×C4—244no—
604C2×C4#89612yes—
605C2×C4C2×#23511yescharacteristic
606C2×C4C2×#23711yescharacteristic
607C2×C4#100411yescharacteristic
608C2×C4C2×D3×Q811yescharacteristic
609C2×C4D3×#4111yescharacteristic
610C2×C4#100911yescharacteristic
611C8—41no—
612C8—41no—
613C8—121no—
614C8—121no—
615C2×C3—24no—
616C2×C3#245011yescharacteristic
617C2×C3C2×#47411yescharacteristic
618C2×C3#308411yescharacteristic
619C2×C3#329411yescharacteristic
620C2×C3#329411yescharacteristic
621C2×C3#331611yescharacteristic
622C2×C3#332011yescharacteristic
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#489014yes—
638C22C2×#89611yescharacteristic
639C22#559111yescharacteristic
640C22#565411yescharacteristic
641C22C2×D3×D811yescharacteristic
642C22C2×#91811yescharacteristic
643C22#577511yescharacteristic
644C22#577811yescharacteristic
645C4—22no—
646C4—42no—
647C4—62no—
648C4—62no—
649C4—81no—
650C4—81no—
651C4—81no—
652C4—122no—
653C4—121no—
654C4—121no—
655C4—122no—
656C4—241no—
657C4#553811yescharacteristic
658C4#556011yescharacteristic
659C4#563511yescharacteristic
660C4#563611yescharacteristic
661C3#2640811yescharacteristic, complemented, Sylow
662C2—24no—
663C2—242no—
664C2#7442311yescharacteristic
665C2C2×#489011yescharacteristic
666C2#7924411yescharacteristic
667C2#8001411yescharacteristic
668C2#8019811yescharacteristic
669C2#8130211yescharacteristic
670C2#8149111yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1852C2
socleC23×C3C22×D4
prime radicalC2×#930C22
hypercenter#1852D3
hypocenterC3#26408
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1852

Complements

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

NH ≅ G/Ntype
2. #73899C2semidirect
3. C3×#1852C2semidirect
4. C2×#4529C2semidirect
5. #70639C2semidirect
6. #70638C2semidirect
7. #70621C2semidirect
8. #70609C2semidirect
9. #69950C2semidirect
12. #68898C2semidirect
13. #68892C2semidirect
14. #68810C2semidirect
15. #68800C2semidirect
71. #1852D3semidirect
95. #930Q8semidirect
661. C3#26408semidirect

Maximal subgroups

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

subgroupindex
#821632
#738992
C3×#18522
C2×#45292
#706392
#706382
#706212
#706092
#699502
#699322
#695152
#688982
#688922
#688102
#688002
#264083