Finite Group Databasebeta

← #10845240

Subgroups — #10845240

Order 768.

928 conjugacy classes of subgroups.

The subgroup lattice has 698 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83259C211yesmaximal, characteristic, complemented
2C2×#5089C211yesmaximal, characteristic, complemented
3C2×#4844C211yesmaximal, characteristic
4#73654C211yesmaximal, characteristic, complemented
5C3×#1987C211yesmaximal, characteristic, complemented
6#70469C211yesmaximal, characteristic, complemented
7#69474C211yesmaximal, characteristic, complemented
8#68914C211yesmaximal, characteristic, complemented
9#68909C211yesmaximal, characteristic, complemented
10#68904C211yesmaximal, characteristic, complemented
11#68902C211yesmaximal, characteristic, complemented
12#68897C211yesmaximal, characteristic
13#68892C211yesmaximal, characteristic, complemented
14#68863C211yesmaximal, characteristic, complemented
15#68862C211yesmaximal, characteristic
16#25334—31nomaximal, Sylow
17C2×C3×C4⋊Q8C2211yescharacteristic
18C2×#946—21no—
19C2×#946—21no—
20C22×D24C2211yescharacteristic
21C22×#204C2211yescharacteristic
22C2×#893C2211yescharacteristic
23C2×#887C2211yescharacteristic
24C2×#875C2211yescharacteristic
25C2×C4×D12C2211yescharacteristic
26C2×C3×C4⋊C8C2211yescharacteristic
27C3×C4×C4⋊C4C2211yescharacteristic
28C2×#821C2211yescharacteristic
29C2×#816C2211yescharacteristic
30C2×Dic3⋊C8C2211yescharacteristic
31C2×Dic3⋊C8C2211yescharacteristic
32#5089—24no—
33#4993—21no—
34#4993—21no—
35#4988C2211yescharacteristic
36#4988C2211yescharacteristic
37#4988C2211yescharacteristic
38#4975C2211yescharacteristic
39#4975C2211yescharacteristic
40#4973C2211yescharacteristic
41C2×#810C2211yescharacteristic
42C2×#810C2211yescharacteristic
43C2×#809C2211yescharacteristic
44C2×#809C2211yescharacteristic
45C2×#809C2211yescharacteristic
46C2×#809C2211yescharacteristic
47#4940—21no—
48C4×(C2×D3)⋊C4—21no—
49C4×C4⋊Dic3C2211yescharacteristic
50C2×#803C2211yescharacteristic
51#4844—24no—
52#4670—21no—
53#4668—21no—
54C3×#340C2211yescharacteristic
55C3×#340C2211yescharacteristic
56C3×#340C2211yescharacteristic
57C3×#340C2211yescharacteristic
58#4550C2211yescharacteristic
59#4550C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#489—31no—
63C2×#463—31no—
64#2542—31no—
65#2367—31no—
66#2072—31no—
67#2071—31no—
68#2055—31no—
69#2054—31no—
70#1987D311yescharacteristic, complemented
71#1940—31no—
72#1880—31no—
73#1879—31no—
74#1803—31no—
75#1802—31no—
76#1762—31no—
77C22×C3×Q8—21no—
78C22×D12C2311yescharacteristic
79C22×D12C2311yescharacteristic
80C22×C4×D3—21no—
81C22×C4×D3—21no—
82C22×Dic6C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C3×C4⋊Q8—22no—
86C3×C4⋊Q8—22no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C42—21no—
93C2×C3×C42C2311yescharacteristic
94#946—44no—
95#946—44no—
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×(C2×D3)⋊C4—21no—
106C2×C4⋊Dic3D411yescharacteristic
107C2×C4⋊Dic3D411yescharacteristic
108C2×C4⋊Dic3C2311yescharacteristic
109C2×Dic3⋊C4—21no—
110C2×Dic3⋊C4—21no—
111C2×Dic3⋊C4—21no—
112C2×Dic3⋊C4—21no—
113C2×Dic3⋊C4—21no—
114C2×C4×Dic3—21no—
115C2×C4×Dic3C2311yescharacteristic
116C2×C4×Dic3C2311yescharacteristic
117C22×C3⋊C8C2311yescharacteristic
118C22×C3⋊C8C2311yescharacteristic
119C2×D24—22no—
120C2×D24—22no—
121C2×D24—41no—
122C2×#204—22no—
123C2×#204—22no—
124C2×#204—41no—
125#893—22no—
126#893—41no—
127#887—22no—
128#887—41no—
129#875—22no—
130#875D414yes—
131C4×D12—22no—
132C4×D12—22no—
133C3×C4⋊C8—22no—
134C3×#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95D411yescharacteristic
138C3⋊#95D411yescharacteristic
139#821—22no—
140#816—22no—
141Dic3⋊C8—41no—
142Dic3⋊C8—41no—
143#810—22no—
144#810—22no—
145#809—22no—
146#809—22no—
147#809—22no—
148#809—22no—
149#803—22no—
150C22×SD16—31no—
151C22×D8—31no—
152C2×C4⋊Q8C2×D311yescharacteristic
153C2×#124—31no—
154C2×C22⋊Q8—61no—
155C2×C22⋊Q8—61no—
156C2×C4×Q8—31no—
157C2×C4×D4—31no—
158C2×C4×D4—31no—
159#489—64no—
160#463—64no—
161C2×C4⋊C8—31no—
162C2×C4⋊C8—31no—
163C2×C4⋊C8C2×D311yescharacteristic
164C2×#103—31no—
165C2×#103—31no—
166C2×#103—31no—
167C2×#102—31no—
168C2×#102—31no—
169C2×#102—31no—
170C2×#102—31no—
171C2×#102—31no—
172C2×#97—31no—
173#399—61no—
174#399—61no—
175#391—31no—
176#390—31no—
177#390—31no—
178#390—31no—
179#388—61no—
180#388—61no—
181#385—61no—
182#384—31no—
183#384—31no—
184C4×C4⋊C4—31no—
185C4×C4⋊C4C2×D311yescharacteristic
186C4×C22⋊C4—61no—
187#340—31no—
188#340—31no—
189#340C2×D311yescharacteristic
190#340C2×D311yescharacteristic
191#340C2×D311yescharacteristic
192#340C2×D311yescharacteristic
193#336—31no—
194#336—31no—
195C23×D3—21no—
196C23×D3—21no—
197C2×C3×Q8—42no—
198C2×C3×Q8—42no—
199C2×C3×Q8—42no—
200C22×C3×C4—21no—
201C22×C3×C4—21no—
202C22×C3×C4—21no—
203C22×C3×C4—21no—
204C22×C3×C4C2×D411yescharacteristic
205C22×C3×C4C2×D411yescharacteristic
206C22×C3×C4C2411yescharacteristic
207C22×Dic3—21no—
208C22×Dic3—21no—
209C22×Dic3C2×D411yescharacteristic
210C22×Dic3C2×D411yescharacteristic
211C22×Dic3C2×D411yescharacteristic
212C22×Dic3C2×D411yescharacteristic
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×D12C2×D412yes—
224C2×D12#4112yes—
225C2×C4×D3—42no—
226C2×C4×D3—42no—
227C2×C4×D3—42no—
228C2×C4×D3—42no—
229C2×C4×D3—42no—
230C2×C4×D3—81no—
231C2×Dic6—21no—
232C2×Dic6—21no—
233C2×Dic6—21no—
234C2×Dic6—21no—
235C2×Dic6—21no—
236C2×Dic6C2×D412yes—
237C2×C3×C8—21no—
238C2×C3×C8—21no—
239C2×C3×C8—21no—
240C2×C3×C8—21no—
241C2×C3×C8—21no—
242C2×C3×C8—21no—
243C3×C4⋊C4—21no—
244C3×C4⋊C4—21no—
245C3×C4⋊C4—21no—
246C3×C4⋊C4—21no—
247C3×C4⋊C4—21no—
248C3×C4⋊C4—21no—
249C3×C4⋊C4—21no—
250C3×C4⋊C4—42no—
251C3×C4⋊C4#4112yes—
252C3×C42—81no—
253C3×C42C2×D412yes—
254C3×C42#4112yes—
255(C2×D3)⋊C4—24no—
256(C2×D3)⋊C4—24no—
257(C2×D3)⋊C4—42no—
258(C2×D3)⋊C4—42no—
259(C2×D3)⋊C4—42no—
260(C2×D3)⋊C4—42no—
261(C2×D3)⋊C4—42no—
262(C2×D3)⋊C4—42no—
263(C2×D3)⋊C4—42no—
264(C2×D3)⋊C4—81no—
265C4⋊Dic3—22no—
266C4⋊Dic3—22no—
267C4⋊Dic3—21no—
268C4⋊Dic3—21no—
269Dic3⋊C4—42no—
270Dic3⋊C4—42no—
271Dic3⋊C4—42no—
272Dic3⋊C4—42no—
273Dic3⋊C4—42no—
274C4×Dic3—21no—
275C4×Dic3—21no—
276C4×Dic3—21no—
277C4×Dic3—21no—
278C4×Dic3—42no—
279C2×C3⋊C8—21no—
280C2×C3⋊C8—21no—
281C2×C3⋊C8—21no—
282C2×C3⋊C8—21no—
283C2×C3⋊C8—21no—
284C2×C3⋊C8—21no—
285D24—44no—
286#204—44no—
287C22×Q8—21no—
288C22×Q8—31no—
289C22×D4—31no—
290C22×D4—31no—
291C23×C4—61no—
292C23×C4—61no—
293C2×SD16—62no—
294C2×SD16—62no—
295C2×SD16—121no—
296C2×D8—62no—
297C2×D8—62no—
298C2×D8—121no—
299C22×C8—31no—
300C22×C8—31no—
301C22×C8C22×D311yescharacteristic
302C22×C8C22×D311yescharacteristic
303C4⋊Q8—22no—
304C4⋊Q8—22no—
305#124—34no—
306#124—62no—
307C22⋊Q8—124no—
308C22⋊Q8—124no—
309C4×Q8—62no—
310C4×Q8—121no—
311C4×D4—62no—
312C4×D4—62no—
313C4×D4—62no—
314C4×D4—121no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—31no—
317C2×C4⋊C4—31no—
318C2×C4⋊C4—31no—
319C2×C4⋊C4—61no—
320C2×C4⋊C4—61no—
321C2×C4⋊C4—61no—
322C2×C4⋊C4—61no—
323C2×C4⋊C4—61no—
324C2×C4⋊C4C22×D311yescharacteristic
325C2×C4⋊C4C22×D311yescharacteristic
326C2×C4⋊C4C22×D311yescharacteristic
327C2×C4⋊C4C22×D311yescharacteristic
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×C22⋊C4—61no—
336C2×C22⋊C4—61no—
337C2×C22⋊C4—61no—
338C2×C42—21no—
339C2×C42—31no—
340C2×C42—31no—
341C2×C42—61no—
342C2×C42C22×D311yescharacteristic
343C4⋊C8—22no—
344C4⋊C8—121no—
345C4⋊C8—121no—
346#103—62no—
347#103—62no—
348#103—62no—
349#102—62no—
350#102—62no—
351#102—62no—
352#102—62no—
353#102—62no—
354#97—62no—
355#95—21no—
356#95—31no—
357#95—31no—
358#95—61no—
359#95—61no—
360C23×C3C22×D411yescharacteristic
361C22×D3—22no—
362C22×D3—22no—
363C22×D3—41no—
364C22×D3—41no—
365C22×D3—41no—
366C22×D3—41no—
367C22×D3—41no—
368C22×D3—41no—
369C22×D3—41no—
370C22×D3—41no—
371C22×D3—41no—
372C22×D3—41no—
373C22×D3—41no—
374C22×D3—41no—
375C3×Q8—84no—
376C2×C3×C4—21no—
377C2×C3×C4—21no—
378C2×C3×C4—21no—
379C2×C3×C4—21no—
380C2×C3×C4—22no—
381C2×C3×C4—41no—
382C2×C3×C4—41no—
383C2×C3×C4—41no—
384C2×C3×C4—41no—
385C2×C3×C4—41no—
386C2×C3×C4—41no—
387C2×C3×C4—41no—
388C2×C3×C4—41no—
389C2×C3×C4—41no—
390C2×C3×C4—41no—
391C2×C3×C4—41no—
392C2×C3×C4#12112yes—
393C2×C3×C4#12112yes—
394C2×C3×C4C22×D411yescharacteristic
395C2×C3×C4C2×#4111yescharacteristic
396C2×C3×C4C2×#4111yescharacteristic
397C2×C3×C4C2×#4111yescharacteristic
398C2×C3×C4H(5,𝔽2)11yescharacteristic
399C2×C3×C42−1+411yescharacteristic
400C2×Dic3—21no—
401C2×Dic3—21no—
402C2×Dic3—21no—
403C2×Dic3—21no—
404C2×Dic3—22no—
405C2×Dic3—21no—
406C2×Dic3—21no—
407C2×Dic3—21no—
408C2×Dic3—21no—
409C2×Dic3—21no—
410C2×Dic3—21no—
411C2×Dic3—41no—
412C2×Dic3—41no—
413C2×Dic3—41no—
414C2×Dic3—41no—
415C2×Dic3—41no—
416C2×Dic3AGL(1,ℤ/8)12yes—
417C2×Dic3AGL(1,ℤ/8)12yes—
418D12—22no—
419D12—22no—
420D12—22no—
421D12—22no—
422D12—41no—
423D12—41no—
424D12—41no—
425D12—41no—
426C4×D3—84no—
427C4×D3—84no—
428Dic6—22no—
429Dic6—22no—
430Dic6—41no—
431Dic6—41no—
432C3×C8—41no—
433C3×C8—41no—
434C3⋊C8—41no—
435C3⋊C8—41no—
436C24—61no—
437C24—61no—
438C2×Q8—32no—
439C2×Q8—42no—
440C2×Q8—42no—
441C2×Q8—42no—
442C2×Q8—61no—
443C2×Q8—61no—
444C2×Q8—61no—
445C2×Q8—61no—
446C2×Q8—61no—
447C2×D4—32no—
448C2×D4—32no—
449C2×D4—61no—
450C2×D4—61no—
451C2×D4—61no—
452C2×D4—61no—
453C2×D4—61no—
454C2×D4—61no—
455C2×D4—61no—
456C2×D4—61no—
457C2×D4—61no—
458C2×D4—61no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—31no—
464C22×C4—31no—
465C22×C4—31no—
466C22×C4—31no—
467C22×C4—61no—
468C22×C4—61no—
469C22×C4—122no—
470C22×C4—122no—
471C22×C4—122no—
472C22×C4—122no—
473C22×C4—122no—
474C22×C4—241no—
475C22×C4D3×D411yescharacteristic
476C22×C4D3×D411yescharacteristic
477C22×C4C23×D311yescharacteristic
478SD16—124no—
479D8—124no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—61no—
487C2×C8—61no—
488C2×C8—61no—
489C2×C8—61no—
490C2×C8—61no—
491C2×C8—61no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—42no—
500C4⋊C4—62no—
501C4⋊C4—62no—
502C4⋊C4—61no—
503C4⋊C4—61no—
504C4⋊C4—122no—
505C4⋊C4—122no—
506C4⋊C4—122no—
507C4⋊C4—122no—
508C4⋊C4—122no—
509C4⋊C4#23912yes—
510C22⋊C4—64no—
511C22⋊C4—64no—
512C22⋊C4—122no—
513C22⋊C4—122no—
514C22⋊C4—122no—
515C22⋊C4—122no—
516C22⋊C4—122no—
517C22⋊C4—122no—
518C22⋊C4—122no—
519C22⋊C4—241no—
520C42—61no—
521C42—61no—
522C42—61no—
523C42—61no—
524C42—81no—
525C42—122no—
526C42#23512yes—
527C42D3×D412yes—
528C22×C3C2×#12111yescharacteristic
529C22×C3#54611yescharacteristic
530C22×C3#54811yescharacteristic
531C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
532C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
533C22×C3#57711yescharacteristic
534C22×C3#57811yescharacteristic
535C2×D3—42no—
536C2×D3—42no—
537C2×D3—42no—
538C2×D3—42no—
539C2×D3—42no—
540C2×D3—42no—
541C2×D3—81no—
542C2×D3—81no—
543C2×D3—81no—
544C2×D3—81no—
545C2×D3—81no—
546C2×D3—81no—
547C2×D3—81no—
548C2×D3—81no—
549C3×C4—22no—
550C3×C4—22no—
551C3×C4—42no—
552C3×C4—81no—
553C3×C4—81no—
554C3×C4—81no—
555C3×C4#52911yescharacteristic
556C3×C4#54211yescharacteristic
557C3×C4#54611yescharacteristic
558C3×C4#54711yescharacteristic
559Dic3—22no—
560Dic3—22no—
561Dic3—42no—
562Dic3—41no—
563Dic3—41no—
564Dic3—81no—
565C23—62no—
566C23—62no—
567C23—121no—
568C23—121no—
569C23—121no—
570C23—121no—
571C23—121no—
572C23—121no—
573C23—121no—
574C23—121no—
575C23—121no—
576C23—121no—
577C23—121no—
578C23—121no—
579C23C2×D3×D411yescharacteristic
580Q8—62no—
581Q8—62no—
582Q8—84no—
583Q8—121no—
584Q8—121no—
585D4—62no—
586D4—62no—
587D4—62no—
588D4—62no—
589D4—121no—
590D4—121no—
591D4—121no—
592D4—121no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—22no—
598C2×C4—32no—
599C2×C4—32no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—61no—
612C2×C4—61no—
613C2×C4—61no—
614C2×C4—61no—
615C2×C4—62no—
616C2×C4—61no—
617C2×C4—61no—
618C2×C4—61no—
619C2×C4—61no—
620C2×C4—61no—
621C2×C4—61no—
622C2×C4—121no—
623C2×C4—121no—
624C2×C4—121no—
625C2×C4—121no—
626C2×C4—121no—
627C2×C4—244no—
628C2×C4—244no—
629C2×C4#88412yes—
630C2×C4#88412yes—
631C2×C4C2×#23511yescharacteristic
632C2×C4C2×D3×D411yescharacteristic
633C2×C4C2×#23911yescharacteristic
634C2×C4#100711yescharacteristic
635C2×C4D3×#4111yescharacteristic
636C2×C4#100911yescharacteristic
637C8—41no—
638C8—41no—
639C8—121no—
640C8—121no—
641C2×C3#242711yescharacteristic
642C2×C3#303811yescharacteristic
643C2×C3#304611yescharacteristic
644C2×C3#330611yescharacteristic
645C2×C3#330811yescharacteristic
646C2×C3#336711yescharacteristic
647C2×C3#337311yescharacteristic
648D3—82no—
649D3—82no—
650C22—122no—
651C22—122no—
652C22—122no—
653C22—122no—
654C22—122no—
655C22—122no—
656C22—241no—
657C22—241no—
658C22—241no—
659C22—241no—
660C22—241no—
661C22—241no—
662C22—241no—
663C22—241no—
664C22C2×#88411yescharacteristic
665C22#563011yescharacteristic
666C22#563311yescharacteristic
667C22#576211yescharacteristic
668C22#576611yescharacteristic
669C22D3×AGL(1,ℤ/8)11yescharacteristic
670C22D3×AGL(1,ℤ/8)11yescharacteristic
671C4—22no—
672C4—22no—
673C4—42no—
674C4—62no—
675C4—62no—
676C4—81no—
677C4—81no—
678C4—81no—
679C4—122no—
680C4—121no—
681C4—121no—
682C4—241no—
683C4#558011yescharacteristic
684C4#558211yescharacteristic
685C4#567611yescharacteristic
686C4#572611yescharacteristic
687C3#2533411yescharacteristic, complemented, Sylow
688C2—242no—
689C2—242no—
690C2#7662411yescharacteristic
691C2#7881511yescharacteristic
692C2#7894211yescharacteristic
693C2#8105211yescharacteristic
694C2#8107711yescharacteristic
695C2#8110011yescharacteristic
696C2#8112111yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1987C2
socleC23×C3C22×D4
prime radicalC22×D24C22
hypercenter#1987D3
hypocenterC3#25334
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1987

Complements

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

NH ≅ G/Ntype
1. #83259C2semidirect
2. C2×#5089C2semidirect
4. #73654C2semidirect
5. C3×#1987C2semidirect
6. #70469C2semidirect
7. #69474C2semidirect
8. #68914C2semidirect
9. #68909C2semidirect
10. #68904C2semidirect
11. #68902C2semidirect
13. #68892C2semidirect
14. #68863C2semidirect
70. #1987D3semidirect
687. C3#25334semidirect

Maximal subgroups

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

subgroupindex
#832592
C2×#50892
C2×#48442
#736542
C3×#19872
#704692
#694742
#689142
#689092
#689042
#689022
#688972
#688922
#688632
#688622
#253343