Finite Group Databasebeta

← #10945059

Subgroups — #10945059

Order 768.

1216 conjugacy classes of subgroups.

The subgroup lattice has 718 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5572C211yesmaximal, characteristic
2C2×#4868C211yesmaximal, characteristic
3C2×#4809C211yesmaximal, characteristic
4#76102C211yesmaximal, characteristic
5C3×#1865C211yesmaximal, characteristic, complemented
6#71632C211yesmaximal, characteristic, complemented
7#70588C211yesmaximal, characteristic, complemented
8#70266C211yesmaximal, characteristic
9#70239C211yesmaximal, characteristic, complemented
10#70235C211yesmaximal, characteristic, complemented
11#69631C211yesmaximal, characteristic, complemented
12#68908C211yesmaximal, characteristic, complemented
13#68873C211yesmaximal, characteristic
14#68685C211yesmaximal, characteristic, complemented
15#68665C211yesmaximal, characteristic
16#33019—31nomaximal, Sylow
17C22×D3×Q8C2211yescharacteristic
18C2×C3×C4×Q8C2211yescharacteristic
19C2×Q8×Dic3C2211yescharacteristic
20C22×#216C2211yescharacteristic
21C22×Dic12C2211yescharacteristic
22#5572—24no—
23#5572—24no—
24C2×#897C2211yescharacteristic
25C2×#892—21no—
26C2×#892C2211yescharacteristic
27C2×#887—21no—
28C2×#887C2211yescharacteristic
29C2×#872—21no—
30C2×#872C2211yescharacteristic
31C2×C4×Dic6—21no—
32C2×C4×Dic6C2211yescharacteristic
33C2×C3×#106C2211yescharacteristic
34C2×C3×#103C2211yescharacteristic
35C2×C3×#103C2211yescharacteristic
36C3×#389C2211yescharacteristic
37C2×#835C2211yescharacteristic
38C2×(C2×D3)⋊C8C2211yescharacteristic
39C2×(C2×D3)⋊C8C2211yescharacteristic
40C2×#816C2211yescharacteristic
41C2×#816C2211yescharacteristic
42#4992—21no—
43#4987C2211yescharacteristic
44#4982—21no—
45#4973C2211yescharacteristic
46#4972C2211yescharacteristic
47C2×#810C2211yescharacteristic
48C2×#810C2211yescharacteristic
49C2×#810C2211yescharacteristic
50C2×#807C2211yescharacteristic
51#4929C2211yescharacteristic
52#4868—24no—
53#4809—24no—
54#4675—21no—
55#4660—21no—
56C3×#340C2211yescharacteristic
57C3×#336C2211yescharacteristic
58#4551C2211yescharacteristic
59#4550C2211yescharacteristic
60#4550C2211yescharacteristic
61#4529C2211yescharacteristic
62#4529C2211yescharacteristic
63#4524C2211yescharacteristic
64C2×#519—31no—
65C2×#485—31no—
66C2×#451—31no—
67#2625—31no—
68#1878—31no—
69#1878—31no—
70#1878—31no—
71#1878—31no—
72#1865—31no—
73#1865D311yescharacteristic, complemented
74#1863—31no—
75#1863—31no—
76#1825—31no—
77#1818—31no—
78#1787—31no—
79C22×C3×Q8C2311yescharacteristic
80C2×D3×Q8—22no—
81C2×D3×Q8—22no—
82C2×D3×Q8—22no—
83C2×D3×Q8—22no—
84C2×D3×Q8—22no—
85C2×D3×Q8D414yes—
86C22×C4×D3—21no—
87C22×C4×D3C2311yescharacteristic
88C22×Dic6—21no—
89C22×Dic6C2311yescharacteristic
90C22×Dic6C2311yescharacteristic
91C22×C3×C8C2311yescharacteristic
92C22×C3×C8C2311yescharacteristic
93C3×C4×Q8—22no—
94C3×C4×Q8—22no—
95C2×C3×C4⋊C4—21no—
96C2×C3×C4⋊C4—21no—
97C2×C3×C4⋊C4C2311yescharacteristic
98C2×C3×C4⋊C4C2311yescharacteristic
99C2×C3×C4⋊C4C2311yescharacteristic
100C2×C3×C42—21no—
101C2×C3×C42C2311yescharacteristic
102Q8×Dic3—22no—
103Q8×Dic3—22no—
104C2×#216—22no—
105C2×#216—22no—
106C2×#216—22no—
107C2×(C2×D3)⋊C4—21no—
108C2×(C2×D3)⋊C4—21no—
109C2×(C2×D3)⋊C4—21no—
110C2×(C2×D3)⋊C4D411yescharacteristic
111C2×(C2×D3)⋊C4D411yescharacteristic
112C2×C4⋊Dic3—21no—
113C2×C4⋊Dic3—21no—
114C2×C4⋊Dic3C2311yescharacteristic
115C2×C4⋊Dic3C2311yescharacteristic
116C2×Dic3⋊C4—21no—
117C2×Dic3⋊C4—21no—
118C2×Dic3⋊C4—21no—
119C2×Dic3⋊C4D411yescharacteristic
120C2×Dic3⋊C4D411yescharacteristic
121C2×C4×Dic3—21no—
122C2×C4×Dic3—21no—
123C2×C4×Dic3—21no—
124C2×C4×Dic3—21no—
125C2×C4×Dic3C2311yescharacteristic
126C22×C3⋊C8C2311yescharacteristic
127C22×C3⋊C8C2311yescharacteristic
128C2×Dic12—22no—
129C2×Dic12—22no—
130C2×Dic12—22no—
131#897—22no—
132#897—22no—
133#892—28no—
134#892—22no—
135#892D414yes—
136#887—28no—
137#887—22no—
138#887—22no—
139#872—28no—
140#872—22no—
141#872—22no—
142C4×Dic6—28no—
143C4×Dic6—22no—
144C4×Dic6—22no—
145C3×#106—22no—
146C3×#103—22no—
147C3×#103—22no—
148C3×#95—21no—
149#835—22no—
150C3⋊#95—21no—
151C3⋊#95—21no—
152C3⋊#95—21no—
153C3⋊#95—21no—
154C3⋊#95—21no—
155C3⋊#95—21no—
156(C2×D3)⋊C8—22no—
157(C2×D3)⋊C8—22no—
158#816—22no—
159#816—22no—
160#810—22no—
161#810—22no—
162#810—22no—
163#807—22no—
164C23×Q8—31no—
165C22×Q16—31no—
166C22×Q16—31no—
167C2×C22⋊Q8—31no—
168#519—64no—
169#519—64no—
170C2×C4×Q8—31no—
171C2×C4×Q8—31no—
172C2×C4×Q8—31no—
173C2×C4×Q8—61no—
174C2×C4×Q8—61no—
175C2×C4×Q8C2×D311yescharacteristic
176C2×#117—31no—
177C2×#117—31no—
178C2×#117—61no—
179C2×#117—61no—
180#485—64no—
181#451—64no—
182C2×#107—31no—
183C2×#106C2×D311yescharacteristic
184C2×#103—31no—
185C2×#103—31no—
186C2×#103—31no—
187C2×#103—31no—
188C2×#103—31no—
189C2×#103—31no—
190C2×#103C2×D311yescharacteristic
191C2×#103C2×D311yescharacteristic
192C2×C22⋊C8—31no—
193C2×C22⋊C8—31no—
194#400—61no—
195#398—61no—
196#397—61no—
197#397—61no—
198#391—31no—
199#391—31no—
200#389—31no—
201#389C2×D311yescharacteristic
202#380—31no—
203#340—31no—
204#340—31no—
205#340—31no—
206#340C2×D311yescharacteristic
207#336—31no—
208#336—31no—
209#336—31no—
210#336C2×D311yescharacteristic
211C23×D3C2×D411yescharacteristic
212C2×C3×Q8—21no—
213C2×C3×Q8—21no—
214C2×C3×Q8—21no—
215C2×C3×Q8C2×D412yes—
216C2×C3×Q8#4112yes—
217C2×C3×Q8#4112yes—
218C22×C3×C4—21no—
219C22×C3×C4—21no—
220C22×C3×C4—21no—
221C22×C3×C4—21no—
222C22×C3×C4C2×D411yescharacteristic
223C22×C3×C4C2×D411yescharacteristic
224C22×C3×C4C2411yescharacteristic
225C22×Dic3—21no—
226C22×Dic3—21no—
227C22×Dic3—21no—
228C22×Dic3—21no—
229C22×Dic3C2×D411yescharacteristic
230C22×Dic3C2×D411yescharacteristic
231C22×Dic3C2×D411yescharacteristic
232D3×Q8—28no—
233D3×Q8—28no—
234D3×Q8—28no—
235D3×Q8—28no—
236C2×C4×D3—24no—
237C2×C4×D3—24no—
238C2×C4×D3—24no—
239C2×C4×D3—21no—
240C2×C4×D3—21no—
241C2×C4×D3—21no—
242C2×C4×D3—21no—
243C2×C4×D3C2×D412yes—
244C2×C4×D3C2×D412yes—
245C2×Dic6—24no—
246C2×Dic6—24no—
247C2×Dic6—21no—
248C2×Dic6—21no—
249C2×Dic6—21no—
250C2×Dic6—24no—
251C2×Dic6—21no—
252C2×Dic6—21no—
253C2×Dic6—21no—
254C2×Dic6—21no—
255C2×Dic6—21no—
256C2×Dic6—21no—
257C2×Dic6C2×D412yes—
258C2×Dic6#4112yes—
259C2×Dic6#4112yes—
260C2×C3×C8—21no—
261C2×C3×C8—21no—
262C2×C3×C8—21no—
263C2×C3×C8—21no—
264C2×C3×C8—21no—
265C2×C3×C8—21no—
266C3×C4⋊C4—24no—
267C3×C4⋊C4—21no—
268C3×C4⋊C4—21no—
269C3×C4⋊C4—21no—
270C3×C4⋊C4—21no—
271C3×C4⋊C4—21no—
272C3×C4⋊C4—81no—
273C3×C4⋊C4C2×D412yes—
274C3×C42—24no—
275C3×C42—21no—
276C3×C42—21no—
277#216—44no—
278(C2×D3)⋊C4—22no—
279(C2×D3)⋊C4—22no—
280(C2×D3)⋊C4—24no—
281(C2×D3)⋊C4—42no—
282(C2×D3)⋊C4—81no—
283C4⋊Dic3—24no—
284C4⋊Dic3—21no—
285C4⋊Dic3—21no—
286C4⋊Dic3—21no—
287C4⋊Dic3—42no—
288C4⋊Dic3C2×D412yes—
289Dic3⋊C4—24no—
290Dic3⋊C4—24no—
291Dic3⋊C4—22no—
292Dic3⋊C4—22no—
293Dic3⋊C4—81no—
294C4×Dic3—24no—
295C4×Dic3—24no—
296C4×Dic3—24no—
297C4×Dic3—21no—
298C4×Dic3—21no—
299C4×Dic3—24no—
300C2×C3⋊C8—21no—
301C2×C3⋊C8—21no—
302C2×C3⋊C8—21no—
303C2×C3⋊C8—21no—
304C2×C3⋊C8—21no—
305C2×C3⋊C8—21no—
306Dic12—44no—
307C22×Q8—34no—
308C22×Q8—31no—
309C22×Q8—31no—
310C22×Q8—61no—
311C22×Q8—62no—
312C22×Q8—62no—
313C22×Q8—62no—
314C22×Q8—62no—
315C22×Q8—62no—
316C22×Q8C22×D311yescharacteristic
317C23×C4—31no—
318C23×C4—61no—
319C2×Q16—62no—
320C2×Q16—62no—
321C2×Q16—62no—
322C2×Q16—62no—
323C2×Q16—62no—
324C2×Q16—62no—
325C22×C8—31no—
326C22×C8—31no—
327C22×C8C22×D311yescharacteristic
328C22×C8C22×D311yescharacteristic
329C22⋊Q8—62no—
330C22⋊Q8—62no—
331C4×Q8—22no—
332C4×Q8—22no—
333C4×Q8—62no—
334C4×Q8—62no—
335C4×Q8—68no—
336C4×Q8—62no—
337C4×Q8—68no—
338C4×Q8—62no—
339C4×Q8—62no—
340C4×Q8—62no—
341#117—34no—
342#117—62no—
343#117—62no—
344#117—68no—
345#117—62no—
346#117—68no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—31no—
350C2×C4⋊C4—31no—
351C2×C4⋊C4—31no—
352C2×C4⋊C4—31no—
353C2×C4⋊C4—61no—
354C2×C4⋊C4—61no—
355C2×C4⋊C4—61no—
356C2×C4⋊C4—61no—
357C2×C4⋊C4—61no—
358C2×C4⋊C4C22×D311yescharacteristic
359C2×C4⋊C4C22×D311yescharacteristic
360C2×C4⋊C4C22×D311yescharacteristic
361C2×C22⋊C4—31no—
362C2×C22⋊C4—31no—
363C2×C22⋊C4—61no—
364C2×C22⋊C4—61no—
365C2×C22⋊C4—61no—
366C2×C42—21no—
367C2×C42—31no—
368C2×C42—61no—
369C2×C42—61no—
370C2×C42—61no—
371C2×C42—61no—
372C2×C42C22×D311yescharacteristic
373#107—62no—
374#106—22no—
375#103—22no—
376#103—22no—
377#103—62no—
378#103—62no—
379#103—62no—
380#103—62no—
381#103—62no—
382#103—62no—
383C22⋊C8—62no—
384C22⋊C8—62no—
385#95—21no—
386#95—61no—
387#95—61no—
388#95—61no—
389#95—61no—
390#95—61no—
391#95—61no—
392C23×C3C22×D411yescharacteristic
393C22×D3—21no—
394C22×D3—21no—
395C22×D3—21no—
396C22×D3—21no—
397C22×D3—21no—
398C22×D3—21no—
399C22×D3#12012yes—
400C3×Q8—22no—
401C3×Q8—22no—
402C3×Q8—22no—
403C3×Q8#12314yes—
404C2×C3×C4—22no—
405C2×C3×C4—22no—
406C2×C3×C4—22no—
407C2×C3×C4—21no—
408C2×C3×C4—22no—
409C2×C3×C4—21no—
410C2×C3×C4—21no—
411C2×C3×C4—21no—
412C2×C3×C4—21no—
413C2×C3×C4—21no—
414C2×C3×C4—41no—
415C2×C3×C4—41no—
416C2×C3×C4—41no—
417C2×C3×C4—41no—
418C2×C3×C4—41no—
419C2×C3×C4—41no—
420C2×C3×C4—41no—
421C2×C3×C4—41no—
422C2×C3×C4C22×D411yescharacteristic
423C2×C3×C4C22×D411yescharacteristic
424C2×C3×C4C2×#4111yescharacteristic
425C2×C3×C4C2×#4111yescharacteristic
426C2×C3×C4C2×#4111yescharacteristic
427C2×C3×C4C2×#4111yescharacteristic
428C2×Dic3—22no—
429C2×Dic3—21no—
430C2×Dic3—21no—
431C2×Dic3—22no—
432C2×Dic3—22no—
433C2×Dic3—21no—
434C2×Dic3—21no—
435C2×Dic3—22no—
436C2×Dic3—21no—
437C2×Dic3—21no—
438C2×Dic3—21no—
439C2×Dic3—21no—
440C2×Dic3—41no—
441C2×Dic3—41no—
442C2×Dic3—41no—
443C2×Dic3—41no—
444C2×Dic3—41no—
445C2×Dic3—41no—
446C2×Dic3—41no—
447C2×Dic3—41no—
448C2×Dic3#12012yes—
449C4×D3—216no—
450C4×D3—22no—
451C4×D3—22no—
452C4×D3—22no—
453C4×D3—22no—
454Dic6—216no—
455Dic6—22no—
456Dic6—22no—
457Dic6—22no—
458Dic6—41no—
459Dic6—41no—
460Dic6—41no—
461Dic6—41no—
462Dic6#12314yes—
463C3×C8—41no—
464C3×C8—41no—
465C3⋊C8—41no—
466C3⋊C8—41no—
467C24—31no—
468C2×Q8—21no—
469C2×Q8—21no—
470C2×Q8—21no—
471C2×Q8—32no—
472C2×Q8—32no—
473C2×Q8—32no—
474C2×Q8—64no—
475C2×Q8—64no—
476C2×Q8—64no—
477C2×Q8—68no—
478C2×Q8—68no—
479C2×Q8—68no—
480C2×Q8—68no—
481C2×Q8—61no—
482C2×Q8—61no—
483C2×Q8—61no—
484C2×Q8—61no—
485C2×Q8—61no—
486C2×Q8—61no—
487C2×Q8—61no—
488C2×Q8—61no—
489C2×Q8—61no—
490C2×Q8#23512yes—
491C2×Q8D3×D412yes—
492C2×Q8#23912yes—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—21no—
496C22×C4—21no—
497C22×C4—31no—
498C22×C4—32no—
499C22×C4—32no—
500C22×C4—31no—
501C22×C4—31no—
502C22×C4—64no—
503C22×C4—64no—
504C22×C4—64no—
505C22×C4—61no—
506C22×C4—61no—
507C22×C4—61no—
508C22×C4—61no—
509C22×C4—61no—
510C22×C4—61no—
511C22×C4—61no—
512C22×C4—61no—
513C22×C4D3×D411yescharacteristic
514C22×C4D3×D411yescharacteristic
515C22×C4C23×D311yescharacteristic
516Q16—124no—
517Q16—124no—
518C2×C8—21no—
519C2×C8—21no—
520C2×C8—21no—
521C2×C8—21no—
522C2×C8—21no—
523C2×C8—21no—
524C2×C8—61no—
525C2×C8—61no—
526C2×C8—61no—
527C2×C8—61no—
528C2×C8—61no—
529C2×C8—61no—
530C4⋊C4—24no—
531C4⋊C4—21no—
532C4⋊C4—21no—
533C4⋊C4—21no—
534C4⋊C4—21no—
535C4⋊C4—21no—
536C4⋊C4—32no—
537C4⋊C4—62no—
538C4⋊C4—62no—
539C4⋊C4—64no—
540C4⋊C4—64no—
541C4⋊C4—64no—
542C4⋊C4—61no—
543C4⋊C4—61no—
544C4⋊C4—61no—
545C4⋊C4—81no—
546C4⋊C4—122no—
547C4⋊C4—241no—
548C4⋊C4D3×D412yes—
549C22⋊C4—62no—
550C22⋊C4—62no—
551C22⋊C4—64no—
552C22⋊C4—122no—
553C22⋊C4—241no—
554C42—24no—
555C42—21no—
556C42—21no—
557C42—64no—
558C42—64no—
559C42—64no—
560C42—64no—
561C42—61no—
562C42—61no—
563C22×C3C2×#12011yescharacteristic
564C22×C3#52511yescharacteristic
565C22×C3#52511yescharacteristic
566C22×C3#57711yescharacteristic
567C22×C3#57811yescharacteristic
568C22×C3#57811yescharacteristic
569C22×C3#57811yescharacteristic
570C2×D3—22no—
571C2×D3—22no—
572C2×D3—22no—
573C2×D3—41no—
574C2×D3—41no—
575C2×D3—41no—
576C2×D3—41no—
577C3×C4—24no—
578C3×C4—41no—
579C3×C4—42no—
580C3×C4—41no—
581C3×C4—81no—
582C3×C4—81no—
583C3×C4C2×#12311yescharacteristic
584C3×C4C2×#12311yescharacteristic
585C3×C4#52511yescharacteristic
586C3×C4#52511yescharacteristic
587Dic3—22no—
588Dic3—24no—
589Dic3—41no—
590Dic3—41no—
591Dic3—42no—
592Dic3—81no—
593Dic3—81no—
594C23—32no—
595C23—61no—
596C23—61no—
597C23—61no—
598C23—61no—
599C23—61no—
600C23—61no—
601C23C2×D3×D411yescharacteristic
602Q8—22no—
603Q8—22no—
604Q8—22no—
605Q8—34no—
606Q8—616no—
607Q8—62no—
608Q8—62no—
609Q8—62no—
610Q8—121no—
611Q8—121no—
612Q8—121no—
613Q8—121no—
614Q8#89414yes—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—22no—
625C2×C4—32no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—616no—
635C2×C4—61no—
636C2×C4—62no—
637C2×C4—62no—
638C2×C4—61no—
639C2×C4—62no—
640C2×C4—62no—
641C2×C4—62no—
642C2×C4—62no—
643C2×C4—62no—
644C2×C4—61no—
645C2×C4—61no—
646C2×C4—61no—
647C2×C4—61no—
648C2×C4—62no—
649C2×C4—61no—
650C2×C4—61no—
651C2×C4—121no—
652C2×C4—121no—
653C2×C4—121no—
654C2×C4—121no—
655C2×C4—121no—
656C2×C4—121no—
657C2×C4—121no—
658C2×C4—121no—
659C2×C4C2×#23511yescharacteristic
660C2×C4C2×D3×D411yescharacteristic
661C2×C4C2×D3×D411yescharacteristic
662C2×C4C2×#23911yescharacteristic
663C2×C4D3×#4111yescharacteristic
664C2×C4D3×#4111yescharacteristic
665C8—41no—
666C8—41no—
667C8—121no—
668C8—121no—
669C2×C3#284911yescharacteristic
670C2×C3#301111yescharacteristic
671C2×C3#301311yescharacteristic
672C2×C3#312111yescharacteristic
673C2×C3#312111yescharacteristic
674C2×C3#312711yescharacteristic
675C2×C3#312711yescharacteristic
676D3—42no—
677C22—62no—
678C22—62no—
679C22—62no—
680C22—121no—
681C22—121no—
682C22—121no—
683C22—121no—
684C22#554711yescharacteristic
685C22D3×#12011yescharacteristic
686C22#562711yescharacteristic
687C22#576211yescharacteristic
688C22#577111yescharacteristic
689C22#577911yescharacteristic
690C22#577911yescharacteristic
691C4—24no—
692C4—41no—
693C4—42no—
694C4—41no—
695C4—62no—
696C4—64no—
697C4—81no—
698C4—81no—
699C4—121no—
700C4—121no—
701C4—122no—
702C4—241no—
703C4—241no—
704C4C2×#89411yescharacteristic
705C4#554711yescharacteristic
706C4#562711yescharacteristic
707C4D3×#12311yescharacteristic
708C3#3301911yescharacteristic, complemented, Sylow
709C2—122no—
710C2#7465611yescharacteristic
711C2#7965811yescharacteristic
712C2#7983611yescharacteristic
713C2#8004211yescharacteristic
714C2#8023011yescharacteristic
715C2#8100011yescharacteristic
716C2#8121111yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1865C2
socleC23×C3C22×D4
prime radicalC23×D3C2×D4
hypercenter#1865D3
hypocenterC3#33019
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. #71632C2semidirect
7. #70588C2semidirect
9. #70239C2semidirect
10. #70235C2semidirect
11. #69631C2semidirect
12. #68908C2semidirect
14. #68685C2semidirect
73. #1865D3semidirect
708. C3#33019semidirect

Maximal subgroups

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

subgroupindex
C2×#55722
C2×#48682
C2×#48092
#761022
C3×#18652
#716322
#705882
#702662
#702392
#702352
#696312
#689082
#688732
#686852
#686652
#330193