Finite Group Databasebeta

← #10828787

Subgroups — #10828787

Order 768.

1164 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5514C211yesmaximal, characteristic
2C22×C3×#106C211yesmaximal, characteristic, complemented
3#82196C211yesmaximal, characteristic, complemented
4C2×#4967C211yesmaximal, characteristic, complemented
5#71975C211yesmaximal, characteristic, complemented
6#71973C211yesmaximal, characteristic, complemented
7#71554C211yesmaximal, characteristic, complemented
8#70584C212yesmaximal, complemented
9#70581C212yesmaximal, complemented
10#70580C212yesmaximal, complemented
11#70566C212yesmaximal, complemented
12#24091—31nomaximal, Sylow
13C23×C3×C8C2211yescharacteristic
14C22×C3×C4⋊C4C2211yescharacteristic
15C22×C3×C4⋊C4C2211yescharacteristic
16C2×#930—21no—
17C2×#930C2211yescharacteristic
18C2×#929—21no—
19C2×C4×#67—21no—
20C2×C4×#67C2211yescharacteristic
21#5514—24no—
22#5514—24no—
23C2×#898—21no—
24C2×#895C2211yescharacteristic
25C2×#894—21no—
26C2×#893C2211yescharacteristic
27C2×#892C2211yescharacteristic
28C2×#890C2211yescharacteristic
29C2×C3×#106—24no—
30C2×C3×#106C2212yes—
31C2×C3×#106C2212yes—
32#5212C2211yescharacteristic
33#5211—21no—
34C2×#830C2211yescharacteristic
35C2×#821C2212yes—
36C2×(C2×D3)⋊C8C2212yes—
37#4992—21no—
38#4987C2212yes—
39#4983—21no—
40#4972C2212yes—
41#4967—24no—
42C2×#809C2212yes—
43C2×#807C2212yes—
44#4551C2212yes—
45#4550C2212yes—
46#4529C2212yes—
47#4529C2212yes—
48C2×#553—31no—
49C2×#480—31no—
50C22×#106D311yescharacteristic, complemented
51#2486—31no—
52#2059—32no—
53#2034—32no—
54#1893—31no—
55#1869—31no—
56#1864—32no—
57#1854—32no—
58#1824—31no—
59C23×C3×C4—21no—
60C23×C3×C4—21no—
61C23×C3×C4C2311yescharacteristic
62C22×#67—21no—
63C22×#67D411yescharacteristic
64C22×D12—21no—
65C22×D12C2311yescharacteristic
66C22×C4×D3—21no—
67C22×C4×D3C2311yescharacteristic
68C22×C3×C8—22no—
69C22×C3×C8—24no—
70C22×C3×C8C2312yes—
71C2×C3×C4⋊C4—21no—
72C2×C3×C4⋊C4—22no—
73C2×C3×C4⋊C4—22no—
74C2×C3×C4⋊C4—22no—
75C2×C3×C4⋊C4—21no—
76C2×C3×C4⋊C4D414yes—
77C2×C3×C4⋊C4C2312yes—
78C2×C3×C4⋊C4C2311yescharacteristic
79C2×C3×C4⋊C4C2311yescharacteristic
80C2×#217—21no—
81C2×#217D411yescharacteristic
82#930—28no—
83#930—22no—
84#930D414yes—
85#929—44no—
86C4×#67—28no—
87C4×#67—22no—
88C4×#67—22no—
89C2×(C2×D3)⋊C4—21no—
90C2×(C2×D3)⋊C4—21no—
91C2×(C2×D3)⋊C4—21no—
92C2×(C2×D3)⋊C4—21no—
93C2×(C2×D3)⋊C4—22no—
94C2×(C2×D3)⋊C4D411yescharacteristic
95C2×C4⋊Dic3—21no—
96C2×C4⋊Dic3C2312yes—
97C2×C4⋊Dic3C2311yescharacteristic
98C2×Dic3⋊C4—21no—
99C2×Dic3⋊C4—22no—
100C2×Dic3⋊C4D411yescharacteristic
101C2×C4×Dic3—21no—
102C2×C4×Dic3C2311yescharacteristic
103C22×C3⋊C8C2312yes—
104#898—28no—
105#895—22no—
106#895—22no—
107#894—28no—
108#893—22no—
109#893—22no—
110#892—22no—
111#892—22no—
112#890—22no—
113#890—22no—
114C3×#106—24no—
115C3×#106—24no—
116C3⋊#95—21no—
117C3⋊#95—22no—
118C3⋊#95—22no—
119#830—22no—
120#821—24no—
121(C2×D3)⋊C8—24no—
122#809—24no—
123#807—24no—
124C23×C8C2×D311yescharacteristic
125#553—64no—
126#553—64no—
127C2×#126—61no—
128C2×#125—31no—
129C2×#123—61no—
130C2×#123—61no—
131C2×#121—31no—
132C2×#121—31no—
133C2×#121—61no—
134C2×C4×D4—31no—
135C2×C4×D4—31no—
136C2×C4×D4—61no—
137C2×#117—31no—
138C22×C4⋊C4C2×D311yescharacteristic
139C22×C4⋊C4C2×D311yescharacteristic
140#480—64no—
141C2×#107—32no—
142C2×#106—24no—
143C2×#106C2×D312yes—
144C2×#106C2×D312yes—
145C2×#102—32no—
146C2×#102—32no—
147C2×C22⋊C8—31no—
148C2×C22⋊C8—32no—
149#398—61no—
150#397—61no—
151#389—32no—
152#381—61no—
153#380—31no—
154#380—32no—
155#340—32no—
156#340—32no—
157#340—32no—
158#336—32no—
159C24×C3C2×D411yescharacteristic
160C23×D3—21no—
161C23×D3C2×D411yescharacteristic
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—21no—
165C22×C3×C4—21no—
166C22×C3×C4—24no—
167C22×C3×C4—22no—
168C22×C3×C4—42no—
169C22×C3×C4—42no—
170C22×C3×C4—42no—
171C22×C3×C4—82no—
172C22×C3×C4C2×D411yescharacteristic
173C22×C3×C4C2×D412yes—
174C22×C3×C4C2×D412yes—
175C22×C3×C4C2411yescharacteristic
176C2×#67—22no—
177C2×#67—24no—
178C2×#67—22no—
179C2×#67—22no—
180C2×#67—42no—
181C2×#67—42no—
182C22×Dic3—21no—
183C22×Dic3—22no—
184C22×Dic3C2×D411yescharacteristic
185C22×Dic3C2×D411yescharacteristic
186C22×Dic3C2×D411yescharacteristic
187C2×D12—22no—
188C2×D12—22no—
189C2×D12—21no—
190C2×D12—24no—
191C2×D12—82no—
192C2×D12C2×D412yes—
193C2×C4×D3—21no—
194C2×C4×D3—21no—
195C2×C4×D3—22no—
196C2×C4×D3—22no—
197C2×C4×D3—24no—
198C2×C4×D3—82no—
199C2×C3×C8—28no—
200C2×C3×C8—22no—
201C2×C3×C8—22no—
202C2×C3×C8#4114yes—
203C3×C4⋊C4—24no—
204C3×C4⋊C4—21no—
205C3×C4⋊C4—21no—
206C3×C4⋊C4—22no—
207C3×C4⋊C4—22no—
208C3×C4⋊C4—42no—
209C3×C4⋊C4C2×D412yes—
210C3×C4⋊C4C2×D412yes—
211#217—22no—
212#217—42no—
213(C2×D3)⋊C4—24no—
214(C2×D3)⋊C4—24no—
215(C2×D3)⋊C4—24no—
216(C2×D3)⋊C4—22no—
217(C2×D3)⋊C4—24no—
218(C2×D3)⋊C4—44no—
219C4⋊Dic3—24no—
220C4⋊Dic3—21no—
221C4⋊Dic3—22no—
222C4⋊Dic3—22no—
223C4⋊Dic3C2×D412yes—
224Dic3⋊C4—24no—
225Dic3⋊C4—22no—
226Dic3⋊C4—44no—
227C4×Dic3—24no—
228C4×Dic3—21no—
229C4×Dic3—21no—
230C2×C3⋊C8—22no—
231C2×C3⋊C8—22no—
232C2×C3⋊C8—22no—
233C22×D4—31no—
234C22×D4—31no—
235C22×D4—61no—
236C22×D4—61no—
237C23×C4—21no—
238C23×C4—21no—
239C23×C4—31no—
240C23×C4—61no—
241C23×C4C22×D311yescharacteristic
242C22×C8—22no—
243C22×C8—24no—
244C22×C8—32no—
245C22×C8C22×D312yes—
246#126—68no—
247#125—62no—
248#125—62no—
249#123—68no—
250#123—124no—
251#121—34no—
252#121—62no—
253#121—68no—
254#121—62no—
255#121—62no—
256C4×D4—62no—
257C4×D4—68no—
258C4×D4—62no—
259C4×D4—62no—
260C4×D4—62no—
261#117—62no—
262#117—62no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—31no—
269C2×C4⋊C4—31no—
270C2×C4⋊C4—32no—
271C2×C4⋊C4—61no—
272C2×C4⋊C4—61no—
273C2×C4⋊C4—62no—
274C2×C4⋊C4#6714yes—
275C2×C4⋊C4C22×D312yes—
276C2×C4⋊C4C22×D311yescharacteristic
277C2×C4⋊C4C22×D311yescharacteristic
278C2×C22⋊C4—31no—
279C2×C22⋊C4—31no—
280C2×C22⋊C4—61no—
281C2×C22⋊C4—61no—
282C2×C22⋊C4—61no—
283C2×C22⋊C4—61no—
284C2×C22⋊C4—62no—
285C2×C22⋊C4—61no—
286C2×C42—31no—
287C2×C42—61no—
288#107—64no—
289#106—24no—
290#106—24no—
291#102—64no—
292#102—64no—
293C22⋊C8—62no—
294C22⋊C8—64no—
295#95—62no—
296#95—62no—
297#95—61no—
298C23×C3—22no—
299C23×C3—22no—
300C23×C3#12012yes—
301C23×C3#13512yes—
302C23×C3AGL(1,ℤ/8)12yes—
303C23×C3C22×D411yescharacteristic
304C22×D3—21no—
305C22×D3—22no—
306C22×D3—22no—
307C22×D3—21no—
308C22×D3—21no—
309C22×D3—22no—
310C22×D3—42no—
311C22×D3—41no—
312C22×D3—41no—
313C22×D3—42no—
314C2×C3×C4—22no—
315C2×C3×C4—22no—
316C2×C3×C4—22no—
317C2×C3×C4—24no—
318C2×C3×C4—22no—
319C2×C3×C4—22no—
320C2×C3×C4—42no—
321C2×C3×C4—42no—
322C2×C3×C4—42no—
323C2×C3×C4—42no—
324C2×C3×C4—42no—
325C2×C3×C4—84no—
326C2×C3×C4—84no—
327C2×C3×C4#12012yes—
328C2×C3×C4C22×D411yescharacteristic
329C2×C3×C4C22×D411yescharacteristic
330C2×C3×C4C2×#4112yes—
331C2×C3×C42−1+412yes—
332#67—44no—
333#67—48no—
334C2×Dic3—21no—
335C2×Dic3—21no—
336C2×Dic3—21no—
337C2×Dic3—22no—
338C2×Dic3—22no—
339C2×Dic3—21no—
340C2×Dic3—21no—
341C2×Dic3—22no—
342C2×Dic3—41no—
343C2×Dic3—41no—
344C2×Dic3—42no—
345C2×Dic3—42no—
346C2×Dic3—42no—
347D12—24no—
348D12—41no—
349D12—41no—
350D12—84no—
351C4×D3—41no—
352C4×D3—41no—
353C4×D3—42no—
354C4×D3—84no—
355C3×C8—24no—
356C3⋊C8—42no—
357C24—31no—
358C24—61no—
359C24D3×D411yescharacteristic
360C2×D4—32no—
361C2×D4—64no—
362C2×D4—62no—
363C2×D4—62no—
364C2×D4—64no—
365C2×D4—62no—
366C2×D4—62no—
367C2×D4—62no—
368C2×D4—61no—
369C2×D4—122no—
370C2×D4—122no—
371C2×D4—242no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—31no—
379C22×C4—31no—
380C22×C4—31no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—64no—
385C22×C4—61no—
386C22×C4—62no—
387C22×C4—61no—
388C22×C4—62no—
389C22×C4—61no—
390C22×C4—62no—
391C22×C4—82no—
392C22×C4—242no—
393C22×C4D3×D411yescharacteristic
394C22×C4D3×D412yes—
395C22×C4C2×#6712yes—
396C22×C4C23×D311yescharacteristic
397C2×C8—28no—
398C2×C8—22no—
399C2×C8—22no—
400C2×C8—62no—
401C2×C8—62no—
402C2×C8—62no—
403C2×C8#23514yes—
404C4⋊C4—24no—
405C4⋊C4—21no—
406C4⋊C4—21no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—32no—
410C4⋊C4—42no—
411C4⋊C4—64no—
412C4⋊C4—64no—
413C4⋊C4—62no—
414C4⋊C4—61no—
415C4⋊C4—62no—
416C4⋊C4—62no—
417C4⋊C4—124no—
418C4⋊C4C2×#6712yes—
419C4⋊C4C2×#6712yes—
420C22⋊C4—64no—
421C22⋊C4—64no—
422C22⋊C4—64no—
423C22⋊C4—62no—
424C22⋊C4—62no—
425C22⋊C4—64no—
426C22⋊C4—124no—
427C22⋊C4—122no—
428C42—64no—
429C42—61no—
430C42—61no—
431C22×C3—24no—
432C22×C3—24no—
433C22×C3—22no—
434C22×C3—22no—
435C22×C3#44914yes—
436C22×C3C2×#12011yescharacteristic
437C22×C3#54712yes—
438C22×C3C2×#13511yescharacteristic
439C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
440C22×C3#57712yes—
441C2×D3—42no—
442C2×D3—42no—
443C2×D3—41no—
444C2×D3—42no—
445C2×D3—41no—
446C2×D3—42no—
447C2×D3—42no—
448C2×D3—41no—
449C2×D3—82no—
450C2×D3—82no—
451C3×C4—22no—
452C3×C4—42no—
453C3×C4—42no—
454C3×C4—82no—
455C3×C4#52511yescharacteristic
456C3×C4#53611yescharacteristic
457C3×C4#54712yes—
458Dic3—41no—
459Dic3—41no—
460Dic3—42no—
461Dic3—41no—
462Dic3—82no—
463C23—22no—
464C23—22no—
465C23—61no—
466C23—61no—
467C23—62no—
468C23—62no—
469C23—62no—
470C23—61no—
471C23—121no—
472C23—121no—
473C23—122no—
474C23—122no—
475C23#91412yes—
476C23#91612yes—
477C23#93712yes—
478C23C2×D3×D411yescharacteristic
479D4—64no—
480D4—124no—
481D4—128no—
482D4—121no—
483D4—121no—
484D4—244no—
485C2×C4—22no—
486C2×C4—24no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—61no—
497C2×C4—61no—
498C2×C4—62no—
499C2×C4—62no—
500C2×C4—61no—
501C2×C4—62no—
502C2×C4—61no—
503C2×C4—61no—
504C2×C4—84no—
505C2×C4—84no—
506C2×C4—121no—
507C2×C4—121no—
508C2×C4—122no—
509C2×C4—121no—
510C2×C4—121no—
511C2×C4—122no—
512C2×C4—122no—
513C2×C4—122no—
514C2×C4—244no—
515C2×C4#93712yes—
516C2×C4C2×#23512yes—
517C2×C4C2×D3×D411yescharacteristic
518C2×C4#100712yes—
519C2×C4C22×#6711yescharacteristic
520C8—24no—
521C8—122no—
522C2×C3—24no—
523C2×C3#240611yescharacteristic
524C2×C3C2×#44911yescharacteristic
525C2×C3#300911yescharacteristic
526C2×C3#333712yes—
527C2×C3#334512yes—
528D3—81no—
529D3—82no—
530C22—22no—
531C22—24no—
532C22—24no—
533C22—22no—
534C22—121no—
535C22—122no—
536C22—122no—
537C22—122no—
538C22—121no—
539C22—122no—
540C22—122no—
541C22—121no—
542C22—242no—
543C22—242no—
544C22#517214yes—
545C22#550712yes—
546C22C2×#91411yescharacteristic
547C22C2×#91611yescharacteristic
548C22#576212yes—
549C22C2×#93711yescharacteristic
550C4—22no—
551C4—42no—
552C4—42no—
553C4—82no—
554C4—121no—
555C4—121no—
556C4—122no—
557C4—121no—
558C4—242no—
559C4#550712yes—
560C4#579111yescharacteristic
561C4#581611yescharacteristic
562C3#2409111yescharacteristic, complemented, Sylow
563C2—24no—
564C2—241no—
565C2—242no—
566C2#7912512yes—
567C2#7912712yes—
568C2#8219211yescharacteristic
569C2C2×#517211yescharacteristic
570C2#8385911yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC22×C3×#106C2
socleC23×C3C22×D4
prime radicalC2×#5514C2
hypercenterC22×#106D3
hypocenterC3#24091
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#106

Complements

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

NH ≅ G/Ntype
2. C22×C3×#106C2semidirect
3. #82196C2semidirect
4. C2×#4967C2semidirect
5. #71975C2semidirect
6. #71973C2semidirect
7. #71554C2semidirect
8. #70584C2semidirect
9. #70581C2semidirect
10. #70580C2semidirect
11. #70566C2semidirect
50. C22×#106D3semidirect
562. C3#24091semidirect

Maximal subgroups

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

subgroupindex
C2×#55142
C22×C3×#1062
#821962
C2×#49672
#719752
#719732
#715542
#705842
#705812
#705802
#705662
#240913