Finite Group Databasebeta

← #10868081

Subgroups — #10868081

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#82163C211yesmaximal, characteristic
2#73921C211yesmaximal, characteristic, complemented
3C3×#1852C211yesmaximal, characteristic, complemented
4#71553C211yesmaximal, characteristic, complemented
5#70639C211yesmaximal, characteristic, complemented
6#70638C211yesmaximal, characteristic, complemented
7#70614C211yesmaximal, characteristic, complemented
8#70610C211yesmaximal, characteristic, complemented
9#69935C211yesmaximal, characteristic
10#69915C211yesmaximal, characteristic, complemented
11#69523C211yesmaximal, characteristic
12#68903C211yesmaximal, characteristic, complemented
13#68797C211yesmaximal, characteristic, complemented
14#68790C211yesmaximal, characteristic, complemented
15#68782C211yesmaximal, characteristic, complemented
16#27046—31nomaximal, Sylow
17C22×C3×C4⋊C4C2211yescharacteristic
18C2×#930C2211yescharacteristic
19C2×C4×#67—21no—
20C2×#893C2211yescharacteristic
21C2×C3×#107C2211yescharacteristic
22C2×C3×#107C2211yescharacteristic
23C2×C3×C22⋊C8C2211yescharacteristic
24C3×#380C2211yescharacteristic
25#5213—21no—
26#5212C2211yescharacteristic
27C4×#217C2211yescharacteristic
28C2×#830C2211yescharacteristic
29C2×#821C2211yescharacteristic
30C2×#821C2211yescharacteristic
31C2×#817C2211yescharacteristic
32C2×#815C2211yescharacteristic
33C2×Dic3⋊C8C2211yescharacteristic
34#4990—21no—
35#4989—21no—
36#4988C2211yescharacteristic
37#4984C2211yescharacteristic
38#4977C2211yescharacteristic
39Dic3×C4⋊C4C2211yescharacteristic
40Dic3×C4⋊C4C2211yescharacteristic
41#4972C2211yescharacteristic
42#4972C2211yescharacteristic
43C2×#809C2211yescharacteristic
44C2×#809C2211yescharacteristic
45C2×#807C2211yescharacteristic
46#4952—21no—
47#4951—21no—
48#4659—21no—
49#4651—21no—
50C3×#340C2211yescharacteristic
51C3×#340C2211yescharacteristic
52#4551C2211yescharacteristic
53#4550C2211yescharacteristic
54#4550C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#2495—31no—
61#2361—31no—
62#2073—31no—
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#1852D311yescharacteristic, complemented
73#1823—31no—
74#1762—31no—
75C23×C3×C4—21no—
76C23×C3×C4C2311yescharacteristic
77C22×#67—21no—
78C22×D12C2311yescharacteristic
79C22×C4×D3—21no—
80C22×C3×C8C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C2×C3×C4⋊C4—22no—
83C2×C3×C4⋊C4—21no—
84C2×C3×C4⋊C4—41no—
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C22⋊C4—21no—
90C2×#217—21no—
91C2×#217D411yescharacteristic
92C2×#217D411yescharacteristic
93#930—22no—
94#930—22no—
95C4×#67—44no—
96C2×(C2×D3)⋊C4—21no—
97C2×(C2×D3)⋊C4—21no—
98C2×(C2×D3)⋊C4—21no—
99C2×(C2×D3)⋊C4—21no—
100C2×C4⋊Dic3—21no—
101C2×C4⋊Dic3C2311yescharacteristic
102C2×C4⋊Dic3C2311yescharacteristic
103C2×C4⋊Dic3C2311yescharacteristic
104C2×Dic3⋊C4—21no—
105C2×Dic3⋊C4—21no—
106C2×C4×Dic3—21no—
107C2×C4×Dic3—21no—
108C2×C4×Dic3C2311yescharacteristic
109C2×C4×Dic3C2311yescharacteristic
110C22×C3⋊C8C2311yescharacteristic
111C22×C3⋊C8C2311yescharacteristic
112#893—22no—
113#893—41no—
114C3×#107—22no—
115C3×#107—41no—
116C3×C22⋊C8—22no—
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#830—41no—
129#821—22no—
130#821—22no—
131#817—22no—
132#815—41no—
133Dic3⋊C8—22no—
134#809—22no—
135#809—41no—
136#807—22no—
137C2×#121—31no—
138C2×C4×D4—31no—
139C2×C4×D4—61no—
140C22×C4⋊C4C2×D311yescharacteristic
141C2×#107—31no—
142C2×#107C2×D311yescharacteristic
143C2×#107C2×D311yescharacteristic
144C2×#106—31no—
145C2×C4⋊C8—31no—
146C2×#102—31no—
147C2×#102—31no—
148C2×#102—31no—
149C2×#102—31no—
150C2×C22⋊C8—31no—
151C2×C22⋊C8C2×D311yescharacteristic
152C2×#97—31no—
153#400—61no—
154#397—61no—
155#390—31no—
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#382—31no—
165#380—31no—
166#380C2×D311yescharacteristic
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—
180C24×C3C2×D411yescharacteristic
181C23×D3—21no—
182C22×C3×C4—21no—
183C22×C3×C4—21no—
184C22×C3×C4—21no—
185C22×C3×C4—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—42no—
193C22×C3×C4—42no—
194C22×C3×C4—81no—
195C22×C3×C4C2×D411yescharacteristic
196C22×C3×C4C2411yescharacteristic
197C2×#67—24no—
198C2×#67—42no—
199C2×#67—42no—
200C22×Dic3—21no—
201C22×Dic3—21no—
202C22×Dic3—21no—
203C22×Dic3C2×D411yescharacteristic
204C22×Dic3C2×D411yescharacteristic
205C22×Dic3C2×D411yescharacteristic
206C22×Dic3C2×D411yescharacteristic
207C2×D12—21no—
208C2×D12—21no—
209C2×D12—21no—
210C2×D12—21no—
211C2×D12—21no—
212C2×D12#4112yes—
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#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#4112yes—
231C3×C22⋊C4—42no—
232#217—22no—
233#217—41no—
234#217—81no—
235(C2×D3)⋊C4—24no—
236(C2×D3)⋊C4—42no—
237(C2×D3)⋊C4—42no—
238(C2×D3)⋊C4—81no—
239C4⋊Dic3—21no—
240C4⋊Dic3—21no—
241C4⋊Dic3—21no—
242C4⋊Dic3—21no—
243C4⋊Dic3—21no—
244C4⋊Dic3—81no—
245C4⋊Dic3#4112yes—
246Dic3⋊C4—42no—
247Dic3⋊C4—81no—
248C4×Dic3—21no—
249C4×Dic3—21no—
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—21no—
263C23×C4—61no—
264C23×C4C22×D311yescharacteristic
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—22no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—31no—
277C2×C4⋊C4—31no—
278C2×C4⋊C4—31no—
279C2×C4⋊C4—41no—
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—21no—
288C2×C22⋊C4—31no—
289C2×C22⋊C4—31no—
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—31no—
297C2×C42—61no—
298C2×C42—61no—
299#107—22no—
300#107—41no—
301#107—62no—
302#106—62no—
303C4⋊C8—62no—
304#102—62no—
305#102—62no—
306#102—62no—
307#102—121no—
308C22⋊C8—22no—
309C22⋊C8—121no—
310#97—121no—
311#95—21no—
312#95—31no—
313#95—31no—
314#95—61no—
315#95—61no—
316#95—61no—
317#95—61no—
318#95—61no—
319#95—61no—
320#95—61no—
321#95—61no—
322C23×C3—21no—
323C23×C3—21no—
324C23×C3—21no—
325C23×C3—21no—
326C23×C3—21no—
327C23×C3—21no—
328C23×C3AGL(1,ℤ/8)12yes—
329C23×C3C22×D411yescharacteristic
330C22×D3—22no—
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—22no—
339C2×C3×C4—21no—
340C2×C3×C4—21no—
341C2×C3×C4—41no—
342C2×C3×C4—41no—
343C2×C3×C4—41no—
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×C4—41no—
356C2×C3×C4—84no—
357C2×C3×C4C2×#4111yescharacteristic
358C2×C3×C4C2×#4111yescharacteristic
359C2×C3×C4C2×#4111yescharacteristic
360C2×C3×C4C2×#4111yescharacteristic
361C2×C3×C4H(5,𝔽2)11yescharacteristic
362C2×C3×C42−1+411yescharacteristic
363#67—48no—
364C2×Dic3—22no—
365C2×Dic3—21no—
366C2×Dic3—21no—
367C2×Dic3—21no—
368C2×Dic3—21no—
369C2×Dic3—21no—
370C2×Dic3—21no—
371C2×Dic3—21no—
372C2×Dic3—21no—
373C2×Dic3—21no—
374C2×Dic3—21no—
375C2×Dic3—21no—
376C2×Dic3—21no—
377C2×Dic3—41no—
378C2×Dic3—41no—
379C2×Dic3—41no—
380C2×Dic3—41no—
381C2×Dic3—41no—
382C2×Dic3—41no—
383C2×Dic3—41no—
384C2×Dic3—41no—
385D12—22no—
386D12—22no—
387D12—41no—
388D12—41no—
389C4×D3—84no—
390C3×C8—22no—
391C3×C8—41no—
392C3⋊C8—41no—
393C3⋊C8—41no—
394C24—61no—
395C24D3×D411yescharacteristic
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—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—31no—
416C22×C4—31no—
417C22×C4—31no—
418C22×C4—31no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—61no—
422C22×C4—61no—
423C22×C4—61no—
424C22×C4—81no—
425C22×C4—122no—
426C22×C4—122no—
427C22×C4—241no—
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#23512yes—
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—122no—
457C4⋊C4—241no—
458C4⋊C4—241no—
459C4⋊C4#23712yes—
460C22⋊C4—42no—
461C22⋊C4—64no—
462C22⋊C4—62no—
463C22⋊C4—122no—
464C22⋊C4—122no—
465C22⋊C4—121no—
466C22⋊C4—241no—
467C22⋊C4—241no—
468C42—61no—
469C42—61no—
470C42—61no—
471C42—61no—
472C42—122no—
473C42—122no—
474C22×C3—22no—
475C22×C3—22no—
476C22×C3—22no—
477C22×C3—41no—
478C22×C3—41no—
479C22×C3—41no—
480C22×C3—41no—
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—42no—
489C2×D3—42no—
490C2×D3—42no—
491C2×D3—81no—
492C2×D3—81no—
493C2×D3—81no—
494C2×D3—81no—
495C3×C4—41no—
496C3×C4—42no—
497C3×C4—81no—
498C3×C4—81no—
499C3×C4—81no—
500C3×C4#52911yescharacteristic
501C3×C4#53911yescharacteristic
502C3×C4#55311yescharacteristic
503C3×C4#55611yescharacteristic
504Dic3—41no—
505Dic3—41no—
506Dic3—42no—
507Dic3—41no—
508Dic3—41no—
509Dic3—81no—
510Dic3—81no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23—62no—
518C23—121no—
519C23—121no—
520C23—121no—
521C23—121no—
522C23—121no—
523C23—121no—
524C23#91112yes—
525C23C2×D3×D411yescharacteristic
526D4—62no—
527D4—62no—
528D4—128no—
529D4—121no—
530D4—121no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—22no—
534C2×C4—21no—
535C2×C4—41no—
536C2×C4—41no—
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—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—62no—
551C2×C4—61no—
552C2×C4—61no—
553C2×C4—61no—
554C2×C4—61no—
555C2×C4—61no—
556C2×C4—61no—
557C2×C4—61no—
558C2×C4—61no—
559C2×C4—61no—
560C2×C4—61no—
561C2×C4—61no—
562C2×C4—61no—
563C2×C4—84no—
564C2×C4—121no—
565C2×C4—121no—
566C2×C4—121no—
567C2×C4—121no—
568C2×C4—121no—
569C2×C4—121no—
570C2×C4—121no—
571C2×C4—121no—
572C2×C4—244no—
573C2×C4C2×#23511yescharacteristic
574C2×C4C2×#23711yescharacteristic
575C2×C4#100411yescharacteristic
576C2×C4#100711yescharacteristic
577C2×C4D3×#4111yescharacteristic
578C2×C4D3×#4111yescharacteristic
579C8—22no—
580C8—41no—
581C8—121no—
582C8—121no—
583C2×C3—42no—
584C2×C3#246611yescharacteristic
585C2×C3#311411yescharacteristic
586C2×C3#312411yescharacteristic
587C2×C3#331811yescharacteristic
588C2×C3#332711yescharacteristic
589C2×C3#334311yescharacteristic
590C2×C3#334711yescharacteristic
591D3—82no—
592C22—22no—
593C22—22no—
594C22—22no—
595C22—41no—
596C22—41no—
597C22—41no—
598C22—41no—
599C22—122no—
600C22—122no—
601C22—122no—
602C22—241no—
603C22—241no—
604C22—241no—
605C22—241no—
606C22#550711yescharacteristic
607C22#559111yescharacteristic
608C22#565611yescharacteristic
609C22C2×#91111yescharacteristic
610C22#576611yescharacteristic
611C22#577411yescharacteristic
612C22#577711yescharacteristic
613C4—42no—
614C4—41no—
615C4—81no—
616C4—81no—
617C4—81no—
618C4—121no—
619C4—121no—
620C4—122no—
621C4—121no—
622C4—121no—
623C4—241no—
624C4—241no—
625C4#554011yescharacteristic
626C4#555711yescharacteristic
627C4#563811yescharacteristic
628C4#564111yescharacteristic
629C3#2704611yescharacteristic, complemented, Sylow
630C2—42no—
631C2—242no—
632C2#7443511yescharacteristic
633C2#7915811yescharacteristic
634C2#7922711yescharacteristic
635C2#8001711yescharacteristic
636C2#8019911yescharacteristic
637C2#8127111yescharacteristic
638C2#8145711yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1852C2
socleC23×C3C22×D4
prime radicalC2×#930C22
hypercenter#1852D3
hypocenterC3#27046
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. #73921C2semidirect
3. C3×#1852C2semidirect
4. #71553C2semidirect
5. #70639C2semidirect
6. #70638C2semidirect
7. #70614C2semidirect
8. #70610C2semidirect
10. #69915C2semidirect
12. #68903C2semidirect
13. #68797C2semidirect
14. #68790C2semidirect
15. #68782C2semidirect
72. #1852D3semidirect
629. C3#27046semidirect

Maximal subgroups

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

subgroupindex
#821632
#739212
C3×#18522
#715532
#706392
#706382
#706142
#706102
#699352
#699152
#695232
#689032
#687972
#687902
#687822
#270463