Finite Group Databasebeta

← #10849322

Subgroups — #10849322

Order 768.

1112 conjugacy classes of subgroups.

The subgroup lattice has 730 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5713C211yesmaximal, characteristic
2C2×#4765C211yesmaximal, characteristic, complemented
3C2×#4761C211yesmaximal, characteristic, complemented
4#76060C211yesmaximal, characteristic, complemented
5C3×#2041C211yesmaximal, characteristic
6#71762C211yesmaximal, characteristic
7#70216C211yesmaximal, characteristic, complemented
8#70178C211yesmaximal, characteristic, complemented
9#70170C211yesmaximal, characteristic, complemented
10#70169C211yesmaximal, characteristic, complemented
11#70162C211yesmaximal, characteristic, complemented
12#70152C211yesmaximal, characteristic, complemented
13#69632C211yesmaximal, characteristic, complemented
14#68684C211yesmaximal, characteristic, complemented
15#68620C211yesmaximal, characteristic, complemented
16#25659—31nomaximal, Sylow
17C2×C3×#127C2211yescharacteristic
18C2×#936—21no—
19C2×#936C2211yescharacteristic
20C2×#936C2211yescharacteristic
21C2×#935—21no—
22C2×#935—21no—
23C2×D4×Dic3—21no—
24C2×D4×Dic3C2211yescharacteristic
25C2×D4×Dic3C2211yescharacteristic
26#5713—24no—
27#5713—24no—
28C2×#888—21no—
29C2×#888C2211yescharacteristic
30C2×C4×Dic6C2211yescharacteristic
31C2×C4×Dic6C2211yescharacteristic
32C2×C3×#102C2211yescharacteristic
33C2×C3×#102C2211yescharacteristic
34C2×C3×#102C2211yescharacteristic
35C2×C3×#102C2211yescharacteristic
36C3×#389C2211yescharacteristic
37C2×#832C2211yescharacteristic
38C2×#832C2211yescharacteristic
39C2×#832C2211yescharacteristic
40C2×#832C2211yescharacteristic
41C2×#817C2211yescharacteristic
42C2×#816C2211yescharacteristic
43C2×#816C2211yescharacteristic
44C2×#815C2211yescharacteristic
45C2×Dic3⋊C8C2211yescharacteristic
46#4984C2211yescharacteristic
47#4982—21no—
48Dic3×C4⋊C4C2211yescharacteristic
49#4972C2211yescharacteristic
50C2×#810C2211yescharacteristic
51C2×#810C2211yescharacteristic
52C2×#808C2211yescharacteristic
53#4928C2211yescharacteristic
54#4765—24no—
55#4761—24no—
56#4657—21no—
57#4646—21no—
58C3×#336C2211yescharacteristic
59#4551C2211yescharacteristic
60#4550C2211yescharacteristic
61#4550C2211yescharacteristic
62#4529C2211yescharacteristic
63#4524C2211yescharacteristic
64C2×#559—31no—
65C2×#489—31no—
66C2×#476—31no—
67#2546—31no—
68#2042—31no—
69#2042—31no—
70#2041—31no—
71#2041D311yescharacteristic
72#1985—31no—
73#1983—31no—
74#1865—31no—
75#1865—31no—
76#1864—31no—
77#1864—31no—
78#1762—31no—
79C22×C3×D4—21no—
80C22×C3×D4—21no—
81C22×C3×D4C2311yescharacteristic
82C22×C3×D4C2311yescharacteristic
83C23×Dic3—21no—
84C23×Dic3—21no—
85C22×Dic6—21no—
86C22×Dic6D411yescharacteristic
87C22×Dic6C2311yescharacteristic
88C22×C3×C8C2311yescharacteristic
89C22×C3×C8C2311yescharacteristic
90C3×#127—22no—
91C3×#127—22no—
92C2×C3×C4⋊C4—21no—
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C42C2311yescharacteristic
96C2×#217—21no—
97C2×#217—21no—
98C2×#217—21no—
99C2×#217—21no—
100C2×#217—21no—
101C2×#217—21no—
102#936—28no—
103#936—22no—
104#936—22no—
105#936—22no—
106#936D414yes—
107#935—28no—
108#935—28no—
109D4×Dic3—28no—
110D4×Dic3—22no—
111D4×Dic3—22no—
112D4×Dic3—22no—
113D4×Dic3D414yes—
114C2×C4⋊Dic3—21no—
115C2×C4⋊Dic3D411yescharacteristic
116C2×C4⋊Dic3C2311yescharacteristic
117C2×C4⋊Dic3C2311yescharacteristic
118C2×C4⋊Dic3C2311yescharacteristic
119C2×Dic3⋊C4—21no—
120C2×Dic3⋊C4—21no—
121C2×Dic3⋊C4—21no—
122C2×Dic3⋊C4—21no—
123C2×Dic3⋊C4D411yescharacteristic
124C2×Dic3⋊C4D411yescharacteristic
125C2×C4×Dic3—21no—
126C2×C4×Dic3—21no—
127C2×C4×Dic3C2311yescharacteristic
128C2×C4×Dic3C2311yescharacteristic
129C22×C3⋊C8C2311yescharacteristic
130C22×C3⋊C8C2311yescharacteristic
131#888—22no—
132#888—22no—
133#888—44no—
134C4×Dic6—22no—
135C4×Dic6—22no—
136C4×Dic6—22no—
137C4×Dic6—22no—
138C3×#102—22no—
139C3×#102—22no—
140C3×#102—22no—
141C3×#102—22no—
142C3×#95—21no—
143#832—22no—
144#832—22no—
145#832—22no—
146#832—22no—
147C3⋊#95—21no—
148C3⋊#95—21no—
149C3⋊#95—21no—
150C3⋊#95—21no—
151#817—22no—
152#816—22no—
153#816—22no—
154#815—22no—
155Dic3⋊C8—22no—
156#810—22no—
157#810—22no—
158#808—22no—
159#559—64no—
160#559—64no—
161C2×C4⋊Q8—31no—
162C2×C4⋊Q8—61no—
163C2×#127C2×D311yescharacteristic
164C2×#124—31no—
165C2×#124—31no—
166C2×#124—61no—
167C2×#123—61no—
168C2×#123—61no—
169C2×C4×Q8—31no—
170C2×C4×Q8—31no—
171C2×C4×D4—31no—
172C2×C4×D4—31no—
173C2×C4×D4—61no—
174#489—64no—
175#476—64no—
176C2×#106—31no—
177C2×#106—31no—
178C2×C4⋊C8—31no—
179C2×#103—31no—
180C2×#103—31no—
181C2×#103—31no—
182C2×#103—31no—
183C2×#102—31no—
184C2×#102—31no—
185C2×#102—31no—
186C2×#102—31no—
187C2×#102C2×D311yescharacteristic
188C2×#102C2×D311yescharacteristic
189C2×#102C2×D311yescharacteristic
190C2×#102C2×D311yescharacteristic
191C2×#97—31no—
192#398—61no—
193#398—61no—
194#391—61no—
195#389—31no—
196#389—31no—
197#389—31no—
198#389C2×D311yescharacteristic
199C4×C4⋊C4—31no—
200#340—31no—
201#340—31no—
202#336—31no—
203#336—31no—
204#336—31no—
205#336C2×D311yescharacteristic
206C24×C3—21no—
207C24×C3—21no—
208C2×C3×D4—24no—
209C2×C3×D4—24no—
210C2×C3×D4—21no—
211C2×C3×D4—21no—
212C2×C3×D4—21no—
213C2×C3×D4—21no—
214C2×C3×D4—21no—
215C2×C3×D4—21no—
216C2×C3×D4—21no—
217C2×C3×D4—21no—
218C2×C3×D4—21no—
219C2×C3×D4—21no—
220C2×C3×D4—42no—
221C2×C3×D4—42no—
222C2×C3×D4—81no—
223C2×C3×D4—81no—
224C2×C3×D4C2×D412yes—
225C2×C3×D4C2×D412yes—
226C22×C3×C4—21no—
227C22×C3×C4—21no—
228C22×C3×C4C2×D411yescharacteristic
229C22×C3×C4C2×D411yescharacteristic
230C22×C3×C4C2411yescharacteristic
231C22×Dic3—21no—
232C22×Dic3—21no—
233C22×Dic3—21no—
234C22×Dic3—24no—
235C22×Dic3—24no—
236C22×Dic3—21no—
237C22×Dic3—81no—
238C22×Dic3—81no—
239C22×Dic3—81no—
240C22×Dic3—81no—
241C22×Dic3C2×D411yescharacteristic
242C22×Dic3C2×D411yescharacteristic
243C22×Dic3C2×D411yescharacteristic
244C22×Dic3C2×D411yescharacteristic
245C2×Dic6—21no—
246C2×Dic6—21no—
247C2×Dic6—21no—
248C2×Dic6—21no—
249C2×Dic6—22no—
250C2×Dic6—21no—
251C2×Dic6—22no—
252C2×Dic6—41no—
253C2×Dic6—42no—
254C2×Dic6—42no—
255C2×Dic6—42no—
256C2×Dic6C2×D412yes—
257C2×C3×C8—21no—
258C2×C3×C8—21no—
259C2×C3×C8—21no—
260C2×C3×C8—21no—
261C2×C3×C8—21no—
262C2×C3×C8—21no—
263C3×C4⋊C4—21no—
264C3×C4⋊C4—21no—
265C3×C4⋊C4—21no—
266C3×C4⋊C4—42no—
267C3×C4⋊C4#4112yes—
268C3×C42—21no—
269C3×C42—21no—
270#217—24no—
271#217—24no—
272#217—24no—
273#217—24no—
274#217—24no—
275#217—24no—
276C4⋊Dic3—21no—
277C4⋊Dic3—21no—
278C4⋊Dic3—22no—
279C4⋊Dic3—21no—
280C4⋊Dic3—21no—
281C4⋊Dic3—42no—
282C4⋊Dic3C2×D412yes—
283C4⋊Dic3#4112yes—
284Dic3⋊C4—24no—
285Dic3⋊C4—22no—
286Dic3⋊C4—22no—
287Dic3⋊C4—42no—
288Dic3⋊C4—42no—
289Dic3⋊C4—42no—
290C4×Dic3—24no—
291C4×Dic3—21no—
292C4×Dic3—21no—
293C4×Dic3—42no—
294C4×Dic3C2×D412yes—
295C4×Dic3C2×D412yes—
296C2×C3⋊C8—21no—
297C2×C3⋊C8—21no—
298C2×C3⋊C8—21no—
299C2×C3⋊C8—21no—
300C2×C3⋊C8—21no—
301C2×C3⋊C8—21no—
302C22×Q8—31no—
303C22×Q8—31no—
304C22×Q8—61no—
305C22×D4—21no—
306C22×D4—21no—
307C22×D4C22×D311yescharacteristic
308C22×D4C22×D311yescharacteristic
309C23×C4—61no—
310C23×C4—61no—
311C22×C8—31no—
312C22×C8—31no—
313C22×C8C22×D311yescharacteristic
314C22×C8C22×D311yescharacteristic
315C4⋊Q8—62no—
316C4⋊Q8—62no—
317C4⋊Q8—124no—
318#127—22no—
319#127—22no—
320#124—34no—
321#124—62no—
322#124—62no—
323#124—68no—
324#124—62no—
325#123—68no—
326#123—68no—
327C4×Q8—62no—
328C4×Q8—62no—
329C4×Q8—62no—
330C4×Q8—62no—
331C4×D4—34no—
332C4×D4—62no—
333C4×D4—62no—
334C4×D4—68no—
335C4×D4—62no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—31no—
338C2×C4⋊C4—31no—
339C2×C4⋊C4—31no—
340C2×C4⋊C4—31no—
341C2×C4⋊C4—31no—
342C2×C4⋊C4—31no—
343C2×C4⋊C4—61no—
344C2×C4⋊C4—61no—
345C2×C4⋊C4—61no—
346C2×C4⋊C4—61no—
347C2×C4⋊C4—61no—
348C2×C4⋊C4C22×D311yescharacteristic
349C2×C4⋊C4C22×D311yescharacteristic
350C2×C22⋊C4—61no—
351C2×C22⋊C4—61no—
352C2×C22⋊C4—61no—
353C2×C22⋊C4—61no—
354C2×C22⋊C4—61no—
355C2×C22⋊C4—61no—
356C2×C42—31no—
357C2×C42—31no—
358C2×C42—61no—
359C2×C42—61no—
360C2×C42C22×D311yescharacteristic
361#106—62no—
362#106—62no—
363C4⋊C8—62no—
364#103—62no—
365#103—62no—
366#103—62no—
367#103—62no—
368#102—22no—
369#102—22no—
370#102—22no—
371#102—22no—
372#102—62no—
373#102—62no—
374#102—62no—
375#102—62no—
376#97—62no—
377#95—21no—
378#95—61no—
379#95—61no—
380#95—61no—
381#95—61no—
382C23×C3—22no—
383C23×C3—22no—
384C23×C3—41no—
385C23×C3—41no—
386C23×C3—41no—
387C23×C3—41no—
388C23×C3—41no—
389C23×C3—41no—
390C23×C3—41no—
391C23×C3—41no—
392C23×C3—41no—
393C23×C3—41no—
394C23×C3—41no—
395C23×C3—41no—
396C23×C3C22×D411yescharacteristic
397C3×D4—22no—
398C3×D4—22no—
399C3×D4—22no—
400C3×D4—22no—
401C3×D4—48no—
402C3×D4—41no—
403C3×D4—41no—
404C3×D4—41no—
405C3×D4—41no—
406C3×D4—84no—
407C2×C3×C4—21no—
408C2×C3×C4—21no—
409C2×C3×C4—21no—
410C2×C3×C4—22no—
411C2×C3×C4—21no—
412C2×C3×C4—21no—
413C2×C3×C4—41no—
414C2×C3×C4—41no—
415C2×C3×C4—41no—
416C2×C3×C4—41no—
417C2×C3×C4—41no—
418C2×C3×C4AGL(1,ℤ/8)12yes—
419C2×C3×C4C22×D411yescharacteristic
420C2×C3×C4C22×D411yescharacteristic
421C2×C3×C4C2×#4111yescharacteristic
422C2×C3×C4C2×#4111yescharacteristic
423C2×C3×C42−1+411yescharacteristic
424C2×C3×C42−1+411yescharacteristic
425C2×Dic3—21no—
426C2×Dic3—21no—
427C2×Dic3—21no—
428C2×Dic3—21no—
429C2×Dic3—21no—
430C2×Dic3—22no—
431C2×Dic3—21no—
432C2×Dic3—22no—
433C2×Dic3—22no—
434C2×Dic3—21no—
435C2×Dic3—21no—
436C2×Dic3—21no—
437C2×Dic3—21no—
438C2×Dic3—41no—
439C2×Dic3—41no—
440C2×Dic3—41no—
441C2×Dic3—41no—
442C2×Dic3—41no—
443C2×Dic3—41no—
444C2×Dic3—41no—
445C2×Dic3—41no—
446C2×Dic3—41no—
447C2×Dic3—84no—
448C2×Dic3—84no—
449C2×Dic3#12012yes—
450C2×Dic3#12012yes—
451Dic6—22no—
452Dic6—22no—
453Dic6—44no—
454Dic6—41no—
455Dic6—41no—
456Dic6—84no—
457C3×C8—41no—
458C3×C8—41no—
459C3⋊C8—41no—
460C3⋊C8—41no—
461C24—21no—
462C24—21no—
463C2×Q8—32no—
464C2×Q8—62no—
465C2×Q8—62no—
466C2×Q8—61no—
467C2×Q8—61no—
468C2×Q8—61no—
469C2×Q8—61no—
470C2×Q8—61no—
471C2×Q8—121no—
472C2×Q8—122no—
473C2×Q8—122no—
474C2×Q8—122no—
475C2×D4—24no—
476C2×D4—24no—
477C2×D4—21no—
478C2×D4—21no—
479C2×D4—21no—
480C2×D4—21no—
481C2×D4—21no—
482C2×D4—21no—
483C2×D4—21no—
484C2×D4—21no—
485C2×D4—21no—
486C2×D4—21no—
487C2×D4—42no—
488C2×D4—42no—
489C2×D4—81no—
490C2×D4—81no—
491C2×D4D3×D412yes—
492C2×D4D3×D412yes—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—31no—
496C22×C4—31no—
497C22×C4—31no—
498C22×C4—31no—
499C22×C4—64no—
500C22×C4—61no—
501C22×C4—61no—
502C22×C4—61no—
503C22×C4—64no—
504C22×C4—61no—
505C22×C4—241no—
506C22×C4—241no—
507C22×C4—241no—
508C22×C4—241no—
509C22×C4D3×D411yescharacteristic
510C22×C4D3×D411yescharacteristic
511C22×C4C23×D311yescharacteristic
512C2×C8—21no—
513C2×C8—21no—
514C2×C8—21no—
515C2×C8—21no—
516C2×C8—21no—
517C2×C8—21no—
518C2×C8—61no—
519C2×C8—61no—
520C2×C8—61no—
521C2×C8—61no—
522C2×C8—61no—
523C2×C8—61no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—32no—
528C4⋊C4—32no—
529C4⋊C4—42no—
530C4⋊C4—62no—
531C4⋊C4—62no—
532C4⋊C4—62no—
533C4⋊C4—64no—
534C4⋊C4—61no—
535C4⋊C4—61no—
536C4⋊C4—61no—
537C4⋊C4—61no—
538C4⋊C4—122no—
539C4⋊C4—122no—
540C4⋊C4—122no—
541C4⋊C4—122no—
542C4⋊C4#23712yes—
543C22⋊C4—64no—
544C22⋊C4—64no—
545C22⋊C4—64no—
546C22⋊C4—64no—
547C22⋊C4—64no—
548C22⋊C4—64no—
549C42—21no—
550C42—21no—
551C42—32no—
552C42—32no—
553C42—64no—
554C42—61no—
555C42—61no—
556C42—122no—
557C22×C3—42no—
558C22×C3—42no—
559C22×C3—42no—
560C22×C3—42no—
561C22×C3—42no—
562C22×C3—42no—
563C22×C3—81no—
564C22×C3—81no—
565C22×C3—81no—
566C22×C3—81no—
567C22×C3—81no—
568C22×C3—81no—
569C22×C3—81no—
570C22×C3—81no—
571C22×C3C2×#12011yescharacteristic
572C22×C3#54811yescharacteristic
573C22×C3#54811yescharacteristic
574C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
575C22×C3#57511yescharacteristic
576C22×C3#57711yescharacteristic
577C22×C3#57711yescharacteristic
578C3×C4—22no—
579C3×C4—42no—
580C3×C4—41no—
581C3×C4—81no—
582C3×C4#53211yescharacteristic
583C3×C4#53611yescharacteristic
584C3×C4#54711yescharacteristic
585C3×C4#54711yescharacteristic
586Dic3—22no—
587Dic3—22no—
588Dic3—42no—
589Dic3—42no—
590Dic3—41no—
591Dic3—41no—
592Dic3—42no—
593Dic3—81no—
594C23—22no—
595C23—22no—
596C23—41no—
597C23—41no—
598C23—41no—
599C23—41no—
600C23—41no—
601C23—41no—
602C23—41no—
603C23—41no—
604C23—41no—
605C23—41no—
606C23—41no—
607C23—41no—
608C23C2×D3×D411yescharacteristic
609Q8—62no—
610Q8—62no—
611Q8—124no—
612Q8—121no—
613Q8—121no—
614Q8—244no—
615D4—22no—
616D4—22no—
617D4—22no—
618D4—22no—
619D4—48no—
620D4—41no—
621D4—41no—
622D4—41no—
623D4—41no—
624D4—84no—
625C2×C4—21no—
626C2×C4—21no—
627C2×C4—21no—
628C2×C4—21no—
629C2×C4—21no—
630C2×C4—22no—
631C2×C4—32no—
632C2×C4—32no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—41no—
637C2×C4—41no—
638C2×C4—61no—
639C2×C4—61no—
640C2×C4—61no—
641C2×C4—61no—
642C2×C4—62no—
643C2×C4—61no—
644C2×C4—61no—
645C2×C4—61no—
646C2×C4—61no—
647C2×C4—61no—
648C2×C4—62no—
649C2×C4—62no—
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×C4—121no—
660C2×C4—244no—
661C2×C4—244no—
662C2×C4#91112yes—
663C2×C4C2×D3×D411yescharacteristic
664C2×C4C2×D3×D411yescharacteristic
665C2×C4C2×#23711yescharacteristic
666C2×C4D3×#4111yescharacteristic
667C2×C4#101011yescharacteristic
668C2×C4#101011yescharacteristic
669C8—41no—
670C8—41no—
671C8—121no—
672C8—121no—
673C2×C3—82no—
674C2×C3—82no—
675C2×C3#243211yescharacteristic
676C2×C3#300611yescharacteristic
677C2×C3#300911yescharacteristic
678C2×C3#337111yescharacteristic
679C2×C3#337111yescharacteristic
680C2×C3#337311yescharacteristic
681C2×C3#337311yescharacteristic
682C22—42no—
683C22—42no—
684C22—42no—
685C22—42no—
686C22—42no—
687C22—42no—
688C22—81no—
689C22—81no—
690C22—81no—
691C22—81no—
692C22—81no—
693C22—81no—
694C22—81no—
695C22—81no—
696C22D3×#12011yescharacteristic
697C22#559811yescharacteristic
698C22#567711yescharacteristic
699C22C2×#91111yescharacteristic
700C22#576211yescharacteristic
701C22#577311yescharacteristic
702C22#577511yescharacteristic
703C4—22no—
704C4—41no—
705C4—42no—
706C4—62no—
707C4—62no—
708C4—81no—
709C4—121no—
710C4—122no—
711C4—122no—
712C4—121no—
713C4—122no—
714C4—241no—
715C4#559611yescharacteristic
716C4#561111yescharacteristic
717C4#564811yescharacteristic
718C4#565511yescharacteristic
719C3#2565911yescharacteristic, complemented, Sylow
720C2—82no—
721C2—82no—
722C2#7690411yescharacteristic
723C2#8000311yescharacteristic
724C2#8014811yescharacteristic
725C2#8032311yescharacteristic
726C2#8037311yescharacteristic
727C2#8158711yescharacteristic
728C2#8171111yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2041C2
socleC23×C3C22×D4
prime radicalC2×C3×#127C22
hypercenter#2041D3
hypocenterC3#25659
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2041

Complements

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

NH ≅ G/Ntype
2. C2×#4765C2semidirect
3. C2×#4761C2semidirect
4. #76060C2semidirect
7. #70216C2semidirect
8. #70178C2semidirect
9. #70170C2semidirect
10. #70169C2semidirect
11. #70162C2semidirect
12. #70152C2semidirect
13. #69632C2semidirect
14. #68684C2semidirect
15. #68620C2semidirect
719. C3#25659semidirect

Maximal subgroups

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

subgroupindex
C2×#57132
C2×#47652
C2×#47612
#760602
C3×#20412
#717622
#702162
#701782
#701702
#701692
#701622
#701522
#696322
#686842
#686202
#256593