Finite Group Databasebeta

← #10828479

Subgroups — #10828479

Order 768.

1144 conjugacy classes of subgroups.

The subgroup lattice has 704 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5578C211yesmaximal, characteristic
2C2×#4862C211yesmaximal, characteristic
3C2×#4805C211yesmaximal, characteristic
4D3×#389C211yesmaximal, characteristic
5C3×#1865C211yesmaximal, characteristic, complemented
6#71629C211yesmaximal, characteristic, complemented
7#70565C211yesmaximal, characteristic, complemented
8#70284C211yesmaximal, characteristic, complemented
9#70268C211yesmaximal, characteristic
10#70246C211yesmaximal, characteristic, complemented
11#69632C211yesmaximal, characteristic, complemented
12#68878C211yesmaximal, characteristic
13#68827C211yesmaximal, characteristic, complemented
14#68684C211yesmaximal, characteristic, complemented
15#68664C211yesmaximal, characteristic
16#24063—31nomaximal, Sylow
17C2×C3×C4×Q8C2211yescharacteristic
18C2×#946C2211yescharacteristic
19C2×#944C2211yescharacteristic
20#5578—24no—
21#5578—24no—
22C2×#897C2211yescharacteristic
23C2×#896—21no—
24C2×D3×C4⋊C4—21no—
25C2×D3×C4⋊C4C2211yescharacteristic
26C2×D3×C4⋊C4C2211yescharacteristic
27C2×#889—21no—
28C2×#888C2211yescharacteristic
29C2×#872—21no—
30C2×C42×D3C2211yescharacteristic
31C2×C4×Dic6—21no—
32C2×C4×Dic6C2211yescharacteristic
33C2×C3×#106C2211yescharacteristic
34C2×C3×#103C2211yescharacteristic
35C2×C3×#103C2211yescharacteristic
36C3×#389C2211yescharacteristic
37C2×#835C2211yescharacteristic
38C2×#835C2211yescharacteristic
39C2×(C2×D3)⋊C8C2211yescharacteristic
40C2×(C2×D3)⋊C8C2211yescharacteristic
41C2×#817C2211yescharacteristic
42C2×#816C2211yescharacteristic
43C2×#816C2211yescharacteristic
44#4984C2211yescharacteristic
45#4976—21no—
46#4972C2211yescharacteristic
47C2×#810C2211yescharacteristic
48C2×#810C2211yescharacteristic
49C2×#808C2211yescharacteristic
50C2×#808C2211yescharacteristic
51#4928C2211yescharacteristic
52#4862—24no—
53#4805—24no—
54D3×#95—21no—
55#4647—21no—
56C3×#340C2211yescharacteristic
57C3×#336C2211yescharacteristic
58#4551C2211yescharacteristic
59#4550C2211yescharacteristic
60#4550C2211yescharacteristic
61#4529C2211yescharacteristic
62#4529C2211yescharacteristic
63#4524C2211yescharacteristic
64C2×#533—31no—
65C2×#483—31no—
66C2×#483—31no—
67C2×#389—31no—
68#2058—31no—
69#2058—31no—
70#2043—31no—
71#2043—31no—
72#1872—31no—
73#1872—31no—
74#1865—31no—
75#1865—31no—
76#1865—31no—
77#1865D311yescharacteristic, complemented
78#1817—31no—
79C22×C3×Q8C2311yescharacteristic
80C22×C4×D3—21no—
81C22×C4×D3—21no—
82C22×C4×D3D411yescharacteristic
83C22×C4×D3D411yescharacteristic
84C22×C4×D3C2311yescharacteristic
85C22×Dic6—21no—
86C22×Dic6C2311yescharacteristic
87C22×C3×C8C2311yescharacteristic
88C22×C3×C8C2311yescharacteristic
89C3×C4×Q8—22no—
90C3×C4×Q8—22no—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4—21no—
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C4⋊C4C2311yescharacteristic
96C2×C3×C42—21no—
97C2×C3×C42C2311yescharacteristic
98#946—22no—
99#946D414yes—
100#944—22no—
101#944—22no—
102C2×(C2×D3)⋊C4—21no—
103C2×(C2×D3)⋊C4—21no—
104C2×C4⋊Dic3—21no—
105C2×C4⋊Dic3—21no—
106C2×C4⋊Dic3C2311yescharacteristic
107C2×C4⋊Dic3C2311yescharacteristic
108C2×C4⋊Dic3C2311yescharacteristic
109C2×Dic3⋊C4—21no—
110C2×Dic3⋊C4—21no—
111C2×Dic3⋊C4—21no—
112C2×Dic3⋊C4—21no—
113C2×Dic3⋊C4—21no—
114C2×Dic3⋊C4—21no—
115C2×Dic3⋊C4—21no—
116C2×Dic3⋊C4—21no—
117C2×Dic3⋊C4—21no—
118C2×C4×Dic3—21no—
119C2×C4×Dic3D411yescharacteristic
120C2×C4×Dic3D411yescharacteristic
121C2×C4×Dic3C2311yescharacteristic
122C22×C3⋊C8C2311yescharacteristic
123C22×C3⋊C8C2311yescharacteristic
124#897—22no—
125#897D414yes—
126#896—28no—
127D3×C4⋊C4—22no—
128D3×C4⋊C4—22no—
129D3×C4⋊C4—22no—
130D3×C4⋊C4—22no—
131D3×C4⋊C4—44no—
132#889—28no—
133#888—22no—
134#888—22no—
135#872—28no—
136C42×D3—22no—
137C42×D3—22no—
138C4×Dic6—28no—
139C4×Dic6—22no—
140C4×Dic6—22no—
141C3×#106—22no—
142C3×#103—22no—
143C3×#103—22no—
144C3×#95—21no—
145#835—22no—
146#835—22no—
147C3⋊#95—21no—
148C3⋊#95—21no—
149C3⋊#95—21no—
150(C2×D3)⋊C8—22no—
151(C2×D3)⋊C8—22no—
152#817—22no—
153#816—22no—
154#816—22no—
155#810—22no—
156#810—22no—
157#808—22no—
158#808—22no—
159#533—64no—
160#533—64no—
161C2×C4⋊Q8—31no—
162C2×C4⋊Q8—31no—
163C2×#125—61no—
164C2×C22⋊Q8—31no—
165C2×C22⋊Q8—31no—
166C2×C22⋊Q8—61no—
167C2×C4×Q8—31no—
168C2×C4×Q8—61no—
169C2×C4×Q8C2×D311yescharacteristic
170C2×#117—61no—
171C22×C4⋊C4—31no—
172C22×C4⋊C4—31no—
173C22×C4⋊C4—61no—
174C22×C42—31no—
175#483—64no—
176#483—64no—
177C2×#106—31no—
178C2×#106—31no—
179C2×#106—31no—
180C2×#106C2×D311yescharacteristic
181C2×#103—31no—
182C2×#103—31no—
183C2×#103—31no—
184C2×#103—31no—
185C2×#103—31no—
186C2×#103—31no—
187C2×#103C2×D311yescharacteristic
188C2×#103C2×D311yescharacteristic
189C2×C22⋊C8—31no—
190C2×C22⋊C8—31no—
191#389—31no—
192#389—31no—
193#389—31no—
194#389—61no—
195#389—61no—
196#389C2×D311yescharacteristic
197C2×#95—61no—
198#340—31no—
199#340—31no—
200#340—31no—
201#340C2×D311yescharacteristic
202#336—31no—
203#336—31no—
204#336—31no—
205#336C2×D311yescharacteristic
206C23×D3C2×D411yescharacteristic
207C2×C3×Q8—21no—
208C2×C3×Q8—21no—
209C2×C3×Q8—21no—
210C2×C3×Q8—21no—
211C2×C3×Q8—21no—
212C2×C3×Q8C2×D412yes—
213C22×C3×C4—21no—
214C22×C3×C4—21no—
215C22×C3×C4—21no—
216C22×C3×C4—21no—
217C22×C3×C4C2×D411yescharacteristic
218C22×C3×C4C2×D411yescharacteristic
219C22×C3×C4C2411yescharacteristic
220C22×Dic3—21no—
221C22×Dic3—21no—
222C22×Dic3—21no—
223C22×Dic3—21no—
224C22×Dic3C2×D411yescharacteristic
225C22×Dic3C2×D411yescharacteristic
226C22×Dic3C2×D411yescharacteristic
227C2×C4×D3—22no—
228C2×C4×D3—22no—
229C2×C4×D3—21no—
230C2×C4×D3—21no—
231C2×C4×D3—21no—
232C2×C4×D3—21no—
233C2×C4×D3—24no—
234C2×C4×D3—22no—
235C2×C4×D3—22no—
236C2×C4×D3—41no—
237C2×C4×D3—42no—
238C2×C4×D3—42no—
239C2×C4×D3—42no—
240C2×C4×D3—42no—
241C2×C4×D3—42no—
242C2×C4×D3SD1614yes—
243C2×C4×D3C2×D412yes—
244C2×C4×D3C2×D412yes—
245C2×Dic6—21no—
246C2×Dic6—21no—
247C2×Dic6—21no—
248C2×Dic6—21no—
249C2×Dic6—24no—
250C2×Dic6—21no—
251C2×Dic6—42no—
252C2×Dic6—42no—
253C2×Dic6C2×D412yes—
254C2×C3×C8—21no—
255C2×C3×C8—21no—
256C2×C3×C8—21no—
257C2×C3×C8—21no—
258C2×C3×C8—21no—
259C2×C3×C8—21no—
260C3×C4⋊C4—24no—
261C3×C4⋊C4—21no—
262C3×C4⋊C4—21no—
263C3×C4⋊C4—21no—
264C3×C4⋊C4—21no—
265C3×C4⋊C4—42no—
266C3×C4⋊C4C2×D412yes—
267C3×C4⋊C4#4112yes—
268C3×C42—24no—
269C3×C42—21no—
270C3×C42—21no—
271(C2×D3)⋊C4—24no—
272(C2×D3)⋊C4—24no—
273C4⋊Dic3—21no—
274C4⋊Dic3—21no—
275C4⋊Dic3—24no—
276C4⋊Dic3—21no—
277C4⋊Dic3—21no—
278C4⋊Dic3—42no—
279C4⋊Dic3C2×D412yes—
280C4⋊Dic3#4112yes—
281Dic3⋊C4—24no—
282Dic3⋊C4—24no—
283Dic3⋊C4—24no—
284Dic3⋊C4—24no—
285Dic3⋊C4—24no—
286Dic3⋊C4—24no—
287Dic3⋊C4—42no—
288Dic3⋊C4—42no—
289Dic3⋊C4—42no—
290C4×Dic3—24no—
291C4×Dic3—21no—
292C4×Dic3—21no—
293C4×Dic3—22no—
294C4×Dic3—22no—
295C2×C3⋊C8—21no—
296C2×C3⋊C8—21no—
297C2×C3⋊C8—21no—
298C2×C3⋊C8—21no—
299C2×C3⋊C8—21no—
300C2×C3⋊C8—21no—
301C22×Q8—31no—
302C22×Q8—61no—
303C22×Q8C22×D311yescharacteristic
304C23×C4—31no—
305C23×C4—31no—
306C23×C4—31no—
307C23×C4—61no—
308C23×C4—61no—
309C22×C8—31no—
310C22×C8—31no—
311C22×C8C22×D311yescharacteristic
312C22×C8C22×D311yescharacteristic
313C4⋊Q8—62no—
314C4⋊Q8—62no—
315C4⋊Q8—62no—
316C4⋊Q8—62no—
317#125—68no—
318C22⋊Q8—34no—
319C22⋊Q8—34no—
320C22⋊Q8—62no—
321C22⋊Q8—68no—
322C22⋊Q8—62no—
323C4×Q8—22no—
324C4×Q8—22no—
325C4×Q8—62no—
326C4×Q8—62no—
327C4×Q8—68no—
328#117—68no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—31no—
332C2×C4⋊C4—31no—
333C2×C4⋊C4—31no—
334C2×C4⋊C4—62no—
335C2×C4⋊C4—62no—
336C2×C4⋊C4—62no—
337C2×C4⋊C4—61no—
338C2×C4⋊C4—61no—
339C2×C4⋊C4—61no—
340C2×C4⋊C4—61no—
341C2×C4⋊C4—61no—
342C2×C4⋊C4—61no—
343C2×C4⋊C4—61no—
344C2×C4⋊C4—62no—
345C2×C4⋊C4—61no—
346C2×C4⋊C4—61no—
347C2×C4⋊C4—61no—
348C2×C4⋊C4—61no—
349C2×C4⋊C4—124no—
350C2×C4⋊C4C22×D311yescharacteristic
351C2×C4⋊C4C22×D311yescharacteristic
352C2×C4⋊C4C22×D311yescharacteristic
353C2×C22⋊C4—61no—
354C2×C22⋊C4—61no—
355C2×C42—21no—
356C2×C42—31no—
357C2×C42—31no—
358C2×C42—31no—
359C2×C42—62no—
360C2×C42—62no—
361C2×C42—61no—
362C2×C42C22×D311yescharacteristic
363#106—22no—
364#106—62no—
365#106—62no—
366#106—62no—
367#103—22no—
368#103—22no—
369#103—62no—
370#103—62no—
371#103—62no—
372#103—62no—
373#103—62no—
374#103—62no—
375C22⋊C8—62no—
376C22⋊C8—62no—
377#95—21no—
378#95—61no—
379#95—61no—
380#95—61no—
381C23×C3C22×D411yescharacteristic
382C22×D3—21no—
383C22×D3—21no—
384C22×D3—21no—
385C22×D3—21no—
386C22×D3#12012yes—
387C22×D3C2×SD1612yes—
388C22×D3#13712yes—
389C3×Q8—22no—
390C3×Q8—22no—
391C3×Q8—41no—
392C3×Q8—41no—
393C2×C3×C4—21no—
394C2×C3×C4—21no—
395C2×C3×C4—21no—
396C2×C3×C4—22no—
397C2×C3×C4—22no—
398C2×C3×C4—22no—
399C2×C3×C4—21no—
400C2×C3×C4—21no—
401C2×C3×C4—41no—
402C2×C3×C4—41no—
403C2×C3×C4—41no—
404C2×C3×C4—41no—
405C2×C3×C4—41no—
406C2×C3×C4—41no—
407C2×C3×C4—41no—
408C2×C3×C4—41no—
409C2×C3×C4—41no—
410C2×C3×C4C2×SD1612yes—
411C2×C3×C4C22×D411yescharacteristic
412C2×C3×C4C22×D411yescharacteristic
413C2×C3×C4C2×#4111yescharacteristic
414C2×C3×C4C2×#4111yescharacteristic
415C2×C3×C42−1+411yescharacteristic
416C2×C3×C42−1+411yescharacteristic
417C2×Dic3—21no—
418C2×Dic3—21no—
419C2×Dic3—21no—
420C2×Dic3—21no—
421C2×Dic3—21no—
422C2×Dic3—21no—
423C2×Dic3—22no—
424C2×Dic3—22no—
425C2×Dic3—22no—
426C2×Dic3—41no—
427C2×Dic3—41no—
428C2×Dic3—41no—
429C2×Dic3—41no—
430C2×Dic3—41no—
431C2×Dic3—41no—
432C2×Dic3—41no—
433C2×Dic3—41no—
434C2×Dic3—41no—
435C2×Dic3#12012yes—
436C2×Dic3C2×SD1612yes—
437C2×Dic3#13712yes—
438C4×D3—28no—
439C4×D3—22no—
440C4×D3—22no—
441C4×D3—22no—
442C4×D3—22no—
443C4×D3—44no—
444C4×D3—48no—
445C4×D3—84no—
446Dic6—22no—
447Dic6—22no—
448Dic6—48no—
449Dic6—41no—
450Dic6—41no—
451C3×C8—41no—
452C3×C8—41no—
453C3⋊C8—41no—
454C3⋊C8—41no—
455C24—31no—
456C2×Q8—21no—
457C2×Q8—21no—
458C2×Q8—21no—
459C2×Q8—21no—
460C2×Q8—21no—
461C2×Q8—32no—
462C2×Q8—64no—
463C2×Q8—61no—
464C2×Q8—61no—
465C2×Q8—61no—
466C2×Q8—61no—
467C2×Q8—61no—
468C2×Q8—122no—
469C2×Q8—122no—
470C2×Q8D3×D412yes—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—31no—
476C22×C4—31no—
477C22×C4—31no—
478C22×C4—34no—
479C22×C4—32no—
480C22×C4—32no—
481C22×C4—62no—
482C22×C4—62no—
483C22×C4—62no—
484C22×C4—62no—
485C22×C4—64no—
486C22×C4—61no—
487C22×C4—61no—
488C22×C4—61no—
489C22×C4—61no—
490C22×C4—61no—
491C22×C4—61no—
492C22×C4—61no—
493C22×C4—61no—
494C22×C4—121no—
495C22×C4—122no—
496C22×C4—122no—
497C22×C4—122no—
498C22×C4—122no—
499C22×C4—122no—
500C22×C4D3×D411yescharacteristic
501C22×C4D3×D411yescharacteristic
502C22×C4C23×D311yescharacteristic
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—21no—
508C2×C8—21no—
509C2×C8—61no—
510C2×C8—61no—
511C2×C8—61no—
512C2×C8—61no—
513C2×C8—61no—
514C2×C8—61no—
515C4⋊C4—24no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—32no—
521C4⋊C4—32no—
522C4⋊C4—42no—
523C4⋊C4—64no—
524C4⋊C4—64no—
525C4⋊C4—64no—
526C4⋊C4—64no—
527C4⋊C4—64no—
528C4⋊C4—64no—
529C4⋊C4—64no—
530C4⋊C4—61no—
531C4⋊C4—61no—
532C4⋊C4—61no—
533C4⋊C4—61no—
534C4⋊C4—122no—
535C4⋊C4—122no—
536C4⋊C4—122no—
537C4⋊C4—122no—
538C4⋊C4#23512yes—
539C4⋊C4D3×D412yes—
540C22⋊C4—64no—
541C22⋊C4—64no—
542C42—24no—
543C42—21no—
544C42—21no—
545C42—62no—
546C42—62no—
547C42—64no—
548C42—61no—
549C42—61no—
550C22×C3C2×#12011yescharacteristic
551C22×C3#52511yescharacteristic
552C22×C3#53611yescharacteristic
553C22×C3C22×SD1611yescharacteristic
554C22×C3C2×#13711yescharacteristic
555C22×C3C2×#13711yescharacteristic
556C22×C3#57511yescharacteristic
557C2×D3—22no—
558C2×D3—22no—
559C2×D3—22no—
560C2×D3—22no—
561C2×D3—22no—
562C2×D3—22no—
563C2×D3#44814yes—
564C3×C4—22no—
565C3×C4—42no—
566C3×C4—42no—
567C3×C4—42no—
568C3×C4—41no—
569C3×C4—81no—
570C3×C4#54711yescharacteristic
571C3×C4#54711yescharacteristic
572C3×C4#54711yescharacteristic
573C3×C4#54711yescharacteristic
574Dic3—22no—
575Dic3—22no—
576Dic3—22no—
577Dic3—42no—
578Dic3—42no—
579Dic3—42no—
580Dic3—81no—
581C23—32no—
582C23—32no—
583C23—32no—
584C23—61no—
585C23—61no—
586C23—61no—
587C23—61no—
588C23C2×D3×D411yescharacteristic
589Q8—22no—
590Q8—22no—
591Q8—41no—
592Q8—41no—
593Q8—62no—
594Q8—62no—
595Q8—128no—
596Q8—121no—
597Q8—121no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—22no—
603C2×C4—21no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—32no—
607C2×C4—32no—
608C2×C4—32no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—68no—
619C2×C4—61no—
620C2×C4—61no—
621C2×C4—61no—
622C2×C4—61no—
623C2×C4—61no—
624C2×C4—61no—
625C2×C4—62no—
626C2×C4—62no—
627C2×C4—62no—
628C2×C4—62no—
629C2×C4—62no—
630C2×C4—62no—
631C2×C4—62no—
632C2×C4—124no—
633C2×C4—128no—
634C2×C4—121no—
635C2×C4—121no—
636C2×C4—121no—
637C2×C4—121no—
638C2×C4—121no—
639C2×C4—121no—
640C2×C4—121no—
641C2×C4—121no—
642C2×C4—121no—
643C2×C4—244no—
644C2×C4D3×SD1612yes—
645C2×C4C2×#23511yescharacteristic
646C2×C4C2×D3×D411yescharacteristic
647C2×C4C2×D3×D411yescharacteristic
648C2×C4#100711yescharacteristic
649C2×C4D3×#4111yescharacteristic
650C2×C4#101011yescharacteristic
651C8—41no—
652C8—41no—
653C8—121no—
654C8—121no—
655C2×C3#240611yescharacteristic
656C2×C3C2×#44811yescharacteristic
657C2×C3#300711yescharacteristic
658C2×C3#310111yescharacteristic
659C2×C3#311311yescharacteristic
660C2×C3#322911yescharacteristic
661C2×C3#323611yescharacteristic
662D3—24no—
663C22—34no—
664C22—62no—
665C22—62no—
666C22—62no—
667C22—62no—
668C22—62no—
669C22—62no—
670C22#554711yescharacteristic
671C22D3×#12011yescharacteristic
672C22#562811yescharacteristic
673C22C2×D3×SD1611yescharacteristic
674C22#576811yescharacteristic
675C22D3×#13711yescharacteristic
676C22D3×#13711yescharacteristic
677C4—22no—
678C4—42no—
679C4—41no—
680C4—42no—
681C4—42no—
682C4—62no—
683C4—62no—
684C4—62no—
685C4—81no—
686C4—122no—
687C4—122no—
688C4—122no—
689C4—241no—
690C4#550711yescharacteristic
691C4#555111yescharacteristic
692C4#563111yescharacteristic
693C4#565511yescharacteristic
694C3#2406311yescharacteristic, complemented, Sylow
695C2—64no—
696C2#7460011yescharacteristic
697C2#7965111yescharacteristic
698C2#7983311yescharacteristic
699C2D3×#44811yescharacteristic
700C2#8022811yescharacteristic
701C2#8101511yescharacteristic
702C2#8123811yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1865C2
socleC23×C3C22×D4
prime radicalC23×D3C2×D4
hypercenter#1865D3
hypocenterC3#24063
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1865

Complements

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

NH ≅ G/Ntype
5. C3×#1865C2semidirect
6. #71629C2semidirect
7. #70565C2semidirect
8. #70284C2semidirect
10. #70246C2semidirect
11. #69632C2semidirect
13. #68827C2semidirect
14. #68684C2semidirect
77. #1865D3semidirect
694. C3#24063semidirect

Maximal subgroups

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

subgroupindex
C2×#55782
C2×#48622
C2×#48052
D3×#3892
C3×#18652
#716292
#705652
#702842
#702682
#702462
#696322
#688782
#688272
#686842
#686642
#240633