Finite Group Databasebeta

← #10845941

Subgroups — #10845941

Order 768.

1154 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82500C211yesmaximal, characteristic, complemented
2C2×#5027C211yesmaximal, characteristic, complemented
3C2×#4807C212yesmaximal, complemented
4C2×C8×D12C211yesmaximal, characteristic, complemented
5#73799C211yesmaximal, characteristic
6C3×C4×#103C211yesmaximal, characteristic, complemented
7#70272C212yesmaximal, complemented
8#69772C211yesmaximal, characteristic, complemented
9#69467C211yesmaximal, characteristic, complemented
10#68883C212yesmaximal, complemented
11#68823C212yesmaximal, complemented
12#25387—31nomaximal, Sylow
13C2×C3×C4×Q8C2211yescharacteristic
14C2×#946C2212yes—
15C22×#215C2212yes—
16C22×D3×C8C2212yes—
17C2×#895C2212yes—
18C2×#895C2212yes—
19C2×#874—21no—
20C2×#874C2211yescharacteristic
21C2×C4×D12C2211yescharacteristic
22C2×C3×#103C2211yescharacteristic
23C2×C3×#103C2211yescharacteristic
24C2×C3×C4×C8C2211yescharacteristic
25C3×C4×C4⋊C4C2211yescharacteristic
26C2×#821C2212yes—
27C2×#821C2212yes—
28C2×(C2×D3)⋊C8C2212yes—
29#5027—24no—
30#4994—22no—
31#4987C2212yes—
32C2×#809C2212yes—
33C2×#808C2212yes—
34#4939—21no—
35#4939—21no—
36#4935C2211yescharacteristic
37#4928—21no—
38C2×C4⋊(C3⋊C8)C2211yescharacteristic
39#4807—28no—
40C8×D12—24no—
41#4671—22no—
42C3×#340C2211yescharacteristic
43C3×#340C2211yescharacteristic
44#4529C2212yes—
45C2×#465—32no—
46C2×#460—31no—
47C2×C8×D4—31no—
48#2541—31no—
49#2538—31no—
50#2034—32no—
51#2000—32no—
52#1957—31no—
53#1948—31no—
54#1893—32no—
55C4×#103D311yescharacteristic, complemented
56C22×C3×Q8C2311yescharacteristic
57C22×D12—21no—
58C22×D12—22no—
59C22×D12—22no—
60C22×D12—21no—
61C22×D12D411yescharacteristic
62C22×D12C2312yes—
63C22×C4×D3C2312yes—
64C22×C3×C8C2311yescharacteristic
65C22×C3×C8C2311yescharacteristic
66C3×C4×Q8—22no—
67C3×C4×Q8—22no—
68C2×C3×C4⋊C4—21no—
69C2×C3×C4⋊C4C2311yescharacteristic
70C2×C3×C4⋊C4C2311yescharacteristic
71C2×C3×C4⋊C4C2311yescharacteristic
72C2×C3×C42—21no—
73C2×C3×C42—21no—
74C2×C3×C42C2311yescharacteristic
75#946—24no—
76#946—42no—
77C2×#215—24no—
78C2×#215—24no—
79C2×#215—42no—
80C2×(C2×D3)⋊C4—22no—
81C2×(C2×D3)⋊C4—22no—
82C2×(C2×D3)⋊C4—22no—
83C2×(C2×D3)⋊C4—22no—
84C2×(C2×D3)⋊C4D412yes—
85C2×C4⋊Dic3—21no—
86C2×C4⋊Dic3—21no—
87C2×C4⋊Dic3D411yescharacteristic
88C2×C4⋊Dic3C2312yes—
89C2×Dic3⋊C4—22no—
90C22×C3⋊C8C2312yes—
91C2×D3×C8—24no—
92C2×D3×C8—24no—
93C2×D3×C8—42no—
94#895—24no—
95#895—24no—
96#895—42no—
97#895—42no—
98#874—22no—
99#874—22no—
100#874—44no—
101C4×D12—22no—
102C4×D12—22no—
103C3×#103—22no—
104C3×#103D414yescomplemented
105C3×C4×C8—22no—
106C3×#95—21no—
107C3⋊#95—22no—
108#821—24no—
109#821—42no—
110(C2×D3)⋊C8—24no—
111#809—24no—
112#808—24no—
113C4⋊(C3⋊C8)D414yescomplemented
114C22×SD16—32no—
115C23×C8—32no—
116C2×#127—31no—
117C2×#127—61no—
118C2×C22⋊Q8—32no—
119C2×#121—32no—
120C2×#121—32no—
121C2×C4×Q8C2×D311yescharacteristic
122C2×C4×D4—31no—
123#465—68no—
124#460—64no—
125C8×D4—64no—
126C2×#106—32no—
127C2×C4⋊C8—31no—
128C2×#103C2×D311yescharacteristic
129C2×#103C2×D311yescharacteristic
130C2×#102—32no—
131C2×#102—32no—
132C2×#102—32no—
133C2×C22⋊C8—32no—
134C2×C4×C8C2×D311yescharacteristic
135#396—62no—
136#392—62no—
137#390—61no—
138#390—61no—
139#389—61no—
140#384—31no—
141#380—32no—
142C4×C4⋊C4C2×D311yescharacteristic
143#340—32no—
144#340C2×D311yescharacteristic
145#340C2×D311yescharacteristic
146C23×D3—22no—
147C23×D3C2×D412yes—
148C2×C3×Q8—21no—
149C2×C3×Q8—21no—
150C2×C3×Q8—21no—
151C2×C3×Q8—21no—
152C2×C3×Q8—21no—
153C2×C3×Q8C2×D412yes—
154C22×C3×C4—21no—
155C22×C3×C4—21no—
156C22×C3×C4—21no—
157C22×C3×C4—21no—
158C22×C3×C4C2×D411yescharacteristic
159C22×C3×C4C2×D411yescharacteristic
160C22×C3×C4C2411yescharacteristic
161C22×Dic3—22no—
162C22×Dic3C2×D412yes—
163C2×D12—22no—
164C2×D12—22no—
165C2×D12—22no—
166C2×D12—22no—
167C2×D12—22no—
168C2×D12—22no—
169C2×D12—22no—
170C2×D12—22no—
171C2×D12—44no—
172C2×D12—44no—
173C2×D12—44no—
174C2×D12—44no—
175C2×D12—42no—
176C2×D12—42no—
177C2×D12—42no—
178C2×D12—42no—
179C2×D12—81no—
180C2×D12—81no—
181C2×D12—82no—
182C2×D12—82no—
183C2×D12SD1614yescomplemented
184C2×C4×D3—22no—
185C2×C4×D3—22no—
186C2×C4×D3—22no—
187C2×C4×D3—22no—
188C2×C4×D3—22no—
189C2×C4×D3—22no—
190C2×C3×C8—21no—
191C2×C3×C8—21no—
192C2×C3×C8—21no—
193C2×C3×C8—21no—
194C2×C3×C8—21no—
195C2×C3×C8C2×D412yes—
196C3×C4⋊C4—21no—
197C3×C4⋊C4—21no—
198C3×C4⋊C4—21no—
199C3×C4⋊C4—21no—
200C3×C4⋊C4—42no—
201C3×C4⋊C4C2×D412yes—
202C3×C4⋊C4#4112yes—
203C3×C42—42no—
204C3×C42—42no—
205C3×C42C2×D412yes—
206C3×C42#4112yes—
207#215—48no—
208(C2×D3)⋊C4—24no—
209(C2×D3)⋊C4—44no—
210(C2×D3)⋊C4—44no—
211(C2×D3)⋊C4—44no—
212(C2×D3)⋊C4—82no—
213C4⋊Dic3—22no—
214C4⋊Dic3—22no—
215C4⋊Dic3—22no—
216C4⋊Dic3—42no—
217C4⋊Dic3—81no—
218Dic3⋊C4—44no—
219C2×C3⋊C8—22no—
220C2×C3⋊C8—22no—
221C2×C3⋊C8C2×D414yes—
222D3×C8—48no—
223C22×Q8C22×D311yescharacteristic
224C22×D4—31no—
225C22×D4—32no—
226C22×D4—62no—
227C22×D4—62no—
228C22×D4—61no—
229C22×D4—61no—
230C23×C4—32no—
231C2×SD16—64no—
232C2×SD16—64no—
233C2×SD16—122no—
234C22×C8—32no—
235C22×C8—64no—
236C22×C8—64no—
237C22×C8—122no—
238C22×C8C22×D311yescharacteristic
239C22×C8C22×D311yescharacteristic
240#127—62no—
241#127—62no—
242#127—124no—
243C22⋊Q8—64no—
244C22⋊Q8—122no—
245#121—64no—
246#121—64no—
247#121—122no—
248#121—122no—
249C4×Q8—22no—
250C4×Q8—22no—
251C4×D4—62no—
252C4×D4—62no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—31no—
255C2×C4⋊C4—32no—
256C2×C4⋊C4—62no—
257C2×C4⋊C4—61no—
258C2×C4⋊C4—61no—
259C2×C4⋊C4C22×D311yescharacteristic
260C2×C4⋊C4C22×D311yescharacteristic
261C2×C4⋊C4C22×D311yescharacteristic
262C2×C22⋊C4—32no—
263C2×C22⋊C4—62no—
264C2×C22⋊C4—62no—
265C2×C22⋊C4—62no—
266C2×C22⋊C4—62no—
267C2×C42—21no—
268C2×C42—21no—
269C2×C42C22×D311yescharacteristic
270#106—64no—
271C4⋊C8—34no—
272#103—22no—
273#103D1214yescomplemented
274#102—64no—
275#102—64no—
276#102—122no—
277C22⋊C8—64no—
278C4×C8—22no—
279#95—21no—
280#95—62no—
281C23×C3C22×D411yescharacteristic
282C22×D3—22no—
283C22×D3—22no—
284C22×D3—22no—
285C22×D3—22no—
286C22×D3—22no—
287C22×D3—22no—
288C22×D3—42no—
289C22×D3—42no—
290C22×D3—42no—
291C22×D3—42no—
292C22×D3—42no—
293C22×D3—42no—
294C22×D3—42no—
295C22×D3C2×SD1614yes—
296C3×Q8—22no—
297C3×Q8—22no—
298C3×Q8—41no—
299C3×Q8—41no—
300C2×C3×C4—21no—
301C2×C3×C4—21no—
302C2×C3×C4—22no—
303C2×C3×C4—22no—
304C2×C3×C4—22no—
305C2×C3×C4—41no—
306C2×C3×C4—41no—
307C2×C3×C4—41no—
308C2×C3×C4—41no—
309C2×C3×C4—41no—
310C2×C3×C4—41no—
311C2×C3×C4—41no—
312C2×C3×C4—41no—
313C2×C3×C4—41no—
314C2×C3×C4#12112yes—
315C2×C3×C4#12112yes—
316C2×C3×C4C2×SD1612yes—
317C2×C3×C4#13512yes—
318C2×C3×C4C22×D411yescharacteristic
319C2×C3×C4C22×D411yescharacteristic
320C2×C3×C4C2×#4111yescharacteristic
321C2×C3×C4C2×#4111yescharacteristic
322C2×C3×C4H(5,𝔽2)11yescharacteristic
323C2×C3×C4H(5,𝔽2)11yescharacteristic
324C2×Dic3—22no—
325C2×Dic3—22no—
326C2×Dic3—22no—
327C2×Dic3—42no—
328C2×Dic3—42no—
329C2×Dic3—42no—
330D12—28no—
331D12—42no—
332D12—42no—
333D12—42no—
334D12—42no—
335D12—84no—
336D12—88no—
337D12—88no—
338D12—84no—
339C4×D3—42no—
340C4×D3—42no—
341C4×D3—42no—
342C4×D3—42no—
343C3×C8—22no—
344C3×C8—41no—
345C3⋊C8—24no—
346C24—32no—
347C24—62no—
348C2×Q8—21no—
349C2×Q8—21no—
350C2×Q8—21no—
351C2×Q8—21no—
352C2×Q8—21no—
353C2×Q8C2×D1212yes—
354C2×D4—34no—
355C2×D4—62no—
356C2×D4—62no—
357C2×D4—62no—
358C2×D4—62no—
359C2×D4—62no—
360C2×D4—62no—
361C2×D4—62no—
362C2×D4—62no—
363C2×D4—124no—
364C2×D4—124no—
365C2×D4—124no—
366C2×D4—124no—
367C2×D4—122no—
368C2×D4—122no—
369C2×D4—122no—
370C2×D4—122no—
371C2×D4—241no—
372C2×D4—241no—
373C2×D4—242no—
374C2×D4—242no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—32no—
380C22×C4—62no—
381C22×C4—62no—
382C22×C4—62no—
383C22×C4—62no—
384C22×C4—62no—
385C22×C4—62no—
386C22×C4—62no—
387C22×C4D3×D411yescharacteristic
388C22×C4D3×D411yescharacteristic
389C22×C4C23×D311yescharacteristic
390SD16—128no—
391C2×C8—21no—
392C2×C8—21no—
393C2×C8—21no—
394C2×C8—21no—
395C2×C8—21no—
396C2×C8—34no—
397C2×C8—62no—
398C2×C8—62no—
399C2×C8—128no—
400C2×C8C2×D1212yes—
401C4⋊C4—21no—
402C4⋊C4—21no—
403C4⋊C4—21no—
404C4⋊C4—21no—
405C4⋊C4—42no—
406C4⋊C4—62no—
407C4⋊C4—62no—
408C4⋊C4—62no—
409C4⋊C4—124no—
410C4⋊C4—122no—
411C4⋊C4—241no—
412C4⋊C4C2×D1212yes—
413C4⋊C4#23512yes—
414C22⋊C4—64no—
415C22⋊C4—124no—
416C22⋊C4—124no—
417C22⋊C4—124no—
418C22⋊C4—242no—
419C42—42no—
420C42—42no—
421C42#23512yes—
422C42D3×D412yes—
423C22×C3C2×#12111yescharacteristic
424C22×C3D4211yescharacteristic
425C22×C3#54611yescharacteristic
426C22×C3C22×SD1611yescharacteristic
427C22×C3C2×#13511yescharacteristic
428C22×C3#57611yescharacteristic
429C22×C3#57711yescharacteristic
430C2×D3—24no—
431C2×D3—24no—
432C2×D3—24no—
433C2×D3—42no—
434C2×D3—42no—
435C2×D3—42no—
436C2×D3—42no—
437C2×D3—82no—
438C2×D3—82no—
439C2×D3—82no—
440C2×D3—82no—
441C2×D3—82no—
442C2×D3—82no—
443C2×D3—82no—
444C3×C4—22no—
445C3×C4—42no—
446C3×C4—42no—
447C3×C4—42no—
448C3×C4—81no—
449C3×C4#46514yescomplemented
450C3×C4#53211yescharacteristic
451C3×C4#54011yescharacteristic
452C3×C4D4211yescharacteristic
453C3×C4#55011yescharacteristic
454Dic3—42no—
455Dic3—82no—
456C23—34no—
457C23—62no—
458C23—62no—
459C23—62no—
460C23—62no—
461C23—62no—
462C23—62no—
463C23—122no—
464C23—122no—
465C23—122no—
466C23—122no—
467C23—122no—
468C23—122no—
469C23—122no—
470C23C2×D3×D411yescharacteristic
471Q8—22no—
472Q8—22no—
473Q8—41no—
474Q8—41no—
475D4—68no—
476D4—122no—
477D4—122no—
478D4—122no—
479D4—122no—
480D4—244no—
481D4—248no—
482D4—248no—
483D4—244no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—62no—
499C2×C4—62no—
500C2×C4—62no—
501C2×C4—122no—
502C2×C4—122no—
503C2×C4—122no—
504C2×C4—122no—
505C2×C4—122no—
506C2×C4—122no—
507C2×C4—122no—
508C2×C4#88412yes—
509C2×C4#88412yes—
510C2×C4D3×SD1612yes—
511C2×C4#91912yes—
512C2×C4C22×D1211yescharacteristic
513C2×C4C2×#23511yescharacteristic
514C2×C4C2×#23511yescharacteristic
515C2×C4C2×D3×D411yescharacteristic
516C2×C4#100911yescharacteristic
517C2×C4#100911yescharacteristic
518C8—22no—
519C8—41no—
520C8—64no—
521C2×C3#242811yescharacteristic
522C2×C3C2×#46511yescharacteristic
523C2×C3#306211yescharacteristic
524C2×C3D4×SD1611yescharacteristic
525C2×C3#328711yescharacteristic
526C2×C3#329511yescharacteristic
527C2×C3#330411yescharacteristic
528D3—44no—
529D3—162no—
530C22—64no—
531C22—64no—
532C22—64no—
533C22—122no—
534C22—122no—
535C22—122no—
536C22—122no—
537C22—242no—
538C22—242no—
539C22—242no—
540C22—242no—
541C22—242no—
542C22—242no—
543C22—242no—
544C22C2×#88411yescharacteristic
545C22D4×D1211yescharacteristic
546C22#555011yescharacteristic
547C22C2×D3×SD1611yescharacteristic
548C22C2×#91911yescharacteristic
549C22#576911yescharacteristic
550C22#577011yescharacteristic
551C4—22no—
552C4—42no—
553C4—42no—
554C4—42no—
555C4—81no—
556C4—122no—
557C4—242no—
558C4#480714yescomplemented
559C4#547711yescharacteristic
560C4#553511yescharacteristic
561C4D4×D1211yescharacteristic
562C4#555311yescharacteristic
563C3#2538711yescharacteristic, complemented, Sylow
564C2—124no—
565C2—482no—
566C2#7348711yescharacteristic
567C2C2×#480711yescharacteristic
568C2#7884011yescharacteristic
569C2SD16×D1211yescharacteristic
570C2#7967611yescharacteristic
571C2#7973411yescharacteristic
572C2#7973711yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×C4×#103C2
socleC23×C3C22×D4
prime radical#73799C2
hypercenterC4×#103D3
hypocenterC3#25387
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C4×#103

Complements

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

NH ≅ G/Ntype
1. #82500C2semidirect
2. C2×#5027C2semidirect
3. C2×#4807C2semidirect
4. C2×C8×D12C2semidirect
6. C3×C4×#103C2semidirect
7. #70272C2semidirect
8. #69772C2semidirect
9. #69467C2semidirect
10. #68883C2semidirect
11. #68823C2semidirect
55. C4×#103D3semidirect
104. C3×#103D4semidirect
113. C4⋊(C3⋊C8)D4semidirect
183. C2×D12SD16semidirect
273. #103D12semidirect
449. C3×C4#465semidirect
558. C4#4807semidirect
563. C3#25387semidirect

Maximal subgroups

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

subgroupindex
#825002
C2×#50272
C2×#48072
C2×C8×D122
#737992
C3×C4×#1032
#702722
#697722
#694672
#688832
#688232
#253873