Finite Group Databasebeta

← #10698043

Subgroups — #10698043

Order 768.

1066 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 1066 conjugacy classes of subgroups — too many to draw. Only its 105 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2847C211yesmaximal, characteristic, complemented
2#84032C211yesmaximal, characteristic, complemented
3#82625C211yesmaximal, characteristic, complemented
4#72126C211yesmaximal, characteristic, complemented
5#71675C211yesmaximal, characteristic, complemented
6#69509C211yesmaximal, characteristic, complemented
7#67625C211yesmaximal, characteristic, complemented
8#10320—31nomaximal, Sylow
9C24×C3×C4C2211yescharacteristic
10C2×C3×#121—21no—
11C2×C3×#121C2211yescharacteristic
12C2×C3×#120—21no—
13C2×#953—21no—
14C2×#930—21no—
15C2×#930C2211yescharacteristic
16C3×#386—21no—
17C3×#386—22no—
18C3×#380—21no—
19C3×#380C2211yescharacteristic
20#5217—21no—
21#5213—21no—
22#5213—22no—
23#5212—21no—
24#5212C2211yescharacteristic
25C2×#830—22no—
26C2×#830C2211yescharacteristic
27C2×#830C2211yescharacteristic
28#5038—22no—
29#5036—21no—
30#5033—21no—
31#4967—21no—
32#4537—22no—
33#4451—22no—
34#3186—31no—
35#2847—31no—
36#2847D311yescharacteristic, complemented
37#1779—31no—
38#1519—31no—
39#1519—31no—
40#1289—31no—
41C25×C3D411yescharacteristic
42C22×C3×D4—22no—
43C22×C3×D4—21no—
44C22×C3×D4D411yescharacteristic, complemented
45C23×C3×C4—21no—
46C23×C3×C4—21no—
47C23×C3×C4—21no—
48C23×C3×C4—21no—
49C23×C3×C4—21no—
50C23×C3×C4—21no—
51C23×C3×C4—44no—
52C23×C3×C4D411yescharacteristic
53C23×C3×C4C2311yescharacteristic
54C22×#67—22no—
55C22×#67—21no—
56C22×D12D411yescharacteristic, complemented
57C3×#121—21no—
58C3×#121—21no—
59C3×#121—21no—
60C3×#121—21no—
61C3×#121—44no—
62C3×#120—42no—
63C3×#120—42no—
64C2×C3×C4⋊C4—21no—
65C2×C3×C4⋊C4D411yescharacteristic, complemented
66C2×C3×C22⋊C4—22no—
67C2×C3×C22⋊C4—22no—
68C2×C3×C22⋊C4—21no—
69C2×C3×C22⋊C4—21no—
70#953—42no—
71#953—42no—
72C2×#217—22no—
73C2×#217—21no—
74C2×#213—22no—
75#930—21no—
76#930—21no—
77#930—21no—
78#930—21no—
79#930—44no—
80C2×(C2×D3)⋊C4—22no—
81C2×(C2×D3)⋊C4—21no—
82C2×C4⋊Dic3—21no—
83C2×C4⋊Dic3D411yescharacteristic, complemented
84C22×C3⋊C8—21no—
85C22×C3⋊C8D412yescomplemented
86C3×#95—22no—
87C3×#95D412yescomplemented
88#832—22no—
89C3⋊#95—22no—
90C3⋊#95D412yescomplemented
91#830—21no—
92#830—21no—
93#830—21no—
94#830—21no—
95#830—44no—
96#809—22no—
97#807—22no—
98C24×C4C2×D311yescharacteristic
99C2×#121—21no—
100C2×#121—31no—
101C2×#121—61no—
102C2×#121C2×D311yescharacteristic
103C2×#120—21no—
104C2×#120—61no—
105#480—61no—
106#480—61no—
107#466—62no—
108#447—61no—
109#447—61no—
110C2×C22⋊C8—31no—
111C2×C22⋊C8—31no—
112C2×C22⋊C8—62no—
113#386—21no—
114#386—22no—
115#386—62no—
116#386—61no—
117#380—21no—
118#380—31no—
119#380—61no—
120#380C2×D311yescharacteristic
121#327—62no—
122#327—62no—
123C24×C3—21no—
124C24×C3—21no—
125C24×C3—21no—
126C24×C3—21no—
127C24×C3—21no—
128C24×C3—21no—
129C24×C3—21no—
130C24×C3—42no—
131C24×C3—42no—
132C24×C3D812yescomplemented
133C24×C3C2×D411yescharacteristic
134C23×D3—21no—
135C2×C3×D4—21no—
136C2×C3×D4—21no—
137C2×C3×D4—42no—
138C2×C3×D4—44no—
139C2×C3×D4—44no—
140C2×C3×D4—44no—
141C2×C3×D4—41no—
142C2×C3×D4—41no—
143C2×C3×D4—41no—
144C2×C3×D4—41no—
145C2×C3×D4—41no—
146C2×C3×D4—41no—
147C22×C3×C4—24no—
148C22×C3×C4—24no—
149C22×C3×C4—24no—
150C22×C3×C4—21no—
151C22×C3×C4—21no—
152C22×C3×C4—21no—
153C22×C3×C4—21no—
154C22×C3×C4—22no—
155C22×C3×C4—42no—
156C22×C3×C4—42no—
157C22×C3×C4—42no—
158C22×C3×C4—42no—
159C22×C3×C4—42no—
160C22×C3×C4—42no—
161C22×C3×C4—42no—
162C22×C3×C4—42no—
163C22×C3×C4—42no—
164C22×C3×C4—42no—
165C22×C3×C4—42no—
166C22×C3×C4—41no—
167C22×C3×C4—41no—
168C22×C3×C4—41no—
169C22×C3×C4—41no—
170C22×C3×C4C2×D411yescharacteristic
171C22×C3×C4C2×D411yescharacteristic
172C22×C3×C4C2×D411yescharacteristic
173C22×C3×C4C2×D411yescharacteristic
174C22×C3×C4C2×D411yescharacteristic
175C22×C3×C4#4111yescharacteristic
176C2×#67—44no—
177C2×#67—44no—
178C2×#67—44no—
179C2×#67—41no—
180C2×#67—41no—
181C2×#67—41no—
182C2×#67—41no—
183C2×#67—41no—
184C2×#67—41no—
185C22×Dic3—21no—
186C22×Dic3—22no—
187C2×D12—21no—
188C2×D12—21no—
189C2×D12—42no—
190C3×C4⋊C4—21no—
191C3×C4⋊C4—21no—
192C3×C4⋊C4—42no—
193C3×C22⋊C4—44no—
194C3×C22⋊C4—44no—
195C3×C22⋊C4—41no—
196C3×C22⋊C4—41no—
197C3×C22⋊C4—41no—
198C3×C22⋊C4—41no—
199#217—44no—
200#217—41no—
201#217—41no—
202#213—82no—
203(C2×D3)⋊C4—44no—
204(C2×D3)⋊C4—41no—
205(C2×D3)⋊C4—41no—
206C4⋊Dic3—21no—
207C4⋊Dic3—21no—
208C4⋊Dic3—42no—
209C2×C3⋊C8—22no—
210C2×C3⋊C8—42no—
211C2×C3⋊C8—42no—
212C2×C3⋊C8—41no—
213C25#6711yescharacteristic
214C22×D4—22no—
215C22×D4—21no—
216C22×D4—31no—
217C22×D4—62no—
218C22×D4—61no—
219C22×D4#6711yescharacteristic, complemented
220C23×C4—21no—
221C23×C4—21no—
222C23×C4—21no—
223C23×C4—21no—
224C23×C4—21no—
225C23×C4—21no—
226C23×C4—44no—
227C23×C4#6711yescharacteristic
228C23×C4C22×D311yescharacteristic
229C2×D8—62no—
230C22×C8—32no—
231C22×C8—61no—
232#121—21no—
233#121—21no—
234#121—21no—
235#121—21no—
236#121—44no—
237#121—61no—
238#121—61no—
239#121—61no—
240#121—61no—
241#121—124no—
242#120—42no—
243#120—42no—
244#120—122no—
245#120—122no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—31no—
248C2×C4⋊C4—61no—
249C2×C4⋊C4#6711yescharacteristic, complemented
250C2×C22⋊C4—22no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—62no—
255C2×C22⋊C4—62no—
256C2×C22⋊C4—61no—
257C2×C22⋊C4—61no—
258#107—62no—
259#102—62no—
260#102—62no—
261C22⋊C8—61no—
262C22⋊C8—61no—
263C22⋊C8—61no—
264C22⋊C8—61no—
265C22⋊C8—124no—
266#95—22no—
267#95—32no—
268#95—62no—
269#95#6712yescomplemented
270C23×C3—22no—
271C23×C3—22no—
272C23×C3—22no—
273C23×C3—22no—
274C23×C3—22no—
275C23×C3—22no—
276C23×C3—22no—
277C23×C3—22no—
278C23×C3—22no—
279C23×C3—41no—
280C23×C3—44no—
281C23×C3—41no—
282C23×C3—41no—
283C23×C3—41no—
284C23×C3—41no—
285C23×C3—41no—
286C23×C3—41no—
287C23×C3—41no—
288C23×C3—44no—
289C23×C3—44no—
290C23×C3—44no—
291C23×C3—42no—
292C23×C3—42no—
293C23×C3—41no—
294C23×C3—41no—
295C23×C3—41no—
296C23×C3—41no—
297C23×C3—41no—
298C23×C3—41no—
299C23×C3—41no—
300C23×C3#12011yescharacteristic
301C23×C3#12011yescharacteristic
302C23×C3#12111yescharacteristic
303C23×C3C2×D811yescharacteristic
304C23×C3#13511yescharacteristic
305C23×C3AGL(1,ℤ/8)11yescharacteristic
306C23×C3AGL(1,ℤ/8)11yescharacteristic
307C22×D3—42no—
308C22×D3—42no—
309C22×D3—41no—
310C22×D3—41no—
311C22×D3—41no—
312C3×D4—42no—
313C3×D4—81no—
314C3×D4—88no—
315C3×D4—81no—
316C3×D4—81no—
317C3×D4—81no—
318C3×D4—81no—
319C2×C3×C4—22no—
320C2×C3×C4—22no—
321C2×C3×C4—22no—
322C2×C3×C4—22no—
323C2×C3×C4—22no—
324C2×C3×C4—22no—
325C2×C3×C4—41no—
326C2×C3×C4—41no—
327C2×C3×C4—41no—
328C2×C3×C4—44no—
329C2×C3×C4—44no—
330C2×C3×C4—44no—
331C2×C3×C4—41no—
332C2×C3×C4—41no—
333C2×C3×C4—41no—
334C2×C3×C4—41no—
335C2×C3×C4—44no—
336C2×C3×C4—42no—
337C2×C3×C4—42no—
338C2×C3×C4—41no—
339C2×C3×C4—42no—
340C2×C3×C4—41no—
341C2×C3×C4—41no—
342C2×C3×C4#12011yescharacteristic
343C2×C3×C4#12111yescharacteristic
344C2×C3×C4#12111yescharacteristic
345C2×C3×C4#12711yescharacteristic
346#67—88no—
347#67—81no—
348#67—81no—
349#67—81no—
350#67—81no—
351C2×Dic3—42no—
352C2×Dic3—42no—
353C2×Dic3—41no—
354C2×Dic3—42no—
355C2×Dic3—41no—
356C2×Dic3—41no—
357D12—42no—
358D12—81no—
359C3⋊C8—42no—
360C3⋊C8—81no—
361C24—21no—
362C24—21no—
363C24—21no—
364C24—21no—
365C24—21no—
366C24—21no—
367C24—21no—
368C24—42no—
369C24—42no—
370C24—61no—
371C24#21312yescomplemented
372C24C2×#6711yescharacteristic
373C2×D4—21no—
374C2×D4—21no—
375C2×D4—42no—
376C2×D4—44no—
377C2×D4—44no—
378C2×D4—44no—
379C2×D4—41no—
380C2×D4—41no—
381C2×D4—41no—
382C2×D4—41no—
383C2×D4—41no—
384C2×D4—41no—
385C2×D4—61no—
386C2×D4—61no—
387C2×D4—122no—
388C2×D4—124no—
389C2×D4—124no—
390C2×D4—124no—
391C2×D4—121no—
392C2×D4—121no—
393C2×D4—121no—
394C2×D4—121no—
395C2×D4—121no—
396C2×D4—121no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—22no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—41no—
417C22×C4—41no—
418C22×C4—41no—
419C22×C4—41no—
420C22×C4—62no—
421C22×C4—61no—
422C22×C4D3×D411yescharacteristic
423C22×C4D3×D411yescharacteristic
424C22×C4D3×D411yescharacteristic
425C22×C4#23711yescharacteristic
426C22×C4C2×#6711yescharacteristic
427C22×C4C2×#6711yescharacteristic
428D8—242no—
429C2×C8—62no—
430C2×C8—122no—
431C2×C8—122no—
432C2×C8—121no—
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4—42no—
436C4⋊C4—61no—
437C4⋊C4—61no—
438C4⋊C4—122no—
439C22⋊C4—44no—
440C22⋊C4—44no—
441C22⋊C4—41no—
442C22⋊C4—41no—
443C22⋊C4—41no—
444C22⋊C4—41no—
445C22⋊C4—124no—
446C22⋊C4—124no—
447C22⋊C4—121no—
448C22⋊C4—121no—
449C22⋊C4—121no—
450C22⋊C4—121no—
451C22×C3—21no—
452C22×C3—21no—
453C22×C3—21no—
454C22×C3—21no—
455C22×C3—21no—
456C22×C3—21no—
457C22×C3—24no—
458C22×C3—24no—
459C22×C3—24no—
460C22×C3—41no—
461C22×C3—41no—
462C22×C3—41no—
463C22×C3—41no—
464C22×C3—42no—
465C22×C3—42no—
466C22×C3—42no—
467C22×C3—42no—
468C22×C3—42no—
469C22×C3—42no—
470C22×C3—42no—
471C22×C3—42no—
472C22×C3—42no—
473C22×C3—42no—
474C22×C3—42no—
475C22×C3—42no—
476C22×C3—82no—
477C22×C3—82no—
478C22×C3—81no—
479C22×C3—81no—
480C22×C3—81no—
481C22×C3#39211yescharacteristic
482C22×C3#44711yescharacteristic
483C22×C3#44711yescharacteristic
484C22×C3#44911yescharacteristic
485C22×C3#44911yescharacteristic
486C22×C3#46611yescharacteristic
487C22×C3#46911yescharacteristic
488C2×D3—82no—
489C2×D3—82no—
490C2×D3—81no—
491C2×D3—81no—
492C2×D3—81no—
493C3×C4—21no—
494C3×C4—21no—
495C3×C4—21no—
496C3×C4—42no—
497C3×C4—82no—
498C3×C4—81no—
499C3×C4B(4,𝔽2)12yes—
500Dic3—82no—
501Dic3—81no—
502C23—22no—
503C23—22no—
504C23—22no—
505C23—22no—
506C23—22no—
507C23—22no—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—41no—
512C23—44no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—44no—
521C23—44no—
522C23—44no—
523C23—42no—
524C23—42no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23—122no—
533C23—122no—
534C23—121no—
535C23—121no—
536C23—121no—
537C23C2×#21311yescharacteristic
538C23#93211yescharacteristic
539C23#93711yescharacteristic
540C23#93911yescharacteristic
541C23#94911yescharacteristic
542C23#95011yescharacteristic
543C23#95311yescharacteristic
544D4—42no—
545D4—81no—
546D4—88no—
547D4—81no—
548D4—81no—
549D4—81no—
550D4—81no—
551D4—122no—
552D4—241no—
553D4—248no—
554D4—241no—
555D4—241no—
556D4—241no—
557D4—241no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—122no—
582C2×C4—122no—
583C2×C4—121no—
584C2×C4—122no—
585C2×C4—121no—
586C2×C4—121no—
587C2×C4#93711yescharacteristic
588C2×C4#93811yescharacteristic
589C2×C4#93911yescharacteristic
590C2×C4#94011yescharacteristic
591C8—122no—
592C8—241no—
593C2×C3—21no—
594C2×C3—21no—
595C2×C3—21no—
596C2×C3—21no—
597C2×C3—21no—
598C2×C3—21no—
599C2×C3—44no—
600C2×C3—161no—
601C2×C3#199911yescharacteristic
602C2×C3#201011yescharacteristic
603C2×C3#202111yescharacteristic
604D3—161no—
605C22—21no—
606C22—21no—
607C22—21no—
608C22—21no—
609C22—21no—
610C22—21no—
611C22—24no—
612C22—24no—
613C22—24no—
614C22—41no—
615C22—41no—
616C22—41no—
617C22—41no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22—42no—
622C22—42no—
623C22—42no—
624C22—42no—
625C22—42no—
626C22—42no—
627C22—42no—
628C22—42no—
629C22—42no—
630C22—82no—
631C22—82no—
632C22—81no—
633C22—81no—
634C22—81no—
635C22—242no—
636C22—242no—
637C22—241no—
638C22—241no—
639C22—241no—
640C22#503611yescharacteristic
641C22#503711yescharacteristic
642C22#503811yescharacteristic
643C22#503911yescharacteristic
644C22#521711yescharacteristic
645C22#522311yescharacteristic
646C22#523811yescharacteristic
647C4—21no—
648C4—21no—
649C4—21no—
650C4—42no—
651C4—82no—
652C4—81no—
653C4—242no—
654C4—241no—
655C4#503212yes—
656C3#1032011yescharacteristic, complemented, Sylow
657C2—21no—
658C2—21no—
659C2—21no—
660C2—21no—
661C2—21no—
662C2—21no—
663C2—44no—
664C2—161no—
665C2—481no—
666C2#7167111yescharacteristic
667C2#7168411yescharacteristic
668C2#7171511yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#2847C2
socleC22×C3#392
hypercenter#2847D3
hypocenterC3#10320
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2847

Complements

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

NH ≅ G/Ntype
1. C3×#2847C2semidirect
2. #84032C2semidirect
3. #82625C2semidirect
4. #72126C2semidirect
5. #71675C2semidirect
6. #69509C2semidirect
7. #67625C2semidirect
36. #2847D3semidirect
44. C22×C3×D4D4semidirect
56. C22×D12D4semidirect
65. C2×C3×C4⋊C4D4semidirect
83. C2×C4⋊Dic3D4semidirect
85. C22×C3⋊C8D4semidirect
87. C3×#95D4semidirect
90. C3⋊#95D4semidirect
132. C24×C3D8semidirect
219. C22×D4#67semidirect
249. C2×C4⋊C4#67semidirect
269. #95#67semidirect
371. C24#213semidirect
656. C3#10320semidirect

Maximal subgroups

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

subgroupindex
C3×#28472
#840322
#826252
#721262
#716752
#695092
#676252
#103203