Finite Group Databasebeta

← #10896370

Subgroups — #10896370

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#74153C211yesmaximal, characteristic
2#73627C211yesmaximal, characteristic, complemented
3C3×#1949C211yesmaximal, characteristic, complemented
4#70640C211yesmaximal, characteristic, complemented
5#70638C211yesmaximal, characteristic, complemented
6#70586C211yesmaximal, characteristic, complemented
7#70584C211yesmaximal, characteristic, complemented
8#70457C211yesmaximal, characteristic
9#70444C211yesmaximal, characteristic, complemented
10#69540C211yesmaximal, characteristic, complemented
11#69487C211yesmaximal, characteristic
12#68898C211yesmaximal, characteristic, complemented
13#68898C211yesmaximal, characteristic, complemented
14#68895C211yesmaximal, characteristic, complemented
15#68894C211yesmaximal, characteristic, complemented
16#29263—31nomaximal, Sylow
17C2×#893C2211yescharacteristic
18C2×#893C2211yescharacteristic
19C2×#888C2211yescharacteristic
20C2×#874C2211yescharacteristic
21C2×C3×#107C2211yescharacteristic
22C2×C3×#106C2211yescharacteristic
23C2×C3×C4⋊C8C2211yescharacteristic
24C3×#382C2211yescharacteristic
25C3×C4×C4⋊C4C2211yescharacteristic
26C2×#821C2211yescharacteristic
27C2×#821C2211yescharacteristic
28C2×#821C2211yescharacteristic
29C2×#821C2211yescharacteristic
30#4988C2211yescharacteristic
31#4988C2211yescharacteristic
32#4984C2211yescharacteristic
33#4977C2211yescharacteristic
34#4972C2211yescharacteristic
35C2×#809C2211yescharacteristic
36C2×#809C2211yescharacteristic
37C2×#809C2211yescharacteristic
38C2×#809C2211yescharacteristic
39C2×#808C2211yescharacteristic
40C2×#807C2211yescharacteristic
41C4×C4⋊Dic3C2211yescharacteristic
42#4933—21no—
43#4932C2211yescharacteristic
44#4929—21no—
45C2×C4⋊(C3⋊C8)C2211yescharacteristic
46#4665—21no—
47#4665—21no—
48#4665—21no—
49#4665—21no—
50#4655—21no—
51#4646—21no—
52C3×#340C2211yescharacteristic
53C3×#340C2211yescharacteristic
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#1982—31no—
68#1949—31no—
69#1949D311yescharacteristic, complemented
70#1864—31no—
71#1864—31no—
72#1862—31no—
73#1862—31no—
74#1832—31no—
75C22×D12—21no—
76C22×D12—21no—
77C22×D12C2311yescharacteristic
78C22×D12C2311yescharacteristic
79C22×C4×D3—21no—
80C22×C4×D3—21no—
81C22×Dic6—21no—
82C22×C3×C8C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C42—21no—
89C2×C3×C42C2311yescharacteristic
90C2×(C2×D3)⋊C4—21no—
91C2×(C2×D3)⋊C4—21no—
92C2×(C2×D3)⋊C4—21no—
93C2×(C2×D3)⋊C4—21no—
94C2×(C2×D3)⋊C4—21no—
95C2×(C2×D3)⋊C4—21no—
96C2×C4⋊Dic3—21no—
97C2×C4⋊Dic3C2311yescharacteristic
98C2×C4⋊Dic3C2311yescharacteristic
99C2×Dic3⋊C4—21no—
100C2×Dic3⋊C4—21no—
101C2×C4×Dic3—21no—
102C2×C4×Dic3C2311yescharacteristic
103C2×C4×Dic3C2311yescharacteristic
104C22×C3⋊C8C2311yescharacteristic
105C22×C3⋊C8C2311yescharacteristic
106#893—41no—
107#893—41no—
108#893—41no—
109#893—41no—
110#888—22no—
111#888—22no—
112#874—22no—
113#874Q814yescomplemented
114C3×#107—41no—
115C3×#106—41no—
116C3×C4⋊C8—22no—
117C3×#95—21no—
118C3×#95—21no—
119C3×#95—21no—
120C3⋊#95—21no—
121C3⋊#95—21no—
122C3⋊#95—21no—
123C3⋊#95—21no—
124C3⋊#95—21no—
125C3⋊#95D411yescharacteristic
126C3⋊#95D411yescharacteristic
127C3⋊#95D411yescharacteristic
128C3⋊#95D411yescharacteristic
129#821—22no—
130#821—22no—
131#821—41no—
132#821—41no—
133#809—22no—
134#809—22no—
135#809—41no—
136#809—41no—
137#808—41no—
138#807—41no—
139C4⋊(C3⋊C8)—22no—
140C2×C4⋊Q8—31no—
141C2×#127—31no—
142C2×C4×D4—31no—
143C2×C4×D4—31no—
144C2×#107—31no—
145C2×#107C2×D311yescharacteristic
146C2×#106—31no—
147C2×#106C2×D311yescharacteristic
148C2×C4⋊C8—31no—
149C2×C4⋊C8C2×D311yescharacteristic
150C2×#102—31no—
151C2×#102—31no—
152C2×#102—31no—
153C2×#102—31no—
154C2×#102—31no—
155C2×#102—31no—
156C2×#102—31no—
157C2×#102—31no—
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#382C2×D311yescharacteristic
172C4×C4⋊C4—31no—
173C4×C4⋊C4C2×D311yescharacteristic
174#376—31no—
175#340—31no—
176#340—31no—
177#340C2×D311yescharacteristic
178#340C2×D311yescharacteristic
179#336—31no—
180#336—31no—
181#336—31no—
182#336—31no—
183C23×D3—21no—
184C23×D3—21no—
185C22×C3×C4—21no—
186C22×C3×C4—21no—
187C22×C3×C4—21no—
188C22×C3×C4—21no—
189C22×C3×C4C2×D411yescharacteristic
190C22×C3×C4C2×D411yescharacteristic
191C22×C3×C4C2411yescharacteristic
192C22×Dic3—21no—
193C22×Dic3—21no—
194C22×Dic3C2×D411yescharacteristic
195C22×Dic3C2×D411yescharacteristic
196C22×Dic3C2×D411yescharacteristic
197C22×Dic3C2×D411yescharacteristic
198C2×D12—21no—
199C2×D12—21no—
200C2×D12—21no—
201C2×D12—21no—
202C2×D12—21no—
203C2×D12—21no—
204C2×D12—21no—
205C2×D12—21no—
206C2×D12—21no—
207C2×D12—21no—
208C2×D12—24no—
209C2×D12—24no—
210C2×D12—81no—
211C2×D12—81no—
212C2×D12—81no—
213C2×D12—81no—
214C2×D12C2×Q812yes—
215C2×D12C2×Q812yes—
216C2×C4×D3—42no—
217C2×C4×D3—42no—
218C2×C4×D3—81no—
219C2×C4×D3—81no—
220C2×C4×D3—81no—
221C2×C4×D3—81no—
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—21no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4#4112yes—
239C3×C42—81no—
240C3×C42C2×Q812yes—
241C3×C42#4112yes—
242(C2×D3)⋊C4—42no—
243(C2×D3)⋊C4—42no—
244(C2×D3)⋊C4—81no—
245(C2×D3)⋊C4—81no—
246(C2×D3)⋊C4—81no—
247(C2×D3)⋊C4—81no—
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—31no—
268C22×D4—31no—
269C22×D4—61no—
270C22×D4—61no—
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—34no—
280#127—62no—
281C4×D4—121no—
282C4×D4—121no—
283C4×D4—121no—
284C4×D4—121no—
285C2×C4⋊C4—31no—
286C2×C4⋊C4—31no—
287C2×C4⋊C4—61no—
288C2×C4⋊C4—61no—
289C2×C4⋊C4—61no—
290C2×C4⋊C4C22×D311yescharacteristic
291C2×C4⋊C4C22×D311yescharacteristic
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—41no—
306#107—121no—
307#106—41no—
308#106—121no—
309C4⋊C8—22no—
310C4⋊C8—62no—
311#102—62no—
312#102—62no—
313#102—62no—
314#102—62no—
315#102—121no—
316#102—121no—
317#102—121no—
318#102—121no—
319#95—21no—
320#95—21no—
321#95—21no—
322#95—31no—
323#95—31no—
324#95—31no—
325#95—31no—
326#95—61no—
327#95—61no—
328#95—61no—
329#95—61no—
330#95—61no—
331C23×C3C22×D411yescharacteristic
332C22×D3—22no—
333C22×D3—22no—
334C22×D3—41no—
335C22×D3—41no—
336C22×D3—41no—
337C22×D3—41no—
338C22×D3—41no—
339C22×D3—41no—
340C22×D3—41no—
341C22×D3—41no—
342C22×D3—41no—
343C22×D3—41no—
344C22×D3—41no—
345C22×D3—41no—
346C2×C3×C4—21no—
347C2×C3×C4—21no—
348C2×C3×C4—21no—
349C2×C3×C4—21no—
350C2×C3×C4—22no—
351C2×C3×C4—41no—
352C2×C3×C4—41no—
353C2×C3×C4—41no—
354C2×C3×C4—41no—
355C2×C3×C4—41no—
356C2×C3×C4—41no—
357C2×C3×C4—41no—
358C2×C3×C4—41no—
359C2×C3×C4—41no—
360C2×C3×C4—41no—
361C2×C3×C4—41no—
362C2×C3×C4C22⋊Q812yes—
363C2×C3×C4C22⋊Q812yes—
364C2×C3×C4C22×Q811yescharacteristic
365C2×C3×C4C2×#4111yescharacteristic
366C2×C3×C4C2×#4111yescharacteristic
367C2×C3×C4C2×#4111yescharacteristic
368C2×C3×C4H(5,𝔽2)11yescharacteristic
369C2×C3×C4H(5,𝔽2)11yescharacteristic
370C2×Dic3—21no—
371C2×Dic3—21no—
372C2×Dic3—22no—
373C2×Dic3—21no—
374C2×Dic3—21no—
375C2×Dic3—21no—
376C2×Dic3—21no—
377C2×Dic3—21no—
378C2×Dic3—21no—
379C2×Dic3—21no—
380C2×Dic3—21no—
381C2×Dic3—21no—
382C2×Dic3—21no—
383C2×Dic3—41no—
384C2×Dic3—41no—
385C2×Dic3—41no—
386C2×Dic3—41no—
387C2×Dic3—41no—
388D12—22no—
389D12—22no—
390D12—22no—
391D12—22no—
392D12—41no—
393D12—41no—
394D12—41no—
395D12—41no—
396D12—84no—
397D12—84no—
398C4×D3—84no—
399C4×D3—84no—
400Dic6—84no—
401C3×C8—41no—
402C3×C8—41no—
403C3⋊C8—41no—
404C3⋊C8—41no—
405C24—61no—
406C24—61no—
407C2×Q8—64no—
408C2×Q8—241no—
409C2×Q8—241no—
410C2×D4—32no—
411C2×D4—32no—
412C2×D4—64no—
413C2×D4—64no—
414C2×D4—61no—
415C2×D4—61no—
416C2×D4—61no—
417C2×D4—61no—
418C2×D4—61no—
419C2×D4—61no—
420C2×D4—61no—
421C2×D4—61no—
422C2×D4—61no—
423C2×D4—61no—
424C2×D4—241no—
425C2×D4—241no—
426C2×D4—241no—
427C2×D4—241no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—31no—
433C22×C4—31no—
434C22×C4—31no—
435C22×C4—31no—
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—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—32no—
467C4⋊C4—64no—
468C4⋊C4—61no—
469C4⋊C4—61no—
470C4⋊C4—61no—
471C4⋊C4—241no—
472C4⋊C4—241no—
473C4⋊C4#23912yes—
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—
486C42#23512yes—
487C42D3×Q812yes—
488C22×C3C2×C22⋊Q811yescharacteristic
489C22×C3#52511yescharacteristic
490C22×C3#53511yescharacteristic
491C22×C3#57611yescharacteristic
492C22×C3#57611yescharacteristic
493C22×C3#57711yescharacteristic
494C22×C3#57711yescharacteristic
495C2×D3—42no—
496C2×D3—42no—
497C2×D3—42no—
498C2×D3—42no—
499C2×D3—42no—
500C2×D3—42no—
501C2×D3—81no—
502C2×D3—81no—
503C2×D3—81no—
504C2×D3—81no—
505C2×D3—81no—
506C2×D3—81no—
507C2×D3—81no—
508C2×D3—81no—
509C3×C4—22no—
510C3×C4—22no—
511C3×C4—42no—
512C3×C4—81no—
513C3×C4—81no—
514C3×C4—81no—
515C3×C4#55411yescharacteristic
516C3×C4#55411yescharacteristic
517C3×C4#55911yescharacteristic
518C3×C4#55911yescharacteristic
519Dic3—41no—
520Dic3—42no—
521Dic3—41no—
522Dic3—41no—
523Dic3—41no—
524Dic3—81no—
525C23—62no—
526C23—62no—
527C23—121no—
528C23—121no—
529C23—121no—
530C23—121no—
531C23—121no—
532C23—121no—
533C23—121no—
534C23—121no—
535C23—121no—
536C23—121no—
537C23—121no—
538C23—121no—
539C23C2×D3×D411yescharacteristic
540Q8—244no—
541D4—62no—
542D4—62no—
543D4—62no—
544D4—62no—
545D4—121no—
546D4—121no—
547D4—121no—
548D4—121no—
549D4—244no—
550D4—244no—
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—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—61no—
568C2×C4—61no—
569C2×C4—61no—
570C2×C4—61no—
571C2×C4—61no—
572C2×C4—61no—
573C2×C4—61no—
574C2×C4—61no—
575C2×C4—61no—
576C2×C4—61no—
577C2×C4—61no—
578C2×C4—61no—
579C2×C4—62no—
580C2×C4—121no—
581C2×C4—121no—
582C2×C4—121no—
583C2×C4—121no—
584C2×C4—121no—
585C2×C4—244no—
586C2×C4—244no—
587C2×C4#89612yes—
588C2×C4#89612yes—
589C2×C4C2×#23511yescharacteristic
590C2×C4#100411yescharacteristic
591C2×C4C2×D3×Q811yescharacteristic
592C2×C4C2×#23911yescharacteristic
593C2×C4D3×#4111yescharacteristic
594C2×C4#100911yescharacteristic
595C8—41no—
596C8—41no—
597C8—121no—
598C8—121no—
599C2×C3#258011yescharacteristic
600C2×C3#308111yescharacteristic
601C2×C3#308111yescharacteristic
602C2×C3#312211yescharacteristic
603C2×C3#312611yescharacteristic
604C2×C3#322411yescharacteristic
605C2×C3#322411yescharacteristic
606D3—82no—
607D3—82no—
608C22—122no—
609C22—122no—
610C22—122no—
611C22—122no—
612C22—122no—
613C22—122no—
614C22—241no—
615C22—241no—
616C22—241no—
617C22—241no—
618C22—241no—
619C22—241no—
620C22—241no—
621C22—241no—
622C22C2×#89611yescharacteristic
623C22#560711yescharacteristic
624C22#562711yescharacteristic
625C22#576911yescharacteristic
626C22#577011yescharacteristic
627C22#577411yescharacteristic
628C22#577811yescharacteristic
629C4—22no—
630C4—22no—
631C4—42no—
632C4—81no—
633C4—81no—
634C4—81no—
635C4—121no—
636C4—121no—
637C4—121no—
638C4—121no—
639C4—122no—
640C4—241no—
641C4#555911yescharacteristic
642C4#558511yescharacteristic
643C4#569011yescharacteristic
644C4#573011yescharacteristic
645C3#2926311yescharacteristic, complemented, Sylow
646C2—242no—
647C2—242no—
648C2#7607511yescharacteristic
649C2#7919711yescharacteristic
650C2#7924211yescharacteristic
651C2#8054611yescharacteristic
652C2#8061011yescharacteristic
653C2#8114611yescharacteristic
654C2#8121011yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1949C2
socleC23×C3C22×D4
prime radicalC2×#874C22
hypercenter#1949D3
hypocenterC3#29263
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1949

Complements

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

NH ≅ G/Ntype
2. #73627C2semidirect
3. C3×#1949C2semidirect
4. #70640C2semidirect
5. #70638C2semidirect
6. #70586C2semidirect
7. #70584C2semidirect
9. #70444C2semidirect
10. #69540C2semidirect
12. #68898C2semidirect
13. #68898C2semidirect
14. #68895C2semidirect
15. #68894C2semidirect
69. #1949D3semidirect
113. #874Q8semidirect
645. C3#29263semidirect

Maximal subgroups

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

subgroupindex
#741532
#736272
C3×#19492
#706402
#706382
#705862
#705842
#704572
#704442
#695402
#694872
#688982
#688952
#688942
#292633