Finite Group Databasebeta

← #10845943

Subgroups — #10845943

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
1C3×#2538C211yesmaximal, characteristic, complemented
2C2×#5048C212yesmaximal, complemented
3C2×#5027C211yesmaximal, characteristic, complemented
4#82378C211yesmaximal, characteristic
5C2×D4×C3⋊C8C211yesmaximal, characteristic, complemented
6#71699C212yesmaximal, complemented
7C4×#835C211yesmaximal, characteristic, complemented
8#69580C211yesmaximal, characteristic, complemented
9#69568C211yesmaximal, characteristic, complemented
10#69524C212yesmaximal, complemented
11#69515C212yesmaximal, complemented
12#25387—31nomaximal, Sylow
13C2×C3×C22⋊Q8C2212yes—
14C2×C3×C4×Q8C2211yescharacteristic
15C2×C3×C4×D4C2211yescharacteristic
16C22×#215C2212yes—
17C2×#940—21no—
18C2×#930C2212yes—
19C2×#930C2212yes—
20C23×C3⋊C8C2212yes—
21C2×#874C2211yescharacteristic
22C3×#396—22no—
23C3×#390—21no—
24C3×#389—21no—
25C3×#384C2211yescharacteristic
26C3×#380C2212yes—
27C2×#835C2211yescharacteristic
28C2×#835C2211yescharacteristic
29C2×#830C2212yes—
30#5048—28no—
31#5027—24no—
32D4×C3⋊C8—24no—
33#4988—21no—
34C2×#809C2212yes—
35C2×#809C2212yes—
36C2×#809C2212yes—
37C2×#808C2212yes—
38#4960—22no—
39C4×C4⋊Dic3C2211yescharacteristic
40C2×C4⋊(C3⋊C8)C2211yescharacteristic
41C2×C4×C3⋊C8C2211yescharacteristic
42#4529C2212yes—
43#4529C2211yescharacteristic
44#4529C2211yescharacteristic
45C2×#465—32no—
46C2×#460—31no—
47C2×C8×D4—31no—
48#2541—31no—
49#2538D311yescharacteristic, complemented
50#2034—32no—
51#2000—32no—
52#1957—31no—
53#1948—31no—
54#1893—32no—
55C4×#103—31no—
56C22×C3×Q8C2311yescharacteristic
57C22×C3×D4D411yescharacteristic
58C23×C3×C4C2312yes—
59C22×#67—22no—
60C22×#67—22no—
61C22×D12—21no—
62C22×D12—21no—
63C22×D12C2312yes—
64C3×C22⋊Q8—24no—
65C3×C22⋊Q8—42no—
66C3×C4×Q8—22no—
67C3×C4×Q8—22no—
68C3×C4×D4—22no—
69C3×C4×D4—22no—
70C2×C3×C4⋊C4—21no—
71C2×C3×C4⋊C4—22no—
72C2×C3×C4⋊C4—21no—
73C2×C3×C4⋊C4—21no—
74C2×C3×C4⋊C4D411yescharacteristic
75C2×C3×C4⋊C4C2312yes—
76C2×C3×C4⋊C4C2311yescharacteristic
77C2×C3×C22⋊C4—22no—
78C2×C3×C22⋊C4—22no—
79C2×C3×C22⋊C4D412yes—
80C2×C3×C42—21no—
81C2×C3×C42C2311yescharacteristic
82C2×#215—24no—
83C2×#215—24no—
84C2×#215—42no—
85#940—44no—
86#930—24no—
87#930—24no—
88#930—42no—
89#930—42no—
90C2×(C2×D3)⋊C4—22no—
91C2×(C2×D3)⋊C4—22no—
92C2×C4⋊Dic3C2311yescharacteristic
93C2×C4⋊Dic3C2311yescharacteristic
94C2×C4×Dic3—21no—
95C22×C3⋊C8—24no—
96C22×C3⋊C8—24no—
97C22×C3⋊C8—42no—
98C22×C3⋊C8C2311yescharacteristic
99C22×C3⋊C8C2311yescharacteristic
100C22×C3⋊C8C2312yes—
101#874—22no—
102#874—22no—
103C3×#95—22no—
104#835—22no—
105#835D414yescomplemented
106C3⋊#95—21no—
107#830—24no—
108#809—24no—
109#809—24no—
110#809—42no—
111#808—24no—
112C4⋊(C3⋊C8)D414yescomplemented
113C4×C3⋊C8—22no—
114C22×SD16—32no—
115C23×C8—32no—
116C2×#127—31no—
117C2×#127—61no—
118C2×C22⋊Q8C2×D312yes—
119C2×#121—32no—
120C2×#121—32no—
121C2×C4×Q8C2×D311yescharacteristic
122C2×C4×D4C2×D311yescharacteristic
123#465—68no—
124#460—64no—
125C8×D4—64no—
126C2×#106—32no—
127C2×C4⋊C8—31no—
128C2×#103—31no—
129C2×#103—31no—
130C2×#102—32no—
131C2×#102—32no—
132C2×#102—32no—
133C2×C22⋊C8—32no—
134C2×C4×C8—31no—
135#396—22no—
136#392—62no—
137#390—21no—
138#390—61no—
139#389—21no—
140#384C2×D311yescharacteristic
141#380C2×D312yes—
142C4×C4⋊C4—31no—
143#340—31no—
144#340—32no—
145#340—31no—
146C24×C3C2×D412yes—
147C23×D3—22no—
148C2×C3×Q8—21no—
149C2×C3×Q8—21no—
150C2×C3×Q8—21no—
151C2×C3×Q8—21no—
152C2×C3×Q8—21no—
153C2×C3×Q8C2×D412yes—
154C2×C3×D4—22no—
155C2×C3×D4—22no—
156C2×C3×D4SD1614yescomplemented
157C22×C3×C4—21no—
158C22×C3×C4—22no—
159C22×C3×C4—22no—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—21no—
165C22×C3×C4—22no—
166C22×C3×C4C2×D412yes—
167C22×C3×C4C2×D411yescharacteristic
168C22×C3×C4C2×D411yescharacteristic
169C22×C3×C4C2411yescharacteristic
170C2×#67—44no—
171C2×#67—44no—
172C2×#67—44no—
173C2×#67—44no—
174C2×#67—82no—
175C2×#67—82no—
176C22×Dic3—21no—
177C22×Dic3—21no—
178C2×D12—22no—
179C2×D12—22no—
180C2×D12—22no—
181C2×D12—22no—
182C2×D12—22no—
183C2×D12—22no—
184C2×D12—42no—
185C2×D12—42no—
186C2×D12—42no—
187C2×D12—42no—
188C2×D12—81no—
189C2×D12—81no—
190C3×C4⋊C4—22no—
191C3×C4⋊C4—22no—
192C3×C4⋊C4—21no—
193C3×C4⋊C4—21no—
194C3×C4⋊C4—22no—
195C3×C4⋊C4—44no—
196C3×C4⋊C4—42no—
197C3×C4⋊C4—42no—
198C3×C4⋊C4—81no—
199C3×C22⋊C4—24no—
200C3×C22⋊C4—44no—
201C3×C22⋊C4—82no—
202C3×C42—42no—
203C3×C42C2×D412yes—
204C3×C42#4112yes—
205#215—48no—
206(C2×D3)⋊C4—44no—
207(C2×D3)⋊C4—44no—
208C4⋊Dic3—21no—
209C4⋊Dic3—21no—
210C4⋊Dic3C2×D412yes—
211C4⋊Dic3#4112yes—
212C4×Dic3—42no—
213C2×C3⋊C8—21no—
214C2×C3⋊C8—22no—
215C2×C3⋊C8—21no—
216C2×C3⋊C8—22no—
217C2×C3⋊C8—21no—
218C2×C3⋊C8—21no—
219C2×C3⋊C8—21no—
220C2×C3⋊C8—48no—
221C2×C3⋊C8C2×D414yes—
222C2×C3⋊C8C2×D412yes—
223C22×Q8C22×D311yescharacteristic
224C22×D4—32no—
225C22×D4—62no—
226C22×D4—62no—
227C22×D4—61no—
228C22×D4—61no—
229C22×D4#6711yescharacteristic
230C23×C4C22×D312yes—
231C2×SD16—64no—
232C2×SD16—64no—
233C2×SD16—122no—
234C22×C8—31no—
235C22×C8—32no—
236C22×C8—31no—
237C22×C8—64no—
238C22×C8—64no—
239C22×C8—122no—
240#127—62no—
241#127—62no—
242#127—124no—
243C22⋊Q8—24no—
244C22⋊Q8—42no—
245#121—64no—
246#121—64no—
247#121—122no—
248#121—122no—
249C4×Q8—22no—
250C4×Q8—22no—
251C4×D4—22no—
252C4×D4—22no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—31no—
258C2×C4⋊C4—31no—
259C2×C4⋊C4#6711yescharacteristic
260C2×C4⋊C4C22×D311yescharacteristic
261C2×C4⋊C4C22×D312yes—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—62no—
265C2×C22⋊C4—62no—
266C2×C22⋊C4#6712yes—
267C2×C42—21no—
268C2×C42—61no—
269C2×C42C22×D311yescharacteristic
270#106—64no—
271C4⋊C8—34no—
272#103—34no—
273#103—62no—
274#102—64no—
275#102—64no—
276#102—122no—
277C22⋊C8—64no—
278C4×C8—62no—
279#95—22no—
280#95—61no—
281C23×C3—22no—
282C23×C3—22no—
283C23×C3—22no—
284C23×C3—22no—
285C23×C3—22no—
286C23×C3—22no—
287C23×C3C2×SD1614yes—
288C23×C3C22×D411yescharacteristic
289C22×D3—42no—
290C22×D3—42no—
291C22×D3—42no—
292C22×D3—42no—
293C22×D3—42no—
294C22×D3—42no—
295C22×D3—42no—
296C3×Q8—22no—
297C3×Q8—22no—
298C3×Q8—41no—
299C3×Q8—41no—
300C3×D4—28no—
301C2×C3×C4—21no—
302C2×C3×C4—21no—
303C2×C3×C4—22no—
304C2×C3×C4—22no—
305C2×C3×C4—22no—
306C2×C3×C4—22no—
307C2×C3×C4—41no—
308C2×C3×C4—41no—
309C2×C3×C4—42no—
310C2×C3×C4—42no—
311C2×C3×C4—42no—
312C2×C3×C4—42no—
313C2×C3×C4—41no—
314C2×C3×C4—42no—
315C2×C3×C4—42no—
316C2×C3×C4—41no—
317C2×C3×C4—42no—
318C2×C3×C4—41no—
319C2×C3×C4#12112yes—
320C2×C3×C4#12112yes—
321C2×C3×C4C2×SD1612yes—
322C2×C3×C4#13512yes—
323C2×C3×C4C22×D411yescharacteristic
324C2×C3×C4C22×D411yescharacteristic
325C2×C3×C4C2×#4111yescharacteristic
326C2×C3×C4C2×#4111yescharacteristic
327C2×C3×C4H(5,𝔽2)11yescharacteristic
328C2×C3×C4H(5,𝔽2)11yescharacteristic
329#67—88no—
330#67—88no—
331C2×Dic3—22no—
332C2×Dic3—22no—
333C2×Dic3—41no—
334C2×Dic3—41no—
335C2×Dic3—41no—
336C2×Dic3—41no—
337D12—42no—
338D12—42no—
339D12—42no—
340D12—42no—
341D12—84no—
342D12—84no—
343C3⋊C8—22no—
344C3⋊C8—24no—
345C3⋊C8—41no—
346C24—62no—
347C24C2×#6712yes—
348C2×Q8—21no—
349C2×Q8—21no—
350C2×Q8—21no—
351C2×Q8—21no—
352C2×Q8—21no—
353C2×Q8D3×D412yes—
354C2×D4—22no—
355C2×D4—22no—
356C2×D4—62no—
357C2×D4—62no—
358C2×D4—62no—
359C2×D4—62no—
360C2×D4—62no—
361C2×D4—62no—
362C2×D4—124no—
363C2×D4—124no—
364C2×D4—124no—
365C2×D4—124no—
366C2×D4—122no—
367C2×D4—122no—
368C2×D4—122no—
369C2×D4—122no—
370C2×D4—241no—
371C2×D4—241no—
372C2×D4—242no—
373C2×D4—242no—
374C2×D4#21514yescomplemented
375C22×C4—21no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—21no—
383C22×C4—22no—
384C22×C4—61no—
385C22×C4—61no—
386C22×C4C2×#6712yes—
387C22×C4C2×#6711yescharacteristic
388C22×C4C2×#6711yescharacteristic
389C22×C4C23×D311yescharacteristic
390SD16—128no—
391C2×C8—34no—
392C2×C8—32no—
393C2×C8—61no—
394C2×C8—62no—
395C2×C8—61no—
396C2×C8—62no—
397C2×C8—61no—
398C2×C8—61no—
399C2×C8—61no—
400C2×C8—128no—
401C4⋊C4—22no—
402C4⋊C4—21no—
403C4⋊C4—22no—
404C4⋊C4—21no—
405C4⋊C4—22no—
406C4⋊C4—32no—
407C4⋊C4—32no—
408C4⋊C4—44no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4—61no—
412C4⋊C4—61no—
413C4⋊C4—81no—
414C22⋊C4—24no—
415C22⋊C4—44no—
416C22⋊C4—82no—
417C22⋊C4—124no—
418C22⋊C4—124no—
419C42—42no—
420C42—122no—
421C42D3×D412yes—
422C42#23712yes—
423C22×C3—24no—
424C22×C3—24no—
425C22×C3—24no—
426C22×C3—42no—
427C22×C3—42no—
428C22×C3—42no—
429C22×C3—42no—
430C22×C3C2×#12111yescharacteristic
431C22×C3D4211yescharacteristic
432C22×C3#54611yescharacteristic
433C22×C3C22×SD1611yescharacteristic
434C22×C3C2×#13511yescharacteristic
435C22×C3#57611yescharacteristic
436C22×C3#57711yescharacteristic
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—82no—
448C3×C4—81no—
449C3×C4#46514yescomplemented
450C3×C4#53211yescharacteristic
451C3×C4#54011yescharacteristic
452C3×C4D4211yescharacteristic
453C3×C4#55011yescharacteristic
454Dic3—42no—
455Dic3—42no—
456C23—22no—
457C23—22no—
458C23—22no—
459C23—22no—
460C23—22no—
461C23—22no—
462C23—122no—
463C23—122no—
464C23—122no—
465C23—122no—
466C23—122no—
467C23—122no—
468C23—122no—
469C23C2×#21514yes—
470C23C22×#6711yescharacteristic
471Q8—22no—
472Q8—22no—
473Q8—41no—
474Q8—41no—
475D4—28no—
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—22no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—41no—
498C2×C4—42no—
499C2×C4—41no—
500C2×C4—42no—
501C2×C4—41no—
502C2×C4—62no—
503C2×C4—62no—
504C2×C4—121no—
505C2×C4—121no—
506C2×C4—121no—
507C2×C4—121no—
508C2×C4#93912yes—
509C2×C4#93912yes—
510C2×C4C2×#21512yes—
511C2×C4#95012yes—
512C2×C4C2×D3×D411yescharacteristic
513C2×C4C2×D3×D411yescharacteristic
514C2×C4C2×#23711yescharacteristic
515C2×C4#100411yescharacteristic
516C2×C4D3×#4111yescharacteristic
517C2×C4#100911yescharacteristic
518C8—62no—
519C8—64no—
520C8—121no—
521C2×C3—44no—
522C2×C3#242811yescharacteristic
523C2×C3C2×#46511yescharacteristic
524C2×C3#306211yescharacteristic
525C2×C3D4×SD1611yescharacteristic
526C2×C3#328711yescharacteristic
527C2×C3#329511yescharacteristic
528C2×C3#330411yescharacteristic
529D3—162no—
530C22—24no—
531C22—24no—
532C22—24no—
533C22—42no—
534C22—42no—
535C22—42no—
536C22—42no—
537C22—242no—
538C22—242no—
539C22—242no—
540C22—242no—
541C22—242no—
542C22—242no—
543C22—242no—
544C22D4×#6711yescharacteristic
545C22C2×#93911yescharacteristic
546C22C22×#21511yescharacteristic
547C22C2×#95011yescharacteristic
548C22#582911yescharacteristic
549C22#583511yescharacteristic
550C22#583711yescharacteristic
551C4—22no—
552C4—42no—
553C4—42no—
554C4—82no—
555C4—81no—
556C4—122no—
557C4—122no—
558C4#504814yescomplemented
559C4#559611yescharacteristic
560C4#559911yescharacteristic
561C4#560911yescharacteristic
562C4#561911yescharacteristic
563C3#2538711yescharacteristic, complemented, Sylow
564C2—44no—
565C2—482no—
566C2#8264911yescharacteristic
567C2C2×#504811yescharacteristic
568C2#8342211yescharacteristic
569C2D4×#21511yescharacteristic
570C2#8358511yescharacteristic
571C2#8363411yescharacteristic
572C2#8367311yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2538C2
socleC23×C3C22×D4
prime radical#82378C2
hypercenter#2538D3
hypocenterC3#25387
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2538

Complements

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

NH ≅ G/Ntype
1. C3×#2538C2semidirect
2. C2×#5048C2semidirect
3. C2×#5027C2semidirect
5. C2×D4×C3⋊C8C2semidirect
6. #71699C2semidirect
7. C4×#835C2semidirect
8. #69580C2semidirect
9. #69568C2semidirect
10. #69524C2semidirect
11. #69515C2semidirect
49. #2538D3semidirect
105. #835D4semidirect
112. C4⋊(C3⋊C8)D4semidirect
156. C2×C3×D4SD16semidirect
374. C2×D4#215semidirect
449. C3×C4#465semidirect
558. C4#5048semidirect
563. C3#25387semidirect

Maximal subgroups

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

subgroupindex
C3×#25382
C2×#50482
C2×#50272
#823782
C2×D4×C3⋊C82
#716992
C4×#8352
#695802
#695682
#695242
#695152
#253873