Finite Group Databasebeta

← #10828508

Subgroups — #10828508

Order 768.

1176 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5479C211yesmaximal, characteristic
2C2×C3×#389C211yesmaximal, characteristic, complemented
3C2×#4971C211yesmaximal, characteristic, complemented
4C2×#4967C211yesmaximal, characteristic
5#71560C211yesmaximal, characteristic
6#71559C211yesmaximal, characteristic, complemented
7#71531C211yesmaximal, characteristic, complemented
8#69631C212yesmaximal, complemented
9#69628C212yesmaximal, complemented
10#69626C212yesmaximal, complemented
11#69606C212yesmaximal, complemented
12#24066—31nomaximal, Sylow
13C22×C3×C4⋊C4—21no—
14C22×C3×C4⋊C4C2211yescharacteristic
15C22×C3×C4⋊C4C2211yescharacteristic
16C22×C3×C42C2211yescharacteristic
17C2×#930C2211yescharacteristic
18C2×#929—21no—
19C2×C4×#67—21no—
20C2×#924C2211yescharacteristic
21#5479—24no—
22#5479—24no—
23C2×#876—21no—
24C2×#875C2211yescharacteristic
25C2×C4×D12C2211yescharacteristic
26C2×#872—21no—
27C2×#870C2211yescharacteristic
28C2×C4×Dic6C2211yescharacteristic
29C3×#389—21no—
30C3×#389—21no—
31C3×#389C2212yes—
32C3×#389C2212yes—
33C2×C3×#95—21no—
34C2×#830C2211yescharacteristic
35C2×#830C2211yescharacteristic
36#4971—24no—
37C2×#810C2212yes—
38C2×#810C2212yes—
39#4967—24no—
40C2×#809C2212yes—
41C2×#809C2212yes—
42C2×#808C2212yes—
43C2×#807C2212yes—
44#4529C2212yes—
45#4529C2212yes—
46#4524C2212yes—
47#4524C2212yes—
48C2×#529—31no—
49C2×#483—31no—
50C2×#480—31no—
51C2×#389D311yescharacteristic, complemented
52#2059—32no—
53#2042—32no—
54#1874—31no—
55#1872—31no—
56#1864—32no—
57#1863—32no—
58#1817—31no—
59C23×C3×C4—21no—
60C23×C3×C4—21no—
61C23×C3×C4D411yescharacteristic
62C23×C3×C4D411yescharacteristic
63C23×C3×C4C2311yescharacteristic
64C22×#67—21no—
65C22×D12C2311yescharacteristic
66C22×C4×D3—21no—
67C22×Dic6C2311yescharacteristic
68C2×C3×C4⋊C4—21no—
69C2×C3×C4⋊C4—24no—
70C2×C3×C4⋊C4—22no—
71C2×C3×C4⋊C4—22no—
72C2×C3×C4⋊C4—21no—
73C2×C3×C4⋊C4—22no—
74C2×C3×C4⋊C4—22no—
75C2×C3×C4⋊C4—44no—
76C2×C3×C4⋊C4C2312yes—
77C2×C3×C4⋊C4C2312yes—
78C2×C3×C42—22no—
79C2×C3×C42—22no—
80C2×C3×C42D411yescharacteristic
81C2×C3×C42D411yescharacteristic
82C2×C3×C42C2311yescharacteristic
83C2×C3×C42C2311yescharacteristic
84C2×#217—21no—
85#930—22no—
86#930D414yes—
87#929—28no—
88C4×#67—28no—
89C2×(C2×D3)⋊C4—21no—
90C2×(C2×D3)⋊C4—21no—
91C2×(C2×D3)⋊C4—21no—
92C2×(C2×D3)⋊C4—21no—
93C2×C4⋊Dic3C2311yescharacteristic
94C2×C4⋊Dic3C2311yescharacteristic
95#924—22no—
96#924D414yes—
97C2×Dic3⋊C4—21no—
98C2×Dic3⋊C4—21no—
99C2×Dic3⋊C4—21no—
100C2×Dic3⋊C4—21no—
101C2×C4×Dic3—21no—
102C22×C3⋊C8C2312yes—
103C22×C3⋊C8C2312yes—
104#876—28no—
105#875—22no—
106#875—22no—
107C4×D12—22no—
108C4×D12—22no—
109#872—28no—
110#870—22no—
111#870—22no—
112C4×Dic6—22no—
113C4×Dic6—22no—
114C3×#95—22no—
115C3×#95—22no—
116#830—22no—
117#830—22no—
118#810—24no—
119#810—24no—
120#809—24no—
121#809—24no—
122#808—24no—
123#807—24no—
124#529—64no—
125#529—64no—
126C2×#126—61no—
127C2×#125—31no—
128C2×#124—31no—
129C2×#123—61no—
130C2×C22⋊Q8—31no—
131C2×#121—31no—
132C2×C4×Q8—31no—
133C2×C4×D4—31no—
134C2×C4×D4—61no—
135C2×#117—61no—
136C22×C4⋊C4—21no—
137C22×C4⋊C4C2×D311yescharacteristic
138C22×C4⋊C4C2×D311yescharacteristic
139C22×C42C2×D311yescharacteristic
140#483—64no—
141#480—64no—
142C2×#107—32no—
143C2×#106—32no—
144C2×#103—32no—
145C2×#103—32no—
146C2×#102—32no—
147C2×#102—32no—
148C2×C22⋊C8—31no—
149C2×C22⋊C8—31no—
150#389—21no—
151#389—21no—
152#389C2×D312yes—
153#389C2×D312yes—
154C2×#95—21no—
155#340—32no—
156#340—32no—
157#336—32no—
158#336—32no—
159C24×C3C2×D411yescharacteristic
160C23×D3—21no—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4—24no—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4—22no—
167C22×C3×C4—22no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—22no—
171C22×C3×C4—41no—
172C22×C3×C4—42no—
173C22×C3×C4—42no—
174C22×C3×C4—42no—
175C22×C3×C4—42no—
176C22×C3×C4—42no—
177C22×C3×C4C2×D412yes—
178C22×C3×C4C2×D412yes—
179C22×C3×C4C2×D411yescharacteristic
180C22×C3×C4C2×D411yescharacteristic
181C22×C3×C4C2×D411yescharacteristic
182C22×C3×C4C2×D411yescharacteristic
183C22×C3×C4C2×D411yescharacteristic
184C22×C3×C4C2411yescharacteristic
185C2×#67—24no—
186C2×#67—42no—
187C2×#67—42no—
188C22×Dic3—21no—
189C22×Dic3—21no—
190C22×Dic3—21no—
191C2×D12—22no—
192C2×D12—22no—
193C2×D12—21no—
194C2×D12C2×D412yes—
195C2×C4×D3—24no—
196C2×C4×D3—42no—
197C2×C4×D3—42no—
198C2×Dic6—22no—
199C2×Dic6—22no—
200C2×Dic6—21no—
201C2×Dic6C2×D412yes—
202C3×C4⋊C4—24no—
203C3×C4⋊C4—22no—
204C3×C4⋊C4—22no—
205C3×C4⋊C4—22no—
206C3×C4⋊C4—44no—
207C3×C4⋊C4—44no—
208C3×C4⋊C4—42no—
209C3×C4⋊C4#4114yes—
210C3×C42—22no—
211C3×C42—22no—
212C3×C42—21no—
213C3×C42—21no—
214C3×C42—21no—
215C3×C42—21no—
216#217—24no—
217(C2×D3)⋊C4—24no—
218(C2×D3)⋊C4—24no—
219(C2×D3)⋊C4—24no—
220(C2×D3)⋊C4—24no—
221C4⋊Dic3—21no—
222C4⋊Dic3—21no—
223C4⋊Dic3C2×D412yes—
224C4⋊Dic3C2×D412yes—
225Dic3⋊C4—24no—
226Dic3⋊C4—24no—
227Dic3⋊C4—24no—
228Dic3⋊C4—24no—
229C4×Dic3—24no—
230C2×C3⋊C8—22no—
231C2×C3⋊C8—22no—
232C2×C3⋊C8—22no—
233C2×C3⋊C8—22no—
234C2×C3⋊C8—22no—
235C2×C3⋊C8—22no—
236C22×Q8—31no—
237C22×D4—31no—
238C22×D4—61no—
239C23×C4—21no—
240C23×C4—21no—
241C23×C4—61no—
242C23×C4#6711yescharacteristic
243C23×C4#6711yescharacteristic
244C23×C4C22×D311yescharacteristic
245C22×C8—32no—
246C22×C8—32no—
247#126—68no—
248#125—62no—
249#125—62no—
250#124—62no—
251#124—62no—
252#123—68no—
253C22⋊Q8—34no—
254C22⋊Q8—62no—
255#121—34no—
256#121—62no—
257C4×Q8—62no—
258C4×Q8—62no—
259C4×D4—62no—
260C4×D4—62no—
261C4×D4—68no—
262#117—68no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—31no—
271C2×C4⋊C4—31no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—61no—
274C2×C4⋊C4—61no—
275C2×C4⋊C4—61no—
276C2×C4⋊C4—61no—
277C2×C4⋊C4C22×D312yes—
278C2×C4⋊C4C22×D312yes—
279C2×C22⋊C4—61no—
280C2×C22⋊C4—61no—
281C2×C22⋊C4—61no—
282C2×C22⋊C4—61no—
283C2×C22⋊C4—61no—
284C2×C42—22no—
285C2×C42—22no—
286C2×C42—61no—
287C2×C42#6711yescharacteristic
288C2×C42#6711yescharacteristic
289C2×C42C22×D311yescharacteristic
290C2×C42C22×D311yescharacteristic
291#107—64no—
292#106—64no—
293#103—64no—
294#103—64no—
295#102—64no—
296#102—64no—
297C22⋊C8—62no—
298C22⋊C8—62no—
299#95—22no—
300#95—22no—
301C23×C3—21no—
302C23×C3—21no—
303C23×C3—21no—
304C23×C3—21no—
305C23×C3#12012yes—
306C23×C3#13512yes—
307C23×C3AGL(1,ℤ/8)12yes—
308C23×C3C22×D411yescharacteristic
309C22×D3—22no—
310C22×D3—41no—
311C22×D3—41no—
312C22×D3—41no—
313C22×D3—41no—
314C22×D3—41no—
315C22×D3—41no—
316C2×C3×C4—21no—
317C2×C3×C4—21no—
318C2×C3×C4—21no—
319C2×C3×C4—21no—
320C2×C3×C4—24no—
321C2×C3×C4—24no—
322C2×C3×C4—21no—
323C2×C3×C4—21no—
324C2×C3×C4—21no—
325C2×C3×C4—24no—
326C2×C3×C4—21no—
327C2×C3×C4—21no—
328C2×C3×C4—21no—
329C2×C3×C4—21no—
330C2×C3×C4—21no—
331C2×C3×C4—21no—
332C2×C3×C4—44no—
333C2×C3×C4—48no—
334C2×C3×C4—42no—
335C2×C3×C4—42no—
336C2×C3×C4—44no—
337C2×C3×C4—42no—
338C2×C3×C4—42no—
339C2×C3×C4—42no—
340C2×C3×C4—84no—
341C2×C3×C4#12012yes—
342C2×C3×C4#13712yes—
343C2×C3×C4C22×D411yescharacteristic
344C2×C3×C4C22×D411yescharacteristic
345C2×C3×C4C2×#4112yes—
346C2×C3×C42−1+412yes—
347#67—48no—
348C2×Dic3—22no—
349C2×Dic3—22no—
350C2×Dic3—22no—
351C2×Dic3—41no—
352C2×Dic3—41no—
353C2×Dic3—41no—
354C2×Dic3—41no—
355C2×Dic3—41no—
356C2×Dic3—41no—
357D12—24no—
358D12—42no—
359C4×D3—48no—
360Dic6—24no—
361Dic6—42no—
362C3⋊C8—42no—
363C3⋊C8—42no—
364C24—61no—
365C24C2×#6711yescharacteristic
366C2×Q8—32no—
367C2×Q8—61no—
368C2×Q8—62no—
369C2×Q8—62no—
370C2×D4—32no—
371C2×D4—64no—
372C2×D4—62no—
373C2×D4—61no—
374C2×D4—62no—
375C2×D4—122no—
376C2×D4—122no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—41no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—64no—
394C22×C4—61no—
395C22×C4—61no—
396C22×C4—61no—
397C22×C4—122no—
398C22×C4—122no—
399C22×C4D3×D412yes—
400C22×C4D3×D412yes—
401C22×C4C2×#6711yescharacteristic
402C22×C4C2×#6711yescharacteristic
403C22×C4C2×#6711yescharacteristic
404C22×C4C2×#6711yescharacteristic
405C22×C4C2×#6711yescharacteristic
406C22×C4C23×D311yescharacteristic
407C2×C8—62no—
408C2×C8—62no—
409C2×C8—62no—
410C2×C8—62no—
411C2×C8—62no—
412C2×C8—62no—
413C4⋊C4—24no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—32no—
418C4⋊C4—32no—
419C4⋊C4—44no—
420C4⋊C4—44no—
421C4⋊C4—42no—
422C4⋊C4—64no—
423C4⋊C4—64no—
424C4⋊C4—64no—
425C4⋊C4—64no—
426C4⋊C4—61no—
427C4⋊C4—61no—
428C4⋊C4#23514yes—
429C22⋊C4—64no—
430C22⋊C4—64no—
431C22⋊C4—64no—
432C22⋊C4—64no—
433C22⋊C4—64no—
434C42—22no—
435C42—22no—
436C42—21no—
437C42—21no—
438C42—21no—
439C42—21no—
440C42—64no—
441C22×C3—22no—
442C22×C3—22no—
443C22×C3—22no—
444C22×C3—22no—
445C22×C3—22no—
446C22×C3—22no—
447C22×C3#44914yes—
448C22×C3C2×#12011yescharacteristic
449C22×C3#52511yescharacteristic
450C22×C3#53611yescharacteristic
451C22×C3C2×#13511yescharacteristic
452C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
453C22×C3C2×#13711yescharacteristic
454C22×C3#57511yescharacteristic
455C2×D3—42no—
456C2×D3—42no—
457C2×D3—42no—
458C2×D3—81no—
459C2×D3—81no—
460C2×D3—81no—
461C2×D3—81no—
462C3×C4—22no—
463C3×C4—22no—
464C3×C4—44no—
465C3×C4—41no—
466C3×C4—41no—
467C3×C4—41no—
468C3×C4—82no—
469C3×C4#54712yes—
470C3×C4#54712yes—
471Dic3—42no—
472Dic3—42no—
473Dic3—42no—
474C23—21no—
475C23—21no—
476C23—21no—
477C23—21no—
478C23—62no—
479C23—121no—
480C23—121no—
481C23—121no—
482C23—121no—
483C23—121no—
484C23—121no—
485C23#93212yes—
486C23#93712yes—
487C23#95012yes—
488C23C22×#6711yescharacteristic
489Q8—64no—
490Q8—122no—
491D4—64no—
492D4—128no—
493D4—122no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—44no—
511C2×C4—48no—
512C2×C4—42no—
513C2×C4—44no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—62no—
519C2×C4—62no—
520C2×C4—62no—
521C2×C4—84no—
522C2×C4—128no—
523C2×C4—121no—
524C2×C4—121no—
525C2×C4—121no—
526C2×C4—121no—
527C2×C4—121no—
528C2×C4—121no—
529C2×C4#93712yes—
530C2×C4#94212yes—
531C2×C4C2×#23512yes—
532C2×C4C2×D3×D411yescharacteristic
533C2×C4C2×D3×D411yescharacteristic
534C2×C4#100712yes—
535C8—122no—
536C8—122no—
537C2×C3—24no—
538C2×C3#240611yescharacteristic
539C2×C3C2×#44911yescharacteristic
540C2×C3#300711yescharacteristic
541C2×C3#310811yescharacteristic
542C2×C3#311411yescharacteristic
543C2×C3#323211yescharacteristic
544C2×C3#323811yescharacteristic
545D3—82no—
546C22—22no—
547C22—22no—
548C22—22no—
549C22—22no—
550C22—22no—
551C22—22no—
552C22—122no—
553C22—122no—
554C22—122no—
555C22—241no—
556C22—241no—
557C22—241no—
558C22—241no—
559C22#503714yes—
560C22#579111yescharacteristic
561C22C2×#93211yescharacteristic
562C22C2×#93711yescharacteristic
563C22C2×#94211yescharacteristic
564C22#581611yescharacteristic
565C22C2×#95011yescharacteristic
566C22#582211yescharacteristic
567C4—22no—
568C4—22no—
569C4—41no—
570C4—44no—
571C4—41no—
572C4—41no—
573C4—82no—
574C4—122no—
575C4—122no—
576C4—122no—
577C4#550712yes—
578C4#550712yes—
579C3#2406611yescharacteristic, complemented, Sylow
580C2—24no—
581C2—242no—
582C2#8219211yescharacteristic
583C2#8231111yescharacteristic
584C2#8244311yescharacteristic
585C2C2×#503711yescharacteristic
586C2#8273511yescharacteristic
587C2#8301811yescharacteristic
588C2#8321711yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#389C2
socleC23×C3C22×D4
prime radicalC2×#930C22
hypercenterC2×#389D3
hypocenterC3#24066
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#389

Complements

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

NH ≅ G/Ntype
2. C2×C3×#389C2semidirect
3. C2×#4971C2semidirect
6. #71559C2semidirect
7. #71531C2semidirect
8. #69631C2semidirect
9. #69628C2semidirect
10. #69626C2semidirect
11. #69606C2semidirect
51. C2×#389D3semidirect
579. C3#24066semidirect

Maximal subgroups

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

subgroupindex
C2×#54792
C2×C3×#3892
C2×#49712
C2×#49672
#715602
#715592
#715312
#696312
#696282
#696262
#696062
#240663