Finite Group Databasebeta

← #10830935

Subgroups — #10830935

Order 768.

1452 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#919C211yesmaximal, characteristic, complemented
2C2×C3×#470C211yesmaximal, characteristic, complemented
3C2×#5135C211yesmaximal, characteristic, complemented
4C2×#5127C211yesmaximal, characteristic, complemented
5#82812C212yesmaximal, complemented
6#72067C211yesmaximal, characteristic, complemented
7#71699C212yesmaximal, complemented
8#70577C211yesmaximal, characteristic, complemented
9#70576C211yesmaximal, characteristic, complemented
10#70280C212yesmaximal, complemented
11#70267C212yesmaximal, complemented
12#24271—31nomaximal, Sylow
13C22×#239C2212yes—
14C22×C3×Q16C2211yescharacteristic
15C22×C3×M16C2211yescharacteristic
16C2×C3×C22⋊Q8C2212yes—
17C22×#215C2212yes—
18C2×#930C2212yes—
19C2×C4×#67C2211yescharacteristic
20C2×#919—24no—
21C2×#919—24no—
22C2×#919—24no—
23C22×D24C2211yescharacteristic
24C22×D3×C8C2211yescharacteristic
25C3×#470—24no—
26C2×C3×#106C2211yescharacteristic
27C2×C3×#103C2212yes—
28#5229C2212yes—
29C2×#835C2212yes—
30C2×#830C2211yescharacteristic
31#5135—24no—
32#5127—24no—
33C2×#821C2212yes—
34C2×(C2×D3)⋊C8C2211yescharacteristic
35C2×#817C2211yescharacteristic
36C2×#815C2211yescharacteristic
37C2×Dic3⋊C8C2211yescharacteristic
38#4991—22no—
39#4989—22no—
40#4988C2212yes—
41#4987C2212yes—
42C2×#809C2212yes—
43#4960—22no—
44#4956—22no—
45#4529C2212yes—
46C22×#135—31no—
47C2×#470D311yescharacteristic, complemented
48C2×#469—31no—
49C2×#435—31no—
50#2407—32no—
51#2055—32no—
52#2000—32no—
53#1972—31no—
54#1971—31no—
55#1807—31no—
56#1788—32no—
57C22×C3×Q8C2312yes—
58C23×C3×C4C2311yescharacteristic
59C22×#67—22no—
60C22×#67D411yescharacteristic
61C2×#239—24no—
62C2×#239—24no—
63C2×#239—24no—
64C2×#239—24no—
65C2×#239—42no—
66C2×#239—42no—
67C22×D12—22no—
68C22×D12C2312yes—
69C22×C4×D3—22no—
70C22×C4×D3C2311yescharacteristic
71C2×C3×Q16—22no—
72C2×C3×Q16—22no—
73C2×C3×Q16D414yescomplemented
74C2×C3×M16—22no—
75C2×C3×M16—22no—
76C2×C3×M16—22no—
77C22×C3×C8C2311yescharacteristic
78C22×C3×C8C2311yescharacteristic
79C3×C22⋊Q8—24no—
80C3×C22⋊Q8—42no—
81C2×C3×C4⋊C4—22no—
82C2×C3×C4⋊C4C2312yes—
83C2×C3×C22⋊C4—22no—
84C2×#217D411yescharacteristic
85C2×#215—24no—
86C2×#215—24no—
87C2×#215—24no—
88#930—24no—
89#930—42no—
90C4×#67—22no—
91C4×#67—22no—
92C2×(C2×D3)⋊C4—22no—
93C2×(C2×D3)⋊C4—22no—
94C2×(C2×D3)⋊C4—22no—
95C2×(C2×D3)⋊C4—22no—
96C2×(C2×D3)⋊C4D411yescharacteristic
97C2×C4⋊Dic3C2312yes—
98C2×Dic3⋊C4D411yescharacteristic
99C2×C4×Dic3C2311yescharacteristic
100C22×C3⋊C8C2311yescharacteristic
101C22×C3⋊C8C2311yescharacteristic
102#919—232no—
103C2×D24—22no—
104C2×D24—22no—
105C2×D24D414yescomplemented
106C2×D3×C8—22no—
107C2×D3×C8—22no—
108C2×D3×C8D414yescomplemented
109C3×#106—22no—
110C3×#103—24no—
111#835—42no—
112C3⋊#95—22no—
113C3⋊#95—22no—
114C3⋊#95—22no—
115#830—22no—
116#821—24no—
117(C2×D3)⋊C8—22no—
118#817—22no—
119#815—22no—
120Dic3⋊C8—22no—
121#809—42no—
122C22×#41—32no—
123C2×#135—64no—
124C2×#135—64no—
125C2×#135—64no—
126C22×Q16C2×D311yescharacteristic
127C22×SD16—32no—
128C22×D8—31no—
129C22×M16C2×D311yescharacteristic
130C23×C8—31no—
131C2×C22⋊Q8C2×D312yes—
132C2×#121—32no—
133C2×C4×D4—31no—
134#470—24no—
135#469—64no—
136#435—64no—
137C2×#106—31no—
138C2×#106C2×D311yescharacteristic
139C2×C4⋊C8—31no—
140C2×#103—32no—
141C2×#103C2×D312yes—
142C2×#102—32no—
143C2×#102—32no—
144C2×C22⋊C8—31no—
145C2×C22⋊C8—31no—
146C2×#97—31no—
147#394—62no—
148#392—62no—
149#391—32no—
150#390—32no—
151#386—62no—
152#385—62no—
153#380—32no—
154#340—32no—
155C24×C3C2×D411yescharacteristic
156C23×D3—22no—
157C23×D3C2×D411yescharacteristic
158C2×C3×Q8—22no—
159C2×C3×Q8—22no—
160C2×C3×Q8—22no—
161C2×C3×Q8—22no—
162C2×C3×Q8—22no—
163C2×C3×Q8C2×D414yes—
164C22×C3×C4—22no—
165C22×C3×C4—21no—
166C22×C3×C4—21no—
167C22×C3×C4—21no—
168C22×C3×C4—21no—
169C22×C3×C4—21no—
170C22×C3×C4—21no—
171C22×C3×C4—22no—
172C22×C3×C4C2×D411yescharacteristic
173C22×C3×C4C2411yescharacteristic
174C2×#67—22no—
175C2×#67—42no—
176C2×#67—44no—
177C2×#67—44no—
178C2×#67—82no—
179C22×Dic3—22no—
180C22×Dic3C2×D411yescharacteristic
181C22×Dic3C2×D411yescharacteristic
182C22×Dic3C2×D411yescharacteristic
183#239—216no—
184#239—48no—
185#239—48no—
186#239—48no—
187C2×D12—22no—
188C2×D12—22no—
189C2×D12—22no—
190C2×D12—22no—
191C2×D12—22no—
192C2×D12—44no—
193C2×D12—44no—
194C2×D12—44no—
195C2×D12C2×D414yes—
196C2×C4×D3—21no—
197C2×C4×D3—21no—
198C2×C4×D3—21no—
199C2×C4×D3—21no—
200C2×C4×D3—44no—
201C2×C4×D3—44no—
202C2×C4×D3—44no—
203C2×C4×D3C2×D412yes—
204C2×C4×D3#4112yes—
205C3×Q16—28no—
206C3×M16—44no—
207C2×C3×C8—21no—
208C2×C3×C8—21no—
209C2×C3×C8—21no—
210C2×C3×C8C2×D412yes—
211C2×C3×C8C2×D412yes—
212C2×C3×C8C2×D412yes—
213C3×C4⋊C4—22no—
214C3×C4⋊C4—22no—
215C3×C4⋊C4—44no—
216C3×C22⋊C4—44no—
217#217—22no—
218#215—216no—
219(C2×D3)⋊C4—22no—
220(C2×D3)⋊C4—44no—
221(C2×D3)⋊C4—44no—
222(C2×D3)⋊C4—82no—
223(C2×D3)⋊C4—82no—
224C4⋊Dic3—22no—
225C4⋊Dic3—22no—
226Dic3⋊C4—22no—
227C4×Dic3—21no—
228C4×Dic3—21no—
229C2×C3⋊C8—21no—
230C2×C3⋊C8—21no—
231C2×C3⋊C8—21no—
232C2×C3⋊C8—21no—
233C2×C3⋊C8—21no—
234C2×C3⋊C8C2×D412yes—
235D24—28no—
236D3×C8—28no—
237C2×#41—64no—
238C2×#41—64no—
239C2×#41—64no—
240C2×#41—64no—
241C2×#41—122no—
242C2×#41—122no—
243C22×Q8C22×D312yes—
244C22×D4—31no—
245C22×D4—32no—
246C22×D4—62no—
247C22×D4—62no—
248C23×C4—31no—
249C23×C4—62no—
250C23×C4C22×D311yescharacteristic
251#135—632no—
252C2×Q16—22no—
253C2×Q16—22no—
254C2×Q16#6714yescomplemented
255C2×SD16—64no—
256C2×SD16—64no—
257C2×SD16—64no—
258C2×D8—34no—
259C2×D8—62no—
260C2×D8—62no—
261C2×M16—22no—
262C2×M16—22no—
263C2×M16—22no—
264C22×C8—31no—
265C22×C8—34no—
266C22×C8—31no—
267C22×C8—62no—
268C22×C8—62no—
269C22×C8C22×D311yescharacteristic
270C22×C8C22×D311yescharacteristic
271C22⋊Q8—24no—
272C22⋊Q8—42no—
273#121—64no—
274#121—122no—
275C4×D4—62no—
276C4×D4—62no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—31no—
279C2×C4⋊C4—32no—
280C2×C4⋊C4C22×D312yes—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—31no—
283C2×C22⋊C4—31no—
284C2×C22⋊C4—62no—
285C2×C22⋊C4—62no—
286C2×C22⋊C4—62no—
287C2×C22⋊C4—62no—
288C2×C42—31no—
289#106—22no—
290#106—62no—
291C4⋊C8—62no—
292#103—24no—
293#103—122no—
294#102—64no—
295#102—122no—
296C22⋊C8—62no—
297C22⋊C8—62no—
298#97—62no—
299#95—62no—
300#95—62no—
301#95—62no—
302C23×C3—21no—
303C23×C3—21no—
304C23×C3—21no—
305C23×C3—22no—
306C23×C3—22no—
307C23×C3C22×D411yescharacteristic
308C22×D3—21no—
309C22×D3—21no—
310C22×D3—22no—
311C22×D3—22no—
312C22×D3—24no—
313C22×D3—42no—
314C22×D3—42no—
315C22×D3—42no—
316C22×D3—42no—
317C22×D3—42no—
318C22×D3—42no—
319C22×D3#12112yes—
320C3×Q8—24no—
321C3×Q8—24no—
322C3×Q8—42no—
323C3×Q8—42no—
324C2×C3×C4—21no—
325C2×C3×C4—21no—
326C2×C3×C4—24no—
327C2×C3×C4—21no—
328C2×C3×C4—42no—
329C2×C3×C4—42no—
330C2×C3×C4—41no—
331C2×C3×C4—41no—
332C2×C3×C4—41no—
333C2×C3×C4—41no—
334C2×C3×C4—42no—
335C2×C3×C4—42no—
336C2×C3×C4—42no—
337C2×C3×C4C22×D411yescharacteristic
338C2×C3×C4C22×D411yescharacteristic
339C2×C3×C4C22×D411yescharacteristic
340C2×C3×C4C2×#4111yescharacteristic
341C2×C3×C4H(5,𝔽2)11yescharacteristic
342C2×C3×C4H(5,𝔽2)11yescharacteristic
343#67—44no—
344#67—88no—
345C2×Dic3—21no—
346C2×Dic3—21no—
347C2×Dic3—21no—
348C2×Dic3—21no—
349C2×Dic3—22no—
350C2×Dic3—22no—
351C2×Dic3—42no—
352C2×Dic3—42no—
353C2×Dic3—42no—
354C2×Dic3#12112yes—
355D12—24no—
356D12—24no—
357D12—416no—
358D12—42no—
359D12—42no—
360C4×D3—22no—
361C4×D3—22no—
362C4×D3—22no—
363C4×D3—22no—
364C4×D3—416no—
365C3×C8—41no—
366C3×C8#12014yes—
367C3⋊C8—22no—
368C3⋊C8—41no—
369C24—31no—
370C24—62no—
371C24D3×D411yescharacteristic
372#41—616no—
373#41—128no—
374#41—128no—
375#41—128no—
376C2×Q8—22no—
377C2×Q8—22no—
378C2×Q8—22no—
379C2×Q8—22no—
380C2×Q8—22no—
381C2×Q8C2×#6714yes—
382C2×D4—34no—
383C2×D4—62no—
384C2×D4—62no—
385C2×D4—62no—
386C2×D4—62no—
387C2×D4—62no—
388C2×D4—62no—
389C2×D4—124no—
390C2×D4—124no—
391C2×D4—124no—
392C2×D4—122no—
393C2×D4—124no—
394C2×D4—124no—
395C2×D4—242no—
396C22×C4—22no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—31no—
405C22×C4—32no—
406C22×C4—32no—
407C22×C4—31no—
408C22×C4—31no—
409C22×C4—61no—
410C22×C4—61no—
411C22×C4—61no—
412C22×C4—61no—
413C22×C4—62no—
414C22×C4—124no—
415C22×C4—124no—
416C22×C4—124no—
417C22×C4D3×D411yescharacteristic
418C22×C4C23×D311yescharacteristic
419Q16—28no—
420SD16—616no—
421D8—68no—
422M16—44no—
423C2×C8—21no—
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—32no—
427C2×C8—61no—
428C2×C8—61no—
429C2×C8—68no—
430C2×C8—61no—
431C2×C8—61no—
432C2×C8—61no—
433C2×C8D3×D412yes—
434C2×C8D3×D412yes—
435C2×C8C2×#6712yes—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—44no—
439C4⋊C4—62no—
440C4⋊C4—62no—
441C4⋊C4—62no—
442C22⋊C4—44no—
443C22⋊C4—62no—
444C22⋊C4—62no—
445C22⋊C4—124no—
446C22⋊C4—124no—
447C22⋊C4—242no—
448C22⋊C4—242no—
449C42—61no—
450C42—61no—
451C22×C3—41no—
452C22×C3—42no—
453C22×C3—42no—
454C22×C3—41no—
455C22×C3—41no—
456C22×C3C2×#12111yescharacteristic
457C22×C3D4211yescharacteristic
458C22×C3D4211yescharacteristic
459C22×C3#57512yes—
460C22×C3#57712yes—
461C2×D3—22no—
462C2×D3—22no—
463C2×D3—22no—
464C2×D3—42no—
465C2×D3—42no—
466C2×D3—44no—
467C2×D3—44no—
468C2×D3—44no—
469C2×D3—82no—
470C2×D3—82no—
471C2×D3—82no—
472C2×D3—82no—
473C3×C4—44no—
474C3×C4—41no—
475C3×C4—82no—
476C3×C4C2×#12011yescharacteristic
477C3×C4#53511yescharacteristic
478C3×C4#54611yescharacteristic
479C3×C4#54611yescharacteristic
480Dic3—22no—
481Dic3—41no—
482Dic3—41no—
483Dic3—82no—
484C23—22no—
485C23—22no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—32no—
490C23—62no—
491C23—62no—
492C23—61no—
493C23—61no—
494C23—64no—
495C23—122no—
496C23—122no—
497C23—122no—
498C23—122no—
499C23—122no—
500C23—122no—
501C23C2×D3×D411yescharacteristic
502Q8—24no—
503Q8—24no—
504Q8—42no—
505Q8—42no—
506D4—64no—
507D4—64no—
508D4—1216no—
509D4—124no—
510D4—122no—
511D4—122no—
512D4—248no—
513C2×C4—24no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—32no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—61no—
528C2×C4—61no—
529C2×C4—62no—
530C2×C4—62no—
531C2×C4—62no—
532C2×C4—62no—
533C2×C4—62no—
534C2×C4—62no—
535C2×C4—61no—
536C2×C4—61no—
537C2×C4—1216no—
538C2×C4—122no—
539C2×C4—122no—
540C2×C4—122no—
541C2×C4C2×D3×D411yescharacteristic
542C2×C4C2×D3×D411yescharacteristic
543C2×C4#100411yescharacteristic
544C2×C4D3×#4111yescharacteristic
545C2×C4#100911yescharacteristic
546C2×C4C22×#6711yescharacteristic
547C8—41no—
548C8—62no—
549C8—121no—
550C8#93714yes—
551C2×C3—81no—
552C2×C3#240811yescharacteristic
553C2×C3#305411yescharacteristic
554C2×C3#306011yescharacteristic
555C2×C3#326712yes—
556C2×C3#327112yes—
557D3—42no—
558D3—84no—
559C22—41no—
560C22—41no—
561C22—42no—
562C22—42no—
563C22—41no—
564C22—62no—
565C22—62no—
566C22—62no—
567C22—122no—
568C22—122no—
569C22—124no—
570C22—124no—
571C22—124no—
572C22—242no—
573C22—242no—
574C22—242no—
575C22—242no—
576C22D3×#12111yescharacteristic
577C22#560911yescharacteristic
578C22#577712yes—
579C22#577812yes—
580C22D4×#6711yescharacteristic
581C4—41no—
582C4—44no—
583C4—62no—
584C4—82no—
585C4—121no—
586C4—121no—
587C4—242no—
588C4#560711yescharacteristic
589C4#561211yescharacteristic
590C4C2×#93711yescharacteristic
591C4#582911yescharacteristic
592C3#2427111yescharacteristic, complemented, Sylow
593C2—81no—
594C2—122no—
595C2—244no—
596C2#8090311yescharacteristic
597C2#8090611yescharacteristic
598C2#8092112yes—
599C2#8264011yescharacteristic
600C2#8399012yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#470C2
socleC23×C3C22×D4
hypercenterC2×#470D3
hypocenterC3#24271
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#470

Complements

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

NH ≅ G/Ntype
1. C22×#919C2semidirect
2. C2×C3×#470C2semidirect
3. C2×#5135C2semidirect
4. C2×#5127C2semidirect
5. #82812C2semidirect
6. #72067C2semidirect
7. #71699C2semidirect
8. #70577C2semidirect
9. #70576C2semidirect
10. #70280C2semidirect
11. #70267C2semidirect
47. C2×#470D3semidirect
73. C2×C3×Q16D4semidirect
105. C2×D24D4semidirect
108. C2×D3×C8D4semidirect
254. C2×Q16#67semidirect
592. C3#24271semidirect

Maximal subgroups

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

subgroupindex
C22×#9192
C2×C3×#4702
C2×#51352
C2×#51272
#828122
#720672
#716992
#705772
#705762
#702802
#702672
#242713