Finite Group Databasebeta

← #10824744

Subgroups — #10824744

Order 768.

1042 conjugacy classes of subgroups.

The subgroup lattice has 734 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4869C211yesmaximal, characteristic, complemented
2C2×#4835C211yesmaximal, characteristic, complemented
3#75964C211yesmaximal, characteristic, complemented
4#75964C211yesmaximal, characteristic, complemented
5C3×#2088C211yesmaximal, characteristic, complemented
6#70583C211yesmaximal, characteristic, complemented
7#70481C211yesmaximal, characteristic, complemented
8#69630C211yesmaximal, characteristic, complemented
9#69629C211yesmaximal, characteristic, complemented
10#68886C211yesmaximal, characteristic
11#68823C211yesmaximal, characteristic, complemented
12#68822C211yesmaximal, characteristic, complemented
13#68786C211yesmaximal, characteristic, complemented
14#68782C211yesmaximal, characteristic, complemented
15#68782C211yesmaximal, characteristic, complemented
16#23708—31nomaximal, Sylow
17C22×D24C2211yescharacteristic
18C2×#895C2211yescharacteristic
19C2×#895C2211yescharacteristic
20C2×#893—21no—
21C2×#893—21no—
22C2×#893C2211yescharacteristic
23C2×#893C2211yescharacteristic
24C2×C42×D3C2211yescharacteristic
25C2×C3×#106C2211yescharacteristic
26C2×C3×C4⋊C8C2211yescharacteristic
27C3×#389C2211yescharacteristic
28C3×#389C2211yescharacteristic
29C2×#821C2211yescharacteristic
30C2×#821C2211yescharacteristic
31C2×(C2×D3)⋊C8C2211yescharacteristic
32C2×(C2×D3)⋊C8C2211yescharacteristic
33C2×#815C2211yescharacteristic
34C2×#815C2211yescharacteristic
35#4990—21no—
36#4990—21no—
37#4987C2211yescharacteristic
38#4987C2211yescharacteristic
39Dic3×C4⋊C4C2211yescharacteristic
40Dic3×C4⋊C4C2211yescharacteristic
41C2×#809C2211yescharacteristic
42C2×#809C2211yescharacteristic
43C2×#809C2211yescharacteristic
44C2×#809C2211yescharacteristic
45C2×#808C2211yescharacteristic
46C2×#807C2211yescharacteristic
47#4939C2211yescharacteristic
48#4939C2211yescharacteristic
49C2×C4⋊(C3⋊C8)C2211yescharacteristic
50#4869—24no—
51#4835—24no—
52#4666—21no—
53#4666—21no—
54#4665—21no—
55#4665—21no—
56C3×#340C2211yescharacteristic
57C3×#340C2211yescharacteristic
58C3×#340C2211yescharacteristic
59#4551C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#442—31no—
63C2×#432—31no—
64C4×#121—31no—
65C4×#121—31no—
66#2088D311yescharacteristic, complemented
67#2077—31no—
68#2067—31no—
69#2065—31no—
70#2054—31no—
71#2034—31no—
72#2033—31no—
73#1999—31no—
74#1776—31no—
75#1762—31no—
76#1762—31no—
77C22×D12—21no—
78C22×D12—21no—
79C22×D12C2311yescharacteristic
80C22×D12C2311yescharacteristic
81C22×C4×D3—21no—
82C22×C4×D3—21no—
83C22×C4×D3D411yescharacteristic
84C22×C4×D3D411yescharacteristic
85C22×C4×D3C2311yescharacteristic
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⋊Dic3C2311yescharacteristic
106C2×C4⋊Dic3C2311yescharacteristic
107C2×C4×Dic3—21no—
108C2×C4×Dic3—21no—
109C2×C4×Dic3D411yescharacteristic
110C2×C4×Dic3D411yescharacteristic
111C2×C4×Dic3C2311yescharacteristic
112C22×C3⋊C8C2311yescharacteristic
113C22×C3⋊C8C2311yescharacteristic
114C2×D24—22no—
115C2×D24—22no—
116C2×D24—22no—
117#895—22no—
118#895—22no—
119#895—22no—
120#895—22no—
121#893—22no—
122#893—22no—
123#893—22no—
124#893—22no—
125#893—44no—
126#893—44no—
127C42×D3—22no—
128C42×D3—22no—
129C3×#106—22no—
130C3×C4⋊C8—22no—
131C3×#95—21no—
132C3×#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137#821—22no—
138#821—22no—
139(C2×D3)⋊C8—22no—
140(C2×D3)⋊C8—22no—
141#815—22no—
142#815D414yescomplemented
143#809—22no—
144#809—22no—
145#809—22no—
146#809—22no—
147#808—22no—
148#807—22no—
149C4⋊(C3⋊C8)—22no—
150C22×D8—31no—
151C2×#121—31no—
152C2×#121—31no—
153C2×C4×D4—31no—
154C2×C4×D4—31no—
155C2×C4×D4—61no—
156C2×C4×D4—61no—
157C22×C42—31no—
158#442—64no—
159#432—64no—
160C2×#107—31no—
161C2×#106—31no—
162C2×#106C2×D311yescharacteristic
163C2×C4⋊C8—31no—
164C2×C4⋊C8C2×D311yescharacteristic
165C2×#102—31no—
166C2×#102—31no—
167C2×#102—31no—
168C2×#102—31no—
169C2×#102—31no—
170C2×#102—31no—
171C2×C22⋊C8—31no—
172C2×C22⋊C8—31no—
173C2×#97—31no—
174C2×#97—31no—
175#390—31no—
176#390—31no—
177#389C2×D311yescharacteristic
178#389C2×D311yescharacteristic
179#388—61no—
180#388—61no—
181#386—61no—
182#386—61no—
183#380—31no—
184#380—31no—
185C4×C4⋊C4—31no—
186C4×C4⋊C4—31no—
187C4×C22⋊C4—61no—
188C4×C22⋊C4—61no—
189#340—31no—
190#340—31no—
191#340—31no—
192#340C2×D311yescharacteristic
193#340C2×D311yescharacteristic
194#340C2×D311yescharacteristic
195C23×D3—21no—
196C23×D3—21no—
197C23×D3C2×D411yescharacteristic
198C22×C3×C4—21no—
199C22×C3×C4—21no—
200C22×C3×C4—21no—
201C22×C3×C4—21no—
202C22×C3×C4C2×D411yescharacteristic
203C22×C3×C4C2×D411yescharacteristic
204C22×C3×C4C2411yescharacteristic
205C22×Dic3—21no—
206C22×Dic3—21no—
207C22×Dic3C2×D411yescharacteristic
208C22×Dic3C2×D411yescharacteristic
209C22×Dic3C2×D411yescharacteristic
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—24no—
219C2×D12—24no—
220C2×D12—21no—
221C2×D12—21no—
222C2×D12—42no—
223C2×D12—42no—
224C2×D12—42no—
225C2×D12—42no—
226C2×D12#4112yes—
227C2×D12#4112yes—
228C2×C4×D3—22no—
229C2×C4×D3—22no—
230C2×C4×D3—21no—
231C2×C4×D3—21no—
232C2×C4×D3—21no—
233C2×C4×D3—21no—
234C2×C4×D3—21no—
235C2×C4×D3—21no—
236C2×C4×D3—22no—
237C2×C4×D3—22no—
238C2×C4×D3—41no—
239C2×C4×D3—41no—
240C2×C4×D3—42no—
241C2×C4×D3—42no—
242C2×C4×D3—42no—
243C2×C4×D3—42no—
244C2×C4×D3—42no—
245C2×C4×D3—42no—
246C2×C3×C8—21no—
247C2×C3×C8—21no—
248C2×C3×C8—21no—
249C2×C3×C8—21no—
250C2×C3×C8—21no—
251C2×C3×C8C2×D412yes—
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—42no—
260C3×C4⋊C4#4112yes—
261C3×C4⋊C4#4112yes—
262C3×C42—21no—
263C3×C42—21no—
264(C2×D3)⋊C4—24no—
265(C2×D3)⋊C4—24no—
266(C2×D3)⋊C4—42no—
267(C2×D3)⋊C4—42no—
268(C2×D3)⋊C4—42no—
269(C2×D3)⋊C4—42no—
270(C2×D3)⋊C4—42no—
271(C2×D3)⋊C4—42no—
272(C2×D3)⋊C4—42no—
273(C2×D3)⋊C4—42no—
274C4⋊Dic3—21no—
275C4⋊Dic3—21no—
276C4⋊Dic3—21no—
277C4⋊Dic3—21no—
278C4×Dic3—22no—
279C4×Dic3—22no—
280C4×Dic3—42no—
281C4×Dic3—42no—
282C4×Dic3C2×D412yes—
283C4×Dic3#4112yes—
284C2×C3⋊C8—21no—
285C2×C3⋊C8—21no—
286C2×C3⋊C8—21no—
287C2×C3⋊C8—21no—
288C2×C3⋊C8—21no—
289C2×C3⋊C8C2×D412yes—
290D24—28no—
291C22×D4—31no—
292C22×D4—31no—
293C22×D4—61no—
294C22×D4—61no—
295C23×C4—31no—
296C23×C4—31no—
297C23×C4—31no—
298C23×C4—61no—
299C23×C4—61no—
300C2×D8—62no—
301C2×D8—62no—
302C2×D8—62no—
303C22×C8—31no—
304C22×C8—31no—
305C22×C8C22×D311yescharacteristic
306C22×C8C22×D311yescharacteristic
307#121—62no—
308#121—62no—
309#121—62no—
310#121—62no—
311C4×D4—62no—
312C4×D4—62no—
313C4×D4—62no—
314C4×D4—62no—
315C4×D4—124no—
316C4×D4—124no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—31no—
320C2×C4⋊C4—31no—
321C2×C4⋊C4C22×D311yescharacteristic
322C2×C4⋊C4C22×D311yescharacteristic
323C2×C4⋊C4C22×D311yescharacteristic
324C2×C4⋊C4C22×D311yescharacteristic
325C2×C22⋊C4—61no—
326C2×C22⋊C4—61no—
327C2×C22⋊C4—61no—
328C2×C22⋊C4—61no—
329C2×C22⋊C4—61no—
330C2×C22⋊C4—61no—
331C2×C22⋊C4—61no—
332C2×C22⋊C4—61no—
333C2×C22⋊C4—61no—
334C2×C22⋊C4—61no—
335C2×C42—31no—
336C2×C42—31no—
337C2×C42—31no—
338C2×C42—62no—
339C2×C42—62no—
340C2×C42—61no—
341C2×C42—61no—
342C2×C42C22×D311yescharacteristic
343#107—62no—
344#106—22no—
345#106—62no—
346C4⋊C8—22no—
347C4⋊C8—62no—
348#102—62no—
349#102—62no—
350#102—62no—
351#102—62no—
352#102—62no—
353#102—62no—
354C22⋊C8—62no—
355C22⋊C8—62no—
356#97—34no—
357#97—62no—
358#95—21no—
359#95—21no—
360#95—61no—
361#95—61no—
362#95—61no—
363#95—61no—
364C23×C3C22×D411yescharacteristic
365C22×D3—21no—
366C22×D3—21no—
367C22×D3—21no—
368C22×D3—21no—
369C22×D3—21no—
370C22×D3—21no—
371C22×D3—22no—
372C22×D3—22no—
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—
384C22×D3—41no—
385C22×D3AGL(1,ℤ/8)12yes—
386C2×C3×C4—21no—
387C2×C3×C4—21no—
388C2×C3×C4—21no—
389C2×C3×C4—22no—
390C2×C3×C4—22no—
391C2×C3×C4—21no—
392C2×C3×C4—21no—
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—41no—
402C2×C3×C4—41no—
403C2×C3×C4AGL(1,ℤ/8)12yes—
404C2×C3×C4C22×D411yescharacteristic
405C2×C3×C4C2×#4111yescharacteristic
406C2×C3×C4C2×#4111yescharacteristic
407C2×C3×C4C2×#4111yescharacteristic
408C2×C3×C4C2×#4111yescharacteristic
409C2×C3×C4C2×#4111yescharacteristic
410C2×Dic3—21no—
411C2×Dic3—21no—
412C2×Dic3—21no—
413C2×Dic3—21no—
414C2×Dic3—21no—
415C2×Dic3—21no—
416C2×Dic3—21no—
417C2×Dic3—41no—
418C2×Dic3—41no—
419C2×Dic3—41no—
420C2×Dic3—41no—
421C2×Dic3—41no—
422C2×Dic3—41no—
423C2×Dic3#12112yes—
424C2×Dic3#12112yes—
425D12—22no—
426D12—22no—
427D12—22no—
428D12—22no—
429D12—48no—
430D12—48no—
431D12—41no—
432D12—41no—
433D12—41no—
434D12—41no—
435C4×D3—44no—
436C4×D3—44no—
437C4×D3—41no—
438C4×D3—41no—
439C4×D3—41no—
440C4×D3—41no—
441C4×D3—84no—
442C4×D3—84no—
443C3×C8—22no—
444C3×C8—41no—
445C3⋊C8—22no—
446C3⋊C8—41no—
447C24—31no—
448C24—61no—
449C24—61no—
450C2×D4—32no—
451C2×D4—32no—
452C2×D4—64no—
453C2×D4—64no—
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—122no—
465C2×D4—122no—
466C2×D4—122no—
467C2×D4—122no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—31no—
473C22×C4—31no—
474C22×C4—31no—
475C22×C4—62no—
476C22×C4—62no—
477C22×C4—62no—
478C22×C4—62no—
479C22×C4—61no—
480C22×C4—61no—
481C22×C4—61no—
482C22×C4—61no—
483C22×C4—61no—
484C22×C4—61no—
485C22×C4—61no—
486C22×C4—61no—
487C22×C4—121no—
488C22×C4—121no—
489C22×C4—122no—
490C22×C4—122no—
491C22×C4—122no—
492C22×C4—122no—
493C22×C4—122no—
494C22×C4—122no—
495C22×C4D3×D411yescharacteristic
496C22×C4D3×D411yescharacteristic
497C22×C4C23×D311yescharacteristic
498D8—68no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—32no—
505C2×C8—61no—
506C2×C8—61no—
507C2×C8—61no—
508C2×C8—61no—
509C2×C8—61no—
510C2×C8D3×D412yes—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—42no—
518C4⋊C4—42no—
519C4⋊C4—61no—
520C4⋊C4—61no—
521C4⋊C4—61no—
522C4⋊C4—61no—
523C4⋊C4#23912yes—
524C4⋊C4#23912yes—
525C22⋊C4—64no—
526C22⋊C4—64no—
527C22⋊C4—122no—
528C22⋊C4—122no—
529C22⋊C4—122no—
530C22⋊C4—122no—
531C22⋊C4—122no—
532C22⋊C4—122no—
533C22⋊C4—122no—
534C22⋊C4—122no—
535C42—21no—
536C42—21no—
537C42—32no—
538C42—32no—
539C42—62no—
540C42—62no—
541C42—122no—
542C42—122no—
543C22×C3C2×#12111yescharacteristic
544C22×C3#52511yescharacteristic
545C22×C3#52511yescharacteristic
546C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
547C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
548C22×C3#57511yescharacteristic
549C22×C3#57511yescharacteristic
550C2×D3—22no—
551C2×D3—22no—
552C2×D3—22no—
553C2×D3—41no—
554C2×D3—41no—
555C2×D3—41no—
556C2×D3—41no—
557C2×D3—42no—
558C2×D3—42no—
559C2×D3—42no—
560C2×D3—42no—
561C2×D3—42no—
562C2×D3—42no—
563C2×D3—81no—
564C2×D3—81no—
565C2×D3—81no—
566C2×D3—81no—
567C2×D3—81no—
568C2×D3—81no—
569C2×D3—81no—
570C2×D3—81no—
571C3×C4—22no—
572C3×C4—42no—
573C3×C4—42no—
574C3×C4—41no—
575C3×C4—81no—
576C3×C4—81no—
577C3×C4#52511yescharacteristic
578C3×C4#52911yescharacteristic
579C3×C4#52911yescharacteristic
580C3×C4#53211yescharacteristic
581Dic3—22no—
582Dic3—22no—
583Dic3—41no—
584Dic3—81no—
585Dic3—81no—
586C23—32no—
587C23—61no—
588C23—61no—
589C23—61no—
590C23—61no—
591C23—61no—
592C23—61no—
593C23—62no—
594C23—62no—
595C23—121no—
596C23—121no—
597C23—121no—
598C23—121no—
599C23—121no—
600C23—121no—
601C23—121no—
602C23—121no—
603C23—121no—
604C23—121no—
605C23—121no—
606C23—121no—
607C23C2×D3×D411yescharacteristic
608D4—62no—
609D4—62no—
610D4—62no—
611D4—62no—
612D4—128no—
613D4—128no—
614D4—121no—
615D4—121no—
616D4—121no—
617D4—121no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—32no—
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—61no—
638C2×C4—61no—
639C2×C4—61no—
640C2×C4—61no—
641C2×C4—61no—
642C2×C4—61no—
643C2×C4—61no—
644C2×C4—124no—
645C2×C4—124no—
646C2×C4—121no—
647C2×C4—121no—
648C2×C4—121no—
649C2×C4—121no—
650C2×C4—121no—
651C2×C4—121no—
652C2×C4—121no—
653C2×C4—121no—
654C2×C4—121no—
655C2×C4—121no—
656C2×C4—244no—
657C2×C4—244no—
658C2×C4#91412yes—
659C2×C4C2×D3×D411yescharacteristic
660C2×C4C2×#23911yescharacteristic
661C2×C4C2×#23911yescharacteristic
662C2×C4D3×#4111yescharacteristic
663C2×C4D3×#4111yescharacteristic
664C2×C4D3×#4111yescharacteristic
665C8—22no—
666C8—41no—
667C8—62no—
668C8—121no—
669C2×C3#239211yescharacteristic
670C2×C3#305511yescharacteristic
671C2×C3#305511yescharacteristic
672C2×C3#310911yescharacteristic
673C2×C3#311111yescharacteristic
674C2×C3#311411yescharacteristic
675C2×C3#311511yescharacteristic
676D3—42no—
677D3—82no—
678D3—82no—
679C22—62no—
680C22—62no—
681C22—62no—
682C22—121no—
683C22—121no—
684C22—121no—
685C22—121no—
686C22—122no—
687C22—122no—
688C22—122no—
689C22—122no—
690C22—122no—
691C22—122no—
692C22—241no—
693C22—241no—
694C22—241no—
695C22—241no—
696C22—241no—
697C22—241no—
698C22—241no—
699C22—241no—
700C22D3×#12111yescharacteristic
701C22#562711yescharacteristic
702C22#562711yescharacteristic
703C22C2×#91411yescharacteristic
704C22#576811yescharacteristic
705C22D3×AGL(1,ℤ/8)11yescharacteristic
706C22#577711yescharacteristic
707C4—22no—
708C4—42no—
709C4—41no—
710C4—42no—
711C4—62no—
712C4—62no—
713C4—81no—
714C4—81no—
715C4—121no—
716C4—241no—
717C4—241no—
718C4#565611yescharacteristic
719C4#569111yescharacteristic
720C4#569111yescharacteristic
721C4#572511yescharacteristic
722C3#2370811yescharacteristic, complemented, Sylow
723C2—122no—
724C2—242no—
725C2—242no—
726C2#7773711yescharacteristic
727C2#8065711yescharacteristic
728C2#8080511yescharacteristic
729C2#8099911yescharacteristic
730C2#8104611yescharacteristic
731C2#8114311yescharacteristic
732C2#8120611yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2088C2
socleC23×C3C22×D4
prime radical#68886C2
hypercenter#2088D3
hypocenterC3#23708
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2088

Complements

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

NH ≅ G/Ntype
1. C2×#4869C2semidirect
2. C2×#4835C2semidirect
3. #75964C2semidirect
4. #75964C2semidirect
5. C3×#2088C2semidirect
6. #70583C2semidirect
7. #70481C2semidirect
8. #69630C2semidirect
9. #69629C2semidirect
11. #68823C2semidirect
12. #68822C2semidirect
13. #68786C2semidirect
14. #68782C2semidirect
15. #68782C2semidirect
66. #2088D3semidirect
142. #815D4semidirect
722. C3#23708semidirect

Maximal subgroups

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

subgroupindex
C2×#48692
C2×#48352
#759642
C3×#20882
#705832
#704812
#696302
#696292
#688862
#688232
#688222
#687862
#687822
#237083