Finite Group Databasebeta

← #10901350

Subgroups — #10901350

Order 768.

1348 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5548C211yesmaximal, characteristic
2C2×C3×#474C211yesmaximal, characteristic, complemented
3#83269C211yesmaximal, characteristic, complemented
4C2×#5087C211yesmaximal, characteristic, complemented
5#71604C211yesmaximal, characteristic, complemented
6#70634C212yesmaximal, complemented
7#70617C212yesmaximal, complemented
8#70479C211yesmaximal, characteristic, complemented
9#70475C211yesmaximal, characteristic, complemented
10#70211C212yesmaximal, complemented
11#70206C212yesmaximal, complemented
12#29687—31nomaximal, Sylow
13C22×#235—21no—
14C2×C3×C4⋊Q8C2211yescharacteristic
15C2×C3×C4×D4C2211yescharacteristic
16C2×#946—21no—
17C2×#940C2211yescharacteristic
18C2×#938C2211yescharacteristic
19C2×C4×#67—21no—
20#5548—24no—
21#5548—24no—
22C2×#895C2211yescharacteristic
23C2×#888C2211yescharacteristic
24C2×#885—21no—
25C2×#884—21no—
26C2×C4×D12C2211yescharacteristic
27C2×C42×D3C2211yescharacteristic
28C3×#474—24no—
29C2×C3×#107C2212yes—
30C2×C3×C4⋊C8C2211yescharacteristic
31C2×C3×#102C2212yes—
32#5229—21no—
33C2×#832C2212yes—
34C2×#821C2212yes—
35C2×(C2×D3)⋊C8C2212yes—
36#5087—24no—
37#4987—21no—
38#4987C2212yes—
39#4972—21no—
40#4972C2212yes—
41C2×#809C2212yes—
42C2×#807C2212yes—
43#4935C2211yescharacteristic
44C2×C4⋊(C3⋊C8)C2211yescharacteristic
45#4551C2212yes—
46#4550C2212yes—
47#4529C2212yes—
48C2×#532—31no—
49C2×#474—31no—
50C2×#474D311yescharacteristic, complemented
51#2857—31no—
52#2057—32no—
53#2041—32no—
54#2033—32no—
55#2033—32no—
56#1802—31no—
57#1802—31no—
58#1797—31no—
59C22×C3×Q8—21no—
60C22×C3×D4C2311yescharacteristic
61C23×C3×C4—21no—
62C22×#67—21no—
63C22×#67—21no—
64C22×#67—21no—
65C2×#235—28no—
66C2×#235—44no—
67C2×#235—44no—
68C22×D12—21no—
69C22×D12C2311yescharacteristic
70C22×C4×D3—21no—
71C22×C4×D3D411yescharacteristic
72C22×C4×D3D411yescharacteristic
73C22×C4×D3C2311yescharacteristic
74C22×Dic6—21no—
75C22×C3×C8C2312yes—
76C3×C4⋊Q8—22no—
77C3×C4⋊Q8—22no—
78C3×C4×D4—22no—
79C3×C4×D4D414yes—
80C2×C3×C4⋊C4—21no—
81C2×C3×C4⋊C4C2312yes—
82C2×C3×C4⋊C4C2311yescharacteristic
83C2×C3×C22⋊C4—21no—
84C2×C3×C42C2311yescharacteristic
85C2×#217—21no—
86#946—44no—
87#940—22no—
88#940—22no—
89#938—22no—
90#938—22no—
91C4×#67—28no—
92C2×(C2×D3)⋊C4—21no—
93C2×(C2×D3)⋊C4—21no—
94C2×(C2×D3)⋊C4—22no—
95C2×C4⋊Dic3—21no—
96C2×C4⋊Dic3—21no—
97C2×C4⋊Dic3C2312yes—
98C2×C4⋊Dic3C2311yescharacteristic
99C2×Dic3⋊C4—21no—
100C2×Dic3⋊C4—22no—
101C2×C4×Dic3D411yescharacteristic
102C2×C4×Dic3D411yescharacteristic
103C2×C4×Dic3C2311yescharacteristic
104C22×C3⋊C8C2312yes—
105#895—22no—
106#895—22no—
107#888—22no—
108#888—22no—
109#885—28no—
110#884—28no—
111C4×D12—22no—
112C4×D12D414yes—
113C42×D3—22no—
114C42×D3—22no—
115C3×#107—24no—
116C3×C4⋊C8—22no—
117C3×#102—24no—
118#832—42no—
119C3⋊#95—22no—
120C3⋊#95—22no—
121#821—42no—
122(C2×D3)⋊C8—42no—
123#809—24no—
124#807—24no—
125C4⋊(C3⋊C8)—22no—
126C22×#41—61no—
127#532—64no—
128#532—64no—
129C2×C4⋊Q8—31no—
130C2×C4⋊Q8C2×D311yescharacteristic
131C2×#127—31no—
132C2×#124—61no—
133C2×C22⋊Q8—61no—
134C2×#121—31no—
135C2×#121—31no—
136C2×#121—61no—
137C2×C4×D4—31no—
138C2×C4×D4—61no—
139C2×C4×D4C2×D311yescharacteristic
140C22×C42—31no—
141#474—24no—
142#474—64no—
143C2×#107—32no—
144C2×#107C2×D312yes—
145C2×C4⋊C8—31no—
146C2×C4⋊C8C2×D311yescharacteristic
147C2×#102—32no—
148C2×#102—32no—
149C2×#102—32no—
150C2×#102C2×D312yes—
151C2×C22⋊C8—32no—
152#391—61no—
153#389—32no—
154#389—61no—
155#384—31no—
156#380—32no—
157#380—61no—
158#340—32no—
159#340—32no—
160#336—32no—
161C24×C3—21no—
162C23×D3—21no—
163C23×D3C2×D411yescharacteristic
164C2×C3×Q8—42no—
165C2×C3×Q8—42no—
166C2×C3×Q8—42no—
167C2×C3×D4—21no—
168C2×C3×D4—21no—
169C2×C3×D4—21no—
170C2×C3×D4—21no—
171C2×C3×D4—21no—
172C2×C3×D4C2×D412yes—
173C22×C3×C4—21no—
174C22×C3×C4—24no—
175C22×C3×C4—22no—
176C22×C3×C4—42no—
177C22×C3×C4—42no—
178C22×C3×C4C2×D411yescharacteristic
179C22×C3×C4C2×D411yescharacteristic
180C22×C3×C4C2411yescharacteristic
181C2×#67—24no—
182C2×#67—24no—
183C2×#67—24no—
184C2×#67—42no—
185C2×#67—42no—
186C2×#67—42no—
187C2×#67—42no—
188C2×#67—42no—
189C2×#67—42no—
190C22×Dic3—21no—
191C22×Dic3—22no—
192C22×Dic3C2×D411yescharacteristic
193C22×Dic3C2×D411yescharacteristic
194C22×Dic3C2×D411yescharacteristic
195#235—432no—
196C2×D12—21no—
197C2×D12—21no—
198C2×D12—21no—
199C2×D12—21no—
200C2×D12—24no—
201C2×D12—21no—
202C2×D12—42no—
203C2×D12—42no—
204C2×D12C2×D412yes—
205C2×C4×D3—22no—
206C2×C4×D3—22no—
207C2×C4×D3—22no—
208C2×C4×D3—24no—
209C2×C4×D3—21no—
210C2×C4×D3—21no—
211C2×C4×D3—21no—
212C2×C4×D3—21no—
213C2×C4×D3—21no—
214C2×C4×D3—21no—
215C2×C4×D3—22no—
216C2×C4×D3—41no—
217C2×C4×D3—42no—
218C2×C4×D3—42no—
219C2×C4×D3D814yes—
220C2×Dic6—24no—
221C2×Dic6—42no—
222C2×Dic6—42no—
223C2×C3×C8—22no—
224C2×C3×C8—22no—
225C2×C3×C8—22no—
226C3×C4⋊C4—21no—
227C3×C4⋊C4—22no—
228C3×C4⋊C4—22no—
229C3×C4⋊C4—42no—
230C3×C4⋊C4C2×D412yes—
231C3×C22⋊C4—24no—
232C3×C42C2×D412yes—
233C3×C42C2×D412yes—
234#217—24no—
235(C2×D3)⋊C4—24no—
236(C2×D3)⋊C4—24no—
237(C2×D3)⋊C4—44no—
238C4⋊Dic3—21no—
239C4⋊Dic3—22no—
240C4⋊Dic3—22no—
241C4⋊Dic3—42no—
242C4⋊Dic3—42no—
243C4⋊Dic3C2×D412yes—
244Dic3⋊C4—24no—
245Dic3⋊C4—44no—
246C4×Dic3—22no—
247C4×Dic3—22no—
248C4×Dic3—21no—
249C4×Dic3—21no—
250C2×C3⋊C8—22no—
251C2×C3⋊C8—22no—
252C2×C3⋊C8—22no—
253C2×#41—68no—
254C2×#41—124no—
255C2×#41—124no—
256C22×Q8—21no—
257C22×Q8—61no—
258C22×D4—31no—
259C22×D4—61no—
260C22×D4—61no—
261C22×D4—61no—
262C22×D4—61no—
263C22×D4C22×D311yescharacteristic
264C23×C4—21no—
265C23×C4—31no—
266C23×C4—31no—
267C23×C4—31no—
268C23×C4—61no—
269C22×C8—32no—
270C22×C8C22×D312yes—
271C4⋊Q8—22no—
272C4⋊Q8—22no—
273C4⋊Q8—62no—
274C4⋊Q8—62no—
275#127—62no—
276#127—62no—
277#124—68no—
278C22⋊Q8—124no—
279#121—68no—
280#121—62no—
281#121—62no—
282#121—62no—
283#121—62no—
284C4×D4—22no—
285C4×D4—34no—
286C4×D4—68no—
287C4×D4—62no—
288C4×D4#6714yes—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—31no—
291C2×C4⋊C4—32no—
292C2×C4⋊C4—61no—
293C2×C4⋊C4—62no—
294C2×C4⋊C4—61no—
295C2×C4⋊C4—61no—
296C2×C4⋊C4C22×D312yes—
297C2×C4⋊C4C22×D311yescharacteristic
298C2×C22⋊C4—21no—
299C2×C22⋊C4—61no—
300C2×C22⋊C4—61no—
301C2×C22⋊C4—61no—
302C2×C22⋊C4—62no—
303C2×C42—31no—
304C2×C42—31no—
305C2×C42—31no—
306C2×C42—62no—
307C2×C42—62no—
308C2×C42C22×D311yescharacteristic
309#107—24no—
310#107—64no—
311C4⋊C8—22no—
312C4⋊C8—62no—
313#102—24no—
314#102—64no—
315#102—122no—
316#102—122no—
317C22⋊C8—122no—
318#95—62no—
319#95—62no—
320C23×C3—22no—
321C23×C3—41no—
322C23×C3—41no—
323C23×C3—41no—
324C23×C3—41no—
325C23×C3—41no—
326C23×C3—41no—
327C23×C3C22×D411yescharacteristic
328C22×D3—21no—
329C22×D3—21no—
330C22×D3—21no—
331C22×D3—21no—
332C22×D3—21no—
333C22×D3—21no—
334C22×D3—22no—
335C22×D3—41no—
336C22×D3—41no—
337C22×D3—41no—
338C22×D3—41no—
339C22×D3—41no—
340C22×D3—41no—
341C22×D3C2×D812yes—
342C3×Q8—84no—
343C3×D4—22no—
344C3×D4—22no—
345C3×D4—41no—
346C3×D4—41no—
347C2×C3×C4—21no—
348C2×C3×C4—21no—
349C2×C3×C4—22no—
350C2×C3×C4—41no—
351C2×C3×C4—41no—
352C2×C3×C4—48no—
353C2×C3×C4—42no—
354C2×C3×C4—42no—
355C2×C3×C4—42no—
356C2×C3×C4#12012yes—
357C2×C3×C4#12012yes—
358C2×C3×C4C2×D812yes—
359C2×C3×C4AGL(1,ℤ/8)12yes—
360C2×C3×C4C22×D411yescharacteristic
361C2×C3×C4C22×D411yescharacteristic
362C2×C3×C4H(5,𝔽2)11yescharacteristic
363C2×C3×C42−1+411yescharacteristic
364C2×C3×C42−1+411yescharacteristic
365C2×C3×C42−1+411yescharacteristic
366#67—48no—
367#67—48no—
368#67—48no—
369C2×Dic3—21no—
370C2×Dic3—21no—
371C2×Dic3—21no—
372C2×Dic3—21no—
373C2×Dic3—21no—
374C2×Dic3—22no—
375C2×Dic3—21no—
376C2×Dic3—21no—
377C2×Dic3—41no—
378C2×Dic3—41no—
379C2×Dic3—42no—
380C2×Dic3—42no—
381C2×Dic3—42no—
382C2×Dic3C2×D812yes—
383C2×Dic3#13712yes—
384D12—22no—
385D12—22no—
386D12—48no—
387D12—41no—
388D12—41no—
389C4×D3—28no—
390C4×D3—44no—
391C4×D3—41no—
392C4×D3—41no—
393C4×D3—41no—
394C4×D3—41no—
395C4×D3—48no—
396Dic6—48no—
397C3×C8—42no—
398C3⋊C8—42no—
399C24—21no—
400C24—31no—
401C24—61no—
402#41—1232no—
403C2×Q8—42no—
404C2×Q8—42no—
405C2×Q8—42no—
406C2×Q8—64no—
407C2×Q8—122no—
408C2×Q8—122no—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4—21no—
414C2×D4—32no—
415C2×D4—64no—
416C2×D4—64no—
417C2×D4—64no—
418C2×D4—64no—
419C2×D4—61no—
420C2×D4—61no—
421C2×D4—61no—
422C2×D4—61no—
423C2×D4—61no—
424C2×D4—122no—
425C2×D4—122no—
426C2×D4—122no—
427C2×D4—122no—
428C2×D4—122no—
429C2×D4—122no—
430C2×D4—122no—
431C2×D4—122no—
432C2×D4C2×#6712yes—
433C22×C4—21no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—31no—
437C22×C4—31no—
438C22×C4—34no—
439C22×C4—31no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—62no—
443C22×C4—62no—
444C22×C4—62no—
445C22×C4—61no—
446C22×C4—62no—
447C22×C4—64no—
448C22×C4—61no—
449C22×C4—61no—
450C22×C4—61no—
451C22×C4—61no—
452C22×C4—61no—
453C22×C4—61no—
454C22×C4—62no—
455C22×C4—121no—
456C22×C4—122no—
457C22×C4—122no—
458C22×C4D3×D411yescharacteristic
459C22×C4D3×D411yescharacteristic
460C22×C4C23×D311yescharacteristic
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—62no—
465C2×C8—62no—
466C2×C8—62no—
467C4⋊C4—21no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—32no—
471C4⋊C4—42no—
472C4⋊C4—64no—
473C4⋊C4—61no—
474C4⋊C4—62no—
475C4⋊C4—62no—
476C4⋊C4—122no—
477C4⋊C4—124no—
478C4⋊C4—122no—
479C4⋊C4C2×#6712yes—
480C22⋊C4—24no—
481C22⋊C4—64no—
482C22⋊C4—64no—
483C22⋊C4—64no—
484C22⋊C4—124no—
485C42—62no—
486C42—62no—
487C42—61no—
488C42—61no—
489C42D3×D412yes—
490C42C2×#6712yes—
491C22×C3—42no—
492C22×C3—42no—
493C22×C3—42no—
494C22×C3—81no—
495C22×C3—81no—
496C22×C3—81no—
497C22×C3—81no—
498C22×C3C2×#12011yescharacteristic
499C22×C3#53611yescharacteristic
500C22×C3#53711yescharacteristic
501C22×C3C22×D811yescharacteristic
502C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
503C22×C3C2×#13711yescharacteristic
504C22×C3#57511yescharacteristic
505C2×D3—22no—
506C2×D3—22no—
507C2×D3—22no—
508C2×D3—41no—
509C2×D3—41no—
510C2×D3—41no—
511C2×D3—41no—
512C2×D3—42no—
513C2×D3—42no—
514C2×D3—42no—
515C2×D3—81no—
516C2×D3—81no—
517C2×D3—81no—
518C2×D3—81no—
519C3×C4—22no—
520C3×C4—42no—
521C3×C4—82no—
522C3×C4#44714yes—
523C3×C4#53611yescharacteristic
524C3×C4#53611yescharacteristic
525C3×C4#53711yescharacteristic
526C3×C4#53711yescharacteristic
527Dic3—22no—
528Dic3—22no—
529Dic3—41no—
530Dic3—42no—
531Dic3—82no—
532C23—22no—
533C23—32no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23—61no—
541C23—61no—
542C23—61no—
543C23—61no—
544C23—61no—
545C23—61no—
546C23—62no—
547C23—121no—
548C23—121no—
549C23—121no—
550C23—121no—
551C23—121no—
552C23—121no—
553C23C2×D3×D411yescharacteristic
554Q8—84no—
555Q8—128no—
556D4—22no—
557D4—22no—
558D4—41no—
559D4—41no—
560D4—62no—
561D4—62no—
562D4—128no—
563D4—128no—
564D4—128no—
565D4—128no—
566D4—121no—
567D4—121no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—22no—
571C2×C4—32no—
572C2×C4—32no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—48no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—61no—
580C2×C4—61no—
581C2×C4—61no—
582C2×C4—61no—
583C2×C4—68no—
584C2×C4—61no—
585C2×C4—62no—
586C2×C4—61no—
587C2×C4—61no—
588C2×C4—124no—
589C2×C4—121no—
590C2×C4—121no—
591C2×C4—128no—
592C2×C4—121no—
593C2×C4—121no—
594C2×C4—121no—
595C2×C4—121no—
596C2×C4—122no—
597C2×C4—122no—
598C2×C4—122no—
599C2×C4D3×D812yes—
600C2×C4#91112yes—
601C2×C4#93712yes—
602C2×C4#93712yes—
603C2×C4C2×D3×D411yescharacteristic
604C2×C4#100711yescharacteristic
605C2×C4#100711yescharacteristic
606C2×C4#100911yescharacteristic
607C2×C4#101011yescharacteristic
608C2×C4C22×#6711yescharacteristic
609C8—42no—
610C8—122no—
611C2×C3—82no—
612C2×C3#261611yescharacteristic
613C2×C3C2×#44711yescharacteristic
614C2×C3#300711yescharacteristic
615C2×C3#322611yescharacteristic
616C2×C3#323511yescharacteristic
617C2×C3#324511yescharacteristic
618C2×C3#325411yescharacteristic
619D3—42no—
620D3—82no—
621C22—42no—
622C22—42no—
623C22—42no—
624C22—62no—
625C22—62no—
626C22—62no—
627C22—81no—
628C22—81no—
629C22—81no—
630C22—81no—
631C22—121no—
632C22—121no—
633C22—121no—
634C22—121no—
635C22—122no—
636C22—122no—
637C22—122no—
638C22—241no—
639C22—241no—
640C22—241no—
641C22—241no—
642C22#562811yescharacteristic
643C22#562911yescharacteristic
644C22C2×D3×D811yescharacteristic
645C22C2×#91111yescharacteristic
646C22D3×#13711yescharacteristic
647C22#577711yescharacteristic
648C22C2×#93711yescharacteristic
649C4—22no—
650C4—42no—
651C4—62no—
652C4—62no—
653C4—82no—
654C4—122no—
655C4—121no—
656C4—242no—
657C4#515614yes—
658C4#562811yescharacteristic
659C4#562911yescharacteristic
660C4#581611yescharacteristic
661C4#583311yescharacteristic
662C3#2968711yescharacteristic, complemented, Sylow
663C2—82no—
664C2—122no—
665C2—242no—
666C2#8096911yescharacteristic
667C2#8097111yescharacteristic
668C2#8097511yescharacteristic
669C2#8097811yescharacteristic
670C2#8282011yescharacteristic
671C2C2×#515611yescharacteristic
672C2#8398111yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#474C2
socleC23×C3C22×D4
prime radicalC2×#5548C2
hypercenterC2×#474D3
hypocenterC3#29687
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#474

Complements

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

NH ≅ G/Ntype
2. C2×C3×#474C2semidirect
3. #83269C2semidirect
4. C2×#5087C2semidirect
5. #71604C2semidirect
6. #70634C2semidirect
7. #70617C2semidirect
8. #70479C2semidirect
9. #70475C2semidirect
10. #70211C2semidirect
11. #70206C2semidirect
50. C2×#474D3semidirect
662. C3#29687semidirect

Maximal subgroups

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

subgroupindex
C2×#55482
C2×C3×#4742
#832692
C2×#50872
#716042
#706342
#706172
#704792
#704752
#702112
#702062
#296873