Finite Group Databasebeta

← #10868090

Subgroups — #10868090

Order 768.

778 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#76235C211yesmaximal, characteristic, complemented
2#76206C211yesmaximal, characteristic
3C3×#1944C211yesmaximal, characteristic, complemented
4#70639C211yesmaximal, characteristic, complemented
5#70632C211yesmaximal, characteristic, complemented
6#70581C211yesmaximal, characteristic
7#70564C211yesmaximal, characteristic, complemented
8#69835C211yesmaximal, characteristic, complemented
9#69812C211yesmaximal, characteristic, complemented
10#69625C211yesmaximal, characteristic, complemented
11#69603C211yesmaximal, characteristic, complemented
12#68903C211yesmaximal, characteristic, complemented
13#68871C211yesmaximal, characteristic, complemented
14#68822C211yesmaximal, characteristic
15#68821C211yesmaximal, characteristic, complemented
16#27046—31nomaximal, Sylow
17C2×#895C2211yescharacteristic
18C2×#893—21no—
19C2×D3×C4⋊C4C2211yescharacteristic
20C2×C4×D12C2211yescharacteristic
21C2×C3×#107C2211yescharacteristic
22C2×C3×#106C2211yescharacteristic
23C2×C3×#97C2211yescharacteristic
24C3×#389C2211yescharacteristic
25C3×#389C2211yescharacteristic
26C2×#821C2211yescharacteristic
27C2×#821C2211yescharacteristic
28C2×(C2×D3)⋊C8C2211yescharacteristic
29C2×(C2×D3)⋊C8C2211yescharacteristic
30C2×Dic3⋊C8C2211yescharacteristic
31#4992—21no—
32#4989—21no—
33#4988C2211yescharacteristic
34#4987C2211yescharacteristic
35#4987C2211yescharacteristic
36#4986—21no—
37#4984C2211yescharacteristic
38#4977C2211yescharacteristic
39Dic3×C4⋊C4C2211yescharacteristic
40C2×#809C2211yescharacteristic
41C2×#809C2211yescharacteristic
42C2×#807C2211yescharacteristic
43C2×#807C2211yescharacteristic
44C4×(C2×D3)⋊C4C2211yescharacteristic
45C4×C4⋊Dic3C2211yescharacteristic
46#4670—21no—
47#4669—21no—
48#4665—21no—
49#4651—21no—
50C3×#340C2211yescharacteristic
51C3×#340C2211yescharacteristic
52#4551C2211yescharacteristic
53#4551C2211yescharacteristic
54#4550C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4524C2211yescharacteristic
60#2495—31no—
61#2361—31no—
62#2073—31no—
63#2067—31no—
64#2034—31no—
65#2033—31no—
66#1949—31no—
67#1944D311yescharacteristic, complemented
68#1924—31no—
69#1921—31no—
70#1921—31no—
71#1862—31no—
72#1852—31no—
73#1823—31no—
74#1762—31no—
75C22×D12—21no—
76C22×D12C2311yescharacteristic
77C22×C4×D3—21no—
78C22×C4×D3—21no—
79C22×C4×D3C2311yescharacteristic
80C22×C3×C8C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C2×C3×C4⋊C4—21no—
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)⋊C4—21no—
95C2×(C2×D3)⋊C4D411yescharacteristic
96C2×(C2×D3)⋊C4D411yescharacteristic
97C2×C4⋊Dic3—21no—
98C2×C4⋊Dic3C2311yescharacteristic
99C2×C4⋊Dic3C2311yescharacteristic
100C2×C4⋊Dic3C2311yescharacteristic
101C2×Dic3⋊C4—21no—
102C2×C4×Dic3—21no—
103C2×C4×Dic3—21no—
104C2×C4×Dic3C2311yescharacteristic
105C22×C3⋊C8C2311yescharacteristic
106C22×C3⋊C8C2311yescharacteristic
107#895—22no—
108#895—22no—
109#893—44no—
110D3×C4⋊C4—22no—
111D3×C4⋊C4—41no—
112C4×D12—22no—
113C4×D12—41no—
114C3×#107—41no—
115C3×#106—22no—
116C3×#97—41no—
117C3×#95—21no—
118C3×#95—21no—
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—22no—
130(C2×D3)⋊C8—22no—
131(C2×D3)⋊C8—41no—
132Dic3⋊C8—22no—
133#809—22no—
134#809—41no—
135#807—22no—
136#807—22no—
137C2×#121—31no—
138C2×C4×D4—31no—
139C2×C4×D4—61no—
140C22×C4⋊C4—31no—
141C2×#107—31no—
142C2×#107—31no—
143C2×#107C2×D311yescharacteristic
144C2×#106C2×D311yescharacteristic
145C2×C4⋊C8—31no—
146C2×#102—31no—
147C2×#102—31no—
148C2×#102—31no—
149C2×#102—31no—
150C2×C22⋊C8—31no—
151C2×C22⋊C8—31no—
152C2×#97C2×D311yescharacteristic
153#400—61no—
154#397—61no—
155#390—31no—
156#389—31no—
157#389C2×D311yescharacteristic
158#389C2×D311yescharacteristic
159#388—61no—
160#388—61no—
161#386—61no—
162#385—61no—
163#383—61no—
164#382—31no—
165#380—31no—
166#380—31no—
167C4×C4⋊C4—31no—
168C4×C4⋊C4—31no—
169C4×C22⋊C4—31no—
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—24no—
200C2×D12—21no—
201C2×D12—42no—
202C2×D12—42no—
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—42no—
213C2×C4×D3—42no—
214C2×C4×D3—81no—
215C2×C4×D3—81no—
216C2×C3×C8—21no—
217C2×C3×C8—21no—
218C2×C3×C8—21no—
219C2×C3×C8—21no—
220C2×C3×C8—21no—
221C2×C3×C8#4112yes—
222C3×C4⋊C4—21no—
223C3×C4⋊C4—21no—
224C3×C4⋊C4—21no—
225C3×C4⋊C4—21no—
226C3×C4⋊C4—21no—
227C3×C4⋊C4—21no—
228C3×C4⋊C4—21no—
229C3×C4⋊C4—42no—
230C3×C4⋊C4—81no—
231C3×C4⋊C4#4112yes—
232C3×C42—21no—
233C3×C42—21no—
234(C2×D3)⋊C4—24no—
235(C2×D3)⋊C4—22no—
236(C2×D3)⋊C4—41no—
237(C2×D3)⋊C4—42no—
238(C2×D3)⋊C4—42no—
239(C2×D3)⋊C4—42no—
240(C2×D3)⋊C4—81no—
241(C2×D3)⋊C4—81no—
242C4⋊Dic3—21no—
243C4⋊Dic3—21no—
244C4⋊Dic3—21no—
245C4⋊Dic3—21no—
246C4⋊Dic3—21no—
247C4⋊Dic3—81no—
248C4⋊Dic3#4112yes—
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—
264C23×C4—61no—
265C22×C8—31no—
266C22×C8—31no—
267C22×C8C22×D311yescharacteristic
268C22×C8C22×D311yescharacteristic
269#121—62no—
270#121—62no—
271C4×D4—62no—
272C4×D4—121no—
273C4×D4—124no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—31no—
277C2×C4⋊C4—31no—
278C2×C4⋊C4—31no—
279C2×C4⋊C4—62no—
280C2×C4⋊C4—61no—
281C2×C4⋊C4—61no—
282C2×C4⋊C4—121no—
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×C42—31no—
296C2×C42—61no—
297C2×C42—61no—
298C2×C42C22×D311yescharacteristic
299#107—41no—
300#107—62no—
301#107—62no—
302#106—22no—
303C4⋊C8—62no—
304#102—62no—
305#102—62no—
306#102—62no—
307#102—121no—
308C22⋊C8—62no—
309C22⋊C8—121no—
310#97—41no—
311#95—21no—
312#95—21no—
313#95—31no—
314#95—31no—
315#95—61no—
316#95—61no—
317#95—61no—
318#95—61no—
319#95—61no—
320#95—61no—
321#95—61no—
322C23×C3C22×D411yescharacteristic
323C22×D3—21no—
324C22×D3—21no—
325C22×D3—21no—
326C22×D3—21no—
327C22×D3—21no—
328C22×D3—21no—
329C22×D3—22no—
330C22×D3—41no—
331C22×D3—41no—
332C22×D3—41no—
333C22×D3—41no—
334C22×D3—41no—
335C22×D3—41no—
336C22×D3AGL(1,ℤ/8)12yes—
337C2×C3×C4—22no—
338C2×C3×C4—21no—
339C2×C3×C4—21no—
340C2×C3×C4—21no—
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—41no—
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—22no—
364C2×Dic3—21no—
365C2×Dic3—21no—
366C2×Dic3—21no—
367C2×Dic3—21no—
368C2×Dic3—21no—
369C2×Dic3—21no—
370C2×Dic3—21no—
371C2×Dic3—41no—
372C2×Dic3—41no—
373C2×Dic3—41no—
374C2×Dic3—41no—
375C2×Dic3—41no—
376C2×Dic3—41no—
377C2×Dic3—41no—
378C2×Dic3—41no—
379D12—22no—
380D12—22no—
381D12—48no—
382D12—41no—
383D12—41no—
384C4×D3—41no—
385C4×D3—41no—
386C4×D3—41no—
387C4×D3—41no—
388C4×D3—84no—
389C4×D3—84no—
390C3×C8—22no—
391C3×C8—41no—
392C3⋊C8—41no—
393C3⋊C8—41no—
394C24—31no—
395C24—61no—
396C2×D4—32no—
397C2×D4—64no—
398C2×D4—61no—
399C2×D4—61no—
400C2×D4—61no—
401C2×D4—61no—
402C2×D4—61no—
403C2×D4—122no—
404C2×D4—122no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—31no—
410C22×C4—31no—
411C22×C4—31no—
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—61no—
421C22×C4—122no—
422C22×C4—122no—
423C22×C4—122no—
424C22×C4—122no—
425C22×C4—241no—
426C22×C4—241no—
427C22×C4D3×D411yescharacteristic
428C22×C4D3×D411yescharacteristic
429C22×C4C23×D311yescharacteristic
430C2×C8—21no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—61no—
436C2×C8—61no—
437C2×C8—61no—
438C2×C8—61no—
439C2×C8—61no—
440C2×C8—61no—
441C2×C8#23912yes—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—32no—
450C4⋊C4—42no—
451C4⋊C4—61no—
452C4⋊C4—61no—
453C4⋊C4—61no—
454C4⋊C4—61no—
455C4⋊C4—61no—
456C4⋊C4—81no—
457C4⋊C4—122no—
458C4⋊C4—241no—
459C4⋊C4#23512yes—
460C22⋊C4—62no—
461C22⋊C4—64no—
462C22⋊C4—121no—
463C22⋊C4—122no—
464C22⋊C4—122no—
465C22⋊C4—122no—
466C22⋊C4—241no—
467C22⋊C4—241no—
468C42—21no—
469C42—21no—
470C42—61no—
471C42—61no—
472C42—122no—
473C42—122no—
474C22×C3#52511yescharacteristic
475C22×C3#54611yescharacteristic
476C22×C3#54711yescharacteristic
477C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
478C22×C3#57511yescharacteristic
479C22×C3#57611yescharacteristic
480C22×C3#57811yescharacteristic
481C2×D3—22no—
482C2×D3—22no—
483C2×D3—22no—
484C2×D3—41no—
485C2×D3—41no—
486C2×D3—41no—
487C2×D3—41no—
488C2×D3—42no—
489C2×D3—42no—
490C2×D3—42no—
491C2×D3—81no—
492C2×D3—81no—
493C2×D3—81no—
494C2×D3—81no—
495C3×C4—42no—
496C3×C4—41no—
497C3×C4—41no—
498C3×C4—81no—
499C3×C4—81no—
500C3×C4—81no—
501C3×C4#52911yescharacteristic
502C3×C4#53911yescharacteristic
503C3×C4#55311yescharacteristic
504C3×C4#55611yescharacteristic
505Dic3—41no—
506Dic3—41no—
507Dic3—41no—
508Dic3—42no—
509Dic3—81no—
510Dic3—81no—
511C23—32no—
512C23—61no—
513C23—61no—
514C23—61no—
515C23—61no—
516C23—61no—
517C23—61no—
518C23—62no—
519C23—121no—
520C23—121no—
521C23—121no—
522C23—121no—
523C23—121no—
524C23—121no—
525C23C2×D3×D411yescharacteristic
526D4—62no—
527D4—62no—
528D4—128no—
529D4—121no—
530D4—121no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—22no—
536C2×C4—21no—
537C2×C4—21no—
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—41no—
549C2×C4—61no—
550C2×C4—61no—
551C2×C4—61no—
552C2×C4—61no—
553C2×C4—61no—
554C2×C4—61no—
555C2×C4—61no—
556C2×C4—61no—
557C2×C4—62no—
558C2×C4—61no—
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—121no—
570C2×C4—121no—
571C2×C4—244no—
572C2×C4—244no—
573C2×C4C2×#23511yescharacteristic
574C2×C4#100411yescharacteristic
575C2×C4C2×#23911yescharacteristic
576C2×C4#100711yescharacteristic
577C2×C4D3×#4111yescharacteristic
578C2×C4D3×#4111yescharacteristic
579C8—22no—
580C8—41no—
581C8—121no—
582C8—121no—
583C2×C3#246611yescharacteristic
584C2×C3#311411yescharacteristic
585C2×C3#312411yescharacteristic
586C2×C3#331811yescharacteristic
587C2×C3#332711yescharacteristic
588C2×C3#334311yescharacteristic
589C2×C3#334711yescharacteristic
590D3—42no—
591D3—82no—
592C22—62no—
593C22—62no—
594C22—62no—
595C22—121no—
596C22—121no—
597C22—121no—
598C22—121no—
599C22—122no—
600C22—122no—
601C22—122no—
602C22—241no—
603C22—241no—
604C22—241no—
605C22—241no—
606C22#554711yescharacteristic
607C22#561011yescharacteristic
608C22#572611yescharacteristic
609C22D3×AGL(1,ℤ/8)11yescharacteristic
610C22#577411yescharacteristic
611C22#577711yescharacteristic
612C22#577911yescharacteristic
613C4—41no—
614C4—42no—
615C4—41no—
616C4—81no—
617C4—81no—
618C4—81no—
619C4—121no—
620C4—121no—
621C4—122no—
622C4—121no—
623C4—241no—
624C4—241no—
625C4#551411yescharacteristic
626C4#556811yescharacteristic
627C4#563811yescharacteristic
628C4#569111yescharacteristic
629C3#2704611yescharacteristic, complemented, Sylow
630C2—122no—
631C2—242no—
632C2#7601911yescharacteristic
633C2#7978211yescharacteristic
634C2#7981311yescharacteristic
635C2#8087611yescharacteristic
636C2#8091811yescharacteristic
637C2#8199011yescharacteristic
638C2#8199411yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1944C2
socleC23×C3C22×D4
prime radicalC2×#895C22
hypercenter#1944D3
hypocenterC3#27046
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1944

Complements

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

NH ≅ G/Ntype
1. #76235C2semidirect
3. C3×#1944C2semidirect
4. #70639C2semidirect
5. #70632C2semidirect
7. #70564C2semidirect
8. #69835C2semidirect
9. #69812C2semidirect
10. #69625C2semidirect
11. #69603C2semidirect
12. #68903C2semidirect
13. #68871C2semidirect
15. #68821C2semidirect
67. #1944D3semidirect
629. C3#27046semidirect

Maximal subgroups

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

subgroupindex
#762352
#762062
C3×#19442
#706392
#706322
#705812
#705642
#698352
#698122
#696252
#696032
#689032
#688712
#688222
#688212
#270463