Finite Group Databasebeta

← #10844595

Subgroups — #10844595

Order 768.

822 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83237C211yesmaximal, characteristic
2C2×#4794C211yesmaximal, characteristic
3C2×#4793C211yesmaximal, characteristic, complemented
4#73652C211yesmaximal, characteristic, complemented
5C3×#1981C211yesmaximal, characteristic, complemented
6#71781C211yesmaximal, characteristic, complemented
7#70484C211yesmaximal, characteristic
8#70452C211yesmaximal, characteristic, complemented
9#70262C211yesmaximal, characteristic, complemented
10#70245C211yesmaximal, characteristic
11#70230C211yesmaximal, characteristic
12#70230C211yesmaximal, characteristic
13#70224C211yesmaximal, characteristic, complemented
14#70220C211yesmaximal, characteristic, complemented
15#69462C211yesmaximal, characteristic
16#25286—31nomaximal, Sylow
17C2×C3×C4⋊Q8C2211yescharacteristic
18C2×#947C2211yescharacteristic
19C2×#947C2211yescharacteristic
20C2×Q8×Dic3C2211yescharacteristic
21C2×#888C2211yescharacteristic
22C2×C4×Dic6—21no—
23C2×C3×C4⋊C8C2211yescharacteristic
24C2×C3×#103C2211yescharacteristic
25C2×C3×#103C2211yescharacteristic
26C2×C3×#103C2211yescharacteristic
27C2×C3×#103C2211yescharacteristic
28C3×C4×C4⋊C4C2211yescharacteristic
29#5229C2211yescharacteristic
30C2×#835C2211yescharacteristic
31C2×#835C2211yescharacteristic
32C2×#835C2211yescharacteristic
33C2×#835C2211yescharacteristic
34C2×#821C2211yescharacteristic
35C2×#821C2211yescharacteristic
36C2×#818C2211yescharacteristic
37C2×#815C2211yescharacteristic
38C2×Dic3⋊C8C2211yescharacteristic
39C2×Dic3⋊C8C2211yescharacteristic
40#4994—21no—
41#4994—21no—
42#4990—21no—
43#4990—21no—
44#4984C2211yescharacteristic
45Dic3×C4⋊C4C2211yescharacteristic
46Dic3×C4⋊C4C2211yescharacteristic
47C2×#809C2211yescharacteristic
48C2×#809C2211yescharacteristic
49C2×#807C2211yescharacteristic
50#4939C2211yescharacteristic
51#4933—21no—
52C4×Dic3⋊C4C2211yescharacteristic
53#4928C2211yescharacteristic
54#4794—24no—
55#4793—24no—
56#4655—21no—
57#4646—21no—
58#4551C2211yescharacteristic
59#4550C2211yescharacteristic
60#4529C2211yescharacteristic
61#4524C2211yescharacteristic
62C2×#489—31no—
63C2×#477—31no—
64#2545—31no—
65#2365—31no—
66#2068—31no—
67#2067—31no—
68#2043—31no—
69#2042—31no—
70#1983—31no—
71#1983—31no—
72#1981D311yescharacteristic, complemented
73#1940—31no—
74#1928—31no—
75#1925—31no—
76#1763—31no—
77C22×C3×Q8C2311yescharacteristic
78C22×C3×Q8C2311yescharacteristic
79C22×D12C2311yescharacteristic
80C22×Dic6—21no—
81C22×C3×C8C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C3×C4⋊Q8—22no—
84C3×C4⋊Q8—41no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C42—21no—
90C2×C3×C42C2311yescharacteristic
91#947—22no—
92#947—22no—
93#947—22no—
94#947—41no—
95Q8×Dic3—22no—
96Q8×Dic3—22no—
97C2×(C2×D3)⋊C4—21no—
98C2×(C2×D3)⋊C4—21no—
99C2×(C2×D3)⋊C4—21no—
100C2×(C2×D3)⋊C4—21no—
101C2×(C2×D3)⋊C4—21no—
102C2×(C2×D3)⋊C4—21no—
103C2×C4⋊Dic3—21no—
104C2×C4⋊Dic3—21no—
105C2×C4⋊Dic3C2311yescharacteristic
106C2×C4⋊Dic3C2311yescharacteristic
107C2×C4⋊Dic3C2311yescharacteristic
108C2×Dic3⋊C4—21no—
109C2×Dic3⋊C4—21no—
110C2×Dic3⋊C4D411yescharacteristic
111C2×Dic3⋊C4D411yescharacteristic
112C2×C4×Dic3—21no—
113C2×C4×Dic3—21no—
114C2×C4×Dic3—21no—
115C2×C4×Dic3C2311yescharacteristic
116C2×C4×Dic3C2311yescharacteristic
117C22×C3⋊C8C2311yescharacteristic
118C22×C3⋊C8C2311yescharacteristic
119#888—22no—
120#888D414yes—
121C4×Dic6—44no—
122C3×C4⋊C8—41no—
123C3×#103—22no—
124C3×#103—22no—
125C3×#103—22no—
126C3×#103—22no—
127C3×#95—21no—
128#835—22no—
129#835—22no—
130#835—22no—
131#835—41no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95D411yescharacteristic
139C3⋊#95D411yescharacteristic
140#821—22no—
141#821—41no—
142#818—22no—
143#815—22no—
144Dic3⋊C8—22no—
145Dic3⋊C8—41no—
146#809—22no—
147#809—22no—
148#807—22no—
149C2×C4⋊Q8—31no—
150C2×C4⋊Q8C2×D311yescharacteristic
151C2×#124—31no—
152C2×#124—31no—
153C2×C4×Q8—31no—
154C2×C4×Q8—61no—
155#489—64no—
156#477—64no—
157C2×#107—31no—
158C2×#107—31no—
159C2×C4⋊C8—31no—
160C2×C4⋊C8—31no—
161C2×C4⋊C8C2×D311yescharacteristic
162C2×#103—31no—
163C2×#103—31no—
164C2×#103—31no—
165C2×#103—31no—
166C2×#103C2×D311yescharacteristic
167C2×#103C2×D311yescharacteristic
168C2×#103C2×D311yescharacteristic
169C2×#103C2×D311yescharacteristic
170C2×#102—31no—
171C2×#102—31no—
172C2×#102—31no—
173C2×#102—31no—
174C2×#97—31no—
175#396—61no—
176#396—61no—
177#391—31no—
178#391—61no—
179#390—31no—
180#389—31no—
181#389—31no—
182#388—61no—
183#388—61no—
184#387—61no—
185#387—61no—
186C4×C4⋊C4—31no—
187C4×C4⋊C4—31no—
188C4×C4⋊C4—31no—
189C4×C4⋊C4C2×D311yescharacteristic
190#340—31no—
191#340—31no—
192#336—31no—
193#336—31no—
194C23×D3—21no—
195C2×C3×Q8—21no—
196C2×C3×Q8—21no—
197C2×C3×Q8—21no—
198C2×C3×Q8—21no—
199C2×C3×Q8—21no—
200C2×C3×Q8—21no—
201C2×C3×Q8—21no—
202C2×C3×Q8—21no—
203C2×C3×Q8—21no—
204C2×C3×Q8—21no—
205C2×C3×Q8—21no—
206C2×C3×Q8#4112yes—
207C22×C3×C4—21no—
208C22×C3×C4—21no—
209C22×C3×C4—21no—
210C22×C3×C4—21no—
211C22×C3×C4C2×D411yescharacteristic
212C22×C3×C4C2×D411yescharacteristic
213C22×C3×C4C2411yescharacteristic
214C22×Dic3—21no—
215C22×Dic3—21no—
216C22×Dic3—21no—
217C22×Dic3C2×D411yescharacteristic
218C22×Dic3C2×D411yescharacteristic
219C22×Dic3C2×D411yescharacteristic
220C22×Dic3C2×D411yescharacteristic
221C2×D12—21no—
222C2×D12—21no—
223C2×D12—21no—
224C2×D12—21no—
225C2×D12—21no—
226C2×D12—21no—
227C2×Dic6—24no—
228C2×Dic6—42no—
229C2×Dic6—42no—
230C2×C3×C8—21no—
231C2×C3×C8—21no—
232C2×C3×C8—21no—
233C2×C3×C8—21no—
234C2×C3×C8—21no—
235C2×C3×C8—21no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—42no—
239C3×C4⋊C4—42no—
240C3×C4⋊C4C2×D412yes—
241C3×C4⋊C4#4112yes—
242C3×C42—21no—
243C3×C42—21no—
244C3×C42—42no—
245(C2×D3)⋊C4—42no—
246(C2×D3)⋊C4—42no—
247(C2×D3)⋊C4—42no—
248(C2×D3)⋊C4—42no—
249(C2×D3)⋊C4—42no—
250(C2×D3)⋊C4—42no—
251C4⋊Dic3—21no—
252C4⋊Dic3—21no—
253C4⋊Dic3—21no—
254C4⋊Dic3—21no—
255C4⋊Dic3—21no—
256C4⋊Dic3—42no—
257C4⋊Dic3—42no—
258C4⋊Dic3C2×D412yes—
259Dic3⋊C4—22no—
260Dic3⋊C4—24no—
261Dic3⋊C4—41no—
262Dic3⋊C4—42no—
263C4×Dic3—21no—
264C4×Dic3—21no—
265C4×Dic3—42no—
266C4×Dic3—42no—
267C4×Dic3—42no—
268C4×Dic3C2×D412yes—
269C4×Dic3#4112yes—
270C2×C3⋊C8—21no—
271C2×C3⋊C8—21no—
272C2×C3⋊C8—21no—
273C2×C3⋊C8—21no—
274C2×C3⋊C8—21no—
275C2×C3⋊C8—21no—
276C22×Q8—61no—
277C22×Q8C22×D311yescharacteristic
278C22×Q8C22×D311yescharacteristic
279C22×D4—31no—
280C22×C8—31no—
281C22×C8—31no—
282C22×C8C22×D311yescharacteristic
283C22×C8C22×D311yescharacteristic
284C4⋊Q8—22no—
285C4⋊Q8—34no—
286C4⋊Q8—41no—
287C4⋊Q8—62no—
288#124—62no—
289#124—62no—
290#124—62no—
291#124—121no—
292C4×Q8—62no—
293C4×Q8—62no—
294C4×Q8—124no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—31no—
298C2×C4⋊C4—31no—
299C2×C4⋊C4—31no—
300C2×C4⋊C4—31no—
301C2×C4⋊C4—31no—
302C2×C4⋊C4—61no—
303C2×C4⋊C4—61no—
304C2×C4⋊C4—61no—
305C2×C4⋊C4—61no—
306C2×C4⋊C4C22×D311yescharacteristic
307C2×C4⋊C4C22×D311yescharacteristic
308C2×C22⋊C4—61no—
309C2×C22⋊C4—61no—
310C2×C22⋊C4—61no—
311C2×C22⋊C4—61no—
312C2×C22⋊C4—61no—
313C2×C22⋊C4—61no—
314C2×C42—21no—
315C2×C42—31no—
316C2×C42—31no—
317C2×C42—61no—
318C2×C42—61no—
319C2×C42—61no—
320C2×C42C22×D311yescharacteristic
321#107—62no—
322#107—62no—
323C4⋊C8—41no—
324C4⋊C8—62no—
325C4⋊C8—121no—
326#103—22no—
327#103—22no—
328#103—22no—
329#103—22no—
330#103—62no—
331#103—62no—
332#103—62no—
333#103—121no—
334#102—62no—
335#102—62no—
336#102—62no—
337#102—121no—
338#97—62no—
339#95—21no—
340#95—31no—
341#95—31no—
342#95—61no—
343#95—61no—
344#95—61no—
345#95—61no—
346#95—61no—
347#95—61no—
348C23×C3C22×D411yescharacteristic
349C22×D3—41no—
350C22×D3—41no—
351C22×D3—41no—
352C22×D3—41no—
353C22×D3—41no—
354C22×D3—41no—
355C22×D3—41no—
356C3×Q8—22no—
357C3×Q8—22no—
358C3×Q8—41no—
359C3×Q8—41no—
360C3×Q8—41no—
361C3×Q8—41no—
362C3×Q8—41no—
363C3×Q8—41no—
364C2×C3×C4—21no—
365C2×C3×C4—21no—
366C2×C3×C4—22no—
367C2×C3×C4—22no—
368C2×C3×C4—21no—
369C2×C3×C4—22no—
370C2×C3×C4—21no—
371C2×C3×C4—21no—
372C2×C3×C4—41no—
373C2×C3×C4—41no—
374C2×C3×C4—41no—
375C2×C3×C4—41no—
376C2×C3×C4—41no—
377C2×C3×C4—41no—
378C2×C3×C4—41no—
379C2×C3×C4—41no—
380C2×C3×C4—41no—
381C2×C3×C4AGL(1,ℤ/8)12yes—
382C2×C3×C4C22×D411yescharacteristic
383C2×C3×C4C2×#4111yescharacteristic
384C2×C3×C4C2×#4111yescharacteristic
385C2×C3×C4C2×#4111yescharacteristic
386C2×C3×C4H(5,𝔽2)11yescharacteristic
387C2×C3×C42−1+411yescharacteristic
388C2×Dic3—21no—
389C2×Dic3—21no—
390C2×Dic3—21no—
391C2×Dic3—21no—
392C2×Dic3—22no—
393C2×Dic3—21no—
394C2×Dic3—21no—
395C2×Dic3—21no—
396C2×Dic3—21no—
397C2×Dic3—21no—
398C2×Dic3—21no—
399C2×Dic3—41no—
400C2×Dic3—41no—
401C2×Dic3—41no—
402C2×Dic3—41no—
403C2×Dic3—41no—
404C2×Dic3—41no—
405C2×Dic3—41no—
406C2×Dic3—41no—
407C2×Dic3#12112yes—
408C2×Dic3#12112yes—
409D12—41no—
410D12—41no—
411D12—41no—
412D12—41no—
413Dic6—48no—
414C3×C8—41no—
415C3×C8—41no—
416C3⋊C8—41no—
417C3⋊C8—41no—
418C24—61no—
419C2×Q8—21no—
420C2×Q8—21no—
421C2×Q8—21no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8—21no—
428C2×Q8—21no—
429C2×Q8—21no—
430C2×Q8—64no—
431C2×Q8—122no—
432C2×Q8—122no—
433C2×Q8#23512yes—
434C2×D4—61no—
435C2×D4—61no—
436C2×D4—61no—
437C2×D4—61no—
438C2×D4—61no—
439C2×D4—61no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—31no—
445C22×C4—31no—
446C22×C4—31no—
447C22×C4—31no—
448C22×C4—61no—
449C22×C4—61no—
450C22×C4—61no—
451C22×C4D3×D411yescharacteristic
452C22×C4D3×D411yescharacteristic
453C22×C4C23×D311yescharacteristic
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—61no—
461C2×C8—61no—
462C2×C8—61no—
463C2×C8—61no—
464C2×C8—61no—
465C2×C8—61no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—32no—
469C4⋊C4—42no—
470C4⋊C4—42no—
471C4⋊C4—64no—
472C4⋊C4—62no—
473C4⋊C4—61no—
474C4⋊C4—61no—
475C4⋊C4—61no—
476C4⋊C4—61no—
477C4⋊C4—61no—
478C4⋊C4—121no—
479C4⋊C4—122no—
480C4⋊C4—122no—
481C4⋊C4—122no—
482C4⋊C4D3×D412yes—
483C4⋊C4#23712yes—
484C22⋊C4—122no—
485C22⋊C4—122no—
486C22⋊C4—122no—
487C22⋊C4—122no—
488C22⋊C4—122no—
489C22⋊C4—122no—
490C42—21no—
491C42—21no—
492C42—32no—
493C42—32no—
494C42—42no—
495C42—61no—
496C42—61no—
497C42—122no—
498C42—122no—
499C42—122no—
500C22×C3C2×#12111yescharacteristic
501C22×C3#54611yescharacteristic
502C22×C3#54811yescharacteristic
503C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
504C22×C3#57511yescharacteristic
505C22×C3#57811yescharacteristic
506C22×C3#57811yescharacteristic
507C2×D3—81no—
508C2×D3—81no—
509C2×D3—81no—
510C2×D3—81no—
511C2×D3—81no—
512C2×D3—81no—
513C2×D3—81no—
514C3×C4—22no—
515C3×C4—42no—
516C3×C4—42no—
517C3×C4—42no—
518C3×C4—41no—
519C3×C4—81no—
520C3×C4#52911yescharacteristic
521C3×C4#54211yescharacteristic
522C3×C4#54811yescharacteristic
523C3×C4#55011yescharacteristic
524Dic3—22no—
525Dic3—22no—
526Dic3—42no—
527Dic3—41no—
528Dic3—41no—
529Dic3—81no—
530Dic3—81no—
531C23—121no—
532C23—121no—
533C23—121no—
534C23—121no—
535C23—121no—
536C23—121no—
537C23—121no—
538C23C2×D3×D411yescharacteristic
539Q8—22no—
540Q8—22no—
541Q8—41no—
542Q8—41no—
543Q8—41no—
544Q8—41no—
545Q8—41no—
546Q8—41no—
547Q8—128no—
548D4—121no—
549D4—121no—
550D4—121no—
551D4—121no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—32no—
561C2×C4—32no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
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—61no—
580C2×C4—61no—
581C2×C4—62no—
582C2×C4—121no—
583C2×C4—121no—
584C2×C4—121no—
585C2×C4—121no—
586C2×C4—121no—
587C2×C4—121no—
588C2×C4—121no—
589C2×C4—121no—
590C2×C4#91412yes—
591C2×C4C2×#23511yescharacteristic
592C2×C4C2×D3×D411yescharacteristic
593C2×C4C2×#23711yescharacteristic
594C2×C4#100411yescharacteristic
595C2×C4D3×#4111yescharacteristic
596C2×C4#101011yescharacteristic
597C8—41no—
598C8—41no—
599C8—121no—
600C8—121no—
601C2×C3#242611yescharacteristic
602C2×C3#303611yescharacteristic
603C2×C3#304411yescharacteristic
604C2×C3#330811yescharacteristic
605C2×C3#331311yescharacteristic
606C2×C3#336711yescharacteristic
607C2×C3#337711yescharacteristic
608D3—161no—
609C22—241no—
610C22—241no—
611C22—241no—
612C22—241no—
613C22—241no—
614C22—241no—
615C22—241no—
616C22#549011yescharacteristic
617C22#559811yescharacteristic
618C22D3×#12111yescharacteristic
619C22C2×#91411yescharacteristic
620C22#576611yescharacteristic
621C22#577711yescharacteristic
622C22#577911yescharacteristic
623C4—22no—
624C4—42no—
625C4—42no—
626C4—42no—
627C4—41no—
628C4—62no—
629C4—62no—
630C4—81no—
631C4—122no—
632C4—121no—
633C4—121no—
634C4—241no—
635C4—241no—
636C4#553811yescharacteristic
637C4#555211yescharacteristic
638C4#566911yescharacteristic
639C4#571811yescharacteristic
640C3#2528611yescharacteristic, complemented, Sylow
641C2—481no—
642C2#7631211yescharacteristic
643C2#7880911yescharacteristic
644C2#7896711yescharacteristic
645C2#8041611yescharacteristic
646C2#8042811yescharacteristic
647C2#8046711yescharacteristic
648C2#8047811yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1981C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenter#1981D3
hypocenterC3#25286
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1981

Complements

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

NH ≅ G/Ntype
3. C2×#4793C2semidirect
4. #73652C2semidirect
5. C3×#1981C2semidirect
6. #71781C2semidirect
8. #70452C2semidirect
9. #70262C2semidirect
13. #70224C2semidirect
14. #70220C2semidirect
72. #1981D3semidirect
640. C3#25286semidirect

Maximal subgroups

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

subgroupindex
#832372
C2×#47942
C2×#47932
#736522
C3×#19812
#717812
#704842
#704522
#702622
#702452
#702302
#702242
#702202
#694622
#252863