Finite Group Databasebeta

← #10877271

Subgroups — #10877271

Order 768.

770 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#946C211yesmaximal, characteristic
2#74148C211yesmaximal, characteristic
3C3×#1805C211yesmaximal, characteristic, complemented
4#71627C211yesmaximal, characteristic, complemented
5#70475C211yesmaximal, characteristic
6#70455C211yesmaximal, characteristic, complemented
7#70269C211yesmaximal, characteristic
8#70269C211yesmaximal, characteristic
9#70250C211yesmaximal, characteristic, complemented
10#70249C211yesmaximal, characteristic, complemented
11#69540C211yesmaximal, characteristic, complemented
12#68877C211yesmaximal, characteristic
13#68876C211yesmaximal, characteristic
14#68811C211yesmaximal, characteristic, complemented
15#68810C211yesmaximal, characteristic, complemented
16#27685—31nomaximal, Sylow
17C2×C3×C4×Q8C2211yescharacteristic
18C2×#946C2211yescharacteristic
19C2×#892C2211yescharacteristic
20C2×#892C2211yescharacteristic
21C2×C42×D3C2211yescharacteristic
22C2×C3×C4⋊C8C2211yescharacteristic
23C2×C3×#103C2211yescharacteristic
24C2×C3×#103C2211yescharacteristic
25C3×#382C2211yescharacteristic
26#5229C2211yescharacteristic
27C2×#835C2211yescharacteristic
28C2×#835C2211yescharacteristic
29C2×(C2×D3)⋊C8C2211yescharacteristic
30C2×(C2×D3)⋊C8C2211yescharacteristic
31C2×#818C2211yescharacteristic
32C2×#817C2211yescharacteristic
33#4989—21no—
34#4987C2211yescharacteristic
35#4978—21no—
36#4975C2211yescharacteristic
37#4975C2211yescharacteristic
38#4972C2211yescharacteristic
39C2×#808C2211yescharacteristic
40C2×#807C2211yescharacteristic
41C4×(C2×D3)⋊C4—21no—
42#4936C2211yescharacteristic
43C4×C4⋊Dic3C2211yescharacteristic
44C4×Dic3⋊C4—21no—
45C2×C4⋊(C3⋊C8)C2211yescharacteristic
46#4667—21no—
47#4667—21no—
48#4649—21no—
49#4649—21no—
50C3×#340C2211yescharacteristic
51C3×#340C2211yescharacteristic
52#4551C2211yescharacteristic
53#4551C2211yescharacteristic
54#4550C2211yescharacteristic
55#4550C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60C4×C22⋊Q8—31no—
61#2285—31no—
62#2100—31no—
63#2099—31no—
64#2096—31no—
65#2095—31no—
66#1949—31no—
67#1949—31no—
68#1873—31no—
69#1873—31no—
70#1854—31no—
71#1854—31no—
72#1805—31no—
73#1805D311yescharacteristic, complemented
74#1797—31no—
75C22×C3×Q8C2311yescharacteristic
76C22×C4×D3D411yescharacteristic
77C22×C4×D3D411yescharacteristic
78C22×C4×D3C2311yescharacteristic
79C22×C3×C8C2311yescharacteristic
80C22×C3×C8C2311yescharacteristic
81C3×C4×Q8—22no—
82C3×C4×Q8—22no—
83C2×C3×C4⋊C4—21no—
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C42—21no—
88C2×C3×C42C2311yescharacteristic
89#946—22no—
90#946Q814yes—
91C2×(C2×D3)⋊C4—21no—
92C2×(C2×D3)⋊C4—21no—
93C2×(C2×D3)⋊C4—21no—
94C2×(C2×D3)⋊C4—21no—
95C2×C4⋊Dic3C2311yescharacteristic
96C2×C4⋊Dic3C2311yescharacteristic
97C2×C4⋊Dic3C2311yescharacteristic
98C2×C4⋊Dic3C2311yescharacteristic
99C2×Dic3⋊C4—21no—
100C2×Dic3⋊C4—21no—
101C2×Dic3⋊C4—21no—
102C2×Dic3⋊C4—21no—
103C2×Dic3⋊C4—21no—
104C2×Dic3⋊C4—21no—
105C2×C4×Dic3—21no—
106C2×C4×Dic3—21no—
107C2×C4×Dic3—21no—
108C2×C4×Dic3D411yescharacteristic
109C2×C4×Dic3D411yescharacteristic
110C2×C4×Dic3C2311yescharacteristic
111C22×C3⋊C8C2311yescharacteristic
112C22×C3⋊C8C2311yescharacteristic
113#892—41no—
114#892—41no—
115#892—41no—
116#892—41no—
117C42×D3—22no—
118C42×D3—22no—
119C3×C4⋊C8—22no—
120C3×#103—41no—
121C3×#103—41no—
122C3×#95—21no—
123C3×#95—21no—
124#835—41no—
125#835—41no—
126C3⋊#95—21no—
127C3⋊#95—21no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133(C2×D3)⋊C8—22no—
134(C2×D3)⋊C8—22no—
135#818—41no—
136#817—41no—
137#808—41no—
138#807—41no—
139C4⋊(C3⋊C8)—22no—
140C2×C22⋊Q8—31no—
141C2×C4×Q8C2×D311yescharacteristic
142C2×#117—31no—
143C2×#117—31no—
144C22×C42—31no—
145C2×#107—31no—
146C2×#107—31no—
147C2×#106—31no—
148C2×#106—31no—
149C2×C4⋊C8—31no—
150C2×C4⋊C8C2×D311yescharacteristic
151C2×#103—31no—
152C2×#103—31no—
153C2×#103C2×D311yescharacteristic
154C2×#103C2×D311yescharacteristic
155C2×C22⋊C8—31no—
156C2×C22⋊C8—31no—
157#391—31no—
158#389—31no—
159#387—61no—
160#385—61no—
161#384—31no—
162#384—31no—
163#382—31no—
164#382C2×D311yescharacteristic
165#380—31no—
166#380—61no—
167#380—61no—
168C4×C4⋊C4—31no—
169C4×C4⋊C4—61no—
170C4×C4⋊C4—61no—
171C4×C4⋊C4—61no—
172C4×C22⋊C4—61no—
173#340—31no—
174#340—31no—
175#340—31no—
176#340—31no—
177#340—31no—
178#340—31no—
179#340C2×D311yescharacteristic
180#340C2×D311yescharacteristic
181#336—31no—
182#336—31no—
183C23×D3C2×D411yescharacteristic
184C2×C3×Q8—21no—
185C2×C3×Q8—21no—
186C2×C3×Q8—21no—
187C2×C3×Q8—21no—
188C2×C3×Q8—21no—
189C2×C3×Q8C2×Q812yes—
190C22×C3×C4—21no—
191C22×C3×C4—21no—
192C22×C3×C4—21no—
193C22×C3×C4—21no—
194C22×C3×C4C2×D411yescharacteristic
195C22×C3×C4C2×D411yescharacteristic
196C22×C3×C4C2411yescharacteristic
197C22×Dic3—21no—
198C22×Dic3—21no—
199C22×Dic3—21no—
200C22×Dic3—21no—
201C22×Dic3C2×D411yescharacteristic
202C22×Dic3C2×D411yescharacteristic
203C22×Dic3C2×D411yescharacteristic
204C2×C4×D3—21no—
205C2×C4×D3—21no—
206C2×C4×D3—21no—
207C2×C4×D3—21no—
208C2×C4×D3—22no—
209C2×C4×D3—22no—
210C2×C4×D3—22no—
211C2×C4×D3—22no—
212C2×C4×D3—41no—
213C2×C4×D3—41no—
214C2×C4×D3C2×Q812yes—
215C2×C4×D3#4112yes—
216C2×C3×C8—21no—
217C2×C3×C8—21no—
218C2×C3×C8—21no—
219C2×C3×C8—21no—
220C2×C3×C8—21no—
221C2×C3×C8—21no—
222C3×C4⋊C4—21no—
223C3×C4⋊C4—21no—
224C3×C4⋊C4—21no—
225C3×C4⋊C4—21no—
226C3×C4⋊C4—21no—
227C3×C4⋊C4—21no—
228C3×C4⋊C4—42no—
229C3×C42—42no—
230C3×C42#4112yes—
231C3×C42#4112yes—
232(C2×D3)⋊C4—24no—
233(C2×D3)⋊C4—42no—
234(C2×D3)⋊C4—42no—
235(C2×D3)⋊C4—42no—
236C4⋊Dic3—21no—
237C4⋊Dic3—21no—
238C4⋊Dic3—21no—
239C4⋊Dic3—21no—
240C4⋊Dic3—21no—
241C4⋊Dic3—21no—
242C4⋊Dic3—21no—
243C4⋊Dic3C2×Q812yes—
244Dic3⋊C4—24no—
245Dic3⋊C4—42no—
246Dic3⋊C4—42no—
247Dic3⋊C4—42no—
248Dic3⋊C4—42no—
249Dic3⋊C4—42no—
250C4×Dic3—22no—
251C4×Dic3—22no—
252C4×Dic3—21no—
253C4×Dic3—21no—
254C4×Dic3—42no—
255C4×Dic3—42no—
256C4×Dic3—42no—
257C2×C3⋊C8—21no—
258C2×C3⋊C8—21no—
259C2×C3⋊C8—21no—
260C2×C3⋊C8—21no—
261C2×C3⋊C8—21no—
262C2×C3⋊C8—21no—
263C22×Q8C22×D311yescharacteristic
264C23×C4—31no—
265C23×C4—31no—
266C23×C4—31no—
267C22×C8—31no—
268C22×C8—31no—
269C22×C8C22×D311yescharacteristic
270C22×C8C22×D311yescharacteristic
271C22⋊Q8—34no—
272C22⋊Q8—62no—
273C4×Q8—22no—
274C4×Q8—22no—
275#117—121no—
276#117—121no—
277#117—121no—
278#117—121no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—31no—
281C2×C4⋊C4—31no—
282C2×C4⋊C4—31no—
283C2×C4⋊C4—31no—
284C2×C4⋊C4—61no—
285C2×C4⋊C4—61no—
286C2×C4⋊C4—61no—
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×C22⋊C4—61no—
294C2×C22⋊C4—61no—
295C2×C22⋊C4—61no—
296C2×C22⋊C4—61no—
297C2×C42—21no—
298C2×C42—31no—
299C2×C42—31no—
300C2×C42—31no—
301C2×C42—62no—
302C2×C42—62no—
303C2×C42—61no—
304C2×C42—61no—
305C2×C42—61no—
306C2×C42C22×D311yescharacteristic
307#107—121no—
308#107—121no—
309#106—121no—
310#106—121no—
311C4⋊C8—22no—
312C4⋊C8—62no—
313#103—41no—
314#103—41no—
315#103—121no—
316#103—121no—
317C22⋊C8—62no—
318C22⋊C8—62no—
319#95—21no—
320#95—21no—
321#95—61no—
322#95—61no—
323#95—61no—
324#95—61no—
325#95—61no—
326#95—61no—
327#95—61no—
328C23×C3C22×D411yescharacteristic
329C22×D3—21no—
330C22×D3—21no—
331C22×D3—21no—
332C22×D3—21no—
333C22×D3—21no—
334C22×D3—21no—
335C22×D3C22⋊Q812yes—
336C3×Q8—22no—
337C3×Q8—22no—
338C3×Q8—41no—
339C3×Q8—41no—
340C2×C3×C4—21no—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—21no—
344C2×C3×C4—22no—
345C2×C3×C4—41no—
346C2×C3×C4—41no—
347C2×C3×C4—41no—
348C2×C3×C4—41no—
349C2×C3×C4—41no—
350C2×C3×C4—41no—
351C2×C3×C4—41no—
352C2×C3×C4—41no—
353C2×C3×C4—41no—
354C2×C3×C4—41no—
355C2×C3×C4—41no—
356C2×C3×C4#12312yes—
357C2×C3×C4#12312yes—
358C2×C3×C4C22×Q811yescharacteristic
359C2×C3×C4C2×#4111yescharacteristic
360C2×C3×C4C2×#4111yescharacteristic
361C2×C3×C4C2×#4111yescharacteristic
362C2×C3×C4H(5,𝔽2)11yescharacteristic
363C2×C3×C4H(5,𝔽2)11yescharacteristic
364C2×Dic3—21no—
365C2×Dic3—21no—
366C2×Dic3—21no—
367C2×Dic3—21no—
368C2×Dic3—21no—
369C2×Dic3—21no—
370C2×Dic3—22no—
371C2×Dic3—41no—
372C2×Dic3—41no—
373C2×Dic3—41no—
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×Dic3C22⋊Q812yes—
383C2×Dic3#13512yes—
384C2×Dic3#13712yes—
385C4×D3—22no—
386C4×D3—22no—
387C4×D3—22no—
388C4×D3—22no—
389C4×D3—44no—
390C4×D3—44no—
391C3×C8—41no—
392C3×C8—41no—
393C3⋊C8—41no—
394C3⋊C8—41no—
395C24—31no—
396C2×Q8—21no—
397C2×Q8—21no—
398C2×Q8—21no—
399C2×Q8—21no—
400C2×Q8—21no—
401C2×Q8D3×Q812yes—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—31no—
407C22×C4—31no—
408C22×C4—31no—
409C22×C4—32no—
410C22×C4—32no—
411C22×C4—62no—
412C22×C4—62no—
413C22×C4—61no—
414C22×C4—61no—
415C22×C4—61no—
416C22×C4—61no—
417C22×C4—61no—
418C22×C4—62no—
419C22×C4—62no—
420C22×C4—61no—
421C22×C4—61no—
422C22×C4—61no—
423C22×C4—121no—
424C22×C4—121no—
425C22×C4D3×D411yescharacteristic
426C22×C4D3×D411yescharacteristic
427C22×C4C23×D311yescharacteristic
428C2×C8—21no—
429C2×C8—21no—
430C2×C8—21no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—61no—
435C2×C8—61no—
436C2×C8—61no—
437C2×C8—61no—
438C2×C8—61no—
439C2×C8—61no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—32no—
447C4⋊C4—42no—
448C4⋊C4—64no—
449C4⋊C4—61no—
450C4⋊C4—61no—
451C4⋊C4—61no—
452C4⋊C4—61no—
453C4⋊C4—61no—
454C4⋊C4—61no—
455C4⋊C4—61no—
456C4⋊C4—122no—
457C4⋊C4—122no—
458C4⋊C4—122no—
459C4⋊C4—122no—
460C4⋊C4—122no—
461C22⋊C4—64no—
462C22⋊C4—122no—
463C22⋊C4—122no—
464C22⋊C4—122no—
465C42—42no—
466C42—62no—
467C42—62no—
468C42—61no—
469C42—61no—
470C42—122no—
471C42—122no—
472C42—122no—
473C42#23512yes—
474C42#23712yes—
475C22×C3C2×C22⋊Q811yescharacteristic
476C22×C3C2×#12311yescharacteristic
477C22×C3#53511yescharacteristic
478C22×C3C2×#13511yescharacteristic
479C22×C3C2×#13711yescharacteristic
480C22×C3#57511yescharacteristic
481C22×C3#57511yescharacteristic
482C2×D3—22no—
483C2×D3—22no—
484C2×D3—22no—
485C2×D3—41no—
486C2×D3—41no—
487C2×D3—41no—
488C2×D3—41no—
489C3×C4—22no—
490C3×C4—22no—
491C3×C4—42no—
492C3×C4—81no—
493C3×C4—81no—
494C3×C4—81no—
495C3×C4#55311yescharacteristic
496C3×C4#55311yescharacteristic
497C3×C4#55411yescharacteristic
498C3×C4#55411yescharacteristic
499Dic3—22no—
500Dic3—22no—
501Dic3—22no—
502Dic3—42no—
503Dic3—81no—
504Dic3—81no—
505Dic3—81no—
506C23—32no—
507C23—61no—
508C23—61no—
509C23—61no—
510C23—61no—
511C23—61no—
512C23—61no—
513C23C2×D3×D411yescharacteristic
514Q8—22no—
515Q8—22no—
516Q8—41no—
517Q8—41no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—22no—
523C2×C4—32no—
524C2×C4—32no—
525C2×C4—32no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—61no—
538C2×C4—61no—
539C2×C4—61no—
540C2×C4—61no—
541C2×C4—61no—
542C2×C4—61no—
543C2×C4—62no—
544C2×C4—62no—
545C2×C4—62no—
546C2×C4—62no—
547C2×C4—62no—
548C2×C4—124no—
549C2×C4—124no—
550C2×C4—121no—
551C2×C4—121no—
552C2×C4—121no—
553C2×C4—121no—
554C2×C4—121no—
555C2×C4—121no—
556C2×C4—121no—
557C2×C4—121no—
558C2×C4—121no—
559C2×C4—121no—
560C2×C4—121no—
561C2×C4#88312yes—
562C2×C4#88312yes—
563C2×C4C2×#23511yescharacteristic
564C2×C4C2×#23711yescharacteristic
565C2×C4#100411yescharacteristic
566C2×C4C2×D3×Q811yescharacteristic
567C2×C4D3×#4111yescharacteristic
568C2×C4#100911yescharacteristic
569C8—41no—
570C8—41no—
571C8—121no—
572C8—121no—
573C2×C3#248711yescharacteristic
574C2×C3#307611yescharacteristic
575C2×C3#307811yescharacteristic
576C2×C3#309111yescharacteristic
577C2×C3#309511yescharacteristic
578C2×C3#321011yescharacteristic
579C2×C3#321911yescharacteristic
580D3—42no—
581C22—62no—
582C22—62no—
583C22—62no—
584C22—121no—
585C22—121no—
586C22—121no—
587C22—121no—
588C22C2×#88311yescharacteristic
589C22#560711yescharacteristic
590C22D3×C22⋊Q811yescharacteristic
591C22D3×#13511yescharacteristic
592C22#576811yescharacteristic
593C22D3×#13711yescharacteristic
594C22#577711yescharacteristic
595C4—22no—
596C4—22no—
597C4—42no—
598C4—62no—
599C4—62no—
600C4—62no—
601C4—81no—
602C4—81no—
603C4—81no—
604C4—122no—
605C4—241no—
606C4—241no—
607C4—241no—
608C4#551111yescharacteristic
609C4#553011yescharacteristic
610C4#561411yescharacteristic
611C4#563511yescharacteristic
612C3#2768511yescharacteristic, complemented, Sylow
613C2—122no—
614C2#7391211yescharacteristic
615C2#7884211yescharacteristic
616C2#7896511yescharacteristic
617C2#8060411yescharacteristic
618C2#8060711yescharacteristic
619C2#8122911yescharacteristic
620C2#8123011yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1805C2
socleC23×C3C22×D4
prime radicalC23×D3C2×D4
hypercenter#1805D3
hypocenterC3#27685
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1805

Complements

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

NH ≅ G/Ntype
3. C3×#1805C2semidirect
4. #71627C2semidirect
6. #70455C2semidirect
9. #70250C2semidirect
10. #70249C2semidirect
11. #69540C2semidirect
14. #68811C2semidirect
15. #68810C2semidirect
73. #1805D3semidirect
612. C3#27685semidirect

Maximal subgroups

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

subgroupindex
C4×#9462
#741482
C3×#18052
#716272
#704752
#704552
#702692
#702502
#702492
#695402
#688772
#688762
#688112
#688102
#276853