Finite Group Databasebeta

← #10901352

Subgroups — #10901352

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×#5007C211yesmaximal, characteristic, complemented
3C2×#4846C211yesmaximal, characteristic, complemented
4#74376C211yesmaximal, characteristic, complemented
5C3×#1802C211yesmaximal, characteristic, complemented
6#70475C211yesmaximal, characteristic, complemented
7#70177C212yesmaximal, complemented
8#70175C212yesmaximal, complemented
9#69584C211yesmaximal, characteristic, complemented
10#68826C212yesmaximal, complemented
11#68822C212yesmaximal, complemented
12#29687—31nomaximal, Sylow
13C22×#235—21no—
14C2×C3×C4×D4C2211yescharacteristic
15C2×#940C2211yescharacteristic
16C2×#938C2211yescharacteristic
17C2×C4×#67—21no—
18#5548—24no—
19#5548—24no—
20C2×#897—21no—
21C2×#895C2211yescharacteristic
22C2×#888C2211yescharacteristic
23C2×#885—21no—
24C2×#884—21no—
25C2×C4×D12C2211yescharacteristic
26C2×C42×D3C2211yescharacteristic
27C2×#869C2211yescharacteristic
28C2×C3×C4⋊C8C2211yescharacteristic
29C2×C3×#102C2212yes—
30C3×#384C2211yescharacteristic
31C2×#832C2212yes—
32C2×#821C2212yes—
33C2×(C2×D3)⋊C8C2212yes—
34C2×#818C2212yes—
35#5007—24no—
36#4987—21no—
37#4987C2212yes—
38#4984—21no—
39#4984C2212yes—
40#4973—21no—
41C2×#809C2212yes—
42C2×#807C2212yes—
43C2×C4⋊(C3⋊C8)C2211yescharacteristic
44#4846—24no—
45C3×#340C2212yes—
46#4550C2212yes—
47#4529C2212yes—
48C2×#532—31no—
49C2×#474—31no—
50C2×#474—31no—
51#2857—31no—
52#2057—32no—
53#2041—32no—
54#2033—32no—
55#2033—32no—
56#1802—31no—
57#1802D311yescharacteristic, complemented
58#1797—31no—
59C22×C3×D4C2311yescharacteristic
60C23×C3×C4—21no—
61C22×#67—21no—
62C22×#67—21no—
63C22×#67—21no—
64C2×#235—28no—
65C2×#235—44no—
66C2×#235—44no—
67C22×D12—21no—
68C22×D12C2311yescharacteristic
69C22×C4×D3—21no—
70C22×C4×D3D411yescharacteristic
71C22×C4×D3D411yescharacteristic
72C22×C4×D3C2311yescharacteristic
73C22×Dic6—21no—
74C22×Dic6—21no—
75C22×C3×C8C2312yes—
76C3×C4×D4—22no—
77C3×C4×D4D414yes—
78C2×C3×C4⋊C4—21no—
79C2×C3×C4⋊C4—21no—
80C2×C3×C4⋊C4C2312yes—
81C2×C3×C4⋊C4C2311yescharacteristic
82C2×C3×C22⋊C4—21no—
83C2×C3×C42C2311yescharacteristic
84C2×#217—21no—
85#940—22no—
86#940—22no—
87#938—22no—
88#938—22no—
89C4×#67—28no—
90C2×(C2×D3)⋊C4—21no—
91C2×(C2×D3)⋊C4—21no—
92C2×(C2×D3)⋊C4—22no—
93C2×C4⋊Dic3—21no—
94C2×C4⋊Dic3—22no—
95C2×C4⋊Dic3C2312yes—
96C2×C4⋊Dic3C2311yescharacteristic
97C2×Dic3⋊C4—21no—
98C2×C4×Dic3D411yescharacteristic
99C2×C4×Dic3D411yescharacteristic
100C2×C4×Dic3C2311yescharacteristic
101C22×C3⋊C8C2312yes—
102#897—44no—
103#895—22no—
104#895—22no—
105#888—22no—
106#888—22no—
107#885—28no—
108#884—28no—
109C4×D12—22no—
110C4×D12D414yes—
111C42×D3—22no—
112C42×D3—22no—
113#869—22no—
114#869—22no—
115C3×C4⋊C8—22no—
116C3×#102—42no—
117#832—24no—
118C3⋊#95—22no—
119C3⋊#95—22no—
120#821—24no—
121(C2×D3)⋊C8—42no—
122#818—24no—
123#809—42no—
124#807—24no—
125C4⋊(C3⋊C8)—22no—
126C22×#41—61no—
127#532—64no—
128#532—64no—
129C2×C4⋊Q8—31no—
130C2×C4⋊Q8—31no—
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—64no—
142#474—64no—
143C2×#107—32no—
144C2×#107—32no—
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#384C2×D311yescharacteristic
156#380—32no—
157#380—61no—
158#340—32no—
159#340C2×D312yes—
160#336—32no—
161C24×C3—21no—
162C23×D3—21no—
163C23×D3C2×D411yescharacteristic
164C2×C3×D4—21no—
165C2×C3×D4—21no—
166C2×C3×D4—21no—
167C2×C3×D4—21no—
168C2×C3×D4—21no—
169C2×C3×D4C2×D412yes—
170C22×C3×C4—21no—
171C22×C3×C4—24no—
172C22×C3×C4—22no—
173C22×C3×C4—42no—
174C22×C3×C4—42no—
175C22×C3×C4C2×D411yescharacteristic
176C22×C3×C4C2×D411yescharacteristic
177C22×C3×C4C2411yescharacteristic
178C2×#67—24no—
179C2×#67—24no—
180C2×#67—24no—
181C2×#67—42no—
182C2×#67—42no—
183C2×#67—42no—
184C2×#67—42no—
185C2×#67—42no—
186C2×#67—42no—
187C22×Dic3—21no—
188C22×Dic3—22no—
189C22×Dic3C2×D411yescharacteristic
190C22×Dic3C2×D411yescharacteristic
191C22×Dic3C2×D411yescharacteristic
192#235—432no—
193C2×D12—21no—
194C2×D12—21no—
195C2×D12—21no—
196C2×D12—21no—
197C2×D12—24no—
198C2×D12—21no—
199C2×D12—42no—
200C2×D12—42no—
201C2×D12C2×D412yes—
202C2×C4×D3—22no—
203C2×C4×D3—22no—
204C2×C4×D3—22no—
205C2×C4×D3—24no—
206C2×C4×D3—21no—
207C2×C4×D3—21no—
208C2×C4×D3—21no—
209C2×C4×D3—21no—
210C2×C4×D3—21no—
211C2×C4×D3—21no—
212C2×C4×D3—22no—
213C2×C4×D3—41no—
214C2×C4×D3—42no—
215C2×C4×D3—42no—
216C2×C4×D3D814yes—
217C2×Dic6—24no—
218C2×Dic6—42no—
219C2×Dic6—42no—
220C2×Dic6—42no—
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⋊C4—42no—
231C3×C4⋊C4C2×D412yes—
232C3×C22⋊C4—24no—
233C3×C42C2×D412yes—
234C3×C42C2×D412yes—
235#217—24no—
236(C2×D3)⋊C4—24no—
237(C2×D3)⋊C4—24no—
238(C2×D3)⋊C4—44no—
239C4⋊Dic3—21no—
240C4⋊Dic3—22no—
241C4⋊Dic3—22no—
242C4⋊Dic3—42no—
243C4⋊Dic3—44no—
244C4⋊Dic3C2×D412yes—
245Dic3⋊C4—24no—
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—61no—
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—62no—
272C4⋊Q8—62no—
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×D4D1214yes—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—31no—
292C2×C4⋊C4—32no—
293C2×C4⋊C4—61no—
294C2×C4⋊C4—62no—
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—64no—
310#107—64no—
311C4⋊C8—22no—
312C4⋊C8—62no—
313#102—42no—
314#102—64no—
315#102—64no—
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×D4—22no—
343C3×D4—22no—
344C3×D4—41no—
345C3×D4—41no—
346C2×C3×C4—21no—
347C2×C3×C4—21no—
348C2×C3×C4—22no—
349C2×C3×C4—41no—
350C2×C3×C4—41no—
351C2×C3×C4—48no—
352C2×C3×C4—42no—
353C2×C3×C4—42no—
354C2×C3×C4—42no—
355C2×C3×C4#12012yes—
356C2×C3×C4#12012yes—
357C2×C3×C4C2×D812yes—
358C2×C3×C4AGL(1,ℤ/8)12yes—
359C2×C3×C4C22×D411yescharacteristic
360C2×C3×C4C22×D411yescharacteristic
361C2×C3×C4H(5,𝔽2)11yescharacteristic
362C2×C3×C42−1+411yescharacteristic
363C2×C3×C42−1+411yescharacteristic
364C2×C3×C42−1+411yescharacteristic
365#67—48no—
366#67—48no—
367#67—48no—
368C2×Dic3—21no—
369C2×Dic3—21no—
370C2×Dic3—21no—
371C2×Dic3—21no—
372C2×Dic3—21no—
373C2×Dic3—22no—
374C2×Dic3—21no—
375C2×Dic3—21no—
376C2×Dic3—41no—
377C2×Dic3—41no—
378C2×Dic3—42no—
379C2×Dic3—42no—
380C2×Dic3—42no—
381C2×Dic3C2×D812yes—
382C2×Dic3#13712yes—
383D12—22no—
384D12—22no—
385D12—48no—
386D12—41no—
387D12—41no—
388C4×D3—28no—
389C4×D3—44no—
390C4×D3—41no—
391C4×D3—41no—
392C4×D3—41no—
393C4×D3—41no—
394C4×D3—48no—
395Dic6—48no—
396Dic6—84no—
397C3×C8—42no—
398C3⋊C8—42no—
399C24—21no—
400C24—31no—
401C24—61no—
402#41—1232no—
403C2×Q8—64no—
404C2×Q8—122no—
405C2×Q8—122no—
406C2×Q8—122no—
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×D1212yes—
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—42no—
473C4⋊C4—64no—
474C4⋊C4—61no—
475C4⋊C4—62no—
476C4⋊C4—62no—
477C4⋊C4—124no—
478C4⋊C4—122no—
479C4⋊C4C2×D1212yes—
480C22⋊C4—24no—
481C22⋊C4—64no—
482C22⋊C4—64no—
483C22⋊C4—64no—
484C22⋊C4—124no—
485C42—62no—
486C42—62no—
487C42—61no—
488C42—61no—
489C42C2×D1212yes—
490C42D3×D412yes—
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—128no—
555Q8—244no—
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—121no—
592C2×C4—121no—
593C2×C4—128no—
594C2×C4—121no—
595C2×C4—121no—
596C2×C4—122no—
597C2×C4—122no—
598C2×C4—122no—
599C2×C4#88212yes—
600C2×C4#88212yes—
601C2×C4D3×D812yes—
602C2×C4#91412yes—
603C2×C4C22×D1211yescharacteristic
604C2×C4C2×D3×D411yescharacteristic
605C2×C4#100411yescharacteristic
606C2×C4#100711yescharacteristic
607C2×C4#101011yescharacteristic
608C2×C4#101011yescharacteristic
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—
642C22C2×#88211yescharacteristic
643C22#560811yescharacteristic
644C22#562811yescharacteristic
645C22C2×D3×D811yescharacteristic
646C22C2×#91411yescharacteristic
647C22#577311yescharacteristic
648C22D3×#13711yescharacteristic
649C4—22no—
650C4—42no—
651C4—62no—
652C4—62no—
653C4—82no—
654C4—122no—
655C4—121no—
656C4—242no—
657C4#477314yes—
658C4#551011yescharacteristic
659C4#552611yescharacteristic
660C4#560811yescharacteristic
661C4#562811yescharacteristic
662C3#2968711yescharacteristic, complemented, Sylow
663C2—82no—
664C2—122no—
665C2—242no—
666C2#7395011yescharacteristic
667C2C2×#477311yescharacteristic
668C2#7888511yescharacteristic
669C2#8034211yescharacteristic
670C2#8044111yescharacteristic
671C2#8096911yescharacteristic
672C2#8106611yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1802C2
socleC23×C3C22×D4
prime radicalC2×#5548C2
hypercenter#1802D3
hypocenterC3#29687
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1802

Complements

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

NH ≅ G/Ntype
2. C2×#5007C2semidirect
3. C2×#4846C2semidirect
4. #74376C2semidirect
5. C3×#1802C2semidirect
6. #70475C2semidirect
7. #70177C2semidirect
8. #70175C2semidirect
9. #69584C2semidirect
10. #68826C2semidirect
11. #68822C2semidirect
57. #1802D3semidirect
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×#50072
C2×#48462
#743762
C3×#18022
#704752
#701772
#701752
#695842
#688262
#688222
#296873