Finite Group Databasebeta

← #10868085

Subgroups — #10868085

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#76433C211yesmaximal, characteristic
2#74235C211yesmaximal, characteristic, complemented
3C3×#2073C211yesmaximal, characteristic, complemented
4#70207C211yesmaximal, characteristic
5#70206C211yesmaximal, characteristic
6#70168C211yesmaximal, characteristic, complemented
7#70152C211yesmaximal, characteristic, complemented
8#69653C211yesmaximal, characteristic, complemented
9#69540C211yesmaximal, characteristic, complemented
10#68874C211yesmaximal, characteristic, complemented
11#68871C211yesmaximal, characteristic, complemented
12#68861C211yesmaximal, characteristic, complemented
13#68790C211yesmaximal, characteristic, complemented
14#68624C211yesmaximal, characteristic, complemented
15#68616C211yesmaximal, characteristic, complemented
16#27046—31nomaximal, Sylow
17C2×#938C2211yescharacteristic
18C2×D4×Dic3C2211yescharacteristic
19C2×D3×C4⋊C4C2211yescharacteristic
20C2×#67⋊C4—21no—
21C2×C3×#102C2211yescharacteristic
22C2×C3×#102C2211yescharacteristic
23C3×#390C2211yescharacteristic
24C3×#382C2211yescharacteristic
25C2×#832C2211yescharacteristic
26C2×#832C2211yescharacteristic
27C2×(C2×D3)⋊C8C2211yescharacteristic
28C2×(C2×D3)⋊C8C2211yescharacteristic
29C2×#818C2211yescharacteristic
30C2×#815C2211yescharacteristic
31#4987C2211yescharacteristic
32#4987C2211yescharacteristic
33#4984C2211yescharacteristic
34Dic3×C4⋊C4C2211yescharacteristic
35#4972C2211yescharacteristic
36C2×#808C2211yescharacteristic
37C2×#807C2211yescharacteristic
38C2×#807C2211yescharacteristic
39#4956—21no—
40#4946—21no—
41#4944—21no—
42C4×(C2×D3)⋊C4C2211yescharacteristic
43C4×C4⋊Dic3C2211yescharacteristic
44#4928C2211yescharacteristic
45C2×C4⋊(C3⋊C8)C2211yescharacteristic
46#4674—21no—
47#4669—21no—
48#4660—21no—
49#4651—21no—
50C3×#340C2211yescharacteristic
51C3×#340C2211yescharacteristic
52C3×#336C2211yescharacteristic
53#4551C2211yescharacteristic
54#4551C2211yescharacteristic
55#4551C2211yescharacteristic
56#4550C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#2495—31no—
61#2361—31no—
62#2073D311yescharacteristic, complemented
63#2067—31no—
64#2034—31no—
65#2033—31no—
66#1949—31no—
67#1944—31no—
68#1924—31no—
69#1921—31no—
70#1921—31no—
71#1862—31no—
72#1852—31no—
73#1823—31no—
74#1762—31no—
75C22×C3×D4C2311yescharacteristic
76C22×#67—21no—
77C23×Dic3—21no—
78C22×C4×D3—21no—
79C22×C4×D3C2311yescharacteristic
80C22×C3×C8C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C2×C3×C4⋊C4C2311yescharacteristic
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C22⋊C4—21no—
86C2×C3×C22⋊C4—21no—
87C2×C3×C42C2311yescharacteristic
88C2×#217—21no—
89C2×#217—21no—
90#938—22no—
91#938—22no—
92D4×Dic3—22no—
93D4×Dic3—41no—
94C2×(C2×D3)⋊C4—21no—
95C2×(C2×D3)⋊C4—21no—
96C2×(C2×D3)⋊C4D411yescharacteristic
97C2×(C2×D3)⋊C4D411yescharacteristic
98C2×C4⋊Dic3—21no—
99C2×C4⋊Dic3C2311yescharacteristic
100C2×C4⋊Dic3C2311yescharacteristic
101C2×C4⋊Dic3C2311yescharacteristic
102C2×C4⋊Dic3C2311yescharacteristic
103C2×Dic3⋊C4—21no—
104C2×Dic3⋊C4—21no—
105C2×Dic3⋊C4—21no—
106C2×C4×Dic3—21no—
107C2×C4×Dic3—21no—
108C2×C4×Dic3C2311yescharacteristic
109C22×C3⋊C8C2311yescharacteristic
110C22×C3⋊C8C2311yescharacteristic
111D3×C4⋊C4—22no—
112D3×C4⋊C4—41no—
113#67⋊C4—44no—
114C3×#102—22no—
115C3×#102—22no—
116C3×#95—21no—
117C3×#95—21no—
118#832—22no—
119#832—41no—
120C3⋊#95—21no—
121C3⋊#95—21no—
122C3⋊#95—21no—
123C3⋊#95—21no—
124C3⋊#95—21no—
125C3⋊#95—21no—
126C3⋊#95—21no—
127C3⋊#95D411yescharacteristic
128C3⋊#95D411yescharacteristic
129(C2×D3)⋊C8—22no—
130(C2×D3)⋊C8—41no—
131#818—22no—
132#815—41no—
133#808—22no—
134#807—22no—
135#807—41no—
136C4⋊(C3⋊C8)—22no—
137C2×#121—31no—
138C2×C4×D4—31no—
139C2×C4×D4—61no—
140C22×C4⋊C4—31no—
141C2×#107—31no—
142C2×#107—31no—
143C2×#107—31no—
144C2×#106—31no—
145C2×C4⋊C8—31no—
146C2×#102—31no—
147C2×#102—31no—
148C2×#102C2×D311yescharacteristic
149C2×#102C2×D311yescharacteristic
150C2×C22⋊C8—31no—
151C2×C22⋊C8—31no—
152C2×#97—31no—
153#400—61no—
154#397—61no—
155#390C2×D311yescharacteristic
156#389—31no—
157#389—31no—
158#389—31no—
159#388—61no—
160#388—61no—
161#386—61no—
162#385—61no—
163#383—61no—
164#382C2×D311yescharacteristic
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#336C2×D311yescharacteristic
180C24×C3—21no—
181C23×D3C2×D411yescharacteristic
182C2×C3×D4—21no—
183C2×C3×D4—21no—
184C2×C3×D4—21no—
185C2×C3×D4—21no—
186C2×C3×D4—21no—
187C2×C3×D4#4112yes—
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4—21no—
191C22×C3×C4C2×D411yescharacteristic
192C22×C3×C4C2×D411yescharacteristic
193C22×C3×C4C2411yescharacteristic
194C2×#67—24no—
195C2×#67—42no—
196C2×#67—42no—
197C22×Dic3—21no—
198C22×Dic3—21no—
199C22×Dic3—21no—
200C22×Dic3—21no—
201C22×Dic3—42no—
202C22×Dic3—42no—
203C22×Dic3—81no—
204C22×Dic3C2×D411yescharacteristic
205C22×Dic3C2×D411yescharacteristic
206C22×Dic3C2×D411yescharacteristic
207C2×C4×D3—21no—
208C2×C4×D3—21no—
209C2×C4×D3—21no—
210C2×C4×D3—21no—
211C2×C4×D3—21no—
212C2×C4×D3—21no—
213C2×C4×D3—42no—
214C2×C4×D3—42no—
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—21no—
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×C22⋊C4—42no—
229C3×C22⋊C4—81no—
230C3×C42—21no—
231C3×C42—21no—
232#217—24no—
233#217—42no—
234(C2×D3)⋊C4—22no—
235(C2×D3)⋊C4—41no—
236(C2×D3)⋊C4—42no—
237(C2×D3)⋊C4—81no—
238C4⋊Dic3—21no—
239C4⋊Dic3—21no—
240C4⋊Dic3—21no—
241C4⋊Dic3—21no—
242C4⋊Dic3—21no—
243C4⋊Dic3—21no—
244C4⋊Dic3—81no—
245C4⋊Dic3#4112yes—
246C4⋊Dic3#4112yes—
247Dic3⋊C4—42no—
248Dic3⋊C4—42no—
249Dic3⋊C4—81no—
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#4112yes—
260C22×D4—61no—
261C22×D4C22×D311yescharacteristic
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—31no—
275C2×C4⋊C4—31no—
276C2×C4⋊C4—31no—
277C2×C4⋊C4—31no—
278C2×C4⋊C4—62no—
279C2×C4⋊C4—61no—
280C2×C4⋊C4—61no—
281C2×C4⋊C4—61no—
282C2×C4⋊C4—61no—
283C2×C4⋊C4—121no—
284C2×C4⋊C4C22×D311yescharacteristic
285C2×C4⋊C4C22×D311yescharacteristic
286C2×C4⋊C4C22×D311yescharacteristic
287C2×C22⋊C4—21no—
288C2×C22⋊C4—21no—
289C2×C22⋊C4—31no—
290C2×C22⋊C4—31no—
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—62no—
300#107—62no—
301#107—121no—
302#106—62no—
303C4⋊C8—62no—
304#102—22no—
305#102—22no—
306#102—62no—
307#102—121no—
308C22⋊C8—62no—
309C22⋊C8—121no—
310#97—121no—
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×C3—22no—
323C23×C3—41no—
324C23×C3—41no—
325C23×C3—41no—
326C23×C3—41no—
327C23×C3—41no—
328C23×C3—41no—
329C23×C3C22×D411yescharacteristic
330C22×D3—21no—
331C22×D3—21no—
332C22×D3—21no—
333C22×D3—21no—
334C22×D3—21no—
335C22×D3—21no—
336C22×D3AGL(1,ℤ/8)12yes—
337C3×D4—22no—
338C3×D4—22no—
339C3×D4—41no—
340C3×D4—41no—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—21no—
344C2×C3×C4—21no—
345C2×C3×C4—21no—
346C2×C3×C4—21no—
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×C4—41no—
356C2×C3×C4C2×#4111yescharacteristic
357C2×C3×C4C2×#4111yescharacteristic
358C2×C3×C4C2×#4111yescharacteristic
359C2×C3×C4C2×#4111yescharacteristic
360C2×C3×C4H(5,𝔽2)11yescharacteristic
361C2×C3×C42−1+411yescharacteristic
362#67—48no—
363C2×Dic3—22no—
364C2×Dic3—21no—
365C2×Dic3—21no—
366C2×Dic3—21no—
367C2×Dic3—21no—
368C2×Dic3—22no—
369C2×Dic3—21no—
370C2×Dic3—21no—
371C2×Dic3—21no—
372C2×Dic3—21no—
373C2×Dic3—21no—
374C2×Dic3—41no—
375C2×Dic3—41no—
376C2×Dic3—41no—
377C2×Dic3—41no—
378C2×Dic3—41no—
379C2×Dic3—41no—
380C2×Dic3—41no—
381C2×Dic3—41no—
382C2×Dic3—41no—
383C2×Dic3—41no—
384C2×Dic3—84no—
385C4×D3—41no—
386C4×D3—41no—
387C4×D3—41no—
388C4×D3—41no—
389C4×D3—84no—
390C3×C8—41no—
391C3×C8—41no—
392C3⋊C8—22no—
393C3⋊C8—41no—
394C24—21no—
395C24—31no—
396C2×D4—21no—
397C2×D4—21no—
398C2×D4—21no—
399C2×D4—21no—
400C2×D4—21no—
401C2×D4—64no—
402C2×D4—122no—
403C2×D4—122no—
404C2×D4#23712yes—
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—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—21no—
436C2×C8—32no—
437C2×C8—61no—
438C2×C8—61no—
439C2×C8—61no—
440C2×C8—61no—
441C2×C8—61no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—32no—
449C4⋊C4—32no—
450C4⋊C4—61no—
451C4⋊C4—61no—
452C4⋊C4—61no—
453C4⋊C4—61no—
454C4⋊C4—61no—
455C4⋊C4—61no—
456C4⋊C4—122no—
457C4⋊C4—122no—
458C4⋊C4—241no—
459C4⋊C4—241no—
460C22⋊C4—42no—
461C22⋊C4—62no—
462C22⋊C4—64no—
463C22⋊C4—81no—
464C22⋊C4—121no—
465C22⋊C4—122no—
466C22⋊C4—122no—
467C22⋊C4—241no—
468C42—21no—
469C42—21no—
470C42—61no—
471C42—61no—
472C42—122no—
473C42—122no—
474C22×C3—42no—
475C22×C3—42no—
476C22×C3—42no—
477C22×C3—81no—
478C22×C3—81no—
479C22×C3—81no—
480C22×C3—81no—
481C22×C3#52511yescharacteristic
482C22×C3#54611yescharacteristic
483C22×C3#54711yescharacteristic
484C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
485C22×C3#57511yescharacteristic
486C22×C3#57611yescharacteristic
487C22×C3#57811yescharacteristic
488C2×D3—22no—
489C2×D3—22no—
490C2×D3—22no—
491C2×D3—41no—
492C2×D3—41no—
493C2×D3—41no—
494C2×D3—41no—
495C3×C4—41no—
496C3×C4—41no—
497C3×C4—81no—
498C3×C4—81no—
499C3×C4—81no—
500C3×C4#52911yescharacteristic
501C3×C4#53911yescharacteristic
502C3×C4#55311yescharacteristic
503C3×C4#55611yescharacteristic
504Dic3—42no—
505Dic3—41no—
506Dic3—42no—
507Dic3—41no—
508Dic3—41no—
509Dic3—81no—
510Dic3—81no—
511C23—22no—
512C23—32no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—61no—
520C23—61no—
521C23—61no—
522C23—61no—
523C23—61no—
524C23—61no—
525C23C2×D3×D411yescharacteristic
526D4—22no—
527D4—22no—
528D4—41no—
529D4—41no—
530D4—128no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—41no—
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—61no—
547C2×C4—61no—
548C2×C4—61no—
549C2×C4—62no—
550C2×C4—61no—
551C2×C4—61no—
552C2×C4—61no—
553C2×C4—61no—
554C2×C4—61no—
555C2×C4—61no—
556C2×C4—62no—
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—121no—
570C2×C4—121no—
571C2×C4—244no—
572C2×C4—244no—
573C2×C4C2×#23711yescharacteristic
574C2×C4D3×#4111yescharacteristic
575C2×C4D3×#4111yescharacteristic
576C2×C4D3×#4111yescharacteristic
577C2×C4#100911yescharacteristic
578C2×C4#101011yescharacteristic
579C8—41no—
580C8—41no—
581C8—62no—
582C8—121no—
583C2×C3—82no—
584C2×C3#246611yescharacteristic
585C2×C3#311411yescharacteristic
586C2×C3#312411yescharacteristic
587C2×C3#331811yescharacteristic
588C2×C3#332711yescharacteristic
589C2×C3#334311yescharacteristic
590C2×C3#334711yescharacteristic
591D3—42no—
592C22—42no—
593C22—42no—
594C22—42no—
595C22—62no—
596C22—62no—
597C22—62no—
598C22—81no—
599C22—81no—
600C22—81no—
601C22—81no—
602C22—121no—
603C22—121no—
604C22—121no—
605C22—121no—
606C22#560611yescharacteristic
607C22#565511yescharacteristic
608C22#567611yescharacteristic
609C22#576811yescharacteristic
610C22#577111yescharacteristic
611C22D3×AGL(1,ℤ/8)11yescharacteristic
612C22#577411yescharacteristic
613C4—41no—
614C4—41no—
615C4—81no—
616C4—81no—
617C4—81no—
618C4—121no—
619C4—122no—
620C4—122no—
621C4—121no—
622C4—121no—
623C4—241no—
624C4—241no—
625C4#564911yescharacteristic
626C4#566511yescharacteristic
627C4#570211yescharacteristic
628C4#570911yescharacteristic
629C3#2704611yescharacteristic, complemented, Sylow
630C2—82no—
631C2—122no—
632C2#7747111yescharacteristic
633C2#8050711yescharacteristic
634C2#8051011yescharacteristic
635C2#8139011yescharacteristic
636C2#8139211yescharacteristic
637C2#8165911yescharacteristic
638C2#8172711yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2073C2
socleC23×C3C22×D4
prime radicalC2×#938C22
hypercenter#2073D3
hypocenterC3#27046
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2073

Complements

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

NH ≅ G/Ntype
2. #74235C2semidirect
3. C3×#2073C2semidirect
6. #70168C2semidirect
7. #70152C2semidirect
8. #69653C2semidirect
9. #69540C2semidirect
10. #68874C2semidirect
11. #68871C2semidirect
12. #68861C2semidirect
13. #68790C2semidirect
14. #68624C2semidirect
15. #68616C2semidirect
62. #2073D3semidirect
629. C3#27046semidirect

Maximal subgroups

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

subgroupindex
#764332
#742352
C3×#20732
#702072
#702062
#701682
#701522
#696532
#695402
#688742
#688712
#688612
#687902
#686242
#686162
#270463