Finite Group Databasebeta

← #10855225

Subgroups — #10855225

Order 768.

986 conjugacy classes of subgroups.

The subgroup lattice has 732 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4872C211yesmaximal, characteristic, complemented
2C2×#4843C211yesmaximal, characteristic
3#76203C211yesmaximal, characteristic, complemented
4#76004C211yesmaximal, characteristic, complemented
5C3×#2087C211yesmaximal, characteristic, complemented
6#70640C211yesmaximal, characteristic, complemented
7#70447C211yesmaximal, characteristic, complemented
8#69627C211yesmaximal, characteristic, complemented
9#69626C211yesmaximal, characteristic, complemented
10#68903C211yesmaximal, characteristic, complemented
11#68897C211yesmaximal, characteristic, complemented
12#68897C211yesmaximal, characteristic, complemented
13#68897C211yesmaximal, characteristic, complemented
14#68895C211yesmaximal, characteristic, complemented
15#68853C211yesmaximal, characteristic, complemented
16#26162—31nomaximal, Sylow
17C22×D24C2211yescharacteristic
18C22×D24C2211yescharacteristic
19C2×#894—21no—
20C2×#893C2211yescharacteristic
21C2×#893C2211yescharacteristic
22C2×#893C2211yescharacteristic
23C2×#874C2211yescharacteristic
24C2×C4×D12C2211yescharacteristic
25C2×C3×#107C2211yescharacteristic
26C2×C3×C4⋊C8C2211yescharacteristic
27C3×#389C2211yescharacteristic
28C3×#389C2211yescharacteristic
29C2×#821C2211yescharacteristic
30C2×#821C2211yescharacteristic
31C2×C8×Dic3C2211yescharacteristic
32#4991—21no—
33#4989—21no—
34#4988C2211yescharacteristic
35#4988C2211yescharacteristic
36#4988C2211yescharacteristic
37#4987—21no—
38#4984C2211yescharacteristic
39#4977C2211yescharacteristic
40C2×#809C2211yescharacteristic
41C2×#809C2211yescharacteristic
42C2×#809C2211yescharacteristic
43C2×#809C2211yescharacteristic
44C2×#809C2211yescharacteristic
45C2×#809C2211yescharacteristic
46C2×#808C2211yescharacteristic
47C4×Dic3⋊C4C2211yescharacteristic
48#4872—24no—
49#4843—24no—
50#4672—21no—
51#4668—21no—
52#4665—21no—
53D3×#95—21no—
54C3×#340C2211yescharacteristic
55C3×#340C2211yescharacteristic
56C3×#340C2211yescharacteristic
57#4550C2211yescharacteristic
58#4550C2211yescharacteristic
59#4550C2211yescharacteristic
60#4529C2211yescharacteristic
61#4524C2211yescharacteristic
62C2×#459—31no—
63C2×C4×D8—31no—
64#2445—31no—
65#2349—31no—
66#2092—31no—
67#2087D311yescharacteristic, complemented
68#2073—31no—
69#2041—31no—
70#1934—31no—
71#1916—31no—
72#1879—31no—
73#1879—31no—
74#1879—31no—
75#1864—31no—
76#1864—31no—
77C22×D12—21no—
78C22×D12—21no—
79C22×D12C2311yescharacteristic
80C22×D12C2311yescharacteristic
81C22×D12C2311yescharacteristic
82C22×C4×D3—21no—
83C22×C4×D3—21no—
84C22×C4×D3—21no—
85C22×C4×D3—21no—
86C22×C3×C8C2311yescharacteristic
87C22×C3×C8C2311yescharacteristic
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C42C2311yescharacteristic
95C2×(C2×D3)⋊C4—21no—
96C2×(C2×D3)⋊C4—21no—
97C2×(C2×D3)⋊C4—21no—
98C2×(C2×D3)⋊C4—21no—
99C2×(C2×D3)⋊C4—21no—
100C2×(C2×D3)⋊C4—21no—
101C2×(C2×D3)⋊C4—21no—
102C2×(C2×D3)⋊C4—21no—
103C2×(C2×D3)⋊C4—21no—
104C2×(C2×D3)⋊C4—21no—
105C2×C4⋊Dic3—21no—
106C2×C4⋊Dic3C2311yescharacteristic
107C2×Dic3⋊C4D411yescharacteristic
108C2×Dic3⋊C4D411yescharacteristic
109C2×C4×Dic3C2311yescharacteristic
110C2×C4×Dic3C2311yescharacteristic
111C22×C3⋊C8C2311yescharacteristic
112C22×C3⋊C8C2311yescharacteristic
113C2×D24—22no—
114C2×D24—22no—
115C2×D24—22no—
116C2×D24—22no—
117C2×D24—22no—
118C2×D24D414yes—
119#894—44no—
120#893—22no—
121#893—22no—
122#893—22no—
123#893—22no—
124#893—41no—
125#893—41no—
126#874—22no—
127#874—41no—
128C4×D12—22no—
129C4×D12—41no—
130C3×#107—22no—
131C3×C4⋊C8—22no—
132C3×#95—21no—
133C3×#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95D411yescharacteristic
138C3⋊#95D411yescharacteristic
139#821—22no—
140#821—22no—
141C8×Dic3—22no—
142#809—22no—
143#809—22no—
144#809—22no—
145#809—22no—
146#809—22no—
147#809—41no—
148#808—41no—
149C22×D8—31no—
150C22×D8—31no—
151C2×#127—31no—
152C2×#123—61no—
153C2×C4×D4—31no—
154C2×C4×D4—31no—
155C2×C4×D4—31no—
156C2×C4×D4—31no—
157#459—64no—
158C4×D8—64no—
159C2×#107C2×D311yescharacteristic
160C2×#106—31no—
161C2×C4⋊C8C2×D311yescharacteristic
162C2×#102—31no—
163C2×#102—31no—
164C2×#102—31no—
165C2×#102—31no—
166C2×#102—31no—
167C2×#102—31no—
168C2×#102—31no—
169C2×#102—31no—
170C2×C4×C8—31no—
171#394—61no—
172#394—61no—
173#390—31no—
174#390—31no—
175#390—31no—
176#389—31no—
177#389C2×D311yescharacteristic
178#389C2×D311yescharacteristic
179#386—61no—
180#385—61no—
181#385—61no—
182#382—31no—
183#380—61no—
184C4×C4⋊C4—31no—
185C2×#95—61no—
186#340—31no—
187#340C2×D311yescharacteristic
188#340C2×D311yescharacteristic
189#340C2×D311yescharacteristic
190#336—31no—
191#336—31no—
192#336—31no—
193#336—31no—
194C23×D3—21no—
195C23×D3—21no—
196C23×D3—21no—
197C22×C3×C4—21no—
198C22×C3×C4—21no—
199C22×C3×C4—21no—
200C22×C3×C4—21no—
201C22×C3×C4C2×D411yescharacteristic
202C22×C3×C4C2×D411yescharacteristic
203C22×C3×C4C2411yescharacteristic
204C22×Dic3—21no—
205C22×Dic3C2×D411yescharacteristic
206C22×Dic3C2×D411yescharacteristic
207C22×Dic3C2×D411yescharacteristic
208C22×Dic3C2×D411yescharacteristic
209C2×D12—21no—
210C2×D12—21no—
211C2×D12—21no—
212C2×D12—21no—
213C2×D12—21no—
214C2×D12—21no—
215C2×D12—21no—
216C2×D12—21no—
217C2×D12—21no—
218C2×D12—21no—
219C2×D12—21no—
220C2×D12—21no—
221C2×D12—21no—
222C2×D12—21no—
223C2×D12—42no—
224C2×D12—42no—
225C2×D12—42no—
226C2×D12—42no—
227C2×D12—81no—
228C2×D12—81no—
229C2×D12C2×D412yes—
230C2×D12C2×D412yes—
231C2×D12#4112yes—
232C2×D12#4112yes—
233C2×C4×D3—24no—
234C2×C4×D3—42no—
235C2×C4×D3—42no—
236C2×C4×D3—42no—
237C2×C4×D3—42no—
238C2×C4×D3—42no—
239C2×C4×D3—42no—
240C2×C4×D3—81no—
241C2×C4×D3—81no—
242C2×C4×D3—81no—
243C2×C4×D3—81no—
244C2×C4×D3—81no—
245C2×C3×C8—21no—
246C2×C3×C8—21no—
247C2×C3×C8—21no—
248C2×C3×C8—21no—
249C2×C3×C8—21no—
250C2×C3×C8C2×D412yes—
251C3×C4⋊C4—21no—
252C3×C4⋊C4—21no—
253C3×C4⋊C4—21no—
254C3×C4⋊C4—21no—
255C3×C4⋊C4—21no—
256C3×C4⋊C4—21no—
257C3×C4⋊C4—21no—
258C3×C4⋊C4—42no—
259C3×C4⋊C4—81no—
260C3×C4⋊C4#4112yes—
261C3×C42—21no—
262C3×C42—21no—
263(C2×D3)⋊C4—24no—
264(C2×D3)⋊C4—42no—
265(C2×D3)⋊C4—42no—
266(C2×D3)⋊C4—42no—
267(C2×D3)⋊C4—42no—
268(C2×D3)⋊C4—42no—
269(C2×D3)⋊C4—42no—
270(C2×D3)⋊C4—81no—
271(C2×D3)⋊C4—81no—
272(C2×D3)⋊C4—81no—
273C4⋊Dic3—21no—
274C4⋊Dic3—21no—
275C4⋊Dic3—81no—
276Dic3⋊C4—22no—
277Dic3⋊C4—22no—
278C4×Dic3—21no—
279C4×Dic3—21no—
280C4×Dic3—21no—
281C4×Dic3—21no—
282C2×C3⋊C8—21no—
283C2×C3⋊C8—21no—
284C2×C3⋊C8—21no—
285C2×C3⋊C8—21no—
286C2×C3⋊C8—21no—
287C2×C3⋊C8—21no—
288D24—28no—
289D24—44no—
290C22×D4—31no—
291C22×D4—31no—
292C22×D4—31no—
293C22×D4—61no—
294C22×D4—61no—
295C23×C4—61no—
296C23×C4—61no—
297C23×C4—61no—
298C23×C4—61no—
299C2×D8—34no—
300C2×D8—62no—
301C2×D8—62no—
302C2×D8—62no—
303C2×D8—62no—
304C2×D8—62no—
305C22×C8—31no—
306C22×C8—31no—
307C22×C8C22×D311yescharacteristic
308C22×C8C22×D311yescharacteristic
309#127—62no—
310#127—121no—
311#123—124no—
312C4×D4—62no—
313C4×D4—62no—
314C4×D4—62no—
315C4×D4—62no—
316C4×D4—62no—
317C4×D4—121no—
318C4×D4—121no—
319C4×D4—121no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—31no—
323C2×C4⋊C4—31no—
324C2×C4⋊C4—31no—
325C2×C4⋊C4—61no—
326C2×C4⋊C4C22×D311yescharacteristic
327C2×C4⋊C4C22×D311yescharacteristic
328C2×C4⋊C4C22×D311yescharacteristic
329C2×C4⋊C4C22×D311yescharacteristic
330C2×C22⋊C4—61no—
331C2×C22⋊C4—61no—
332C2×C22⋊C4—61no—
333C2×C22⋊C4—61no—
334C2×C22⋊C4—61no—
335C2×C22⋊C4—61no—
336C2×C22⋊C4—61no—
337C2×C22⋊C4—61no—
338C2×C22⋊C4—61no—
339C2×C22⋊C4—61no—
340C2×C42—31no—
341C2×C42—31no—
342C2×C42C22×D311yescharacteristic
343#107—22no—
344#106—121no—
345C4⋊C8—22no—
346#102—62no—
347#102—62no—
348#102—62no—
349#102—62no—
350#102—62no—
351#102—62no—
352#102—62no—
353#102—121no—
354C4×C8—62no—
355#95—21no—
356#95—21no—
357#95—31no—
358#95—31no—
359#95—61no—
360#95—61no—
361#95—61no—
362C23×C3C22×D411yescharacteristic
363C22×D3—22no—
364C22×D3—22no—
365C22×D3—22no—
366C22×D3—22no—
367C22×D3—41no—
368C22×D3—41no—
369C22×D3—41no—
370C22×D3—41no—
371C22×D3—41no—
372C22×D3—41no—
373C22×D3—41no—
374C22×D3—41no—
375C22×D3—41no—
376C22×D3—41no—
377C22×D3—41no—
378C22×D3—41no—
379C22×D3—41no—
380C22×D3—41no—
381C22×D3—41no—
382C22×D3—41no—
383C22×D3—41no—
384C2×C3×C4—21no—
385C2×C3×C4—21no—
386C2×C3×C4—22no—
387C2×C3×C4—21no—
388C2×C3×C4—21no—
389C2×C3×C4—21no—
390C2×C3×C4—41no—
391C2×C3×C4—41no—
392C2×C3×C4—41no—
393C2×C3×C4—41no—
394C2×C3×C4—41no—
395C2×C3×C4—41no—
396C2×C3×C4—41no—
397C2×C3×C4—41no—
398C2×C3×C4—41no—
399C2×C3×C4—41no—
400C2×C3×C4—41no—
401C2×C3×C4#13512yes—
402C2×C3×C4C22×D411yescharacteristic
403C2×C3×C4C2×#4111yescharacteristic
404C2×C3×C4C2×#4111yescharacteristic
405C2×C3×C4C2×#4111yescharacteristic
406C2×C3×C4H(5,𝔽2)11yescharacteristic
407C2×C3×C42−1+411yescharacteristic
408C2×Dic3—21no—
409C2×Dic3—21no—
410C2×Dic3—21no—
411C2×Dic3—21no—
412C2×Dic3—21no—
413C2×Dic3—21no—
414C2×Dic3—21no—
415C2×Dic3—21no—
416C2×Dic3—21no—
417C2×Dic3—21no—
418C2×Dic3—21no—
419C2×Dic3—41no—
420C2×Dic3—41no—
421C2×Dic3—41no—
422C2×Dic3AGL(1,ℤ/8)12yes—
423D12—22no—
424D12—22no—
425D12—22no—
426D12—22no—
427D12—22no—
428D12—41no—
429D12—41no—
430D12—41no—
431D12—41no—
432D12—41no—
433D12—41no—
434D12—84no—
435D12—84no—
436D12#12314yescomplemented
437C4×D3—84no—
438C4×D3—84no—
439C4×D3—84no—
440C4×D3—84no—
441C3×C8—22no—
442C3×C8—41no—
443C3⋊C8—41no—
444C3⋊C8—41no—
445C24—61no—
446C24—61no—
447C24—61no—
448C2×D4—32no—
449C2×D4—32no—
450C2×D4—32no—
451C2×D4—32no—
452C2×D4—61no—
453C2×D4—61no—
454C2×D4—61no—
455C2×D4—61no—
456C2×D4—61no—
457C2×D4—61no—
458C2×D4—61no—
459C2×D4—61no—
460C2×D4—61no—
461C2×D4—61no—
462C2×D4—61no—
463C2×D4—61no—
464C2×D4—61no—
465C2×D4—61no—
466C2×D4—122no—
467C2×D4—122no—
468C2×D4—122no—
469C2×D4—122no—
470C2×D4—241no—
471C2×D4—241no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—31no—
477C22×C4—31no—
478C22×C4—31no—
479C22×C4—31no—
480C22×C4—64no—
481C22×C4—61no—
482C22×C4—122no—
483C22×C4—122no—
484C22×C4—122no—
485C22×C4—122no—
486C22×C4—122no—
487C22×C4—122no—
488C22×C4—241no—
489C22×C4—241no—
490C22×C4—241no—
491C22×C4—241no—
492C22×C4—241no—
493C22×C4D3×D411yescharacteristic
494C22×C4D3×D411yescharacteristic
495C22×C4C23×D311yescharacteristic
496D8—68no—
497D8—124no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—61no—
504C2×C8—61no—
505C2×C8—61no—
506C2×C8—61no—
507C2×C8—61no—
508C2×C8—61no—
509C2×C8D3×D412yes—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—42no—
518C4⋊C4—62no—
519C4⋊C4—62no—
520C4⋊C4—61no—
521C4⋊C4—61no—
522C4⋊C4—81no—
523C4⋊C4—241no—
524C4⋊C4#23912yes—
525C22⋊C4—64no—
526C22⋊C4—122no—
527C22⋊C4—122no—
528C22⋊C4—122no—
529C22⋊C4—122no—
530C22⋊C4—122no—
531C22⋊C4—122no—
532C22⋊C4—241no—
533C22⋊C4—241no—
534C22⋊C4—241no—
535C42—21no—
536C42—21no—
537C42—61no—
538C42—61no—
539C42—61no—
540C42—61no—
541C22×C3#52511yescharacteristic
542C22×C3D4211yescharacteristic
543C22×C3#54811yescharacteristic
544C22×C3C2×#13511yescharacteristic
545C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
546C22×C3#57611yescharacteristic
547C22×C3#57611yescharacteristic
548C2×D3—24no—
549C2×D3—42no—
550C2×D3—42no—
551C2×D3—42no—
552C2×D3—42no—
553C2×D3—42no—
554C2×D3—42no—
555C2×D3—42no—
556C2×D3—42no—
557C2×D3—42no—
558C2×D3—81no—
559C2×D3—81no—
560C2×D3—81no—
561C2×D3—81no—
562C2×D3—81no—
563C2×D3—81no—
564C2×D3—81no—
565C2×D3—81no—
566C2×D3—81no—
567C2×D3—81no—
568C2×D3—81no—
569C3×C4—22no—
570C3×C4—42no—
571C3×C4—41no—
572C3×C4—81no—
573C3×C4—81no—
574C3×C4—81no—
575C3×C4C2×#12311yescharacteristic
576C3×C4#53911yescharacteristic
577C3×C4#54711yescharacteristic
578C3×C4#55311yescharacteristic
579Dic3—22no—
580Dic3—41no—
581Dic3—41no—
582Dic3—41no—
583Dic3—81no—
584C23—62no—
585C23—62no—
586C23—62no—
587C23—62no—
588C23—121no—
589C23—121no—
590C23—121no—
591C23—121no—
592C23—121no—
593C23—121no—
594C23—121no—
595C23—121no—
596C23—121no—
597C23—121no—
598C23—121no—
599C23—121no—
600C23—121no—
601C23—121no—
602C23—121no—
603C23—121no—
604C23—121no—
605C23C2×D3×D411yescharacteristic
606D4—34no—
607D4—62no—
608D4—62no—
609D4—62no—
610D4—62no—
611D4—62no—
612D4—121no—
613D4—121no—
614D4—121no—
615D4—121no—
616D4—121no—
617D4—121no—
618D4—244no—
619D4—244no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—21no—
625C2×C4—22no—
626C2×C4—32no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—41no—
637C2×C4—41no—
638C2×C4—61no—
639C2×C4—61no—
640C2×C4—61no—
641C2×C4—61no—
642C2×C4—61no—
643C2×C4—61no—
644C2×C4—61no—
645C2×C4—61no—
646C2×C4—61no—
647C2×C4—61no—
648C2×C4—61no—
649C2×C4—121no—
650C2×C4—121no—
651C2×C4—121no—
652C2×C4—244no—
653C2×C4—244no—
654C2×C4—244no—
655C2×C4—244no—
656C2×C4#91912yes—
657C2×C4C2×D3×D411yescharacteristic
658C2×C4C2×#23911yescharacteristic
659C2×C4#100711yescharacteristic
660C2×C4D3×#4111yescharacteristic
661C2×C4D3×#4111yescharacteristic
662C2×C4#100911yescharacteristic
663C8—22no—
664C8—41no—
665C8—121no—
666C8—121no—
667C2×C3#244311yescharacteristic
668C2×C3#310611yescharacteristic
669C2×C3#312211yescharacteristic
670C2×C3#326411yescharacteristic
671C2×C3#328011yescharacteristic
672C2×C3#336111yescharacteristic
673C2×C3#336611yescharacteristic
674D3—44no—
675D3—82no—
676D3—161no—
677C22—64no—
678C22—122no—
679C22—122no—
680C22—122no—
681C22—122no—
682C22—122no—
683C22—122no—
684C22—122no—
685C22—122no—
686C22—122no—
687C22—241no—
688C22—241no—
689C22—241no—
690C22—241no—
691C22—241no—
692C22—241no—
693C22—241no—
694C22—241no—
695C22—241no—
696C22—241no—
697C22—241no—
698C22#560911yescharacteristic
699C22#562711yescharacteristic
700C22#563311yescharacteristic
701C22C2×#91911yescharacteristic
702C22#576911yescharacteristic
703C22D3×AGL(1,ℤ/8)11yescharacteristic
704C22#577411yescharacteristic
705C4—22no—
706C4—41no—
707C4—42no—
708C4—62no—
709C4—81no—
710C4—81no—
711C4—81no—
712C4—121no—
713C4—121no—
714C4—121no—
715C4—241no—
716C4D3×#12311yescharacteristic
717C4#569211yescharacteristic
718C4#569511yescharacteristic
719C4#572611yescharacteristic
720C3#2616211yescharacteristic, complemented, Sylow
721C2—124no—
722C2—242no—
723C2—481no—
724C2#7795111yescharacteristic
725C2#8077311yescharacteristic
726C2#8088111yescharacteristic
727C2#8109311yescharacteristic
728C2#8113411yescharacteristic
729C2#8114611yescharacteristic
730C2#8115211yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2087C2
socleC23×C3C22×D4
prime radicalC2×#4843C2
hypercenter#2087D3
hypocenterC3#26162
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2087

Complements

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

NH ≅ G/Ntype
1. C2×#4872C2semidirect
3. #76203C2semidirect
4. #76004C2semidirect
5. C3×#2087C2semidirect
6. #70640C2semidirect
7. #70447C2semidirect
8. #69627C2semidirect
9. #69626C2semidirect
10. #68903C2semidirect
11. #68897C2semidirect
12. #68897C2semidirect
13. #68897C2semidirect
14. #68895C2semidirect
15. #68853C2semidirect
67. #2087D3semidirect
436. D12#123semidirect
720. C3#26162semidirect

Maximal subgroups

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

subgroupindex
C2×#48722
C2×#48432
#762032
#760042
C3×#20872
#706402
#704472
#696272
#696262
#689032
#688972
#688952
#688532
#261623