Finite Group Databasebeta

← #10877212

Subgroups — #10877212

Order 768.

854 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82224C211yesmaximal, characteristic, complemented
2#73865C211yesmaximal, characteristic
3C3×#1854C211yesmaximal, characteristic, complemented
4#71586C211yesmaximal, characteristic
5#70640C211yesmaximal, characteristic, complemented
6#70615C211yesmaximal, characteristic, complemented
7#70586C211yesmaximal, characteristic, complemented
8#70563C211yesmaximal, characteristic, complemented
9#69933C211yesmaximal, characteristic
10#69908C211yesmaximal, characteristic, complemented
11#69506C211yesmaximal, characteristic, complemented
12#68892C211yesmaximal, characteristic, complemented
13#68892C211yesmaximal, characteristic, complemented
14#68844C211yesmaximal, characteristic, complemented
15#68812C211yesmaximal, characteristic, complemented
16#27679—31nomaximal, Sylow
17C2×C3×#117C2211yescharacteristic
18C2×#930C2211yescharacteristic
19C2×#926C2211yescharacteristic
20C22×C4×Dic3C2211yescharacteristic
21C2×#893C2211yescharacteristic
22C2×#67⋊C4—21no—
23C2×C3×#107C2211yescharacteristic
24C2×C3×#106C2211yescharacteristic
25C2×C3×C22⋊C8C2211yescharacteristic
26C3×#380C2211yescharacteristic
27C2×#830C2211yescharacteristic
28C2×#821C2211yescharacteristic
29C2×#821C2211yescharacteristic
30C2×Dic3⋊C8C2211yescharacteristic
31C2×Dic3⋊C8C2211yescharacteristic
32#4988C2211yescharacteristic
33#4977C2211yescharacteristic
34#4977C2211yescharacteristic
35#4975C2211yescharacteristic
36#4975C2211yescharacteristic
37Dic3×C4⋊C4C2211yescharacteristic
38#4972C2211yescharacteristic
39C2×#809C2211yescharacteristic
40C2×#809C2211yescharacteristic
41C2×#808C2211yescharacteristic
42C2×#807C2211yescharacteristic
43#4958—21no—
44#4942—21no—
45#4942—21no—
46C4×Dic3⋊C4—21no—
47C4×Dic3⋊C4—21no—
48#4669—21no—
49#4666—21no—
50C3×#340C2211yescharacteristic
51C3×#340C2211yescharacteristic
52#4551C2211yescharacteristic
53#4551C2211yescharacteristic
54#4550C2211yescharacteristic
55#4550C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60C4×#121—31no—
61#2285—31no—
62#2101—31no—
63#2101—31no—
64#2092—31no—
65#2092—31no—
66#1948—31no—
67#1947—31no—
68#1875—31no—
69#1875—31no—
70#1854—31no—
71#1854D311yescharacteristic, complemented
72#1802—31no—
73#1802—31no—
74#1797—31no—
75C23×C3×C4C2311yescharacteristic
76C22×#67—21no—
77C23×Dic3D411yescharacteristic
78C23×Dic3D411yescharacteristic
79C22×D12C2311yescharacteristic
80C22×C4×D3—21no—
81C22×C3×C8C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C3×#117—41no—
84C3×#117—41no—
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C22⋊C4—21no—
90C2×C3×C22⋊C4—21no—
91C2×C3×C42—21no—
92C2×#217—21no—
93#930—22no—
94#930Q814yes—
95C2×(C2×D3)⋊C4—21no—
96C2×(C2×D3)⋊C4—21no—
97C2×(C2×D3)⋊C4—21no—
98C2×(C2×D3)⋊C4—21no—
99#926—41no—
100#926—41no—
101C2×C4⋊Dic3C2311yescharacteristic
102C2×C4⋊Dic3C2311yescharacteristic
103C2×C4⋊Dic3C2311yescharacteristic
104C2×Dic3⋊C4—21no—
105C2×Dic3⋊C4—21no—
106C2×Dic3⋊C4—21no—
107C2×Dic3⋊C4—21no—
108C2×Dic3⋊C4—21no—
109C2×C4×Dic3—22no—
110C2×C4×Dic3—22no—
111C2×C4×Dic3—21no—
112C2×C4×Dic3—21no—
113C2×C4×Dic3D411yescharacteristic
114C2×C4×Dic3D411yescharacteristic
115C2×C4×Dic3C2311yescharacteristic
116C2×C4×Dic3C2311yescharacteristic
117C22×C3⋊C8C2311yescharacteristic
118C22×C3⋊C8C2311yescharacteristic
119#893—22no—
120#893—22no—
121#67⋊C4—44no—
122C3×#107—41no—
123C3×#106—41no—
124C3×C22⋊C8—22no—
125C3×#95—21no—
126C3⋊#95—21no—
127C3⋊#95—21no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131C3⋊#95—21no—
132#830—22no—
133#821—41no—
134#821—41no—
135Dic3⋊C8—22no—
136Dic3⋊C8—22no—
137#809—41no—
138#809—41no—
139#808—41no—
140#807—41no—
141C2×#121—31no—
142C2×C4×D4—31no—
143C2×C4×D4—61no—
144C2×#117—31no—
145C2×#117C2×D311yescharacteristic
146C22×C42—31no—
147C2×#107—31no—
148C2×#107C2×D311yescharacteristic
149C2×#106—31no—
150C2×#106C2×D311yescharacteristic
151C2×C4⋊C8—31no—
152C2×C4⋊C8—31no—
153C2×#102—31no—
154C2×#102—31no—
155C2×#102—31no—
156C2×#102—31no—
157C2×C22⋊C8—31no—
158C2×C22⋊C8C2×D311yescharacteristic
159#390—31no—
160#389—31no—
161#388—61no—
162#386—61no—
163#384—31no—
164#384—31no—
165#382—31no—
166#382—31no—
167#380—61no—
168#380—61no—
169#380C2×D311yescharacteristic
170C4×C4⋊C4—31no—
171C4×C4⋊C4—61no—
172C4×C4⋊C4—61no—
173C4×C22⋊C4—61no—
174#340—31no—
175#340—31no—
176#340—31no—
177#340—31no—
178#340—31no—
179#340—31no—
180#340C2×D311yescharacteristic
181#340C2×D311yescharacteristic
182#336—31no—
183#336—31no—
184C24×C3C2×D411yescharacteristic
185C23×D3—21no—
186C22×C3×C4—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×C4C2×D411yescharacteristic
195C22×C3×C4C2×Q812yes—
196C22×C3×C4#4112yes—
197C22×C3×C4C2411yescharacteristic
198C2×#67—24no—
199C2×#67—42no—
200C2×#67—42no—
201C22×Dic3—21no—
202C22×Dic3—22no—
203C22×Dic3—22no—
204C22×Dic3—22no—
205C22×Dic3—22no—
206C22×Dic3—21no—
207C22×Dic3—21no—
208C22×Dic3—41no—
209C22×Dic3—41no—
210C22×Dic3C2×D411yescharacteristic
211C22×Dic3C2×D411yescharacteristic
212C22×Dic3C2×D411yescharacteristic
213C22×Dic3C2×D411yescharacteristic
214C2×D12—21no—
215C2×D12—21no—
216C2×D12—21no—
217C2×D12—21no—
218C2×D12—21no—
219C2×D12C2×Q812yes—
220C2×C4×D3—42no—
221C2×C4×D3—42no—
222C2×C4×D3—42no—
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—21no—
237C3×C22⋊C4—42no—
238C3×C22⋊C4—42no—
239C3×C42—42no—
240#217—42no—
241(C2×D3)⋊C4—24no—
242(C2×D3)⋊C4—42no—
243(C2×D3)⋊C4—42no—
244(C2×D3)⋊C4—42no—
245C4⋊Dic3—21no—
246C4⋊Dic3—21no—
247C4⋊Dic3—21no—
248C4⋊Dic3—21no—
249C4⋊Dic3—21no—
250C4⋊Dic3C2×Q812yes—
251Dic3⋊C4—42no—
252Dic3⋊C4—42no—
253Dic3⋊C4—42no—
254Dic3⋊C4—42no—
255Dic3⋊C4—42no—
256C4×Dic3—21no—
257C4×Dic3—21no—
258C4×Dic3—22no—
259C4×Dic3—22no—
260C4×Dic3—42no—
261C4×Dic3—42no—
262C4×Dic3#4112yes—
263C4×Dic3#4112yes—
264C2×C3⋊C8—21no—
265C2×C3⋊C8—21no—
266C2×C3⋊C8—21no—
267C2×C3⋊C8—21no—
268C2×C3⋊C8—21no—
269C2×C3⋊C8—21no—
270C22×D4—31no—
271C22×D4—61no—
272C23×C4—31no—
273C23×C4—31no—
274C23×C4—61no—
275C23×C4C22×D311yescharacteristic
276C22×C8—31no—
277C22×C8—31no—
278C22×C8C22×D311yescharacteristic
279C22×C8C22×D311yescharacteristic
280#121—34no—
281#121—62no—
282C4×D4—62no—
283C4×D4—62no—
284C4×D4—124no—
285#117—41no—
286#117—41no—
287#117—121no—
288#117—121no—
289C2×C4⋊C4—31no—
290C2×C4⋊C4—31no—
291C2×C4⋊C4—31no—
292C2×C4⋊C4—61no—
293C2×C4⋊C4—61no—
294C2×C4⋊C4—61no—
295C2×C4⋊C4—61no—
296C2×C4⋊C4—61no—
297C2×C4⋊C4C22×D311yescharacteristic
298C2×C4⋊C4C22×D311yescharacteristic
299C2×C4⋊C4C22×D311yescharacteristic
300C2×C4⋊C4C22×D311yescharacteristic
301C2×C22⋊C4—21no—
302C2×C22⋊C4—21no—
303C2×C22⋊C4—61no—
304C2×C22⋊C4—61no—
305C2×C22⋊C4—61no—
306C2×C22⋊C4—61no—
307C2×C22⋊C4—61no—
308C2×C42—21no—
309C2×C42—31no—
310C2×C42—31no—
311C2×C42—31no—
312C2×C42—31no—
313C2×C42—62no—
314C2×C42—62no—
315C2×C42—61no—
316C2×C42—61no—
317#107—41no—
318#107—121no—
319#106—41no—
320#106—121no—
321C4⋊C8—62no—
322C4⋊C8—62no—
323#102—121no—
324#102—121no—
325#102—121no—
326#102—121no—
327C22⋊C8—22no—
328C22⋊C8—62no—
329#95—21no—
330#95—61no—
331#95—61no—
332#95—61no—
333#95—61no—
334#95—61no—
335#95—61no—
336C23×C3—21no—
337C23×C3—21no—
338C23×C3—21no—
339C23×C3—21no—
340C23×C3—21no—
341C23×C3—21no—
342C23×C3C22⋊Q812yes—
343C23×C3C22×D411yescharacteristic
344C22×D3—22no—
345C22×D3—41no—
346C22×D3—41no—
347C22×D3—41no—
348C22×D3—41no—
349C22×D3—41no—
350C22×D3—41no—
351C2×C3×C4—21no—
352C2×C3×C4—21no—
353C2×C3×C4—22no—
354C2×C3×C4—22no—
355C2×C3×C4—22no—
356C2×C3×C4—22no—
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×C4—41no—
369C2×C3×C4C22⋊Q812yes—
370C2×C3×C4C22×Q811yescharacteristic
371C2×C3×C4C2×#4111yescharacteristic
372C2×C3×C4C2×#4111yescharacteristic
373C2×C3×C4C2×#4111yescharacteristic
374C2×C3×C4H(5,𝔽2)11yescharacteristic
375C2×C3×C4H(5,𝔽2)11yescharacteristic
376#67—48no—
377C2×Dic3—21no—
378C2×Dic3—21no—
379C2×Dic3—22no—
380C2×Dic3—21no—
381C2×Dic3—21no—
382C2×Dic3—21no—
383C2×Dic3—21no—
384C2×Dic3—21no—
385C2×Dic3—21no—
386C2×Dic3—44no—
387C2×Dic3—41no—
388C2×Dic3—41no—
389C2×Dic3—44no—
390C2×Dic3—41no—
391C2×Dic3—41no—
392C2×Dic3—41no—
393C2×Dic3—41no—
394C2×Dic3—41no—
395C2×Dic3—41no—
396C2×Dic3#12312yes—
397C2×Dic3#12312yes—
398C2×Dic3#13512yes—
399C2×Dic3AGL(1,ℤ/8)12yes—
400D12—22no—
401D12—22no—
402D12—41no—
403D12—41no—
404C4×D3—84no—
405C3×C8—41no—
406C3×C8—41no—
407C3⋊C8—41no—
408C3⋊C8—41no—
409C24—61no—
410C24D3×D411yescharacteristic
411C2×D4—32no—
412C2×D4—64no—
413C2×D4—61no—
414C2×D4—61no—
415C2×D4—61no—
416C2×D4—61no—
417C2×D4—61no—
418C2×D4—122no—
419C2×D4—122no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—31no—
429C22×C4—31no—
430C22×C4—31no—
431C22×C4—31no—
432C22×C4—62no—
433C22×C4—62no—
434C22×C4—62no—
435C22×C4—62no—
436C22×C4—61no—
437C22×C4—61no—
438C22×C4—61no—
439C22×C4—122no—
440C22×C4—122no—
441C22×C4—122no—
442C22×C4—121no—
443C22×C4—121no—
444C22×C4#23512yes—
445C22×C4D3×D411yescharacteristic
446C22×C4D3×Q812yes—
447C22×C4C23×D311yescharacteristic
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—61no—
455C2×C8—61no—
456C2×C8—61no—
457C2×C8—61no—
458C2×C8—61no—
459C2×C8—61no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—32no—
469C4⋊C4—61no—
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—122no—
479C22⋊C4—42no—
480C22⋊C4—42no—
481C22⋊C4—64no—
482C22⋊C4—122no—
483C22⋊C4—122no—
484C22⋊C4—122no—
485C22⋊C4—122no—
486C42—32no—
487C42—32no—
488C42—42no—
489C42—62no—
490C42—62no—
491C42—61no—
492C42—61no—
493C42—122no—
494C42—122no—
495C22×C3—22no—
496C22×C3—22no—
497C22×C3—22no—
498C22×C3—41no—
499C22×C3—41no—
500C22×C3—41no—
501C22×C3—41no—
502C22×C3C2×C22⋊Q811yescharacteristic
503C22×C3C2×#12311yescharacteristic
504C22×C3#53511yescharacteristic
505C22×C3C2×#13511yescharacteristic
506C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
507C22×C3#57511yescharacteristic
508C22×C3#57511yescharacteristic
509C2×D3—42no—
510C2×D3—42no—
511C2×D3—42no—
512C2×D3—81no—
513C2×D3—81no—
514C2×D3—81no—
515C2×D3—81no—
516C3×C4—22no—
517C3×C4—81no—
518C3×C4—81no—
519C3×C4—81no—
520C3×C4—81no—
521C3×C4#55311yescharacteristic
522C3×C4#55311yescharacteristic
523C3×C4#55411yescharacteristic
524C3×C4#55411yescharacteristic
525Dic3—22no—
526Dic3—42no—
527Dic3—41no—
528Dic3—41no—
529Dic3—81no—
530Dic3—81no—
531Dic3#48214yes—
532C23—21no—
533C23—21no—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—62no—
539C23—121no—
540C23—121no—
541C23—121no—
542C23—121no—
543C23—121no—
544C23—121no—
545C23#89612yes—
546C23C2×D3×D411yescharacteristic
547D4—62no—
548D4—62no—
549D4—128no—
550D4—121no—
551D4—121no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—32no—
559C2×C4—32no—
560C2×C4—32no—
561C2×C4—32no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—61no—
575C2×C4—61no—
576C2×C4—61no—
577C2×C4—61no—
578C2×C4—62no—
579C2×C4—61no—
580C2×C4—61no—
581C2×C4—61no—
582C2×C4—61no—
583C2×C4—124no—
584C2×C4—124no—
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—244no—
594C2×C4#89612yes—
595C2×C4C2×#23511yescharacteristic
596C2×C4#100411yescharacteristic
597C2×C4#100411yescharacteristic
598C2×C4C2×D3×Q811yescharacteristic
599C2×C4D3×#4111yescharacteristic
600C2×C4D3×#4111yescharacteristic
601C8—41no—
602C8—41no—
603C8—121no—
604C8—121no—
605C2×C3—42no—
606C2×C3#248711yescharacteristic
607C2×C3#307611yescharacteristic
608C2×C3#307711yescharacteristic
609C2×C3C2×#48211yescharacteristic
610C2×C3#309311yescharacteristic
611C2×C3#321211yescharacteristic
612C2×C3#321411yescharacteristic
613D3—82no—
614C22—22no—
615C22—22no—
616C22—22no—
617C22—41no—
618C22—41no—
619C22—41no—
620C22—41no—
621C22—122no—
622C22—122no—
623C22—122no—
624C22—241no—
625C22—241no—
626C22—241no—
627C22—241no—
628C22C2×#89611yescharacteristic
629C22#559211yescharacteristic
630C22D3×#12311yescharacteristic
631C22D3×#13511yescharacteristic
632C22#576811yescharacteristic
633C22D3×AGL(1,ℤ/8)11yescharacteristic
634C22#577711yescharacteristic
635C4—22no—
636C4—34no—
637C4—62no—
638C4—81no—
639C4—81no—
640C4—81no—
641C4—81no—
642C4—121no—
643C4—122no—
644C4—121no—
645C4—241no—
646C4—241no—
647C4#554011yescharacteristic
648C4#554011yescharacteristic
649C4#563611yescharacteristic
650C4#563611yescharacteristic
651C3#2767911yescharacteristic, complemented, Sylow
652C2—42no—
653C2—242no—
654C2#7445711yescharacteristic
655C2#7919611yescharacteristic
656C2#7923911yescharacteristic
657C2#8007811yescharacteristic
658C2#8022111yescharacteristic
659C2D3×#48211yescharacteristic
660C2#8151611yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1854C2
socleC23×C3C22×D4
prime radicalC2×#930C22
hypercenter#1854D3
hypocenterC3#27679
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1854

Complements

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

NH ≅ G/Ntype
1. #82224C2semidirect
3. C3×#1854C2semidirect
5. #70640C2semidirect
6. #70615C2semidirect
7. #70586C2semidirect
8. #70563C2semidirect
10. #69908C2semidirect
11. #69506C2semidirect
12. #68892C2semidirect
13. #68892C2semidirect
14. #68844C2semidirect
15. #68812C2semidirect
71. #1854D3semidirect
651. C3#27679semidirect

Maximal subgroups

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

subgroupindex
#822242
#738652
C3×#18542
#715862
#706402
#706152
#705862
#705632
#699332
#699082
#695062
#688922
#688442
#688122
#276793