Finite Group Databasebeta

← #10877314

Subgroups — #10877314

Order 768.

788 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82236C211yesmaximal, characteristic
2#82233C211yesmaximal, characteristic, complemented
3C3×#1798C211yesmaximal, characteristic, complemented
4#70480C211yesmaximal, characteristic, complemented
5#70479C211yesmaximal, characteristic, complemented
6#70455C211yesmaximal, characteristic, complemented
7#70454C211yesmaximal, characteristic, complemented
8#69933C211yesmaximal, characteristic
9#69932C211yesmaximal, characteristic
10#69929C211yesmaximal, characteristic, complemented
11#69928C211yesmaximal, characteristic, complemented
12#68682C211yesmaximal, characteristic, complemented
13#68680C211yesmaximal, characteristic, complemented
14#68644C211yesmaximal, characteristic, complemented
15#68642C211yesmaximal, characteristic, complemented
16#27688—31nomaximal, Sylow
17C2×C3×#117C2211yescharacteristic
18C2×#930C2211yescharacteristic
19C2×#926C2211yescharacteristic
20C22×C4⋊Dic3C2211yescharacteristic
21C2×C4×D12C2211yescharacteristic
22C2×C3×C4⋊C8C2211yescharacteristic
23C2×C3×C4⋊C8C2211yescharacteristic
24C2×C3×C22⋊C8C2211yescharacteristic
25C2×C3×C22⋊C8C2211yescharacteristic
26#5212C2211yescharacteristic
27C2×#821C2211yescharacteristic
28C2×#821C2211yescharacteristic
29C2×#821C2211yescharacteristic
30C2×#821C2211yescharacteristic
31C2×#818C2211yescharacteristic
32C2×#817C2211yescharacteristic
33C2×#817C2211yescharacteristic
34C2×#817C2211yescharacteristic
35#4990—21no—
36#4989—21no—
37#4984—21no—
38#4984—21no—
39#4957—21no—
40#4956—21no—
41#4950—21no—
42#4950—21no—
43#4939C2211yescharacteristic
44#4936C2211yescharacteristic
45#4936C2211yescharacteristic
46#4935C2211yescharacteristic
47#4935C2211yescharacteristic
48C4×C4⋊Dic3C2211yescharacteristic
49#4928C2211yescharacteristic
50C3×#336C2211yescharacteristic
51C3×#336C2211yescharacteristic
52#4551C2211yescharacteristic
53#4551C2211yescharacteristic
54#4551C2211yescharacteristic
55#4551C2211yescharacteristic
56#4551C2211yescharacteristic
57#4551C2211yescharacteristic
58#4551C2211yescharacteristic
59#4551C2211yescharacteristic
60#2324—31no—
61#2317—31no—
62#2101—31no—
63#2099—31no—
64#2094—31no—
65#2092—31no—
66#1949—31no—
67#1948—31no—
68#1875—31no—
69#1874—31no—
70#1854—31no—
71#1853—31no—
72#1804—31no—
73#1802—31no—
74#1798D311yescharacteristic, complemented
75C23×C3×C4C2311yescharacteristic
76C22×#67—21no—
77C23×Dic3—21no—
78C22×D12C2311yescharacteristic
79C22×C4×D3—21no—
80C22×C3×C8C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C3×#117—22no—
85C3×#117—22no—
86C2×C3×C4⋊C4D411yescharacteristic
87C2×C3×C4⋊C4D411yescharacteristic
88C2×C3×C22⋊C4D411yescharacteristic
89C2×C3×C22⋊C4D411yescharacteristic
90C2×C3×C42C2311yescharacteristic
91C2×C3×C42C2311yescharacteristic
92C2×#217—21no—
93C2×#217—21no—
94#930—22no—
95#930Q814yes—
96C2×(C2×D3)⋊C4—21no—
97C2×(C2×D3)⋊C4—21no—
98C2×(C2×D3)⋊C4—21no—
99C2×(C2×D3)⋊C4—21no—
100#926—41no—
101#926—41no—
102C2×C4⋊Dic3—21no—
103C2×C4⋊Dic3—21no—
104C2×C4⋊Dic3—21no—
105C2×C4⋊Dic3—21no—
106C2×C4⋊Dic3—21no—
107C2×C4⋊Dic3—41no—
108C2×C4⋊Dic3—41no—
109C2×C4⋊Dic3C2311yescharacteristic
110C2×C4⋊Dic3C2311yescharacteristic
111C2×C4⋊Dic3C2311yescharacteristic
112C2×C4⋊Dic3C2311yescharacteristic
113C2×C4⋊Dic3C2311yescharacteristic
114C2×C4⋊Dic3C2311yescharacteristic
115C2×C4⋊Dic3C2311yescharacteristic
116C2×Dic3⋊C4—21no—
117C2×C4×Dic3—21no—
118C2×C4×Dic3—21no—
119C4×D12—22no—
120C4×D12—22no—
121C3×C4⋊C8—22no—
122C3×C4⋊C8—22no—
123C3×C22⋊C8—22no—
124C3×C22⋊C8—22no—
125C3⋊#95—21no—
126C3⋊#95—21no—
127C3⋊#95—21no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131C3⋊#95—21no—
132#821—41no—
133#821—41no—
134#821—41no—
135#821—41no—
136#818—41no—
137#817—41no—
138#817—41no—
139#817—41no—
140C2×#121—31no—
141C2×C4×D4—31no—
142C2×#117—31no—
143C2×#117C2×D311yescharacteristic
144C22×C4⋊C4—31no—
145C2×#107—31no—
146C2×#106—31no—
147C2×#106—31no—
148C2×#106—31no—
149C2×C4⋊C8C2×D311yescharacteristic
150C2×C4⋊C8C2×D311yescharacteristic
151C2×#102—31no—
152C2×#102—31no—
153C2×#102—31no—
154C2×#102—31no—
155C2×C22⋊C8C2×D311yescharacteristic
156C2×C22⋊C8C2×D311yescharacteristic
157#390—31no—
158#390—61no—
159#389—31no—
160#389—61no—
161#389—61no—
162#388—61no—
163#386—61no—
164#385—61no—
165#385—61no—
166#385—61no—
167#384—31no—
168#384—31no—
169#382—31no—
170#382—31no—
171#380—31no—
172C4×C4⋊C4—31no—
173#340—31no—
174#340—31no—
175#340—31no—
176#340—31no—
177#340—31no—
178#340—31no—
179#340—31no—
180#340—31no—
181#336C2×D311yescharacteristic
182#336C2×D311yescharacteristic
183C24×C3C2×D411yescharacteristic
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×C4C2×D411yescharacteristic
192C22×C3×C4C2×D411yescharacteristic
193C22×C3×C4C2×D411yescharacteristic
194C22×C3×C4C2×Q812yes—
195C22×C3×C4#4112yes—
196C22×C3×C4C2411yescharacteristic
197C2×#67—24no—
198C2×#67—81no—
199C2×#67—81no—
200C22×Dic3—21no—
201C22×Dic3—21no—
202C22×Dic3—21no—
203C22×Dic3—21no—
204C22×Dic3—21no—
205C22×Dic3—21no—
206C22×Dic3—21no—
207C22×Dic3—42no—
208C22×Dic3—81no—
209C22×Dic3—81no—
210C2×D12—21no—
211C2×D12—21no—
212C2×D12—21no—
213C2×D12—21no—
214C2×D12—21no—
215C2×D12C2×Q812yes—
216C2×C4×D3—42no—
217C2×C4×D3—81no—
218C2×C4×D3—81no—
219C2×C3×C8—21no—
220C2×C3×C8—21no—
221C2×C3×C8—21no—
222C2×C3×C8—21no—
223C2×C3×C8—21no—
224C2×C3×C8—21no—
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—22no—
232C3×C4⋊C4—22no—
233C3×C22⋊C4—22no—
234C3×C22⋊C4—22no—
235C3×C42—21no—
236C3×C42—21no—
237C3×C42#4112yes—
238C3×C42#4112yes—
239#217—42no—
240#217—81no—
241(C2×D3)⋊C4—24no—
242(C2×D3)⋊C4—42no—
243(C2×D3)⋊C4—81no—
244(C2×D3)⋊C4—81no—
245C4⋊Dic3—21no—
246C4⋊Dic3—21no—
247C4⋊Dic3—21no—
248C4⋊Dic3—21no—
249C4⋊Dic3—21no—
250C4⋊Dic3—21no—
251C4⋊Dic3—21no—
252C4⋊Dic3—21no—
253C4⋊Dic3—21no—
254C4⋊Dic3—21no—
255C4⋊Dic3—21no—
256C4⋊Dic3—21no—
257C4⋊Dic3—21no—
258C4⋊Dic3—42no—
259C4⋊Dic3—81no—
260C4⋊Dic3—81no—
261C4⋊Dic3—81no—
262C4⋊Dic3—81no—
263C4⋊Dic3C2×Q812yes—
264Dic3⋊C4—81no—
265C4×Dic3—42no—
266C4×Dic3—81no—
267C22×D4—31no—
268C22×D4—61no—
269C23×C4—61no—
270C23×C4—61no—
271C23×C4C22×D311yescharacteristic
272C22×C8C22×D311yescharacteristic
273C22×C8C22×D311yescharacteristic
274C22×C8C22×D311yescharacteristic
275C22×C8C22×D311yescharacteristic
276#121—34no—
277#121—62no—
278C4×D4—62no—
279C4×D4—62no—
280#117—22no—
281#117—22no—
282#117—121no—
283#117—121no—
284C2×C4⋊C4—31no—
285C2×C4⋊C4—31no—
286C2×C4⋊C4—31no—
287C2×C4⋊C4—31no—
288C2×C4⋊C4—31no—
289C2×C4⋊C4—31no—
290C2×C4⋊C4—31no—
291C2×C4⋊C4—61no—
292C2×C4⋊C4—61no—
293C2×C4⋊C4—61no—
294C2×C4⋊C4—61no—
295C2×C4⋊C4—61no—
296C2×C4⋊C4—61no—
297C2×C4⋊C4—121no—
298C2×C4⋊C4—121no—
299C2×C4⋊C4D1211yescharacteristic
300C2×C4⋊C4D1211yescharacteristic
301C2×C22⋊C4—61no—
302C2×C22⋊C4—61no—
303C2×C22⋊C4—61no—
304C2×C22⋊C4—61no—
305C2×C22⋊C4—61no—
306C2×C22⋊C4—61no—
307C2×C22⋊C4D1211yescharacteristic
308C2×C22⋊C4D1211yescharacteristic
309C2×C42—61no—
310C2×C42—61no—
311C2×C42C22×D311yescharacteristic
312C2×C42C22×D311yescharacteristic
313#107—121no—
314#106—121no—
315#106—121no—
316#106—121no—
317C4⋊C8—22no—
318C4⋊C8—22no—
319#102—121no—
320#102—121no—
321#102—121no—
322#102—121no—
323C22⋊C8—22no—
324C22⋊C8—22no—
325#95—61no—
326#95—61no—
327#95—61no—
328#95—61no—
329#95—61no—
330#95—61no—
331#95—61no—
332C23×C3—21no—
333C23×C3—21no—
334C23×C3—21no—
335C23×C3—21no—
336C23×C3—21no—
337C23×C3—21no—
338C23×C3C22⋊Q812yes—
339C23×C3C22×D411yescharacteristic
340C22×D3—22no—
341C22×D3—41no—
342C22×D3—41no—
343C22×D3—41no—
344C22×D3—41no—
345C22×D3—41no—
346C22×D3—41no—
347C2×C3×C4—21no—
348C2×C3×C4—21no—
349C2×C3×C4—21no—
350C2×C3×C4—21no—
351C2×C3×C4—21no—
352C2×C3×C4—21no—
353C2×C3×C4—22no—
354C2×C3×C4—22no—
355C2×C3×C4—22no—
356C2×C3×C4—22no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—21no—
361C2×C3×C4—21no—
362C2×C3×C4—21no—
363C2×C3×C4C22⋊Q812yes—
364C2×C3×C4#12312yes—
365C2×C3×C4#12312yes—
366C2×C3×C4C22×Q811yescharacteristic
367C2×C3×C4C2×#4111yescharacteristic
368C2×C3×C4C2×#4111yescharacteristic
369C2×C3×C4C2×#4111yescharacteristic
370C2×C3×C4H(5,𝔽2)11yescharacteristic
371C2×C3×C4H(5,𝔽2)11yescharacteristic
372#67—84no—
373C2×Dic3—22no—
374C2×Dic3—41no—
375C2×Dic3—41no—
376C2×Dic3—41no—
377C2×Dic3—41no—
378C2×Dic3—41no—
379C2×Dic3—41no—
380C2×Dic3—41no—
381C2×Dic3—41no—
382C2×Dic3—41no—
383C2×Dic3—41no—
384C2×Dic3—41no—
385C2×Dic3—41no—
386C2×Dic3—41no—
387C2×Dic3—41no—
388C2×Dic3—41no—
389C2×Dic3—41no—
390C2×Dic3—41no—
391C2×Dic3—41no—
392C2×Dic3—41no—
393C2×Dic3—41no—
394C2×Dic3—84no—
395D12—22no—
396D12—22no—
397D12—41no—
398D12—41no—
399C4×D3—84no—
400C3×C8—41no—
401C3×C8—41no—
402C3×C8—41no—
403C3×C8—41no—
404C24—61no—
405C24C2×D1211yescharacteristic
406C2×D4—32no—
407C2×D4—64no—
408C2×D4—61no—
409C2×D4—61no—
410C2×D4—61no—
411C2×D4—61no—
412C2×D4—61no—
413C2×D4—241no—
414C2×D4—241no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—61no—
420C22×C4—61no—
421C22×C4—61no—
422C22×C4—61no—
423C22×C4—61no—
424C22×C4—61no—
425C22×C4—61no—
426C22×C4—122no—
427C22×C4—122no—
428C22×C4—241no—
429C22×C4—241no—
430C22×C4—241no—
431C22×C4—241no—
432C22×C4C2×D1211yescharacteristic
433C22×C4C2×D1211yescharacteristic
434C22×C4C2×D1211yescharacteristic
435C22×C4C2×D1211yescharacteristic
436C22×C4C2×D1211yescharacteristic
437C22×C4#23712yes—
438C22×C4D3×Q812yes—
439C22×C4C23×D311yescharacteristic
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—21no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—32no—
455C4⋊C4—61no—
456C4⋊C4—61no—
457C4⋊C4—61no—
458C4⋊C4—61no—
459C4⋊C4—61no—
460C4⋊C4—61no—
461C4⋊C4—61no—
462C4⋊C4—61no—
463C4⋊C4—61no—
464C4⋊C4—61no—
465C4⋊C4—61no—
466C4⋊C4—61no—
467C4⋊C4—61no—
468C4⋊C4—122no—
469C4⋊C4—241no—
470C4⋊C4—241no—
471C4⋊C4—241no—
472C4⋊C4—241no—
473C4⋊C4—241no—
474C22⋊C4—22no—
475C22⋊C4—22no—
476C22⋊C4—64no—
477C22⋊C4—122no—
478C22⋊C4—122no—
479C22⋊C4—241no—
480C22⋊C4—241no—
481C22⋊C4—241no—
482C42—21no—
483C42—21no—
484C42—122no—
485C42—241no—
486C42#23712yes—
487C42#23712yes—
488C22×C3—22no—
489C22×C3—22no—
490C22×C3—22no—
491C22×C3—41no—
492C22×C3—41no—
493C22×C3—41no—
494C22×C3—41no—
495C22×C3C2×C22⋊Q811yescharacteristic
496C22×C3C2×#12311yescharacteristic
497C22×C3#53511yescharacteristic
498C22×C3#57611yescharacteristic
499C22×C3#57711yescharacteristic
500C22×C3#57711yescharacteristic
501C22×C3#57711yescharacteristic
502C2×D3—42no—
503C2×D3—42no—
504C2×D3—42no—
505C2×D3—81no—
506C2×D3—81no—
507C2×D3—81no—
508C2×D3—81no—
509C3×C4—22no—
510C3×C4—22no—
511C3×C4—22no—
512C3×C4—41no—
513C3×C4—41no—
514C3×C4#55311yescharacteristic
515C3×C4#55311yescharacteristic
516C3×C4#55411yescharacteristic
517C3×C4#55411yescharacteristic
518Dic3—42no—
519Dic3—81no—
520Dic3—81no—
521Dic3—81no—
522Dic3—81no—
523Dic3—81no—
524Dic3—81no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—62no—
532C23—121no—
533C23—121no—
534C23—121no—
535C23—121no—
536C23—121no—
537C23—121no—
538C23#89712yes—
539C23C22×D1211yescharacteristic
540D4—62no—
541D4—62no—
542D4—121no—
543D4—121no—
544D4—244no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—62no—
562C2×C4—121no—
563C2×C4—121no—
564C2×C4—121no—
565C2×C4—121no—
566C2×C4—121no—
567C2×C4—121no—
568C2×C4—121no—
569C2×C4—121no—
570C2×C4—121no—
571C2×C4—121no—
572C2×C4—121no—
573C2×C4—121no—
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—244no—
583C2×C4—244no—
584C2×C4#88612yes—
585C2×C4#88612yes—
586C2×C4#89712yes—
587C2×C4C2×#23711yescharacteristic
588C2×C4C2×#23711yescharacteristic
589C2×C4C2×#23711yescharacteristic
590C2×C4C2×D3×Q811yescharacteristic
591C2×C4#100911yescharacteristic
592C2×C4#100911yescharacteristic
593C8—41no—
594C8—41no—
595C8—41no—
596C8—41no—
597C2×C3—42no—
598C2×C3#248711yescharacteristic
599C2×C3#308111yescharacteristic
600C2×C3#308411yescharacteristic
601C2×C3#309611yescharacteristic
602C2×C3#309911yescharacteristic
603C2×C3#322111yescharacteristic
604C2×C3#322411yescharacteristic
605D3—82no—
606C22—22no—
607C22—22no—
608C22—22no—
609C22—41no—
610C22—41no—
611C22—41no—
612C22—41no—
613C22—122no—
614C22—122no—
615C22—122no—
616C22—241no—
617C22—241no—
618C22—241no—
619C22—241no—
620C22C2×#88611yescharacteristic
621C22C2×#89711yescharacteristic
622C22#552511yescharacteristic
623C22#575111yescharacteristic
624C22#575111yescharacteristic
625C22#575111yescharacteristic
626C22#575211yescharacteristic
627C4—22no—
628C4—22no—
629C4—22no—
630C4—41no—
631C4—41no—
632C4—122no—
633C4—241no—
634C4—241no—
635C4—241no—
636C4—241no—
637C4—241no—
638C4—241no—
639C4#561811yescharacteristic
640C4#561811yescharacteristic
641C4#563511yescharacteristic
642C4#563511yescharacteristic
643C3#2768811yescharacteristic, complemented, Sylow
644C2—42no—
645C2—242no—
646C2#7402811yescharacteristic
647C2#7865111yescharacteristic
648C2#7865411yescharacteristic
649C2#7909011yescharacteristic
650C2#7909211yescharacteristic
651C2#7939211yescharacteristic
652C2#7939711yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1798C2
socleC23×C3C22×D4
prime radicalC2×#930C22
hypercenter#1798D3
hypocenterC3#27688
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117 ◁ #1798

Complements

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

NH ≅ G/Ntype
2. #82233C2semidirect
3. C3×#1798C2semidirect
4. #70480C2semidirect
5. #70479C2semidirect
6. #70455C2semidirect
7. #70454C2semidirect
10. #69929C2semidirect
11. #69928C2semidirect
12. #68682C2semidirect
13. #68680C2semidirect
14. #68644C2semidirect
15. #68642C2semidirect
74. #1798D3semidirect
643. C3#27688semidirect

Maximal subgroups

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

subgroupindex
#822362
#822332
C3×#17982
#704802
#704792
#704552
#704542
#699332
#699322
#699292
#699282
#686822
#686802
#686442
#686422
#276883