Finite Group Databasebeta

← #10867984

Subgroups — #10867984

Order 768.

762 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#74364C211yesmaximal, characteristic, complemented
2#73649C211yesmaximal, characteristic
3C3×#1948C211yesmaximal, characteristic, complemented
4#70581C211yesmaximal, characteristic
5#70581C211yesmaximal, characteristic
6#70560C211yesmaximal, characteristic, complemented
7#70558C211yesmaximal, characteristic, complemented
8#70483C211yesmaximal, characteristic, complemented
9#70449C211yesmaximal, characteristic, complemented
10#69585C211yesmaximal, characteristic, complemented
11#69461C211yesmaximal, characteristic, complemented
12#68894C211yesmaximal, characteristic, complemented
13#68876C211yesmaximal, characteristic, complemented
14#68813C211yesmaximal, characteristic, complemented
15#68780C211yesmaximal, characteristic, complemented
16#27039—31nomaximal, Sylow
17C2×#898—21no—
18C2×#895C2211yescharacteristic
19C2×#893C2211yescharacteristic
20C2×#892C2211yescharacteristic
21C2×C3×#106C2211yescharacteristic
22C2×C3×#106C2211yescharacteristic
23C2×C3×C4⋊C8C2211yescharacteristic
24C3×#384C2211yescharacteristic
25C3×C4×C4⋊C4C2211yescharacteristic
26C2×#821C2211yescharacteristic
27C2×#821C2211yescharacteristic
28C2×(C2×D3)⋊C8C2211yescharacteristic
29C2×(C2×D3)⋊C8C2211yescharacteristic
30C2×#817C2211yescharacteristic
31C2×C8×Dic3C2211yescharacteristic
32#4992—21no—
33#4987C2211yescharacteristic
34#4987C2211yescharacteristic
35#4985—21no—
36#4984C2211yescharacteristic
37#4984C2211yescharacteristic
38#4980—21no—
39Dic3×C4⋊C4C2211yescharacteristic
40C2×#809C2211yescharacteristic
41C2×#809C2211yescharacteristic
42C2×#807C2211yescharacteristic
43#4939—21no—
44#4939C2211yescharacteristic
45C4×(C2×D3)⋊C4—21no—
46C4×(C2×D3)⋊C4C2211yescharacteristic
47#4928C2211yescharacteristic
48#4670—21no—
49#4665—21no—
50C3×#340C2211yescharacteristic
51C3×#340C2211yescharacteristic
52#4551C2211yescharacteristic
53#4550C2211yescharacteristic
54#4529C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4524C2211yescharacteristic
60#2496—31no—
61#2363—31no—
62#2071—31no—
63#2065—31no—
64#2034—31no—
65#2034—31no—
66#1948D311yescharacteristic, complemented
67#1941—31no—
68#1923—31no—
69#1921—31no—
70#1920—31no—
71#1862—31no—
72#1854—31no—
73#1825—31no—
74C4×#102—31no—
75C22×D12—21no—
76C22×D12C2311yescharacteristic
77C22×C4×D3—21no—
78C22×C4×D3C2311yescharacteristic
79C22×C3×C8C2311yescharacteristic
80C22×C3×C8C2311yescharacteristic
81C2×C3×C4⋊C4—21no—
82C2×C3×C4⋊C4—21no—
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C42—21no—
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)⋊C4—21no—
95C2×(C2×D3)⋊C4—21no—
96C2×(C2×D3)⋊C4D411yescharacteristic
97C2×(C2×D3)⋊C4D411yescharacteristic
98C2×C4⋊Dic3—21no—
99C2×C4⋊Dic3—21no—
100C2×C4⋊Dic3C2311yescharacteristic
101C2×C4⋊Dic3C2311yescharacteristic
102C2×C4⋊Dic3C2311yescharacteristic
103C2×Dic3⋊C4—21no—
104C2×C4×Dic3—21no—
105C2×C4×Dic3—21no—
106C2×C4×Dic3C2311yescharacteristic
107C22×C3⋊C8C2311yescharacteristic
108C22×C3⋊C8C2311yescharacteristic
109#898—44no—
110#895—22no—
111#895—22no—
112#893—22no—
113#893—41no—
114#892—22no—
115#892—41no—
116C3×#106—22no—
117C3×#106—22no—
118C3×C4⋊C8—22no—
119C3×#95—21no—
120C3⋊#95—21no—
121C3⋊#95—21no—
122C3⋊#95—21no—
123C3⋊#95—21no—
124C3⋊#95—21no—
125C3⋊#95—21no—
126C3⋊#95D411yescharacteristic
127C3⋊#95D411yescharacteristic
128#821—22no—
129#821—41no—
130(C2×D3)⋊C8—22no—
131(C2×D3)⋊C8—41no—
132#817—41no—
133C8×Dic3—41no—
134#809—22no—
135#809—22no—
136#807—22no—
137C2×#126—61no—
138C2×#121—31no—
139C2×C4×D4—31no—
140C2×#117—31no—
141C2×#107—31no—
142C2×#106—31no—
143C2×#106C2×D311yescharacteristic
144C2×#106C2×D311yescharacteristic
145C2×C4⋊C8C2×D311yescharacteristic
146C2×#102—31no—
147C2×#102—31no—
148C2×#102—31no—
149C2×#102—31no—
150C2×C22⋊C8—31no—
151C2×C22⋊C8—31no—
152C2×C4×C8—31no—
153#400—61no—
154#397—61no—
155#390—31no—
156#390—61no—
157#389—31no—
158#389—31no—
159#389—31no—
160#388—61no—
161#387—61no—
162#386—61no—
163#384C2×D311yescharacteristic
164#380—31no—
165#380—31no—
166C4×C4⋊C4—31no—
167C4×C4⋊C4C2×D311yescharacteristic
168C4×C22⋊C4—31no—
169C4×C22⋊C4—61no—
170#340—31no—
171#340—31no—
172#340—31no—
173#340—31no—
174#340—31no—
175#340—31no—
176#340C2×D311yescharacteristic
177#340C2×D311yescharacteristic
178#336—31no—
179#336—31no—
180C23×D3—21no—
181C23×D3C2×D411yescharacteristic
182C22×C3×C4—21no—
183C22×C3×C4—21no—
184C22×C3×C4—21no—
185C22×C3×C4—21no—
186C22×C3×C4C2×D411yescharacteristic
187C22×C3×C4C2×D411yescharacteristic
188C22×C3×C4C2411yescharacteristic
189C22×Dic3—21no—
190C22×Dic3—21no—
191C22×Dic3—21no—
192C22×Dic3C2×D411yescharacteristic
193C22×Dic3C2×D411yescharacteristic
194C22×Dic3C2×D411yescharacteristic
195C2×D12—21no—
196C2×D12—21no—
197C2×D12—21no—
198C2×D12—21no—
199C2×D12—21no—
200C2×D12—24no—
201C2×D12—81no—
202C2×D12—81no—
203C2×D12#4112yes—
204C2×C4×D3—21no—
205C2×C4×D3—21no—
206C2×C4×D3—21no—
207C2×C4×D3—21no—
208C2×C4×D3—21no—
209C2×C4×D3—21no—
210C2×C4×D3—42no—
211C2×C4×D3—42no—
212C2×C4×D3—81no—
213C2×C3×C8—21no—
214C2×C3×C8—21no—
215C2×C3×C8—21no—
216C2×C3×C8—21no—
217C2×C3×C8—21no—
218C2×C3×C8#4112yes—
219C3×C4⋊C4—21no—
220C3×C4⋊C4—21no—
221C3×C4⋊C4—21no—
222C3×C4⋊C4—21no—
223C3×C4⋊C4—21no—
224C3×C4⋊C4—21no—
225C3×C4⋊C4—42no—
226C3×C4⋊C4—42no—
227C3×C4⋊C4#4112yes—
228C3×C4⋊C4#4112yes—
229C3×C42—21no—
230C3×C42—21no—
231C3×C42—81no—
232(C2×D3)⋊C4—24no—
233(C2×D3)⋊C4—22no—
234(C2×D3)⋊C4—22no—
235(C2×D3)⋊C4—42no—
236(C2×D3)⋊C4—42no—
237(C2×D3)⋊C4—42no—
238(C2×D3)⋊C4—42no—
239(C2×D3)⋊C4—42no—
240(C2×D3)⋊C4—81no—
241C4⋊Dic3—21no—
242C4⋊Dic3—21no—
243C4⋊Dic3—21no—
244C4⋊Dic3—21no—
245C4⋊Dic3—21no—
246C4⋊Dic3—21no—
247C4⋊Dic3—42no—
248C4⋊Dic3—81no—
249Dic3⋊C4—42no—
250C4×Dic3—21no—
251C4×Dic3—21no—
252C4×Dic3—42no—
253C4×Dic3—42no—
254C2×C3⋊C8—21no—
255C2×C3⋊C8—21no—
256C2×C3⋊C8—21no—
257C2×C3⋊C8—21no—
258C2×C3⋊C8—21no—
259C2×C3⋊C8—21no—
260C22×D4—31no—
261C22×D4—61no—
262C23×C4—31no—
263C23×C4—61no—
264C22×C8—31no—
265C22×C8—31no—
266C22×C8C22×D311yescharacteristic
267C22×C8C22×D311yescharacteristic
268#126—124no—
269#121—62no—
270#121—62no—
271C4×D4—62no—
272C4×D4—121no—
273#117—62no—
274#117—121no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—31no—
278C2×C4⋊C4—31no—
279C2×C4⋊C4—31no—
280C2×C4⋊C4—61no—
281C2×C4⋊C4—61no—
282C2×C4⋊C4—61no—
283C2×C4⋊C4C22×D311yescharacteristic
284C2×C4⋊C4C22×D311yescharacteristic
285C2×C4⋊C4C22×D311yescharacteristic
286C2×C4⋊C4C22×D311yescharacteristic
287C2×C22⋊C4—31no—
288C2×C22⋊C4—31no—
289C2×C22⋊C4—61no—
290C2×C22⋊C4—61no—
291C2×C22⋊C4—61no—
292C2×C22⋊C4—61no—
293C2×C22⋊C4—61no—
294C2×C22⋊C4—61no—
295C2×C22⋊C4—61no—
296C2×C42—21no—
297C2×C42—31no—
298C2×C42—61no—
299C2×C42—61no—
300C2×C42C22×D311yescharacteristic
301#107—62no—
302#106—22no—
303#106—22no—
304#106—121no—
305C4⋊C8—22no—
306#102—62no—
307#102—62no—
308#102—62no—
309#102—121no—
310C22⋊C8—62no—
311C22⋊C8—121no—
312C4×C8—121no—
313#95—21no—
314#95—31no—
315#95—31no—
316#95—61no—
317#95—61no—
318#95—61no—
319#95—61no—
320#95—61no—
321#95—61no—
322C23×C3C22×D411yescharacteristic
323C22×D3—22no—
324C22×D3—21no—
325C22×D3—21no—
326C22×D3—21no—
327C22×D3—21no—
328C22×D3—21no—
329C22×D3—21no—
330C22×D3—21no—
331C22×D3—41no—
332C22×D3—41no—
333C22×D3—41no—
334C22×D3—41no—
335C22×D3—41no—
336C22×D3—41no—
337C2×C3×C4—21no—
338C2×C3×C4—21no—
339C2×C3×C4—22no—
340C2×C3×C4—22no—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—21no—
344C2×C3×C4—41no—
345C2×C3×C4—41no—
346C2×C3×C4—41no—
347C2×C3×C4—41no—
348C2×C3×C4—41no—
349C2×C3×C4—41no—
350C2×C3×C4—41no—
351C2×C3×C4—41no—
352C2×C3×C4—41no—
353C2×C3×C4—41no—
354C2×C3×C4#13512yes—
355C2×C3×C4C2×#4111yescharacteristic
356C2×C3×C4C2×#4111yescharacteristic
357C2×C3×C4C2×#4111yescharacteristic
358C2×C3×C4C2×#4111yescharacteristic
359C2×C3×C4H(5,𝔽2)11yescharacteristic
360C2×C3×C42−1+411yescharacteristic
361C2×Dic3—21no—
362C2×Dic3—21no—
363C2×Dic3—21no—
364C2×Dic3—21no—
365C2×Dic3—21no—
366C2×Dic3—21no—
367C2×Dic3—21no—
368C2×Dic3—21no—
369C2×Dic3—41no—
370C2×Dic3—41no—
371C2×Dic3—41no—
372C2×Dic3—41no—
373C2×Dic3—41no—
374C2×Dic3—41no—
375C2×Dic3—41no—
376C2×Dic3—41no—
377C2×Dic3—41no—
378C2×Dic3AGL(1,ℤ/8)12yes—
379D12—22no—
380D12—22no—
381D12—41no—
382D12—41no—
383D12—84no—
384C4×D3—41no—
385C4×D3—41no—
386C4×D3—41no—
387C4×D3—41no—
388C4×D3—84no—
389C3×C8—22no—
390C3×C8—41no—
391C3⋊C8—41no—
392C3⋊C8—41no—
393C24—31no—
394C24—61no—
395C2×D4—32no—
396C2×D4—64no—
397C2×D4—61no—
398C2×D4—61no—
399C2×D4—61no—
400C2×D4—61no—
401C2×D4—61no—
402C2×D4—241no—
403C2×D4—241no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—31no—
409C22×C4—31no—
410C22×C4—31no—
411C22×C4—61no—
412C22×C4—61no—
413C22×C4—61no—
414C22×C4—61no—
415C22×C4—61no—
416C22×C4—61no—
417C22×C4—61no—
418C22×C4—61no—
419C22×C4—61no—
420C22×C4—122no—
421C22×C4—122no—
422C22×C4—241no—
423C22×C4D3×D411yescharacteristic
424C22×C4D3×D411yescharacteristic
425C22×C4C23×D311yescharacteristic
426C2×C8—21no—
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—21no—
430C2×C8—21no—
431C2×C8—61no—
432C2×C8—61no—
433C2×C8—61no—
434C2×C8—61no—
435C2×C8—61no—
436C2×C8—61no—
437C2×C8#23512yes—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—42no—
445C4⋊C4—42no—
446C4⋊C4—61no—
447C4⋊C4—61no—
448C4⋊C4—61no—
449C4⋊C4—61no—
450C4⋊C4—61no—
451C4⋊C4—61no—
452C4⋊C4—122no—
453C4⋊C4—122no—
454C4⋊C4—241no—
455C4⋊C4#23712yes—
456C4⋊C4#23912yes—
457C22⋊C4—62no—
458C22⋊C4—64no—
459C22⋊C4—62no—
460C22⋊C4—122no—
461C22⋊C4—122no—
462C22⋊C4—122no—
463C22⋊C4—122no—
464C22⋊C4—122no—
465C22⋊C4—241no—
466C42—21no—
467C42—21no—
468C42—61no—
469C42—61no—
470C42—81no—
471C42—122no—
472C42—122no—
473C22×C3#52511yescharacteristic
474C22×C3#54611yescharacteristic
475C22×C3#54711yescharacteristic
476C22×C3C2×#13511yescharacteristic
477C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
478C22×C3#57711yescharacteristic
479C22×C3#57711yescharacteristic
480C2×D3—41no—
481C2×D3—41no—
482C2×D3—41no—
483C2×D3—42no—
484C2×D3—41no—
485C2×D3—42no—
486C2×D3—41no—
487C2×D3—42no—
488C2×D3—41no—
489C2×D3—41no—
490C2×D3—81no—
491C2×D3—81no—
492C2×D3—81no—
493C2×D3—81no—
494C3×C4—22no—
495C3×C4—41no—
496C3×C4—42no—
497C3×C4—42no—
498C3×C4—81no—
499C3×C4—81no—
500C3×C4#52911yescharacteristic
501C3×C4#53911yescharacteristic
502C3×C4#55311yescharacteristic
503C3×C4#55611yescharacteristic
504Dic3—22no—
505Dic3—41no—
506Dic3—41no—
507Dic3—81no—
508Dic3—81no—
509Dic3—81no—
510C23—61no—
511C23—61no—
512C23—61no—
513C23—61no—
514C23—61no—
515C23—62no—
516C23—61no—
517C23—61no—
518C23—121no—
519C23—121no—
520C23—121no—
521C23—121no—
522C23—121no—
523C23—121no—
524C23C2×D3×D411yescharacteristic
525D4—62no—
526D4—62no—
527D4—121no—
528D4—121no—
529D4—244no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—32no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—61no—
549C2×C4—61no—
550C2×C4—61no—
551C2×C4—61no—
552C2×C4—61no—
553C2×C4—61no—
554C2×C4—61no—
555C2×C4—61no—
556C2×C4—121no—
557C2×C4—121no—
558C2×C4—121no—
559C2×C4—121no—
560C2×C4—121no—
561C2×C4—121no—
562C2×C4—121no—
563C2×C4—121no—
564C2×C4—121no—
565C2×C4—121no—
566C2×C4—121no—
567C2×C4—121no—
568C2×C4—121no—
569C2×C4—244no—
570C2×C4#91612yes—
571C2×C4C2×#23511yescharacteristic
572C2×C4C2×#23711yescharacteristic
573C2×C4C2×#23911yescharacteristic
574C2×C4#100711yescharacteristic
575C2×C4D3×#4111yescharacteristic
576C2×C4#100911yescharacteristic
577C8—22no—
578C8—41no—
579C8—121no—
580C8—121no—
581C2×C3#246611yescharacteristic
582C2×C3#310611yescharacteristic
583C2×C3#312311yescharacteristic
584C2×C3#330411yescharacteristic
585C2×C3#330611yescharacteristic
586C2×C3#333711yescharacteristic
587C2×C3#335411yescharacteristic
588D3—81no—
589D3—82no—
590C22—121no—
591C22—121no—
592C22—122no—
593C22—121no—
594C22—121no—
595C22—121no—
596C22—121no—
597C22—122no—
598C22—121no—
599C22—122no—
600C22—241no—
601C22—241no—
602C22—241no—
603C22—241no—
604C22#550711yescharacteristic
605C22#560611yescharacteristic
606C22#563011yescharacteristic
607C22C2×#91611yescharacteristic
608C22#576211yescharacteristic
609C22D3×AGL(1,ℤ/8)11yescharacteristic
610C22#577811yescharacteristic
611C4—22no—
612C4—42no—
613C4—41no—
614C4—42no—
615C4—62no—
616C4—81no—
617C4—81no—
618C4—121no—
619C4—121no—
620C4—241no—
621C4—241no—
622C4—241no—
623C4#556511yescharacteristic
624C4#558011yescharacteristic
625C4#569211yescharacteristic
626C4#573211yescharacteristic
627C3#2703911yescharacteristic, complemented, Sylow
628C2—241no—
629C2—242no—
630C2#7602011yescharacteristic
631C2#7912511yescharacteristic
632C2#7923511yescharacteristic
633C2#8036111yescharacteristic
634C2#8041311yescharacteristic
635C2#8102311yescharacteristic
636C2#8107711yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1948C2
socleC23×C3C22×D4
prime radicalC2×#895C22
hypercenter#1948D3
hypocenterC3#27039
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1948

Complements

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

NH ≅ G/Ntype
1. #74364C2semidirect
3. C3×#1948C2semidirect
6. #70560C2semidirect
7. #70558C2semidirect
8. #70483C2semidirect
9. #70449C2semidirect
10. #69585C2semidirect
11. #69461C2semidirect
12. #68894C2semidirect
13. #68876C2semidirect
14. #68813C2semidirect
15. #68780C2semidirect
66. #1948D3semidirect
627. C3#27039semidirect

Maximal subgroups

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

subgroupindex
#743642
#736492
C3×#19482
#705812
#705602
#705582
#704832
#704492
#695852
#694612
#688942
#688762
#688132
#687802
#270393