Finite Group Databasebeta

← #10827389

Subgroups — #10827389

Order 768.

1304 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5565C211yesmaximal, characteristic
2C3×#2409C211yesmaximal, characteristic, complemented
3C2×#5227C211yesmaximal, characteristic, complemented
4C22×#835C211yesmaximal, characteristic, complemented
5#71699C212yesmaximal, complemented
6#71693C212yesmaximal, complemented
7#71610C211yesmaximal, characteristic, complemented
8#69687C212yesmaximal, complemented
9#69669C212yesmaximal, complemented
10#69515C211yesmaximal, characteristic, complemented
11#69502C211yesmaximal, characteristic, complemented
12#23964—31nomaximal, Sylow
13C23×C3×Q8C2211yescharacteristic
14C2×C3×C22⋊Q8C2212yes—
15C2×C3×C4×D4C2211yescharacteristic
16C22×#215C2212yes—
17C2×#939—21no—
18C2×#930C2211yescharacteristic
19C2×#930C2211yescharacteristic
20C2×#926C2211yescharacteristic
21C22×C4⋊Dic3C2211yescharacteristic
22C23×C3⋊C8C2211yescharacteristic
23#5565—24no—
24#5565—24no—
25C2×#898—21no—
26C2×#883—21no—
27C2×#67⋊C4—21no—
28C2×C4×D12C2211yescharacteristic
29C2×#870C2211yescharacteristic
30C3×#395—21no—
31C3×#393—21no—
32C3×#391C2212yes—
33C3×#386—21no—
34C3×#385—21no—
35C3×#380C2211yescharacteristic
36#5227—24no—
37C2×#835—24no—
38C2×#835C2212yes—
39C2×#835C2212yes—
40C2×#835C2212yes—
41C2×#830C2211yescharacteristic
42C2×#830C2212yes—
43C2×#809C2212yes—
44#4529C2212yes—
45#4529C2212yes—
46#4529C2212yes—
47#4524C2212yes—
48C2×#553—31no—
49C2×#448—31no—
50C22×#103—31no—
51#2409D311yescharacteristic, complemented
52#2026—32no—
53#2000—32no—
54#1893—31no—
55#1877—32no—
56#1873—32no—
57#1869—31no—
58#1824—31no—
59C22×C3×Q8—21no—
60C22×C3×Q8—24no—
61C22×C3×Q8—22no—
62C22×C3×Q8—22no—
63C22×C3×Q8—42no—
64C22×C3×Q8C2312yes—
65C22×C3×D4D411yescharacteristic
66C23×C3×C4—21no—
67C23×C3×C4C2311yescharacteristic
68C23×C3×C4C2311yescharacteristic
69C22×#67—21no—
70C22×#67—21no—
71C23×Dic3—21no—
72C22×D12C2311yescharacteristic
73C22×C4×D3—21no—
74C3×C22⋊Q8—24no—
75C3×C22⋊Q8—42no—
76C3×C4×D4—22no—
77C3×C4×D4—22no—
78C2×C3×C4⋊C4—22no—
79C2×C3×C4⋊C4D411yescharacteristic
80C2×C3×C4⋊C4C2312yes—
81C2×C3×C22⋊C4—21no—
82C2×C3×C22⋊C4—22no—
83C2×C3×C22⋊C4D411yescharacteristic
84C2×C3×C22⋊C4D411yescharacteristic
85C2×C3×C42C2311yescharacteristic
86C2×#217—21no—
87C2×#215—24no—
88C2×#215—24no—
89C2×#215—42no—
90#939—28no—
91#930—22no—
92#930—22no—
93#930—22no—
94#930D414yes—
95C2×(C2×D3)⋊C4—21no—
96C2×(C2×D3)⋊C4—21no—
97C2×(C2×D3)⋊C4—21no—
98#926—22no—
99#926—22no—
100C2×C4⋊Dic3—21no—
101C2×C4⋊Dic3—22no—
102C2×C4⋊Dic3D414yes—
103C2×C4⋊Dic3C2311yescharacteristic
104C2×C4⋊Dic3C2311yescharacteristic
105C2×C4⋊Dic3C2311yescharacteristic
106C2×Dic3⋊C4—21no—
107C2×Dic3⋊C4—21no—
108C2×C4×Dic3—21no—
109C22×C3⋊C8—22no—
110C22×C3⋊C8—42no—
111C22×C3⋊C8C2312yes—
112C22×C3⋊C8C2312yes—
113#898—28no—
114#883—28no—
115#67⋊C4—28no—
116C4×D12—22no—
117C4×D12—22no—
118#870—22no—
119#870—22no—
120C3×#95—21no—
121C3×#95—22no—
122C3×#95—22no—
123#835—24no—
124#835—24no—
125#835—24no—
126#830—22no—
127#830—42no—
128#809—42no—
129C23×Q8C2×D311yescharacteristic
130C22×SD16—32no—
131C23×C8—31no—
132#553—64no—
133#553—64no—
134C2×#126—61no—
135C2×#125—31no—
136C2×#123—61no—
137C2×C22⋊Q8C2×D312yes—
138C2×#121—31no—
139C2×#121—31no—
140C2×#121—61no—
141C2×C4×D4—31no—
142C2×C4×D4—61no—
143C2×C4×D4C2×D311yescharacteristic
144C2×#117—31no—
145C22×C4⋊C4—31no—
146#448—64no—
147C2×#103—32no—
148C2×#103—32no—
149C2×#103—32no—
150C2×#103—64no—
151C2×#102—32no—
152C2×C22⋊C8—31no—
153C2×C22⋊C8—32no—
154#395—21no—
155#393—21no—
156#391C2×D312yes—
157#386—21no—
158#385—21no—
159#380C2×D311yescharacteristic
160#340—32no—
161#340—32no—
162#340—32no—
163#336—32no—
164C24×C3C2×D411yescharacteristic
165C24×C3C2×D411yescharacteristic
166C23×D3—21no—
167C2×C3×Q8—22no—
168C2×C3×Q8—28no—
169C2×C3×Q8—22no—
170C2×C3×Q8—22no—
171C2×C3×Q8—22no—
172C2×C3×Q8—22no—
173C2×C3×Q8—44no—
174C2×C3×Q8—48no—
175C2×C3×Q8—44no—
176C2×C3×Q8—42no—
177C2×C3×Q8#4114yes—
178C2×C3×D4—22no—
179C2×C3×D4—22no—
180C2×C3×D4—22no—
181C22×C3×C4—22no—
182C22×C3×C4—22no—
183C22×C3×C4—22no—
184C22×C3×C4—21no—
185C22×C3×C4—21no—
186C22×C3×C4—22no—
187C22×C3×C4—22no—
188C22×C3×C4—22no—
189C22×C3×C4—44no—
190C22×C3×C4—42no—
191C22×C3×C4C2×D412yes—
192C22×C3×C4C2×D412yes—
193C22×C3×C4C2×D411yescharacteristic
194C22×C3×C4C2×D411yescharacteristic
195C22×C3×C4C2×D411yescharacteristic
196C22×C3×C4C2×D411yescharacteristic
197C22×C3×C4C2411yescharacteristic
198C2×#67—24no—
199C2×#67—24no—
200C2×#67—42no—
201C2×#67—42no—
202C2×#67—82no—
203C22×Dic3—21no—
204C22×Dic3—21no—
205C22×Dic3—21no—
206C22×Dic3—24no—
207C22×Dic3—82no—
208C2×D12—22no—
209C2×D12—22no—
210C2×D12—21no—
211C2×D12C2×D412yes—
212C2×C4×D3—24no—
213C2×C4×D3—82no—
214C3×C4⋊C4—22no—
215C3×C4⋊C4—22no—
216C3×C4⋊C4—22no—
217C3×C4⋊C4—44no—
218C3×C22⋊C4—22no—
219C3×C22⋊C4—22no—
220C3×C22⋊C4—44no—
221C3×C22⋊C4—42no—
222C3×C42—21no—
223C3×C42—21no—
224#217—24no—
225#215—48no—
226(C2×D3)⋊C4—24no—
227(C2×D3)⋊C4—24no—
228(C2×D3)⋊C4—24no—
229C4⋊Dic3—24no—
230C4⋊Dic3—21no—
231C4⋊Dic3—21no—
232C4⋊Dic3—21no—
233C4⋊Dic3C2×D412yes—
234C4⋊Dic3C2×D412yes—
235C4⋊Dic3C2×D412yes—
236Dic3⋊C4—24no—
237Dic3⋊C4—24no—
238C4×Dic3—24no—
239C2×C3⋊C8—22no—
240C2×C3⋊C8—22no—
241C2×C3⋊C8—22no—
242C2×C3⋊C8—22no—
243C2×C3⋊C8—22no—
244C2×C3⋊C8—22no—
245C2×C3⋊C8—44no—
246C22×Q8—21no—
247C22×Q8—24no—
248C22×Q8—22no—
249C22×Q8—22no—
250C22×Q8—42no—
251C22×Q8C22×D312yes—
252C22×D4—31no—
253C22×D4—61no—
254C22×D4—61no—
255C22×D4#6711yescharacteristic
256C23×C4—21no—
257C23×C4—61no—
258C23×C4—61no—
259C23×C4C22×D311yescharacteristic
260C23×C4C22×D311yescharacteristic
261C2×SD16—64no—
262C2×SD16—64no—
263C2×SD16—122no—
264C22×C8—32no—
265C22×C8—32no—
266C22×C8—62no—
267C22×C8—122no—
268#126—68no—
269#125—62no—
270#125—62no—
271#123—68no—
272C22⋊Q8—24no—
273C22⋊Q8—42no—
274#121—34no—
275#121—62no—
276#121—62no—
277#121—68no—
278#121—62no—
279C4×D4—22no—
280C4×D4—22no—
281C4×D4—62no—
282C4×D4—62no—
283C4×D4—68no—
284#117—62no—
285#117—62no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—34no—
288C2×C4⋊C4—31no—
289C2×C4⋊C4—31no—
290C2×C4⋊C4—31no—
291C2×C4⋊C4—62no—
292C2×C4⋊C4—61no—
293C2×C4⋊C4—61no—
294C2×C4⋊C4—61no—
295C2×C4⋊C4#6711yescharacteristic
296C2×C4⋊C4C22×D312yes—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—21no—
299C2×C22⋊C4—61no—
300C2×C22⋊C4—61no—
301C2×C22⋊C4—61no—
302C2×C22⋊C4—61no—
303C2×C22⋊C4#6711yescharacteristic
304C2×C22⋊C4#6711yescharacteristic
305C2×C42—61no—
306C2×C42C22×D311yescharacteristic
307#103—64no—
308#103—64no—
309#103—64no—
310#102—122no—
311C22⋊C8—62no—
312C22⋊C8—122no—
313#95—21no—
314#95—22no—
315#95—22no—
316C23×C3—22no—
317C23×C3—22no—
318C23×C3—21no—
319C23×C3—22no—
320C23×C3—21no—
321C23×C3—21no—
322C23×C3—22no—
323C23×C3#12012yes—
324C23×C3#13512yes—
325C23×C3#13712yes—
326C23×C3C22×D411yescharacteristic
327C22×D3—22no—
328C22×D3—42no—
329C22×D3—41no—
330C22×D3—41no—
331C22×D3—42no—
332C3×Q8—24no—
333C3×Q8—24no—
334C3×Q8—48no—
335C3×Q8—42no—
336C3×Q8—42no—
337C3×D4—44no—
338C2×C3×C4—22no—
339C2×C3×C4—22no—
340C2×C3×C4—22no—
341C2×C3×C4—24no—
342C2×C3×C4—21no—
343C2×C3×C4—21no—
344C2×C3×C4—21no—
345C2×C3×C4—24no—
346C2×C3×C4—22no—
347C2×C3×C4—21no—
348C2×C3×C4—21no—
349C2×C3×C4—22no—
350C2×C3×C4—48no—
351C2×C3×C4—42no—
352C2×C3×C4—42no—
353C2×C3×C4—41no—
354C2×C3×C4—41no—
355C2×C3×C4—42no—
356C2×C3×C4—42no—
357C2×C3×C4—42no—
358C2×C3×C4—42no—
359C2×C3×C4#12012yes—
360C2×C3×C4C22×D411yescharacteristic
361C2×C3×C4C22×D411yescharacteristic
362C2×C3×C4C2×#4112yes—
363C2×C3×C4H(5,𝔽2)12yes—
364#67—48no—
365#67—84no—
366C2×Dic3—22no—
367C2×Dic3—22no—
368C2×Dic3—22no—
369C2×Dic3—41no—
370C2×Dic3—41no—
371C2×Dic3—42no—
372C2×Dic3—42no—
373C2×Dic3—84no—
374D12—24no—
375D12—41no—
376D12—41no—
377C4×D3—84no—
378C3⋊C8—42no—
379C3⋊C8—42no—
380C24—61no—
381C24C2×#6711yescharacteristic
382C24C2×#6711yescharacteristic
383C2×Q8—22no—
384C2×Q8—28no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—44no—
390C2×Q8—48no—
391C2×Q8—44no—
392C2×Q8—42no—
393C2×Q8#23714yes—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—22no—
397C2×D4—32no—
398C2×D4—64no—
399C2×D4—64no—
400C2×D4—61no—
401C2×D4—62no—
402C2×D4—62no—
403C2×D4—122no—
404C2×D4—122no—
405C2×D4—242no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—44no—
415C22×C4—42no—
416C22×C4—64no—
417C22×C4—61no—
418C22×C4—61no—
419C22×C4—61no—
420C22×C4—64no—
421C22×C4—242no—
422C22×C4—242no—
423C22×C4D3×D412yes—
424C22×C4D3×D412yes—
425C22×C4C2×#6711yescharacteristic
426C22×C4C2×#6711yescharacteristic
427C22×C4C2×#6711yescharacteristic
428C22×C4C2×#6711yescharacteristic
429C22×C4C23×D311yescharacteristic
430SD16—128no—
431C2×C8—62no—
432C2×C8—62no—
433C2×C8—62no—
434C2×C8—62no—
435C2×C8—62no—
436C2×C8—62no—
437C2×C8—124no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—32no—
442C4⋊C4—32no—
443C4⋊C4—32no—
444C4⋊C4—44no—
445C4⋊C4—64no—
446C4⋊C4—64no—
447C4⋊C4—64no—
448C4⋊C4—61no—
449C4⋊C4—61no—
450C4⋊C4—61no—
451C22⋊C4—22no—
452C22⋊C4—22no—
453C22⋊C4—44no—
454C22⋊C4—42no—
455C22⋊C4—64no—
456C22⋊C4—64no—
457C22⋊C4—64no—
458C22⋊C4—64no—
459C42—21no—
460C42—21no—
461C42—64no—
462C22×C3—22no—
463C22×C3—22no—
464C22×C3—24no—
465C22×C3—24no—
466C22×C3—41no—
467C22×C3—41no—
468C22×C3—41no—
469C22×C3—42no—
470C22×C3—42no—
471C22×C3#45214yes—
472C22×C3C2×#12011yescharacteristic
473C22×C3#54612yes—
474C22×C3C2×#13511yescharacteristic
475C22×C3C2×#13711yescharacteristic
476C22×C3#57712yes—
477C2×D3—42no—
478C2×D3—42no—
479C2×D3—42no—
480C2×D3—82no—
481C2×D3—82no—
482C3×C4—22no—
483C3×C4—41no—
484C3×C4—44no—
485C3×C4—41no—
486C3×C4—41no—
487C3×C4—82no—
488C3×C4#52511yescharacteristic
489C3×C4#53511yescharacteristic
490C3×C4#54612yes—
491Dic3—42no—
492Dic3—42no—
493Dic3—42no—
494C23—22no—
495C23—22no—
496C23—21no—
497C23—22no—
498C23—21no—
499C23—21no—
500C23—22no—
501C23—62no—
502C23—122no—
503C23—121no—
504C23—121no—
505C23—122no—
506C23#93712yes—
507C23#94212yes—
508C23#95012yes—
509C23C22×#6711yescharacteristic
510Q8—24no—
511Q8—24no—
512Q8—48no—
513Q8—42no—
514Q8—42no—
515D4—44no—
516D4—64no—
517D4—128no—
518D4—121no—
519D4—121no—
520D4—244no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—24no—
524C2×C4—22no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—24no—
529C2×C4—22no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—22no—
533C2×C4—48no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—41no—
537C2×C4—42no—
538C2×C4—41no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—62no—
543C2×C4—62no—
544C2×C4—62no—
545C2×C4—121no—
546C2×C4—121no—
547C2×C4—122no—
548C2×C4—122no—
549C2×C4—244no—
550C2×C4—244no—
551C2×C4#93712yes—
552C2×C4C2×D3×D411yescharacteristic
553C2×C4C2×D3×D411yescharacteristic
554C2×C4C2×#23712yes—
555C2×C4#100412yes—
556C8—122no—
557C8—122no—
558C2×C3—24no—
559C2×C3—81no—
560C2×C3#240411yescharacteristic
561C2×C3C2×#45211yescharacteristic
562C2×C3#300911yescharacteristic
563C2×C3#330112yes—
564C2×C3#332012yes—
565D3—82no—
566C22—24no—
567C22—24no—
568C22—22no—
569C22—22no—
570C22—41no—
571C22—42no—
572C22—42no—
573C22—41no—
574C22—41no—
575C22—122no—
576C22—122no—
577C22—122no—
578C22—242no—
579C22—242no—
580C22#504714yes—
581C22C2×#93711yescharacteristic
582C22#580612yes—
583C22C2×#94211yescharacteristic
584C22C2×#95011yescharacteristic
585C22#583712yes—
586C4—22no—
587C4—41no—
588C4—41no—
589C4—44no—
590C4—41no—
591C4—82no—
592C4—122no—
593C4—122no—
594C4—122no—
595C4#558611yescharacteristic
596C4#559112yes—
597C4#559211yescharacteristic
598C3#2396411yescharacteristic, complemented, Sylow
599C2—24no—
600C2—81no—
601C2—242no—
602C2#8258111yescharacteristic
603C2C2×#504711yescharacteristic
604C2#8340411yescharacteristic
605C2#8353512yes—
606C2#8355312yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2409C2
socleC23×C3C22×D4
prime radicalC2×#5565C2
hypercenter#2409D3
hypocenterC3#23964
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2409

Complements

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

NH ≅ G/Ntype
2. C3×#2409C2semidirect
3. C2×#5227C2semidirect
4. C22×#835C2semidirect
5. #71699C2semidirect
6. #71693C2semidirect
7. #71610C2semidirect
8. #69687C2semidirect
9. #69669C2semidirect
10. #69515C2semidirect
11. #69502C2semidirect
51. #2409D3semidirect
598. C3#23964semidirect

Maximal subgroups

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

subgroupindex
C2×#55652
C3×#24092
C2×#52272
C22×#8352
#716992
#716932
#716102
#696872
#696692
#695152
#695022
#239643