Finite Group Databasebeta

← #10896365

Subgroups — #10896365

Order 768.

798 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83172C211yesmaximal, characteristic
2#73627C211yesmaximal, characteristic, complemented
3C3×#1982C211yesmaximal, characteristic
4#71761C211yesmaximal, characteristic
5#70455C211yesmaximal, characteristic, complemented
6#70444C211yesmaximal, characteristic, complemented
7#70182C211yesmaximal, characteristic, complemented
8#70181C211yesmaximal, characteristic, complemented
9#70170C211yesmaximal, characteristic, complemented
10#70169C211yesmaximal, characteristic, complemented
11#70168C211yesmaximal, characteristic, complemented
12#70167C211yesmaximal, characteristic, complemented
13#70163C211yesmaximal, characteristic, complemented
14#70163C211yesmaximal, characteristic, complemented
15#69468C211yesmaximal, characteristic, complemented
16#29263—31nomaximal, Sylow
17C2×C3×#127C2211yescharacteristic
18C2×D4×Dic3C2211yescharacteristic
19C2×D4×Dic3C2211yescharacteristic
20C2×#888C2211yescharacteristic
21C2×C3×C4⋊C8C2211yescharacteristic
22C2×C3×#102C2211yescharacteristic
23C2×C3×#102C2211yescharacteristic
24C2×C3×#102C2211yescharacteristic
25C2×C3×#102C2211yescharacteristic
26C3×C4×C4⋊C4C2211yescharacteristic
27#5221C2211yescharacteristic
28#5221C2211yescharacteristic
29#5220—21no—
30#5220—21no—
31#5220—21no—
32#5220—21no—
33C2×#832C2211yescharacteristic
34C2×#832C2211yescharacteristic
35C2×#832C2211yescharacteristic
36C2×#832C2211yescharacteristic
37C2×#818C2211yescharacteristic
38C2×#817C2211yescharacteristic
39#4984C2211yescharacteristic
40#4977C2211yescharacteristic
41#4972C2211yescharacteristic
42C2×#808C2211yescharacteristic
43C2×#807C2211yescharacteristic
44#4936C2211yescharacteristic
45C4×C4⋊Dic3C2211yescharacteristic
46#4933—21no—
47#4932C2211yescharacteristic
48#4929—21no—
49C2×C4⋊(C3⋊C8)C2211yescharacteristic
50#4655—21no—
51#4646—21no—
52#4551C2211yescharacteristic
53#4551C2211yescharacteristic
54#4550C2211yescharacteristic
55#4550C2211yescharacteristic
56#4550C2211yescharacteristic
57#4550C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#2331—31no—
61#2328—31no—
62#2092—31no—
63#2092—31no—
64#2023—31no—
65#2023—31no—
66#1982—31no—
67#1982D311yescharacteristic
68#1949—31no—
69#1949—31no—
70#1864—31no—
71#1864—31no—
72#1862—31no—
73#1862—31no—
74#1832—31no—
75C22×C3×D4—21no—
76C22×C3×D4—21no—
77C22×C3×D4C2311yescharacteristic
78C22×C3×D4C2311yescharacteristic
79C23×Dic3—21no—
80C23×Dic3—21no—
81C22×Dic6—21no—
82C22×C3×C8C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C3×#127—22no—
85C3×#127Q814yescomplemented
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C42—21no—
89C2×C3×C42C2311yescharacteristic
90C2×#217—21no—
91C2×#217—21no—
92C2×#217—21no—
93C2×#217—21no—
94C2×#217—21no—
95C2×#217—21no—
96D4×Dic3—41no—
97D4×Dic3—41no—
98D4×Dic3—41no—
99D4×Dic3—41no—
100C2×C4⋊Dic3—21no—
101C2×C4⋊Dic3C2311yescharacteristic
102C2×C4⋊Dic3C2311yescharacteristic
103C2×C4⋊Dic3C2311yescharacteristic
104C2×C4⋊Dic3C2311yescharacteristic
105C2×Dic3⋊C4—21no—
106C2×Dic3⋊C4—21no—
107C2×C4×Dic3—21no—
108C2×C4×Dic3C2311yescharacteristic
109C2×C4×Dic3C2311yescharacteristic
110C22×C3⋊C8C2311yescharacteristic
111C22×C3⋊C8C2311yescharacteristic
112#888—22no—
113#888—22no—
114C3×C4⋊C8—22no—
115C3×#102—22no—
116C3×#102—22no—
117C3×#102—41no—
118C3×#102—41no—
119C3×#95—21no—
120#832—22no—
121#832—22no—
122#832—41no—
123#832—41no—
124C3⋊#95—21no—
125C3⋊#95—21no—
126C3⋊#95—21no—
127C3⋊#95—21no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131C3⋊#95D411yescharacteristic
132C3⋊#95D411yescharacteristic
133C3⋊#95D411yescharacteristic
134C3⋊#95D411yescharacteristic
135#818—41no—
136#817—41no—
137#808—41no—
138#807—41no—
139C4⋊(C3⋊C8)—22no—
140C2×C4⋊Q8—31no—
141C2×#127C2×D311yescharacteristic
142C2×C4×D4—31no—
143C2×C4×D4—31no—
144C2×#107—31no—
145C2×#107—31no—
146C2×#106—31no—
147C2×#106—31no—
148C2×C4⋊C8—31no—
149C2×C4⋊C8C2×D311yescharacteristic
150C2×#102—31no—
151C2×#102—31no—
152C2×#102—31no—
153C2×#102—31no—
154C2×#102C2×D311yescharacteristic
155C2×#102C2×D311yescharacteristic
156C2×#102C2×D311yescharacteristic
157C2×#102C2×D311yescharacteristic
158#391—61no—
159#391—61no—
160#390—31no—
161#390—31no—
162#389—31no—
163#389—31no—
164#387—61no—
165#387—61no—
166#386—61no—
167#386—61no—
168#386—61no—
169#386—61no—
170#382—31no—
171#382—31no—
172C4×C4⋊C4—31no—
173C4×C4⋊C4C2×D311yescharacteristic
174#376—31no—
175#340—31no—
176#340—31no—
177#340—31no—
178#340—31no—
179#336—31no—
180#336—31no—
181#336—31no—
182#336—31no—
183C24×C3—21no—
184C24×C3—21no—
185C2×C3×D4—24no—
186C2×C3×D4—24no—
187C2×C3×D4—21no—
188C2×C3×D4—21no—
189C2×C3×D4—21no—
190C2×C3×D4—21no—
191C2×C3×D4—21no—
192C2×C3×D4—21no—
193C2×C3×D4—21no—
194C2×C3×D4—21no—
195C2×C3×D4—21no—
196C2×C3×D4—21no—
197C2×C3×D4—81no—
198C2×C3×D4—81no—
199C2×C3×D4—81no—
200C2×C3×D4—81no—
201C2×C3×D4C2×Q812yes—
202C2×C3×D4C2×Q812yes—
203C22×C3×C4—21no—
204C22×C3×C4—21no—
205C22×C3×C4C2×D411yescharacteristic
206C22×C3×C4C2×D411yescharacteristic
207C22×C3×C4C2411yescharacteristic
208C22×Dic3—21no—
209C22×Dic3—21no—
210C22×Dic3—21no—
211C22×Dic3—21no—
212C22×Dic3—42no—
213C22×Dic3—42no—
214C22×Dic3—81no—
215C22×Dic3—81no—
216C22×Dic3—81no—
217C22×Dic3—81no—
218C22×Dic3C2×D411yescharacteristic
219C22×Dic3C2×D411yescharacteristic
220C22×Dic3C2×D411yescharacteristic
221C22×Dic3C2×D411yescharacteristic
222C2×Dic6—24no—
223C2×Dic6—81no—
224C2×Dic6—81no—
225C2×C3×C8—21no—
226C2×C3×C8—21no—
227C2×C3×C8—21no—
228C2×C3×C8—21no—
229C2×C3×C8—21no—
230C2×C3×C8—21no—
231C3×C4⋊C4—21no—
232C3×C4⋊C4—21no—
233C3×C4⋊C4—21no—
234C3×C4⋊C4#4112yes—
235C3×C42—81no—
236C3×C42C2×Q812yes—
237C3×C42#4112yes—
238#217—42no—
239#217—42no—
240#217—81no—
241#217—81no—
242#217—81no—
243#217—81no—
244C4⋊Dic3—21no—
245C4⋊Dic3—21no—
246C4⋊Dic3—21no—
247C4⋊Dic3—21no—
248C4⋊Dic3—21no—
249C4⋊Dic3—21no—
250C4⋊Dic3—21no—
251C4⋊Dic3—81no—
252C4⋊Dic3#4112yes—
253Dic3⋊C4—24no—
254Dic3⋊C4—81no—
255C4×Dic3—21no—
256C4×Dic3—21no—
257C4×Dic3—21no—
258C4×Dic3—21no—
259C4×Dic3—81no—
260C2×C3⋊C8—21no—
261C2×C3⋊C8—21no—
262C2×C3⋊C8—21no—
263C2×C3⋊C8—21no—
264C2×C3⋊C8—21no—
265C2×C3⋊C8—21no—
266C22×Q8—61no—
267C22×D4—21no—
268C22×D4—21no—
269C22×D4C22×D311yescharacteristic
270C22×D4C22×D311yescharacteristic
271C23×C4—61no—
272C23×C4—61no—
273C22×C8—31no—
274C22×C8—31no—
275C22×C8C22×D311yescharacteristic
276C22×C8C22×D311yescharacteristic
277C4⋊Q8—62no—
278C4⋊Q8—62no—
279#127—22no—
280#127Dic614yescomplemented
281C4×D4—121no—
282C4×D4—121no—
283C4×D4—121no—
284C4×D4—121no—
285C2×C4⋊C4—31no—
286C2×C4⋊C4—31no—
287C2×C4⋊C4—31no—
288C2×C4⋊C4—31no—
289C2×C4⋊C4—61no—
290C2×C4⋊C4—61no—
291C2×C4⋊C4—61no—
292C2×C4⋊C4C22×D311yescharacteristic
293C2×C4⋊C4C22×D311yescharacteristic
294C2×C22⋊C4—61no—
295C2×C22⋊C4—61no—
296C2×C22⋊C4—61no—
297C2×C22⋊C4—61no—
298C2×C22⋊C4—61no—
299C2×C22⋊C4—61no—
300C2×C42—21no—
301C2×C42—31no—
302C2×C42—31no—
303C2×C42—61no—
304C2×C42C22×D311yescharacteristic
305#107—121no—
306#107—121no—
307#106—121no—
308#106—121no—
309C4⋊C8—22no—
310C4⋊C8—62no—
311#102—22no—
312#102—22no—
313#102—41no—
314#102—41no—
315#102—62no—
316#102—62no—
317#102—121no—
318#102—121no—
319#95—21no—
320#95—31no—
321#95—31no—
322#95—31no—
323#95—31no—
324#95—61no—
325#95—61no—
326#95—61no—
327#95—61no—
328#95—61no—
329#95—61no—
330#95—61no—
331C23×C3—22no—
332C23×C3—22no—
333C23×C3—41no—
334C23×C3—41no—
335C23×C3—41no—
336C23×C3—41no—
337C23×C3—41no—
338C23×C3—41no—
339C23×C3—41no—
340C23×C3—41no—
341C23×C3—41no—
342C23×C3—41no—
343C23×C3—41no—
344C23×C3—41no—
345C23×C3C22×D411yescharacteristic
346C3×D4—22no—
347C3×D4—22no—
348C3×D4—22no—
349C3×D4—22no—
350C3×D4—41no—
351C3×D4—41no—
352C3×D4—41no—
353C3×D4—41no—
354C3×D4—84no—
355C3×D4—84no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—22no—
361C2×C3×C4—41no—
362C2×C3×C4—41no—
363C2×C3×C4—41no—
364C2×C3×C4—41no—
365C2×C3×C4—41no—
366C2×C3×C4C22⋊Q812yes—
367C2×C3×C4C22⋊Q812yes—
368C2×C3×C4C22×Q811yescharacteristic
369C2×C3×C4C2×#4111yescharacteristic
370C2×C3×C4C2×#4111yescharacteristic
371C2×C3×C4C2×#4111yescharacteristic
372C2×C3×C4H(5,𝔽2)11yescharacteristic
373C2×C3×C4H(5,𝔽2)11yescharacteristic
374C2×Dic3—21no—
375C2×Dic3—21no—
376C2×Dic3—22no—
377C2×Dic3—21no—
378C2×Dic3—21no—
379C2×Dic3—21no—
380C2×Dic3—21no—
381C2×Dic3—21no—
382C2×Dic3—21no—
383C2×Dic3—21no—
384C2×Dic3—21no—
385C2×Dic3—21no—
386C2×Dic3—21no—
387C2×Dic3—41no—
388C2×Dic3—41no—
389C2×Dic3—41no—
390C2×Dic3—41no—
391C2×Dic3—41no—
392C2×Dic3—41no—
393C2×Dic3—41no—
394C2×Dic3—41no—
395C2×Dic3—41no—
396C2×Dic3—41no—
397C2×Dic3—41no—
398C2×Dic3—84no—
399C2×Dic3—84no—
400Dic6—84no—
401C3×C8—41no—
402C3×C8—41no—
403C3⋊C8—41no—
404C3⋊C8—41no—
405C24—21no—
406C24—21no—
407C2×Q8—64no—
408C2×Q8—241no—
409C2×Q8—241no—
410C2×D4—24no—
411C2×D4—24no—
412C2×D4—21no—
413C2×D4—21no—
414C2×D4—21no—
415C2×D4—21no—
416C2×D4—21no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4—81no—
423C2×D4—81no—
424C2×D4—81no—
425C2×D4—81no—
426C2×D4C2×Dic612yes—
427C2×D4C2×Dic612yes—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—31no—
431C22×C4—31no—
432C22×C4—31no—
433C22×C4—31no—
434C22×C4—61no—
435C22×C4—61no—
436C22×C4—61no—
437C22×C4—61no—
438C22×C4—122no—
439C22×C4—122no—
440C22×C4—241no—
441C22×C4—241no—
442C22×C4—241no—
443C22×C4—241no—
444C22×C4D3×D411yescharacteristic
445C22×C4D3×D411yescharacteristic
446C22×C4C23×D311yescharacteristic
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—61no—
454C2×C8—61no—
455C2×C8—61no—
456C2×C8—61no—
457C2×C8—61no—
458C2×C8—61no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—32no—
463C4⋊C4—64no—
464C4⋊C4—61no—
465C4⋊C4—61no—
466C4⋊C4—61no—
467C4⋊C4—61no—
468C4⋊C4—61no—
469C4⋊C4—61no—
470C4⋊C4—61no—
471C4⋊C4—241no—
472C4⋊C4—241no—
473C4⋊C4#23712yes—
474C22⋊C4—122no—
475C22⋊C4—122no—
476C22⋊C4—241no—
477C22⋊C4—241no—
478C22⋊C4—241no—
479C22⋊C4—241no—
480C42—61no—
481C42—61no—
482C42—61no—
483C42—61no—
484C42—81no—
485C42—241no—
486C42C2×Dic612yes—
487C42#23712yes—
488C22×C3—42no—
489C22×C3—42no—
490C22×C3—42no—
491C22×C3—42no—
492C22×C3—42no—
493C22×C3—42no—
494C22×C3—81no—
495C22×C3—81no—
496C22×C3—81no—
497C22×C3—81no—
498C22×C3—81no—
499C22×C3—81no—
500C22×C3—81no—
501C22×C3—81no—
502C22×C3C2×C22⋊Q811yescharacteristic
503C22×C3#52511yescharacteristic
504C22×C3#53511yescharacteristic
505C22×C3#57611yescharacteristic
506C22×C3#57611yescharacteristic
507C22×C3#57711yescharacteristic
508C22×C3#57711yescharacteristic
509C3×C4—22no—
510C3×C4—22no—
511C3×C4—42no—
512C3×C4—81no—
513C3×C4#55411yescharacteristic
514C3×C4#55411yescharacteristic
515C3×C4#55911yescharacteristic
516C3×C4#55911yescharacteristic
517Dic3—41no—
518Dic3—42no—
519Dic3—41no—
520Dic3—41no—
521Dic3—41no—
522Dic3—81no—
523Dic3—81no—
524Dic3—81no—
525C23—22no—
526C23—22no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23C2×D3×D411yescharacteristic
540Q8—244no—
541D4—22no—
542D4—22no—
543D4—22no—
544D4—22no—
545D4—41no—
546D4—41no—
547D4—41no—
548D4—41no—
549D4—84no—
550D4—84no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—61no—
562C2×C4—61no—
563C2×C4—61no—
564C2×C4—61no—
565C2×C4—61no—
566C2×C4—61no—
567C2×C4—61no—
568C2×C4—61no—
569C2×C4—61no—
570C2×C4—61no—
571C2×C4—62no—
572C2×C4—61no—
573C2×C4—61no—
574C2×C4—121no—
575C2×C4—121no—
576C2×C4—121no—
577C2×C4—121no—
578C2×C4—121no—
579C2×C4—121no—
580C2×C4—121no—
581C2×C4—121no—
582C2×C4—121no—
583C2×C4—121no—
584C2×C4—121no—
585C2×C4—244no—
586C2×C4—244no—
587C2×C4#87812yes—
588C2×C4#87812yes—
589C2×C4C22×Dic611yescharacteristic
590C2×C4C2×#23711yescharacteristic
591C2×C4C2×#23711yescharacteristic
592C2×C4#100411yescharacteristic
593C2×C4D3×#4111yescharacteristic
594C2×C4#100911yescharacteristic
595C8—41no—
596C8—41no—
597C8—121no—
598C8—121no—
599C2×C3—82no—
600C2×C3—82no—
601C2×C3#258011yescharacteristic
602C2×C3#308111yescharacteristic
603C2×C3#308111yescharacteristic
604C2×C3#312211yescharacteristic
605C2×C3#312611yescharacteristic
606C2×C3#322411yescharacteristic
607C2×C3#322411yescharacteristic
608C22—42no—
609C22—42no—
610C22—42no—
611C22—42no—
612C22—42no—
613C22—42no—
614C22—81no—
615C22—81no—
616C22—81no—
617C22—81no—
618C22—81no—
619C22—81no—
620C22—81no—
621C22—81no—
622C22C2×#87811yescharacteristic
623C22#560611yescharacteristic
624C22#560711yescharacteristic
625C22#575711yescharacteristic
626C22#576211yescharacteristic
627C22#577411yescharacteristic
628C22#577511yescharacteristic
629C4—22no—
630C4—22no—
631C4—42no—
632C4—81no—
633C4—121no—
634C4—121no—
635C4—122no—
636C4—121no—
637C4—121no—
638C4—241no—
639C4—241no—
640C4—241no—
641C4#553911yescharacteristic
642C4#554311yescharacteristic
643C4#568111yescharacteristic
644C4#571311yescharacteristic
645C3#2926311yescharacteristic, complemented, Sylow
646C2—82no—
647C2—82no—
648C2#7636411yescharacteristic
649C2#7873111yescharacteristic
650C2#7891811yescharacteristic
651C2#8031711yescharacteristic
652C2#8034111yescharacteristic
653C2#8036511yescharacteristic
654C2#8039211yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1982C2
socleC23×C3C22×D4
prime radicalC2×C3×#127C22
hypercenter#1982D3
hypocenterC3#29263
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1982

Complements

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

NH ≅ G/Ntype
2. #73627C2semidirect
5. #70455C2semidirect
6. #70444C2semidirect
7. #70182C2semidirect
8. #70181C2semidirect
9. #70170C2semidirect
10. #70169C2semidirect
11. #70168C2semidirect
12. #70167C2semidirect
13. #70163C2semidirect
14. #70163C2semidirect
15. #69468C2semidirect
85. C3×#127Q8semidirect
280. #127Dic6semidirect
645. C3#29263semidirect

Maximal subgroups

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

subgroupindex
#831722
#736272
C3×#19822
#717612
#704552
#704442
#701822
#701812
#701702
#701692
#701682
#701672
#701632
#694682
#292633