Finite Group Databasebeta

← #10822566

Subgroups — #10822566

Order 768.

1214 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
1C2×#5519C211yesmaximal, characteristic, complemented
2#83768C211yesmaximal, characteristic
3#82270C211yesmaximal, characteristic
4#82213C211yesmaximal, characteristic
5C3×#1733C211yesmaximal, characteristic, complemented
6C2×#4554C211yesmaximal, characteristic, complemented
7#71578C211yesmaximal, characteristic, complemented
8#68769C214yesmaximal, complemented
9#68763C214yesmaximal, complemented
10#23462—31nomaximal, Sylow
11C22×#235C2211yescharacteristic
12C22×C3×M16C2211yescharacteristic
13C2×C3×#117C2211yescharacteristic
14C2×C3×#117C2211yescharacteristic
15C2×#926C2211yescharacteristic
16C22×C4⋊Dic3C2211yescharacteristic
17C22×#208C2211yescharacteristic
18#5519—22no—
19#5519—22no—
20#5519—22no—
21#5519C2214yes—
22C2×#890—21no—
23C2×#888—21no—
24C2×#878—22no—
25C2×#870—21no—
26C2×#869—21no—
27#5133—21no—
28#5133—21no—
29#5131C2212yes—
30#5120C2212yes—
31#5113—22no—
32#5004C2212yes—
33#5002—22no—
34#5001C2212yes—
35#4987—22no—
36#4966—21no—
37#4966—21no—
38#4958—22no—
39#4939C2212yes—
40#4929C2212yes—
41C3×#339C2214yes—
42#4554C2214yes—
43#4523C2214yes—
44C2×#544—31no—
45#2896—31no—
46#2896—31no—
47#2311—31no—
48#2028—34no—
49#2027—34no—
50C2×#343—31no—
51#1733—31no—
52#1733D311yescharacteristic, complemented
53C23×C3×C4C2311yescharacteristic
54C22×#67—21no—
55C23×Dic3D412yes—
56C2×#235—22no—
57C2×#235—21no—
58C2×#235—21no—
59C2×#235—21no—
60C2×#235—21no—
61C2×#235—22no—
62C2×#235—22no—
63C2×#235Q814yescomplemented
64C22×D12C2311yescharacteristic
65C22×C4×D3—21no—
66C22×Dic6—22no—
67C22×Dic6C2311yescharacteristic
68C2×C3×M16—21no—
69C2×C3×M16—22no—
70C2×C3×M16—22no—
71C2×C3×M16C2312yes—
72C22×C3×C8—21no—
73C3×#117—21no—
74C3×#117—21no—
75C3×#117—22no—
76C3×#117—22no—
77C3×#117D412yes—
78C3×#117C2312yes—
79C2×C3×C4⋊C4—21no—
80C2×C3×C4⋊C4D411yescharacteristic
81C2×C3×C4⋊C4D411yescharacteristic
82C2×C3×C22⋊C4—21no—
83C2×C3×C22⋊C4—21no—
84C2×C3×C42—21no—
85C2×C3×C42C2312yes—
86C2×#217—21no—
87C2×(C2×D3)⋊C4—24no—
88#926—21no—
89#926—21no—
90#926D412yes—
91#926C2312yes—
92C2×C4⋊Dic3—22no—
93C2×C4⋊Dic3—21no—
94C2×C4⋊Dic3—21no—
95C2×C4⋊Dic3—21no—
96C2×C4⋊Dic3—21no—
97C2×C4⋊Dic3D412yes—
98C2×C4⋊Dic3D411yescharacteristic
99C2×C4⋊Dic3D411yescharacteristic
100C2×C4⋊Dic3C2312yes—
101C2×Dic3⋊C4—22no—
102C2×Dic3⋊C4—22no—
103C2×C4×Dic3—21no—
104C2×#208—21no—
105C2×#208—22no—
106C2×#208—22no—
107C2×#208C2312yes—
108C22×C3⋊C8—21no—
109#890—24no—
110#890—24no—
111#888—24no—
112#888—24no—
113#878—28no—
114#878—28no—
115#870—24no—
116#870—24no—
117#869—24no—
118#869—24no—
119C3⋊#95—24no—
120#821—22no—
121#821—22no—
122#816—22no—
123#816—22no—
124#810—22no—
125#810—22no—
126#809—22no—
127#809—22no—
128C22×#41—31no—
129C22×M16—31no—
130C22×M16C2×D311yescharacteristic
131#544—34no—
132#544—62no—
133#544—62no—
134#544—62no—
135C2×C4⋊Q8—61no—
136C2×C4⋊Q8—61no—
137C2×#125—61no—
138C2×#125—61no—
139C2×C22⋊Q8—62no—
140C2×#117—31no—
141C2×#117C2×D311yescharacteristic
142C2×#117C2×D311yescharacteristic
143C22×C4⋊C4—31no—
144#419—32no—
145#419—32no—
146#418—32no—
147#418—32no—
148#417—61no—
149#417—61no—
150#417—61no—
151#417—61no—
152#416—62no—
153#416—62no—
154#391—32no—
155#390—32no—
156#386—62no—
157#380—62no—
158#343—34no—
159#339—34no—
160#339C2×D314yes—
161C24×C3C2×D411yescharacteristic
162C23×D3—21no—
163C22×C3×C4—21no—
164C22×C3×C4—21no—
165C22×C3×C4—22no—
166C22×C3×C4C2×D411yescharacteristic
167C22×C3×C4C2×D411yescharacteristic
168C22×C3×C4C2×D411yescharacteristic
169C22×C3×C4C2×D411yescharacteristic
170C22×C3×C4C2×D411yescharacteristic
171C22×C3×C4C2×Q812yes—
172C22×C3×C4#4112yes—
173C22×C3×C4#4112yes—
174C22×C3×C4#4112yes—
175C22×C3×C4C2411yescharacteristic
176C2×#67—24no—
177C2×#67—41no—
178C2×#67—41no—
179C2×#67—42no—
180C22×Dic3—21no—
181C22×Dic3—21no—
182C22×Dic3—22no—
183C22×Dic3—21no—
184C22×Dic3—22no—
185C22×Dic3—82no—
186C22×Dic3C2×D414yes—
187#235—24no—
188#235—24no—
189#235—24no—
190#235—24no—
191#235—42no—
192#235—42no—
193#235—42no—
194#235—42no—
195C2×D12—21no—
196C2×D12—21no—
197C2×D12—21no—
198C2×D12—21no—
199C2×D12—21no—
200C2×D12C2×Q812yescomplemented
201C2×C4×D3—24no—
202C2×C4×D3—41no—
203C2×C4×D3—41no—
204C2×C4×D3—42no—
205C2×Dic6—21no—
206C2×Dic6—21no—
207C2×Dic6—21no—
208C2×Dic6—21no—
209C2×Dic6—21no—
210C2×Dic6—28no—
211C2×Dic6—84no—
212C2×Dic6C2×Q812yescomplemented
213C3×M16—22no—
214C3×M16—22no—
215C3×M16—42no—
216C2×C3×C8—22no—
217C2×C3×C8—42no—
218C3×C4⋊C4—21no—
219C3×C4⋊C4—21no—
220C3×C4⋊C4—42no—
221C3×C4⋊C4C2×D412yes—
222C3×C4⋊C4C2×D412yes—
223C3×C22⋊C4—22no—
224C3×C22⋊C4—22no—
225C3×C22⋊C4—42no—
226C3×C42—22no—
227C3×C42—22no—
228C3×C42—22no—
229C3×C42—22no—
230#217—22no—
231#217—22no—
232(C2×D3)⋊C4—84no—
233C4⋊Dic3—28no—
234C4⋊Dic3—21no—
235C4⋊Dic3—22no—
236C4⋊Dic3—21no—
237C4⋊Dic3—22no—
238C4⋊Dic3—22no—
239C4⋊Dic3—22no—
240C4⋊Dic3C2×D412yes—
241C4⋊Dic3C2×D412yes—
242Dic3⋊C4—28no—
243Dic3⋊C4—28no—
244C4×Dic3—22no—
245C4×Dic3—22no—
246#208—22no—
247#208—22no—
248#208—42no—
249C2×C3⋊C8—22no—
250C2×C3⋊C8—42no—
251C2×#41—34no—
252C2×#41—62no—
253C2×#41—61no—
254C2×#41—61no—
255C2×#41—62no—
256C2×#41—61no—
257C2×#41—61no—
258C2×#41—62no—
259C22×Q8—31no—
260C22×Q8—62no—
261C22×D4—31no—
262C22×D4—61no—
263C23×C4—32no—
264C23×C4—61no—
265C23×C4C22×D311yescharacteristic
266C2×M16—21no—
267C2×M16—22no—
268C2×M16—22no—
269C2×M16—32no—
270C2×M16—61no—
271C2×M16—62no—
272C2×M16—62no—
273C2×M16C22×D312yes—
274C22×C8—21no—
275C22×C8—61no—
276C4⋊Q8—64no—
277C4⋊Q8—64no—
278C4⋊Q8—64no—
279C4⋊Q8—64no—
280#125—64no—
281#125—64no—
282#125—64no—
283#125—64no—
284C22⋊Q8—68no—
285C22⋊Q8—68no—
286#117—21no—
287#117—21no—
288#117—22no—
289#117—22no—
290#117—32no—
291#117—32no—
292#117—61no—
293#117—61no—
294#117D1212yes—
295#117C22×D312yes—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—31no—
298C2×C4⋊C4—31no—
299C2×C4⋊C4—32no—
300C2×C4⋊C4—32no—
301C2×C4⋊C4—62no—
302C2×C4⋊C4—62no—
303C2×C4⋊C4—62no—
304C2×C4⋊C4—61no—
305C2×C4⋊C4—61no—
306C2×C4⋊C4—61no—
307C2×C4⋊C4—61no—
308C2×C4⋊C4D1211yescharacteristic
309C2×C4⋊C4D1211yescharacteristic
310C2×C22⋊C4—21no—
311C2×C22⋊C4—21no—
312C2×C22⋊C4—61no—
313C2×C22⋊C4—64no—
314C2×C42—21no—
315C2×C42—61no—
316C2×C42C22×D312yes—
317#103—62no—
318#103—62no—
319#103—62no—
320#103—62no—
321#102—62no—
322#102—62no—
323#102—62no—
324#102—62no—
325#95—64no—
326C23×C3—41no—
327C23×C3—41no—
328C23×C3#12011yescharacteristic
329C23×C3#12011yescharacteristic
330C23×C3C22⋊Q812yes—
331C23×C3#12412yes—
332C23×C3C22×D411yescharacteristic
333C22×D3—22no—
334C22×D3—42no—
335C22×D3—42no—
336C22×D3—41no—
337C22×D3—41no—
338C2×C3×C4—22no—
339C2×C3×C4—22no—
340C2×C3×C4—42no—
341C2×C3×C4—42no—
342C2×C3×C4—42no—
343C2×C3×C4—42no—
344C2×C3×C4—42no—
345C2×C3×C4#12011yescharacteristic
346C2×C3×C4#12011yescharacteristic
347C2×C3×C4C22⋊Q812yes—
348C2×C3×C4C22⋊Q812yes—
349C2×C3×C4C22⋊Q812yes—
350C2×C3×C4C22⋊Q812yes—
351C2×C3×C4C22⋊Q812yes—
352C2×C3×C4#12412yes—
353C2×C3×C4#12412yes—
354C2×C3×C4#12412yes—
355C2×C3×C4#12412yes—
356C2×C3×C4#12412yes—
357C2×C3×C4C22×D411yescharacteristic
358C2×C3×C4C22×D411yescharacteristic
359C2×C3×C4C22×Q811yescharacteristic
360C2×C3×C4C2×#4111yescharacteristic
361C2×C3×C4C2×#4111yescharacteristic
362C2×C3×C4C2×#4111yescharacteristic
363#67—44no—
364#67—44no—
365C2×Dic3—22no—
366C2×Dic3—22no—
367C2×Dic3—24no—
368C2×Dic3—22no—
369C2×Dic3—42no—
370C2×Dic3—44no—
371C2×Dic3—42no—
372C2×Dic3—41no—
373C2×Dic3—41no—
374C2×Dic3—44no—
375C2×Dic3—82no—
376D12—22no—
377D12—22no—
378D12—41no—
379D12—41no—
380C4×D3—44no—
381C4×D3—44no—
382Dic6—22no—
383Dic6—22no—
384Dic6—41no—
385Dic6—41no—
386Dic6—88no—
387C3×C8—42no—
388C3⋊C8—42no—
389C24—61no—
390C24D3×D411yescharacteristic
391#41—64no—
392#41—64no—
393#41—64no—
394#41—64no—
395#41—122no—
396#41—122no—
397#41—122no—
398#41—122no—
399C2×Q8—32no—
400C2×Q8—68no—
401C2×Q8—61no—
402C2×Q8—61no—
403C2×Q8—61no—
404C2×Q8—61no—
405C2×Q8—61no—
406C2×Q8—244no—
407C2×D4—32no—
408C2×D4—64no—
409C2×D4—61no—
410C2×D4—61no—
411C2×D4—61no—
412C2×D4—61no—
413C2×D4—61no—
414C2×D4—121no—
415C2×D4—121no—
416C2×D4—122no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—34no—
421C22×C4—61no—
422C22×C4—61no—
423C22×C4—62no—
424C22×C4—61no—
425C22×C4—62no—
426C22×C4—64no—
427C22×C4—121no—
428C22×C4—121no—
429C22×C4—122no—
430C22×C4—242no—
431C22×C4C2×D1211yescharacteristic
432C22×C4C2×D1211yescharacteristic
433C22×C4#23512yes—
434C22×C4#23512yes—
435C22×C4D3×D411yescharacteristic
436C22×C4D3×D411yescharacteristic
437C22×C4D3×D411yescharacteristic
438C22×C4#23712yes—
439C22×C4D3×Q812yes—
440C22×C4C23×D311yescharacteristic
441M16—22no—
442M16—22no—
443M16—42no—
444M16—62no—
445M16—62no—
446M16—122no—
447C2×C8—22no—
448C2×C8—42no—
449C2×C8—62no—
450C2×C8—122no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—32no—
454C4⋊C4—32no—
455C4⋊C4—42no—
456C4⋊C4—68no—
457C4⋊C4—68no—
458C4⋊C4—68no—
459C4⋊C4—62no—
460C4⋊C4—62no—
461C4⋊C4—61no—
462C4⋊C4—61no—
463C4⋊C4—62no—
464C4⋊C4—62no—
465C4⋊C4C2×D1212yes—
466C4⋊C4C2×D1212yes—
467C22⋊C4—22no—
468C22⋊C4—22no—
469C22⋊C4—42no—
470C22⋊C4—62no—
471C22⋊C4—62no—
472C22⋊C4—244no—
473C42—22no—
474C42—22no—
475C42—22no—
476C42—22no—
477C42—62no—
478C42—62no—
479C22×C3—21no—
480C22×C3—21no—
481C22×C3—21no—
482C22×C3—21no—
483C22×C3—21no—
484C22×C3—21no—
485C22×C3—42no—
486C22×C3—42no—
487C22×C3#39312yes—
488C22×C3#39312yes—
489C22×C3C2×#12011yescharacteristic
490C22×C3C2×C22⋊Q811yescharacteristic
491C22×C3C2×#12411yescharacteristic
492C2×D3—44no—
493C2×D3—42no—
494C2×D3—82no—
495C2×D3—81no—
496C2×D3—81no—
497C3×C4—42no—
498C3×C4—42no—
499C3×C4—82no—
500C3×C4#39312yes—
501C3×C4#39312yes—
502C3×C4C2×C22⋊Q811yescharacteristic
503C3×C4C2×C22⋊Q811yescharacteristic
504C3×C4C2×#12411yescharacteristic
505C3×C4C2×#12411yescharacteristic
506Dic3—42no—
507Dic3—42no—
508Dic3—44no—
509Dic3—42no—
510C23—41no—
511C23—41no—
512C23—62no—
513C23—122no—
514C23—122no—
515C23—121no—
516C23—121no—
517C23#88211yescharacteristic
518C23#88211yescharacteristic
519C23#88512yes—
520C23#89612yes—
521C23C2×D3×D411yescharacteristic
522Q8—62no—
523Q8—62no—
524Q8—121no—
525Q8—121no—
526Q8—248no—
527D4—62no—
528D4—62no—
529D4—124no—
530D4—124no—
531D4—121no—
532D4—121no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—62no—
541C2×C4—62no—
542C2×C4—64no—
543C2×C4—62no—
544C2×C4—124no—
545C2×C4—122no—
546C2×C4—122no—
547C2×C4—121no—
548C2×C4—121no—
549C2×C4—124no—
550C2×C4—124no—
551C2×C4—124no—
552C2×C4—242no—
553C2×C4#87512yes—
554C2×C4#87512yes—
555C2×C4#88211yescharacteristic
556C2×C4#88211yescharacteristic
557C2×C4#88512yes—
558C2×C4#88512yes—
559C2×C4#88512yes—
560C2×C4#89612yes—
561C2×C4#89612yes—
562C2×C4#89612yes—
563C2×C4#89712yes—
564C2×C4#89712yes—
565C2×C4C22×D1211yescharacteristic
566C2×C4C2×#23511yescharacteristic
567C2×C4C2×#23511yescharacteristic
568C2×C4C2×D3×D411yescharacteristic
569C2×C4C2×#23711yescharacteristic
570C2×C4C2×D3×Q811yescharacteristic
571C8—42no—
572C8—122no—
573C2×C3—21no—
574C2×C3—21no—
575C2×C3—22no—
576C2×C3—22no—
577C2×C3C2×#39311yescharacteristic
578C2×C3#301911yescharacteristic
579C2×C3#302011yescharacteristic
580D3—82no—
581C22—21no—
582C22—21no—
583C22—21no—
584C22—21no—
585C22—21no—
586C22—21no—
587C22—42no—
588C22—42no—
589C22—124no—
590C22—122no—
591C22—242no—
592C22—241no—
593C22—241no—
594C22#467312yes—
595C22#467312yes—
596C22C2×#88211yescharacteristic
597C22C2×#88511yescharacteristic
598C22C2×#89611yescharacteristic
599C4—42no—
600C4—42no—
601C4—82no—
602C4—124no—
603C4—122no—
604C4—122no—
605C4—122no—
606C4#467312yes—
607C4#467312yes—
608C4C2×#87511yescharacteristic
609C4C2×#88511yescharacteristic
610C4C2×#89611yescharacteristic
611C4C2×#89711yescharacteristic
612C3#2346211yescharacteristic, complemented, Sylow
613C2—21no—
614C2—21no—
615C2—22no—
616C2—22no—
617C2—242no—
618C2C2×#467311yescharacteristic
619C2#7899311yescharacteristic
620C2#7899611yescharacteristic
subgroupG/H
centerC22C2×#882
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1733C2
socleC22×C3C2×#120
prime radicalC22×#235C22
hypercenter#1733D3
hypocenterC3#23462
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1733

Complements

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

NH ≅ G/Ntype
1. C2×#5519C2semidirect
5. C3×#1733C2semidirect
6. C2×#4554C2semidirect
7. #71578C2semidirect
8. #68769C2semidirect
9. #68763C2semidirect
52. #1733D3semidirect
63. C2×#235Q8semidirect
200. C2×D12C2×Q8semidirect
212. C2×Dic6C2×Q8semidirect
612. C3#23462semidirect

Maximal subgroups

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

subgroupindex
C2×#55192
#837682
#822702
#822132
C3×#17332
C2×#45542
#715782
#687692
#687632
#234623