Finite Group Databasebeta

← #10876104

Subgroups — #10876104

Order 768.

772 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#76342C211yesmaximal, characteristic, complemented
2#74228C211yesmaximal, characteristic, complemented
3C3×#2073C211yesmaximal, characteristic, complemented
4#70208C211yesmaximal, characteristic
5#70208C211yesmaximal, characteristic
6#70188C211yesmaximal, characteristic, complemented
7#70180C211yesmaximal, characteristic, complemented
8#69657C211yesmaximal, characteristic
9#69542C211yesmaximal, characteristic, complemented
10#68893C211yesmaximal, characteristic, complemented
11#68814C211yesmaximal, characteristic, complemented
12#68812C211yesmaximal, characteristic, complemented
13#68782C211yesmaximal, characteristic, complemented
14#68670C211yesmaximal, characteristic, complemented
15#68613C211yesmaximal, characteristic, complemented
16#27581—31nomaximal, Sylow
17C22×#213C2211yescharacteristic
18C2×#893C2211yescharacteristic
19C2×#893C2211yescharacteristic
20C2×C4×D12C2211yescharacteristic
21C2×C3×#102C2211yescharacteristic
22C2×C3×#102C2211yescharacteristic
23C3×#390C2211yescharacteristic
24C3×#382C2211yescharacteristic
25#5221C2211yescharacteristic
26#5221C2211yescharacteristic
27C2×#832C2211yescharacteristic
28C2×#821C2211yescharacteristic
29C2×#821C2211yescharacteristic
30C2×#818C2211yescharacteristic
31C2×#815C2211yescharacteristic
32C2×Dic3⋊C8C2211yescharacteristic
33#4977C2211yescharacteristic
34Dic3×C4⋊C4C2211yescharacteristic
35#4972C2211yescharacteristic
36#4972C2211yescharacteristic
37C2×#809C2211yescharacteristic
38#4953—21no—
39#4952—21no—
40#4951—21no—
41#4944—21no—
42C4×C4⋊Dic3C2211yescharacteristic
43#4928C2211yescharacteristic
44#4668—21no—
45#4668—21no—
46#4667—21no—
47D3×#95—21no—
48C3×#340C2211yescharacteristic
49C3×#340C2211yescharacteristic
50C3×#336C2211yescharacteristic
51#4551C2211yescharacteristic
52#4550C2211yescharacteristic
53#4550C2211yescharacteristic
54#4529C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4524C2211yescharacteristic
60#2513—31no—
61#2349—31no—
62#2101—31no—
63#2101—31no—
64#2073D311yescharacteristic, complemented
65#2067—31no—
66#1936—31no—
67#1918—31no—
68#1917—31no—
69#1879—31no—
70#1879—31no—
71#1879—31no—
72#1862—31no—
73#1804—31no—
74#1762—31no—
75C22×C3×D4C2311yescharacteristic
76C22×D12C2311yescharacteristic
77C22×C4×D3—21no—
78C22×C4×D3—21no—
79C22×C4×D3—21no—
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—
90C2×#217—21no—
91C2×#217—21no—
92C2×#213—22no—
93C2×#213—22no—
94C2×#213—22no—
95C2×(C2×D3)⋊C4—21no—
96C2×(C2×D3)⋊C4—21no—
97C2×(C2×D3)⋊C4—21no—
98C2×C4⋊Dic3D411yescharacteristic
99C2×C4⋊Dic3D411yescharacteristic
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×Dic3C2311yescharacteristic
108C2×C4×Dic3C2311yescharacteristic
109C22×C3⋊C8C2311yescharacteristic
110C22×C3⋊C8C2311yescharacteristic
111#893—22no—
112#893—22no—
113#893—41no—
114#893—41no—
115C4×D12—22no—
116C4×D12—22no—
117C3×#102—41no—
118C3×#102—41no—
119C3×#95—21no—
120C3×#95—21no—
121#832—22no—
122C3⋊#95—21no—
123C3⋊#95—21no—
124C3⋊#95—21no—
125C3⋊#95—21no—
126C3⋊#95—21no—
127C3⋊#95—21no—
128C3⋊#95D411yescharacteristic
129C3⋊#95D411yescharacteristic
130#821—22no—
131#821—22no—
132#818—22no—
133#815—41no—
134Dic3⋊C8—41no—
135#809—22no—
136C22×D8—31no—
137C2×C4×D4—31no—
138C2×C4×D4—31no—
139C2×C4×D4—31no—
140C2×#107—31no—
141C2×C4⋊C8—31no—
142C2×#102—31no—
143C2×#102—31no—
144C2×#102—31no—
145C2×#102—31no—
146C2×#102C2×D311yescharacteristic
147C2×#102C2×D311yescharacteristic
148C2×#97—31no—
149#397—61no—
150#397—61no—
151#390—31no—
152#390—31no—
153#390C2×D311yescharacteristic
154#389—31no—
155#389—31no—
156#389—31no—
157#388—61no—
158#388—61no—
159#385—61no—
160#385—61no—
161#382—31no—
162#382C2×D311yescharacteristic
163#380—61no—
164C4×C4⋊C4—31no—
165C4×C4⋊C4—31no—
166C2×#95—61no—
167#340—31no—
168#340—31no—
169#340—31no—
170#340—31no—
171#340—31no—
172#340—31no—
173#340C2×D311yescharacteristic
174#340C2×D311yescharacteristic
175#336—31no—
176#336—31no—
177#336—31no—
178#336C2×D311yescharacteristic
179C24×C3—21no—
180C23×D3—21no—
181C2×C3×D4—21no—
182C2×C3×D4—21no—
183C2×C3×D4—21no—
184C2×C3×D4—21no—
185C2×C3×D4—21no—
186C2×C3×D4—21no—
187C22×C3×C4—21no—
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4C2×D411yescharacteristic
191C22×C3×C4C2×D411yescharacteristic
192C22×C3×C4C2411yescharacteristic
193C22×Dic3—21no—
194C22×Dic3—21no—
195C22×Dic3—21no—
196C22×Dic3C2×D411yescharacteristic
197C22×Dic3C2×D411yescharacteristic
198C22×Dic3C2×D411yescharacteristic
199C22×Dic3C2×D411yescharacteristic
200C2×D12—21no—
201C2×D12—21no—
202C2×D12—21no—
203C2×D12#4112yes—
204C2×D12#4112yes—
205C2×D12#4112yes—
206C2×C4×D3—24no—
207C2×C4×D3—42no—
208C2×C4×D3—42no—
209C2×C4×D3—42no—
210C2×C4×D3—42no—
211C2×C4×D3—81no—
212C2×C4×D3—81no—
213C2×C4×D3—81no—
214C2×C4×D3—81no—
215C2×C3×C8—21no—
216C2×C3×C8—21no—
217C2×C3×C8—21no—
218C2×C3×C8—21no—
219C2×C3×C8—21no—
220C2×C3×C8—21no—
221C3×C4⋊C4—21no—
222C3×C4⋊C4—21no—
223C3×C4⋊C4—21no—
224C3×C4⋊C4—21no—
225C3×C4⋊C4—21no—
226C3×C4⋊C4—21no—
227C3×C22⋊C4—42no—
228C3×C22⋊C4—81no—
229C3×C42—21no—
230C3×C42—21no—
231#217—42no—
232#217—81no—
233#217—81no—
234#217—81no—
235#213—44no—
236(C2×D3)⋊C4—24no—
237(C2×D3)⋊C4—42no—
238(C2×D3)⋊C4—42no—
239C4⋊Dic3—21no—
240C4⋊Dic3—21no—
241C4⋊Dic3—21no—
242C4⋊Dic3—21no—
243C4⋊Dic3—21no—
244C4⋊Dic3—41no—
245C4⋊Dic3—41no—
246C4⋊Dic3#4112yes—
247Dic3⋊C4—81no—
248Dic3⋊C4—81no—
249Dic3⋊C4—81no—
250C4×Dic3—21no—
251C4×Dic3—21no—
252C4×Dic3—21no—
253C4×Dic3—21no—
254C4×Dic3—81no—
255C2×C3⋊C8—21no—
256C2×C3⋊C8—21no—
257C2×C3⋊C8—21no—
258C2×C3⋊C8—21no—
259C2×C3⋊C8—21no—
260C2×C3⋊C8—21no—
261C22×D4—31no—
262C22×D4C22×D311yescharacteristic
263C23×C4—61no—
264C23×C4—61no—
265C23×C4—61no—
266C2×D8—62no—
267C2×D8—62no—
268C2×D8—62no—
269C22×C8—31no—
270C22×C8—31no—
271C22×C8C22×D311yescharacteristic
272C22×C8C22×D311yescharacteristic
273C4×D4—62no—
274C4×D4—62no—
275C4×D4—62no—
276C4×D4—62no—
277C4×D4—121no—
278C4×D4—121no—
279C2×C4⋊C4—31no—
280C2×C4⋊C4—31no—
281C2×C4⋊C4—31no—
282C2×C4⋊C4—31no—
283C2×C4⋊C4—31no—
284C2×C4⋊C4—61no—
285C2×C4⋊C4—61no—
286C2×C4⋊C4—61no—
287C2×C4⋊C4C22×D311yescharacteristic
288C2×C4⋊C4C22×D311yescharacteristic
289C2×C4⋊C4C22×D311yescharacteristic
290C2×C22⋊C4—21no—
291C2×C22⋊C4—21no—
292C2×C22⋊C4—61no—
293C2×C22⋊C4—61no—
294C2×C22⋊C4—61no—
295C2×C22⋊C4—61no—
296C2×C22⋊C4—61no—
297C2×C22⋊C4—61no—
298C2×C22⋊C4—61no—
299C2×C42—31no—
300C2×C42—31no—
301C2×C42—61no—
302C2×C42C22×D311yescharacteristic
303#107—62no—
304C4⋊C8—121no—
305#102—41no—
306#102—41no—
307#102—62no—
308#102—62no—
309#102—62no—
310#102—62no—
311#97—121no—
312#95—21no—
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×C3—41no—
323C23×C3—41no—
324C23×C3—41no—
325C23×C3—41no—
326C23×C3—41no—
327C23×C3—41no—
328C23×C3—41no—
329C23×C3C22×D411yescharacteristic
330C22×D3—22no—
331C22×D3—22no—
332C22×D3—22no—
333C22×D3—41no—
334C22×D3—41no—
335C22×D3—41no—
336C22×D3—41no—
337C3×D4—41no—
338C3×D4—41no—
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
362C2×Dic3—22no—
363C2×Dic3—21no—
364C2×Dic3—21no—
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—41no—
376C2×Dic3—41no—
377C2×Dic3—41no—
378C2×Dic3—41no—
379C2×Dic3—41no—
380C2×Dic3—41no—
381C2×Dic3—41no—
382C2×Dic3—41no—
383D12—22no—
384D12—22no—
385D12—22no—
386D12#12614yes—
387C4×D3—84no—
388C4×D3—84no—
389C4×D3—84no—
390C3×C8—41no—
391C3×C8—41no—
392C3⋊C8—41no—
393C3⋊C8—41no—
394C24—21no—
395C24—61no—
396C2×D4—21no—
397C2×D4—21no—
398C2×D4—21no—
399C2×D4—21no—
400C2×D4—21no—
401C2×D4—21no—
402C2×D4—32no—
403C2×D4—32no—
404C2×D4—32no—
405C2×D4—61no—
406C2×D4—61no—
407C2×D4—61no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—31no—
412C22×C4—31no—
413C22×C4—31no—
414C22×C4—31no—
415C22×C4—64no—
416C22×C4—61no—
417C22×C4—61no—
418C22×C4—61no—
419C22×C4—122no—
420C22×C4—122no—
421C22×C4—122no—
422C22×C4—122no—
423C22×C4—241no—
424C22×C4—241no—
425C22×C4—241no—
426C22×C4—241no—
427C22×C4D3×D411yescharacteristic
428C22×C4D3×D411yescharacteristic
429C22×C4C23×D311yescharacteristic
430D8—124no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—61no—
438C2×C8—61no—
439C2×C8—61no—
440C2×C8—61no—
441C2×C8—61no—
442C2×C8—61no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—32no—
450C4⋊C4—61no—
451C4⋊C4—61no—
452C4⋊C4—61no—
453C4⋊C4—61no—
454C4⋊C4—61no—
455C4⋊C4—121no—
456C4⋊C4—121no—
457C4⋊C4—241no—
458C4⋊C4—241no—
459C4⋊C4—241no—
460C22⋊C4—42no—
461C22⋊C4—64no—
462C22⋊C4—81no—
463C22⋊C4—122no—
464C22⋊C4—122no—
465C22⋊C4—122no—
466C22⋊C4—241no—
467C22⋊C4—241no—
468C22⋊C4—241no—
469C42—21no—
470C42—21no—
471C42—61no—
472C42—61no—
473C42—61no—
474C42—61no—
475C42—241no—
476C22×C3—81no—
477C22×C3—81no—
478C22×C3—81no—
479C22×C3—81no—
480C22×C3—81no—
481C22×C3—81no—
482C22×C3—81no—
483C22×C3#52511yescharacteristic
484C22×C3#54611yescharacteristic
485C22×C3#54811yescharacteristic
486C22×C3#57511yescharacteristic
487C22×C3#57511yescharacteristic
488C22×C3#57611yescharacteristic
489C22×C3#57611yescharacteristic
490C2×D3—24no—
491C2×D3—42no—
492C2×D3—42no—
493C2×D3—42no—
494C2×D3—42no—
495C2×D3—42no—
496C2×D3—42no—
497C3×C4—41no—
498C3×C4—41no—
499C3×C4—81no—
500C3×C4—81no—
501C3×C4—81no—
502C3×C4C2×#12611yescharacteristic
503C3×C4#53911yescharacteristic
504C3×C4#55211yescharacteristic
505C3×C4#55311yescharacteristic
506Dic3—42no—
507Dic3—41no—
508Dic3—41no—
509Dic3—41no—
510Dic3—41no—
511Dic3—81no—
512Dic3—81no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—62no—
521C23—62no—
522C23—62no—
523C23—121no—
524C23—121no—
525C23—121no—
526C23—121no—
527C23C2×D3×D411yescharacteristic
528D4—34no—
529D4—41no—
530D4—41no—
531D4—41no—
532D4—41no—
533D4—62no—
534D4—62no—
535D4—62no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
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—41no—
551C2×C4—62no—
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—61no—
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×C4—244no—
574C2×C4—244no—
575C2×C4#100411yescharacteristic
576C2×C4D3×#4111yescharacteristic
577C2×C4D3×#4111yescharacteristic
578C2×C4D3×#4111yescharacteristic
579C2×C4D3×#4111yescharacteristic
580C2×C4#101011yescharacteristic
581C8—41no—
582C8—41no—
583C8—121no—
584C8—121no—
585C2×C3—161no—
586C2×C3#248111yescharacteristic
587C2×C3#311111yescharacteristic
588C2×C3#312211yescharacteristic
589C2×C3#331911yescharacteristic
590C2×C3#332211yescharacteristic
591C2×C3#337011yescharacteristic
592C2×C3#337011yescharacteristic
593D3—44no—
594C22—64no—
595C22—81no—
596C22—81no—
597C22—81no—
598C22—81no—
599C22—81no—
600C22—81no—
601C22—81no—
602C22—122no—
603C22—122no—
604C22—122no—
605C22—122no—
606C22—122no—
607C22—122no—
608C22#561211yescharacteristic
609C22#565611yescharacteristic
610C22#567711yescharacteristic
611C22#576811yescharacteristic
612C22#576911yescharacteristic
613C22#577311yescharacteristic
614C22#577411yescharacteristic
615C4—41no—
616C4—41no—
617C4—81no—
618C4—81no—
619C4—81no—
620C4—121no—
621C4—121no—
622C4—121no—
623C4—121no—
624C4—122no—
625C4—241no—
626C4—241no—
627C4#566011yescharacteristic
628C4#566111yescharacteristic
629C4D3×#12611yescharacteristic
630C4#570811yescharacteristic
631C3#2758111yescharacteristic, complemented, Sylow
632C2—124no—
633C2—161no—
634C2#7748311yescharacteristic
635C2#8056311yescharacteristic
636C2#8056511yescharacteristic
637C2#8135011yescharacteristic
638C2#8135411yescharacteristic
639C2#8166011yescharacteristic
640C2#8171011yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2073C2
socleC23×C3C22×D4
prime radicalC22×#213C22
hypercenter#2073D3
hypocenterC3#27581
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
1. #76342C2semidirect
2. #74228C2semidirect
3. C3×#2073C2semidirect
6. #70188C2semidirect
7. #70180C2semidirect
9. #69542C2semidirect
10. #68893C2semidirect
11. #68814C2semidirect
12. #68812C2semidirect
13. #68782C2semidirect
14. #68670C2semidirect
15. #68613C2semidirect
64. #2073D3semidirect
631. C3#27581semidirect

Maximal subgroups

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

subgroupindex
#763422
#742282
C3×#20732
#702082
#701882
#701802
#696572
#695422
#688932
#688142
#688122
#687822
#686702
#686132
#275813