Finite Group Databasebeta

← #10876202

Subgroups — #10876202

Order 768.

872 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82070C211yesmaximal, characteristic
2#82060C211yesmaximal, characteristic, complemented
3C3×#1817C211yesmaximal, characteristic, complemented
4#69960C211yesmaximal, characteristic
5#69960C211yesmaximal, characteristic
6#69929C211yesmaximal, characteristic, complemented
7#69929C211yesmaximal, characteristic, complemented
8#68693C211yesmaximal, characteristic, complemented
9#68680C211yesmaximal, characteristic, complemented
10#68638C211yesmaximal, characteristic, complemented
11#68637C211yesmaximal, characteristic, complemented
12#68628C211yesmaximal, characteristic, complemented
13#68627C211yesmaximal, characteristic, complemented
14#68626C211yesmaximal, characteristic, complemented
15#68625C211yesmaximal, characteristic, complemented
16#27590—31nomaximal, Sylow
17C22×#235C2211yescharacteristic
18C22×C3×C42C2211yescharacteristic
19C2×#926C2211yescharacteristic
20C2×C3×C22⋊C8C2211yescharacteristic
21C2×C3×C22⋊C8C2211yescharacteristic
22#5213—21no—
23#5213—21no—
24#5212C2211yescharacteristic
25#5212C2211yescharacteristic
26#5211—21no—
27C4×#217—21no—
28C2×#821C2211yescharacteristic
29C2×#821C2211yescharacteristic
30C2×#821C2211yescharacteristic
31C2×#821C2211yescharacteristic
32C2×#818C2211yescharacteristic
33C2×#818C2211yescharacteristic
34C2×#817C2211yescharacteristic
35C2×#817C2211yescharacteristic
36C2×#816C2211yescharacteristic
37C2×#816C2211yescharacteristic
38C2×#816C2211yescharacteristic
39C2×#816C2211yescharacteristic
40#4939C2211yescharacteristic
41#4939C2211yescharacteristic
42C4×(C2×D3)⋊C4—21no—
43C4×(C2×D3)⋊C4—21no—
44#4937—21no—
45#4936C2211yescharacteristic
46#4936C2211yescharacteristic
47C4×C4⋊Dic3C2211yescharacteristic
48C4×C4⋊Dic3C2211yescharacteristic
49#4932—21no—
50#4929C2211yescharacteristic
51#4929C2211yescharacteristic
52C3×#336C2211yescharacteristic
53C3×#336C2211yescharacteristic
54C3×#336C2211yescharacteristic
55C3×#336C2211yescharacteristic
56#4551C2211yescharacteristic
57#4551C2211yescharacteristic
58#4551C2211yescharacteristic
59#4551C2211yescharacteristic
60#2297—31no—
61#2283—31no—
62#2097—31no—
63#2092—31no—
64#2074—31no—
65#2073—31no—
66#1928—31no—
67#1927—31no—
68#1926—31no—
69#1925—31no—
70#1854—31no—
71#1854—31no—
72#1817D311yescharacteristic, complemented
73#1788—31no—
74#1788—31no—
75C23×C3×C4D411yescharacteristic
76C23×C3×C4D411yescharacteristic
77C23×C3×C4C2311yescharacteristic
78C22×#67—21no—
79C2×#235—22no—
80C2×#235—22no—
81C2×#235—22no—
82C2×#235—22no—
83C2×#235—41no—
84C2×#235—41no—
85C22×D12C2311yescharacteristic
86C22×C4×D3—21no—
87C22×Dic6C2311yescharacteristic
88C22×C3×C8C2311yescharacteristic
89C22×C3×C8C2311yescharacteristic
90C22×C3×C8C2311yescharacteristic
91C22×C3×C8C2311yescharacteristic
92C2×C3×C42—22no—
93C2×C3×C42—22no—
94C2×C3×C42D411yescharacteristic
95C2×C3×C42D411yescharacteristic
96C2×C3×C42C2311yescharacteristic
97C2×C3×C42C2311yescharacteristic
98C2×#217—21no—
99C2×#217—21no—
100C2×#217—21no—
101C2×(C2×D3)⋊C4—21no—
102C2×(C2×D3)⋊C4—21no—
103C2×(C2×D3)⋊C4—21no—
104C2×(C2×D3)⋊C4—21no—
105#926—41no—
106#926—41no—
107C2×C4⋊Dic3C2311yescharacteristic
108C2×C4⋊Dic3C2311yescharacteristic
109C2×C4⋊Dic3C2311yescharacteristic
110C2×C4⋊Dic3C2311yescharacteristic
111C2×C4⋊Dic3C2311yescharacteristic
112C2×C4⋊Dic3C2311yescharacteristic
113C2×C4×Dic3—21no—
114C2×C4×Dic3—21no—
115C2×C4×Dic3—21no—
116C3×C22⋊C8—22no—
117C3×C22⋊C8—22no—
118C3⋊#95—21no—
119C3⋊#95—21no—
120C3⋊#95—21no—
121C3⋊#95—21no—
122C3⋊#95—21no—
123C3⋊#95—21no—
124C3⋊#95—21no—
125C3⋊#95—21no—
126C3⋊#95—21no—
127C3⋊#95—21no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130#821—22no—
131#821—22no—
132#821—41no—
133#821—41no—
134#818—41no—
135#818—41no—
136#817—41no—
137#817—41no—
138#816—22no—
139#816—22no—
140#816—41no—
141#816—41no—
142C22×#41—31no—
143C2×#117—31no—
144C22×C42C2×D311yescharacteristic
145C2×#107—31no—
146C2×#107—31no—
147C2×#106—31no—
148C2×#106—31no—
149C2×#103—31no—
150C2×#103—31no—
151C2×#103—31no—
152C2×#103—31no—
153C2×#102—31no—
154C2×#102—31no—
155C2×#102—31no—
156C2×#102—31no—
157C2×C22⋊C8C2×D311yescharacteristic
158C2×C22⋊C8C2×D311yescharacteristic
159#391—31no—
160#391—31no—
161#390—31no—
162#390—31no—
163#386—61no—
164#386—61no—
165#383—61no—
166#382—31no—
167#382—31no—
168#381—61no—
169#380—31no—
170#380—31no—
171C4×C4⋊C4—31no—
172C4×C4⋊C4—31no—
173C4×C22⋊C4—61no—
174C4×C22⋊C4—61no—
175C4×C22⋊C4—61no—
176#376—61no—
177#340—31no—
178#340—31no—
179#340—31no—
180#340—31no—
181#336C2×D311yescharacteristic
182#336C2×D311yescharacteristic
183#336C2×D311yescharacteristic
184#336C2×D311yescharacteristic
185C24×C3C2×D411yescharacteristic
186C23×D3—21no—
187C22×C3×C4—21no—
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4—21no—
191C22×C3×C4—22no—
192C22×C3×C4—22no—
193C22×C3×C4—22no—
194C22×C3×C4—41no—
195C22×C3×C4—41no—
196C22×C3×C4—41no—
197C22×C3×C4C2×D411yescharacteristic
198C22×C3×C4C2×D411yescharacteristic
199C22×C3×C4C2×D411yescharacteristic
200C22×C3×C4C2×D411yescharacteristic
201C22×C3×C4C2×D411yescharacteristic
202C22×C3×C4#4112yes—
203C22×C3×C4#4112yes—
204C22×C3×C4C2411yescharacteristic
205C2×#67—24no—
206C2×#67—42no—
207C2×#67—42no—
208C22×Dic3—21no—
209C22×Dic3—21no—
210C22×Dic3—21no—
211C22×Dic3—21no—
212C22×Dic3—21no—
213C22×Dic3—21no—
214C22×Dic3—21no—
215#235—44no—
216#235—44no—
217#235—44no—
218#235—44no—
219C2×D12—21no—
220C2×D12—21no—
221C2×D12—21no—
222C2×D12—21no—
223C2×D12—21no—
224C2×D12#4112yes—
225C2×C4×D3—24no—
226C2×C4×D3—42no—
227C2×C4×D3—42no—
228C2×Dic6—21no—
229C2×Dic6—21no—
230C2×Dic6—21no—
231C2×Dic6—21no—
232C2×Dic6—21no—
233C2×Dic6#4112yes—
234C2×C3×C8—21no—
235C2×C3×C8—21no—
236C2×C3×C8—21no—
237C2×C3×C8—21no—
238C2×C3×C8—21no—
239C2×C3×C8—21no—
240C2×C3×C8—21no—
241C2×C3×C8—21no—
242C2×C3×C8—21no—
243C2×C3×C8—21no—
244C2×C3×C8—21no—
245C2×C3×C8—21no—
246C3×C42—22no—
247C3×C42—22no—
248C3×C42—21no—
249C3×C42—21no—
250C3×C42—21no—
251C3×C42—21no—
252#217—42no—
253#217—42no—
254#217—42no—
255(C2×D3)⋊C4—42no—
256(C2×D3)⋊C4—42no—
257(C2×D3)⋊C4—42no—
258(C2×D3)⋊C4—42no—
259C4⋊Dic3—21no—
260C4⋊Dic3—21no—
261C4⋊Dic3—21no—
262C4⋊Dic3—21no—
263C4⋊Dic3—21no—
264C4⋊Dic3—21no—
265C4⋊Dic3—21no—
266C4⋊Dic3—21no—
267C4⋊Dic3—21no—
268C4⋊Dic3—21no—
269C4⋊Dic3—21no—
270C4⋊Dic3—21no—
271C4×Dic3—42no—
272C4×Dic3—42no—
273C4×Dic3—42no—
274C2×#41—62no—
275C2×#41—62no—
276C2×#41—62no—
277C2×#41—62no—
278C2×#41—121no—
279C2×#41—121no—
280C22×Q8—31no—
281C22×D4—31no—
282C22×D4—61no—
283C23×C4—61no—
284C23×C4D1211yescharacteristic
285C23×C4D1211yescharacteristic
286C23×C4C22×D311yescharacteristic
287C22×C8C22×D311yescharacteristic
288C22×C8C22×D311yescharacteristic
289C22×C8C22×D311yescharacteristic
290C22×C8C22×D311yescharacteristic
291#117—121no—
292#117—121no—
293C2×C4⋊C4—31no—
294C2×C4⋊C4—31no—
295C2×C4⋊C4—31no—
296C2×C4⋊C4—31no—
297C2×C4⋊C4—31no—
298C2×C4⋊C4—31no—
299C2×C22⋊C4—61no—
300C2×C22⋊C4—61no—
301C2×C22⋊C4—61no—
302C2×C22⋊C4—61no—
303C2×C22⋊C4—61no—
304C2×C22⋊C4—61no—
305C2×C22⋊C4—61no—
306C2×C42—22no—
307C2×C42—22no—
308C2×C42—61no—
309C2×C42—61no—
310C2×C42—61no—
311C2×C42D1211yescharacteristic
312C2×C42D1211yescharacteristic
313C2×C42C22×D311yescharacteristic
314C2×C42C22×D311yescharacteristic
315#107—121no—
316#107—121no—
317#106—121no—
318#106—121no—
319#103—62no—
320#103—62no—
321#103—121no—
322#103—121no—
323#102—62no—
324#102—62no—
325#102—121no—
326#102—121no—
327C22⋊C8—22no—
328C22⋊C8—22no—
329#95—61no—
330#95—61no—
331#95—61no—
332#95—61no—
333#95—61no—
334#95—61no—
335#95—61no—
336#95—61no—
337#95—61no—
338#95—61no—
339#95—61no—
340#95—61no—
341C23×C3—21no—
342C23×C3—21no—
343C23×C3—21no—
344C23×C3—21no—
345C23×C3—21no—
346C23×C3—21no—
347C23×C3#12312yes—
348C23×C3C22×D411yescharacteristic
349C22×D3—22no—
350C22×D3—41no—
351C22×D3—41no—
352C22×D3—41no—
353C22×D3—41no—
354C22×D3—41no—
355C22×D3—41no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—22no—
361C2×C3×C4—22no—
362C2×C3×C4—22no—
363C2×C3×C4—22no—
364C2×C3×C4—21no—
365C2×C3×C4—21no—
366C2×C3×C4—21no—
367C2×C3×C4—21no—
368C2×C3×C4—21no—
369C2×C3×C4—21no—
370C2×C3×C4—21no—
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—44no—
374C2×C3×C4—44no—
375C2×C3×C4#12312yes—
376C2×C3×C4#13512yes—
377C2×C3×C4C2×#4111yescharacteristic
378C2×C3×C4C2×#4111yescharacteristic
379C2×C3×C4C2×#4111yescharacteristic
380C2×C3×C4C2×#4111yescharacteristic
381C2×C3×C4H(5,𝔽2)11yescharacteristic
382C2×C3×C4H(5,𝔽2)11yescharacteristic
383#67—48no—
384C2×Dic3—22no—
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—41no—
395C2×Dic3—41no—
396C2×Dic3—41no—
397C2×Dic3—41no—
398C2×Dic3—41no—
399C2×Dic3—41no—
400C2×Dic3—41no—
401C2×Dic3—41no—
402C2×Dic3—41no—
403C2×Dic3—41no—
404C2×Dic3—41no—
405D12—22no—
406D12—22no—
407D12—41no—
408D12—41no—
409C4×D3—48no—
410Dic6—22no—
411Dic6—22no—
412Dic6—41no—
413Dic6—41no—
414C3×C8—41no—
415C3×C8—41no—
416C3×C8—41no—
417C3×C8—41no—
418C24—61no—
419C24C2×D1211yescharacteristic
420#41—124no—
421#41—124no—
422#41—124no—
423#41—124no—
424C2×Q8—32no—
425C2×Q8—61no—
426C2×Q8—61no—
427C2×Q8—61no—
428C2×Q8—61no—
429C2×Q8—61no—
430C2×D4—32no—
431C2×D4—64no—
432C2×D4—61no—
433C2×D4—61no—
434C2×D4—61no—
435C2×D4—61no—
436C2×D4—61no—
437C2×D4—122no—
438C2×D4—122no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—41no—
447C22×C4—41no—
448C22×C4—41no—
449C22×C4—64no—
450C22×C4—61no—
451C22×C4—61no—
452C22×C4—61no—
453C22×C4—61no—
454C22×C4—61no—
455C22×C4—61no—
456C22×C4—61no—
457C22×C4—122no—
458C22×C4—122no—
459C22×C4C2×D1211yescharacteristic
460C22×C4C2×D1211yescharacteristic
461C22×C4C2×D1211yescharacteristic
462C22×C4C2×D1211yescharacteristic
463C22×C4C2×D1211yescharacteristic
464C22×C4#23712yes—
465C22×C4#23712yes—
466C22×C4C23×D311yescharacteristic
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C4⋊C4—61no—
480C4⋊C4—61no—
481C4⋊C4—61no—
482C4⋊C4—61no—
483C4⋊C4—61no—
484C4⋊C4—61no—
485C4⋊C4—61no—
486C4⋊C4—61no—
487C4⋊C4—61no—
488C4⋊C4—61no—
489C4⋊C4—61no—
490C4⋊C4—61no—
491C22⋊C4—122no—
492C22⋊C4—122no—
493C22⋊C4—122no—
494C22⋊C4—122no—
495C22⋊C4—122no—
496C22⋊C4—122no—
497C22⋊C4—122no—
498C42—22no—
499C42—22no—
500C42—21no—
501C42—21no—
502C42—21no—
503C42—21no—
504C42—122no—
505C42—122no—
506C42—122no—
507C22×C3—22no—
508C22×C3—22no—
509C22×C3—22no—
510C22×C3—41no—
511C22×C3—41no—
512C22×C3—41no—
513C22×C3—41no—
514C22×C3C2×#12311yescharacteristic
515C22×C3#52511yescharacteristic
516C22×C3#53511yescharacteristic
517C22×C3C2×#13511yescharacteristic
518C22×C3#57511yescharacteristic
519C22×C3#57511yescharacteristic
520C22×C3#57511yescharacteristic
521C2×D3—42no—
522C2×D3—42no—
523C2×D3—42no—
524C2×D3—81no—
525C2×D3—81no—
526C2×D3—81no—
527C2×D3—81no—
528C3×C4—22no—
529C3×C4—22no—
530C3×C4—41no—
531C3×C4—41no—
532C3×C4—41no—
533C3×C4#55111yescharacteristic
534C3×C4#55111yescharacteristic
535C3×C4#55111yescharacteristic
536C3×C4#55111yescharacteristic
537Dic3—42no—
538Dic3—81no—
539Dic3—81no—
540Dic3—81no—
541Dic3—81no—
542Dic3—81no—
543Dic3—81no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—62no—
551C23—121no—
552C23—121no—
553C23—121no—
554C23—121no—
555C23—121no—
556C23—121no—
557C23#88612yes—
558C23C22×D1211yescharacteristic
559Q8—62no—
560Q8—62no—
561Q8—121no—
562Q8—121no—
563D4—62no—
564D4—62no—
565D4—128no—
566D4—121no—
567D4—121no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—62no—
588C2×C4—128no—
589C2×C4—121no—
590C2×C4—121no—
591C2×C4—121no—
592C2×C4—121no—
593C2×C4—121no—
594C2×C4—121no—
595C2×C4—121no—
596C2×C4—121no—
597C2×C4—121no—
598C2×C4—121no—
599C2×C4—121no—
600C2×C4—121no—
601C2×C4—121no—
602C2×C4—121no—
603C2×C4—121no—
604C2×C4—121no—
605C2×C4—121no—
606C2×C4—121no—
607C2×C4—121no—
608C2×C4—121no—
609C2×C4#88612yes—
610C2×C4#90412yes—
611C2×C4C2×#23711yescharacteristic
612C2×C4C2×#23711yescharacteristic
613C2×C4D3×#4111yescharacteristic
614C2×C4D3×#4111yescharacteristic
615C2×C4#100911yescharacteristic
616C2×C4#100911yescharacteristic
617C8—41no—
618C8—41no—
619C8—41no—
620C8—41no—
621C2×C3—42no—
622C2×C3#248211yescharacteristic
623C2×C3#309111yescharacteristic
624C2×C3#309311yescharacteristic
625C2×C3#310211yescharacteristic
626C2×C3#311111yescharacteristic
627C2×C3#321211yescharacteristic
628C2×C3#321911yescharacteristic
629D3—82no—
630C22—22no—
631C22—22no—
632C22—22no—
633C22—41no—
634C22—41no—
635C22—41no—
636C22—41no—
637C22—122no—
638C22—122no—
639C22—122no—
640C22—241no—
641C22—241no—
642C22—241no—
643C22—241no—
644C22C2×#88611yescharacteristic
645C22#552411yescharacteristic
646C22#552511yescharacteristic
647C22C2×#90411yescharacteristic
648C22#574811yescharacteristic
649C22#574811yescharacteristic
650C22#574811yescharacteristic
651C4—22no—
652C4—22no—
653C4—41no—
654C4—41no—
655C4—41no—
656C4—122no—
657C4—241no—
658C4—241no—
659C4—241no—
660C4—241no—
661C4—241no—
662C4—241no—
663C4#565811yescharacteristic
664C4#565811yescharacteristic
665C4#565811yescharacteristic
666C4#565811yescharacteristic
667C3#2759011yescharacteristic, complemented, Sylow
668C2—42no—
669C2—242no—
670C2#7412911yescharacteristic
671C2#7859111yescharacteristic
672C2#7862311yescharacteristic
673C2#7929911yescharacteristic
674C2#7930811yescharacteristic
675C2#7934511yescharacteristic
676C2#7935411yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1817C2
socleC23×C3C22×D4
prime radicalC22×#235C22
hypercenter#1817D3
hypocenterC3#27590
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1817

Complements

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

NH ≅ G/Ntype
2. #82060C2semidirect
3. C3×#1817C2semidirect
6. #69929C2semidirect
7. #69929C2semidirect
8. #68693C2semidirect
9. #68680C2semidirect
10. #68638C2semidirect
11. #68637C2semidirect
12. #68628C2semidirect
13. #68627C2semidirect
14. #68626C2semidirect
15. #68625C2semidirect
72. #1817D3semidirect
667. C3#27590semidirect

Maximal subgroups

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

subgroupindex
#820702
#820602
C3×#18172
#699602
#699292
#686932
#686802
#686382
#686372
#686282
#686272
#686262
#686252
#275903