Finite Group Databasebeta

← #10900966

Subgroups — #10900966

Order 768.

1064 conjugacy classes of subgroups.

The subgroup lattice has 700 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5517C211yesmaximal, characteristic, complemented
2C2×C3×#482C211yesmaximal, characteristic
3#82699C211yesmaximal, characteristic
4C2×#5035C211yesmaximal, characteristic
5#71575C211yesmaximal, characteristic, complemented
6#70618C211yesmaximal, characteristic, complemented
7#70617C211yesmaximal, characteristic, complemented
8#70566C211yesmaximal, characteristic, complemented
9#70565C211yesmaximal, characteristic, complemented
10#70164C211yesmaximal, characteristic, complemented
11#70164C211yesmaximal, characteristic, complemented
12#70163C211yesmaximal, characteristic, complemented
13#70163C211yesmaximal, characteristic, complemented
14#69926C211yesmaximal, characteristic, complemented
15#69910C211yesmaximal, characteristic, complemented
16#29644—31nomaximal, Sylow
17C2×C3×#121C2211yescharacteristic
18C2×C3×#117C2211yescharacteristic
19C2×#935—21no—
20C2×#926C2211yescharacteristic
21C22×C4×Dic3C2211yescharacteristic
22#5517—24no—
23#5517—24no—
24C2×#890C2211yescharacteristic
25C2×#890C2211yescharacteristic
26C2×#888C2211yescharacteristic
27C2×#888C2211yescharacteristic
28C2×#878—21no—
29C2×#878—21no—
30C2×C4×Dic6—21no—
31C2×C4×Dic6—21no—
32C3×#482—24no—
33C2×C3×#107C2211yescharacteristic
34C2×C3×#106C2211yescharacteristic
35C2×C3×#102C2211yescharacteristic
36C2×C3×#102C2211yescharacteristic
37C2×C3×C22⋊C8C2211yescharacteristic
38#5221C2211yescharacteristic
39#5221C2211yescharacteristic
40#5220—21no—
41C2×#832C2211yescharacteristic
42C2×#832C2211yescharacteristic
43#5212C2211yescharacteristic
44C2×#830C2211yescharacteristic
45#5035—24no—
46#4972C2211yescharacteristic
47#4972C2211yescharacteristic
48C2×#808C2211yescharacteristic
49C2×#807C2211yescharacteristic
50#4944—21no—
51#4944—21no—
52#4551C2211yescharacteristic
53#4551C2211yescharacteristic
54#4551C2211yescharacteristic
55#4551C2211yescharacteristic
56#4550C2211yescharacteristic
57#4550C2211yescharacteristic
58#4550C2211yescharacteristic
59#4550C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62#4529C2211yescharacteristic
63#4529C2211yescharacteristic
64C2×#533—31no—
65C2×#482—31no—
66C2×#482D311yescharacteristic
67#2854—31no—
68#2059—31no—
69#2059—31no—
70#2058—31no—
71#2057—31no—
72#2025—31no—
73#2025—31no—
74#2023—31no—
75#2023—31no—
76#1825—31no—
77#1825—31no—
78#1817—31no—
79C22×C3×D4—21no—
80C22×C3×D4C2311yescharacteristic
81C23×C3×C4C2311yescharacteristic
82C23×Dic3D411yescharacteristic
83C23×Dic3D411yescharacteristic
84C22×Dic6—21no—
85C22×Dic6—21no—
86C22×C3×C8C2311yescharacteristic
87C22×C3×C8C2311yescharacteristic
88C3×#121—22no—
89C3×#121—22no—
90C3×#117—22no—
91C3×#117D414yes—
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C22⋊C4—21no—
96C2×C3×C22⋊C4—21no—
97C2×C3×C42—21no—
98C2×#217—21no—
99C2×#217—21no—
100C2×#217—21no—
101C2×#217—21no—
102C2×#217—21no—
103C2×#217—21no—
104#935—44no—
105#926—22no—
106#926D414yes—
107C2×C4⋊Dic3—21no—
108C2×C4⋊Dic3—21no—
109C2×C4⋊Dic3C2311yescharacteristic
110C2×C4⋊Dic3C2311yescharacteristic
111C2×C4⋊Dic3C2311yescharacteristic
112C2×C4⋊Dic3C2311yescharacteristic
113C2×Dic3⋊C4—21no—
114C2×Dic3⋊C4—21no—
115C2×Dic3⋊C4—21no—
116C2×Dic3⋊C4—21no—
117C2×Dic3⋊C4—21no—
118C2×Dic3⋊C4—21no—
119C2×C4×Dic3—21no—
120C2×C4×Dic3—22no—
121C2×C4×Dic3—22no—
122C2×C4×Dic3D411yescharacteristic
123C2×C4×Dic3D411yescharacteristic
124C2×C4×Dic3C2311yescharacteristic
125C2×C4×Dic3C2311yescharacteristic
126C22×C3⋊C8C2311yescharacteristic
127C22×C3⋊C8C2311yescharacteristic
128#890—22no—
129#890—22no—
130#890—22no—
131#890—22no—
132#888—22no—
133#888—22no—
134#888—22no—
135#888—22no—
136#878—28no—
137#878—28no—
138C4×Dic6—28no—
139C4×Dic6—28no—
140C3×#107—22no—
141C3×#106—22no—
142C3×#102—22no—
143C3×#102—22no—
144C3×C22⋊C8—22no—
145#832—22no—
146#832—22no—
147C3⋊#95—21no—
148C3⋊#95—21no—
149C3⋊#95—21no—
150#830—22no—
151#808—22no—
152#807—22no—
153#533—64no—
154#533—64no—
155C2×C4⋊Q8—31no—
156C2×C4⋊Q8—31no—
157C2×#125—31no—
158C2×#125—31no—
159C2×#123—61no—
160C2×C22⋊Q8—61no—
161C2×C22⋊Q8—61no—
162C2×#121C2×D311yescharacteristic
163C2×C4×Q8—61no—
164C2×C4×Q8—61no—
165C2×#117—31no—
166C2×#117C2×D311yescharacteristic
167C22×C42—31no—
168#482—24no—
169#482—64no—
170C2×#107—31no—
171C2×#107C2×D311yescharacteristic
172C2×#106—31no—
173C2×#106C2×D311yescharacteristic
174C2×#102—31no—
175C2×#102—31no—
176C2×#102C2×D311yescharacteristic
177C2×#102C2×D311yescharacteristic
178C2×C22⋊C8—31no—
179C2×C22⋊C8C2×D311yescharacteristic
180#390—31no—
181#390—31no—
182#389—31no—
183#389—31no—
184#388—61no—
185#388—61no—
186#386—61no—
187#380—31no—
188#340—31no—
189#340—31no—
190#340—31no—
191#340—31no—
192#340—31no—
193#340—31no—
194#340—31no—
195#340—31no—
196#336—31no—
197#336—31no—
198#336—31no—
199#336—31no—
200C24×C3—21no—
201C24×C3C2×D411yescharacteristic
202C2×C3×D4—21no—
203C2×C3×D4—21no—
204C2×C3×D4—21no—
205C2×C3×D4—21no—
206C2×C3×D4—21no—
207C2×C3×D4—21no—
208C2×C3×D4—42no—
209C2×C3×D4—42no—
210C2×C3×D4—42no—
211C22×C3×C4—21no—
212C22×C3×C4—21no—
213C22×C3×C4—21no—
214C22×C3×C4—21no—
215C22×C3×C4—21no—
216C22×C3×C4—21no—
217C22×C3×C4—21no—
218C22×C3×C4C2×D411yescharacteristic
219C22×C3×C4C2×D412yes—
220C22×C3×C4C2×D412yes—
221C22×C3×C4C2411yescharacteristic
222C22×Dic3—22no—
223C22×Dic3—22no—
224C22×Dic3—21no—
225C22×Dic3—21no—
226C22×Dic3—22no—
227C22×Dic3—22no—
228C22×Dic3—21no—
229C22×Dic3—21no—
230C22×Dic3—41no—
231C22×Dic3—41no—
232C22×Dic3C2×D411yescharacteristic
233C22×Dic3C2×D411yescharacteristic
234C22×Dic3C2×D411yescharacteristic
235C22×Dic3C2×D411yescharacteristic
236C2×Dic6—24no—
237C2×Dic6—24no—
238C2×Dic6—42no—
239C2×Dic6—42no—
240C2×Dic6—42no—
241C2×Dic6—42no—
242C2×C3×C8—21no—
243C2×C3×C8—21no—
244C2×C3×C8—21no—
245C2×C3×C8—21no—
246C2×C3×C8—21no—
247C2×C3×C8—21no—
248C3×C4⋊C4—21no—
249C3×C4⋊C4—21no—
250C3×C4⋊C4—21no—
251C3×C4⋊C4—21no—
252C3×C4⋊C4C2×D412yes—
253C3×C4⋊C4C2×D412yes—
254C3×C22⋊C4—24no—
255C3×C22⋊C4—42no—
256C3×C42—24no—
257#217—24no—
258#217—42no—
259#217—42no—
260#217—42no—
261#217—42no—
262#217—42no—
263C4⋊Dic3—24no—
264C4⋊Dic3—24no—
265C4⋊Dic3—21no—
266C4⋊Dic3—21no—
267C4⋊Dic3—21no—
268C4⋊Dic3—21no—
269C4⋊Dic3—21no—
270C4⋊Dic3—21no—
271C4⋊Dic3C2×D412yes—
272C4⋊Dic3C2×D412yes—
273Dic3⋊C4—24no—
274Dic3⋊C4—24no—
275Dic3⋊C4—24no—
276Dic3⋊C4—24no—
277Dic3⋊C4—42no—
278Dic3⋊C4—42no—
279C4×Dic3—22no—
280C4×Dic3—22no—
281C4×Dic3—24no—
282C4×Dic3—21no—
283C4×Dic3—21no—
284C4×Dic3—21no—
285C4×Dic3—21no—
286C2×C3⋊C8—21no—
287C2×C3⋊C8—21no—
288C2×C3⋊C8—21no—
289C2×C3⋊C8—21no—
290C2×C3⋊C8—21no—
291C2×C3⋊C8—21no—
292C22×Q8—61no—
293C22×Q8—61no—
294C22×D4—21no—
295C22×D4C22×D311yescharacteristic
296C23×C4—31no—
297C23×C4—31no—
298C23×C4C22×D311yescharacteristic
299C22×C8—31no—
300C22×C8—31no—
301C22×C8C22×D311yescharacteristic
302C22×C8C22×D311yescharacteristic
303C4⋊Q8—62no—
304C4⋊Q8—62no—
305C4⋊Q8—62no—
306C4⋊Q8—62no—
307#125—62no—
308#125—62no—
309#125—62no—
310#125—62no—
311#123—124no—
312C22⋊Q8—68no—
313C22⋊Q8—68no—
314#121—22no—
315#121—22no—
316C4×Q8—68no—
317C4×Q8—68no—
318#117—22no—
319#117—34no—
320#117—62no—
321#117#6714yes—
322C2×C4⋊C4—31no—
323C2×C4⋊C4—31no—
324C2×C4⋊C4—31no—
325C2×C4⋊C4—31no—
326C2×C4⋊C4—61no—
327C2×C4⋊C4—61no—
328C2×C4⋊C4—61no—
329C2×C4⋊C4—61no—
330C2×C4⋊C4—61no—
331C2×C4⋊C4—61no—
332C2×C4⋊C4—61no—
333C2×C4⋊C4—61no—
334C2×C4⋊C4C22×D311yescharacteristic
335C2×C4⋊C4C22×D311yescharacteristic
336C2×C4⋊C4C22×D311yescharacteristic
337C2×C22⋊C4—21no—
338C2×C22⋊C4—21no—
339C2×C22⋊C4—61no—
340C2×C22⋊C4—61no—
341C2×C22⋊C4—61no—
342C2×C22⋊C4—61no—
343C2×C22⋊C4—61no—
344C2×C22⋊C4—61no—
345C2×C42—21no—
346C2×C42—31no—
347C2×C42—31no—
348C2×C42—31no—
349C2×C42—31no—
350C2×C42—61no—
351C2×C42—62no—
352C2×C42—62no—
353#107—22no—
354#107—62no—
355#106—22no—
356#106—62no—
357#102—22no—
358#102—22no—
359#102—62no—
360#102—62no—
361C22⋊C8—22no—
362C22⋊C8—62no—
363#95—61no—
364#95—61no—
365#95—61no—
366C23×C3—21no—
367C23×C3—21no—
368C23×C3—21no—
369C23×C3—21no—
370C23×C3—21no—
371C23×C3—21no—
372C23×C3—41no—
373C23×C3—41no—
374C23×C3—41no—
375C23×C3—41no—
376C23×C3—41no—
377C23×C3—41no—
378C23×C3—41no—
379C23×C3#12012yes—
380C23×C3C22×D411yescharacteristic
381C3×D4—41no—
382C3×D4—41no—
383C3×D4—41no—
384C3×D4—41no—
385C3×D4—84no—
386C2×C3×C4—21no—
387C2×C3×C4—21no—
388C2×C3×C4—22no—
389C2×C3×C4—22no—
390C2×C3×C4—22no—
391C2×C3×C4—22no—
392C2×C3×C4—22no—
393C2×C3×C4—22no—
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#12012yes—
402C2×C3×C4C22×D411yescharacteristic
403C2×C3×C4C22×D411yescharacteristic
404C2×C3×C42−1+411yescharacteristic
405C2×C3×C42−1+411yescharacteristic
406C2×C3×C42−1+411yescharacteristic
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—22no—
415C2×Dic3—22no—
416C2×Dic3—21no—
417C2×Dic3—21no—
418C2×Dic3—21no—
419C2×Dic3—21no—
420C2×Dic3—44no—
421C2×Dic3—44no—
422C2×Dic3—41no—
423C2×Dic3—41no—
424C2×Dic3—41no—
425C2×Dic3—41no—
426C2×Dic3—41no—
427C2×Dic3—41no—
428C2×Dic3—41no—
429C2×Dic3—41no—
430C2×Dic3—41no—
431C2×Dic3—41no—
432C2×Dic3#13512yes—
433C2×Dic3AGL(1,ℤ/8)12yes—
434Dic6—48no—
435Dic6—48no—
436C3×C8—41no—
437C3×C8—41no—
438C3⋊C8—41no—
439C3⋊C8—41no—
440C24—21no—
441C24D3×D411yescharacteristic
442C2×Q8—64no—
443C2×Q8—64no—
444C2×Q8—122no—
445C2×Q8—122no—
446C2×Q8—122no—
447C2×Q8—122no—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—21no—
452C2×D4—21no—
453C2×D4—21no—
454C2×D4—42no—
455C2×D4—42no—
456C2×D4—42no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—31no—
465C22×C4—31no—
466C22×C4—31no—
467C22×C4—31no—
468C22×C4—62no—
469C22×C4—62no—
470C22×C4—62no—
471C22×C4—62no—
472C22×C4—61no—
473C22×C4—61no—
474C22×C4—61no—
475C22×C4—61no—
476C22×C4—121no—
477C22×C4—121no—
478C22×C4D3×D411yescharacteristic
479C22×C4D3×D412yes—
480C22×C4C2×#6712yes—
481C22×C4C23×D311yescharacteristic
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—61no—
489C2×C8—61no—
490C2×C8—61no—
491C2×C8—61no—
492C2×C8—61no—
493C2×C8—61no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—32no—
499C4⋊C4—32no—
500C4⋊C4—64no—
501C4⋊C4—64no—
502C4⋊C4—64no—
503C4⋊C4—64no—
504C4⋊C4—64no—
505C4⋊C4—64no—
506C4⋊C4—61no—
507C4⋊C4—61no—
508C4⋊C4—61no—
509C4⋊C4—61no—
510C4⋊C4—61no—
511C4⋊C4—61no—
512C4⋊C4—122no—
513C4⋊C4—122no—
514C4⋊C4C2×#6712yes—
515C4⋊C4C2×#6712yes—
516C22⋊C4—24no—
517C22⋊C4—42no—
518C22⋊C4—64no—
519C22⋊C4—122no—
520C22⋊C4—122no—
521C22⋊C4—122no—
522C22⋊C4—122no—
523C22⋊C4—122no—
524C42—24no—
525C42—62no—
526C42—62no—
527C42—64no—
528C42—61no—
529C42—61no—
530C42—61no—
531C42—61no—
532C22×C3—22no—
533C22×C3—22no—
534C22×C3—22no—
535C22×C3—41no—
536C22×C3—41no—
537C22×C3—41no—
538C22×C3—41no—
539C22×C3—81no—
540C22×C3—81no—
541C22×C3—81no—
542C22×C3—81no—
543C22×C3—81no—
544C22×C3—81no—
545C22×C3—81no—
546C22×C3C2×#12011yescharacteristic
547C22×C3#53611yescharacteristic
548C22×C3#53611yescharacteristic
549C22×C3C2×#13511yescharacteristic
550C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
551C22×C3#57511yescharacteristic
552C22×C3#57511yescharacteristic
553C3×C4—22no—
554C3×C4—42no—
555C3×C4—42no—
556C3×C4—81no—
557C3×C4#53611yescharacteristic
558C3×C4#53611yescharacteristic
559C3×C4#53611yescharacteristic
560C3×C4#53611yescharacteristic
561Dic3—22no—
562Dic3—22no—
563Dic3—42no—
564Dic3—41no—
565Dic3—42no—
566Dic3—41no—
567Dic3—81no—
568Dic3—81no—
569C23—21no—
570C23—21no—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—21no—
575C23—41no—
576C23—41no—
577C23—41no—
578C23—41no—
579C23—41no—
580C23—41no—
581C23—41no—
582C23#93712yes—
583C23C2×D3×D411yescharacteristic
584Q8—128no—
585Q8—128no—
586D4—41no—
587D4—41no—
588D4—41no—
589D4—41no—
590D4—84no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—32no—
600C2×C4—32no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—61no—
609C2×C4—61no—
610C2×C4—61no—
611C2×C4—61no—
612C2×C4—61no—
613C2×C4—61no—
614C2×C4—61no—
615C2×C4—61no—
616C2×C4—61no—
617C2×C4—61no—
618C2×C4—62no—
619C2×C4—62no—
620C2×C4—124no—
621C2×C4—124no—
622C2×C4—121no—
623C2×C4—121no—
624C2×C4—121no—
625C2×C4—121no—
626C2×C4—121no—
627C2×C4—121no—
628C2×C4—121no—
629C2×C4—121no—
630C2×C4—121no—
631C2×C4—121no—
632C2×C4#93712yes—
633C2×C4C2×D3×D411yescharacteristic
634C2×C4#101011yescharacteristic
635C2×C4#101011yescharacteristic
636C2×C4#101011yescharacteristic
637C2×C4#101011yescharacteristic
638C2×C4C22×#6711yescharacteristic
639C8—41no—
640C8—41no—
641C8—121no—
642C8—121no—
643C2×C3—42no—
644C2×C3—161no—
645C2×C3#261411yescharacteristic
646C2×C3#300211yescharacteristic
647C2×C3#300611yescharacteristic
648C2×C3#322811yescharacteristic
649C2×C3#323311yescharacteristic
650C2×C3#323511yescharacteristic
651C2×C3#324011yescharacteristic
652C22—22no—
653C22—22no—
654C22—22no—
655C22—41no—
656C22—41no—
657C22—41no—
658C22—41no—
659C22—81no—
660C22—81no—
661C22—81no—
662C22—81no—
663C22—81no—
664C22—81no—
665C22—81no—
666C22#564811yescharacteristic
667C22#564811yescharacteristic
668C22D3×#13511yescharacteristic
669C22#576811yescharacteristic
670C22D3×AGL(1,ℤ/8)11yescharacteristic
671C22#577311yescharacteristic
672C22C2×#93711yescharacteristic
673C4—22no—
674C4—42no—
675C4—42no—
676C4—62no—
677C4—62no—
678C4—81no—
679C4—121no—
680C4—122no—
681C4—121no—
682C4—122no—
683C4—241no—
684C4—241no—
685C4#564811yescharacteristic
686C4#564811yescharacteristic
687C4#582511yescharacteristic
688C4#582511yescharacteristic
689C3#2964411yescharacteristic, complemented, Sylow
690C2—42no—
691C2—161no—
692C2#8135611yescharacteristic
693C2#8135711yescharacteristic
694C2#8135811yescharacteristic
695C2#8135911yescharacteristic
696C2#8286411yescharacteristic
697C2#8391111yescharacteristic
698C2#8394611yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#482C2
socleC23×C3C22×D4
prime radicalC2×C3×#121C22
hypercenterC2×#482D3
hypocenterC3#29644
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#482

Complements

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

NH ≅ G/Ntype
1. C2×#5517C2semidirect
5. #71575C2semidirect
6. #70618C2semidirect
7. #70617C2semidirect
8. #70566C2semidirect
9. #70565C2semidirect
10. #70164C2semidirect
11. #70164C2semidirect
12. #70163C2semidirect
13. #70163C2semidirect
14. #69926C2semidirect
15. #69910C2semidirect
689. C3#29644semidirect

Maximal subgroups

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

subgroupindex
C2×#55172
C2×C3×#4822
#826992
C2×#50352
#715752
#706182
#706172
#705662
#705652
#701642
#701632
#699262
#699102
#296443