Finite Group Databasebeta

← #10846013

Subgroups — #10846013

Order 768.

1206 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83188C211yesmaximal, characteristic, complemented
2C2×#5072C211yesmaximal, characteristic, complemented
3C2×#4782C212yesmaximal, complemented
4C2×#4688C211yesmaximal, characteristic, complemented
5#74368C211yesmaximal, characteristic, complemented
6C3×#1957C211yesmaximal, characteristic, complemented
7#70207C212yesmaximal, complemented
8#70204C212yesmaximal, complemented
9#70203C212yesmaximal, complemented
10C4×#821C211yesmaximal, characteristic, complemented
11#69568C211yesmaximal, characteristic, complemented
12#25392—31nomaximal, Sylow
13C2×C3×#127C2211yescharacteristic
14C2×#940—21no—
15C2×#938C2212yes—
16C2×#938C2212yes—
17C22×#213C2212yes—
18C22×ΓL(1,𝔽25)C2212yes—
19C2×#895—21no—
20C2×#895C2212yes—
21C2×C4×D12C2211yescharacteristic
22C2×C4×D12C2211yescharacteristic
23C2×C3×#102C2212yes—
24C2×C3×#102C2212yes—
25C2×C3×C4×C8C2211yescharacteristic
26C3×#384C2211yescharacteristic
27#5223—22no—
28#5221—21no—
29C2×#832C2212yes—
30C2×#821C2211yescharacteristic
31C2×#821C2211yescharacteristic
32C2×(C2×D3)⋊C8C2212yes—
33#5072—24no—
34#4991—22no—
35#4987—21no—
36#4987C2212yes—
37C2×#808C2212yes—
38C4×C4⋊Dic3C2211yescharacteristic
39C2×C4⋊(C3⋊C8)C2211yescharacteristic
40#4782—28no—
41#4688—24no—
42#4551C2211yescharacteristic
43#4551C2211yescharacteristic
44#4529C2212yes—
45C2×#469—32no—
46C2×#459—31no—
47C2×#436—31no—
48#2541—31no—
49#2537—31no—
50#2034—32no—
51#1999—32no—
52#1957D311yescharacteristic, complemented
53#1948—31no—
54#1895—32no—
55C4×#102—31no—
56C22×C3×D4—21no—
57C22×C3×D4—21no—
58C22×C3×D4C2312yes—
59C22×#67—22no—
60C22×#67—22no—
61C22×D12—22no—
62C22×D12D411yescharacteristic
63C22×D12C2311yescharacteristic
64C22×C4×D3—21no—
65C22×C4×D3C2312yes—
66C22×C3×C8C2311yescharacteristic
67C22×C3×C8C2311yescharacteristic
68C3×#127—22no—
69C3×#127—22no—
70C2×C3×C4⋊C4—21no—
71C2×C3×C4⋊C4—21no—
72C2×C3×C4⋊C4C2312yes—
73C2×C3×C42C2311yescharacteristic
74C2×#217—22no—
75C2×#217—22no—
76#940—44no—
77#938—24no—
78#938—24no—
79#938—42no—
80#938—42no—
81C2×#213—24no—
82C2×#213—24no—
83C2×#213—42no—
84C2×(C2×D3)⋊C4—22no—
85C2×(C2×D3)⋊C4—21no—
86C2×(C2×D3)⋊C4—22no—
87C2×(C2×D3)⋊C4D412yes—
88C2×C4⋊Dic3D411yescharacteristic
89C2×C4⋊Dic3C2311yescharacteristic
90C2×C4⋊Dic3C2311yescharacteristic
91C2×C4⋊Dic3C2311yescharacteristic
92C2×C4×Dic3—21no—
93C22×C3⋊C8C2312yes—
94C2×ΓL(1,𝔽25)—24no—
95C2×ΓL(1,𝔽25)—24no—
96C2×ΓL(1,𝔽25)—42no—
97#895—24no—
98#895—42no—
99#895—44no—
100C4×D12—22no—
101C4×D12—22no—
102C4×D12—22no—
103C4×D12—22no—
104C3×#102—24no—
105C3×#102—42no—
106C3×C4×C8—22no—
107#832—24no—
108C3⋊#95—21no—
109C3⋊#95—22no—
110#821—22no—
111#821D414yescomplemented
112(C2×D3)⋊C8—24no—
113#808—24no—
114C4⋊(C3⋊C8)D414yescomplemented
115C22×D8—32no—
116C22×M16—32no—
117C2×#127—61no—
118C2×#127C2×D311yescharacteristic
119C2×#121—32no—
120C2×#121—32no—
121C2×#121—32no—
122C2×#121—61no—
123C2×C4×D4—31no—
124C2×C4×D4—31no—
125#469—68no—
126#459—64no—
127#436—64no—
128C2×#106—32no—
129C2×C4⋊C8—31no—
130C2×#102—31no—
131C2×#102—32no—
132C2×#102—31no—
133C2×#102C2×D312yes—
134C2×#102C2×D312yes—
135C2×C22⋊C8—32no—
136C2×C4×C8C2×D311yescharacteristic
137#394—62no—
138#392—62no—
139#390—61no—
140#384C2×D311yescharacteristic
141#380—32no—
142#380—61no—
143C4×C4⋊C4—31no—
144#340—31no—
145#340—31no—
146#340—32no—
147C24×C3—22no—
148C23×D3—21no—
149C23×D3C2×D412yes—
150C2×C3×D4—22no—
151C2×C3×D4—22no—
152C2×C3×D4—22no—
153C2×C3×D4—22no—
154C2×C3×D4—22no—
155C2×C3×D4—22no—
156C2×C3×D4—42no—
157C2×C3×D4—42no—
158C2×C3×D4—42no—
159C2×C3×D4—42no—
160C2×C3×D4—81no—
161C2×C3×D4—81no—
162C22×C3×C4—22no—
163C22×C3×C4C2×D411yescharacteristic
164C22×C3×C4C2×D411yescharacteristic
165C22×C3×C4C2411yescharacteristic
166C2×#67—44no—
167C2×#67—44no—
168C2×#67—44no—
169C2×#67—44no—
170C2×#67—82no—
171C2×#67—82no—
172C22×Dic3—21no—
173C22×Dic3—21no—
174C22×Dic3—21no—
175C22×Dic3C2×D412yes—
176C2×D12—21no—
177C2×D12—21no—
178C2×D12—21no—
179C2×D12—21no—
180C2×D12—22no—
181C2×D12—22no—
182C2×D12—22no—
183C2×D12—21no—
184C2×D12—44no—
185C2×D12—44no—
186C2×D12—82no—
187C2×D12C2×D412yes—
188C2×C4×D3—22no—
189C2×C4×D3—22no—
190C2×C4×D3—22no—
191C2×C4×D3—22no—
192C2×C4×D3—22no—
193C2×C4×D3—22no—
194C2×C4×D3—42no—
195C2×C4×D3—42no—
196C2×C4×D3—42no—
197C2×C3×C8—21no—
198C2×C3×C8—21no—
199C2×C3×C8—21no—
200C2×C3×C8—21no—
201C2×C3×C8—21no—
202C2×C3×C8C2×D412yes—
203C3×C4⋊C4—22no—
204C3×C4⋊C4—22no—
205C3×C4⋊C4—42no—
206C3×C4⋊C4—81no—
207C3×C42C2×D412yes—
208C3×C42#4112yes—
209#217—44no—
210#217—44no—
211#213—48no—
212(C2×D3)⋊C4—24no—
213(C2×D3)⋊C4—44no—
214(C2×D3)⋊C4—44no—
215(C2×D3)⋊C4—42no—
216C4⋊Dic3—22no—
217C4⋊Dic3—21no—
218C4⋊Dic3—21no—
219C4⋊Dic3—21no—
220C4⋊Dic3—21no—
221C4⋊Dic3C2×D412yes—
222C4⋊Dic3#4112yes—
223C4×Dic3—42no—
224C2×C3⋊C8—22no—
225C2×C3⋊C8—22no—
226C2×C3⋊C8C2×D414yes—
227ΓL(1,𝔽25)—48no—
228C22×D4—21no—
229C22×D4—21no—
230C22×D4—31no—
231C22×D4—31no—
232C22×D4—62no—
233C22×D4—62no—
234C22×D4—62no—
235C22×D4C22×D312yes—
236C23×C4—32no—
237C23×C4—61no—
238C2×D8—64no—
239C2×D8—64no—
240C2×D8—122no—
241C2×M16—64no—
242C2×M16—64no—
243C2×M16—122no—
244C22×C8—32no—
245C22×C8C22×D311yescharacteristic
246C22×C8C22×D311yescharacteristic
247#127—22no—
248#127—22no—
249#127—124no—
250#121—64no—
251#121—64no—
252#121—64no—
253#121—122no—
254#121—124no—
255#121—122no—
256#121—122no—
257C4×D4—62no—
258C4×D4—62no—
259C4×D4—62no—
260C4×D4—62no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—31no—
264C2×C4⋊C4—31no—
265C2×C4⋊C4—31no—
266C2×C4⋊C4—31no—
267C2×C4⋊C4C22×D312yes—
268C2×C22⋊C4—32no—
269C2×C22⋊C4—62no—
270C2×C22⋊C4—62no—
271C2×C22⋊C4—62no—
272C2×C22⋊C4—62no—
273C2×C22⋊C4—61no—
274C2×C42—61no—
275C2×C42C22×D311yescharacteristic
276#106—64no—
277C4⋊C8—34no—
278#102—24no—
279#102—34no—
280#102—42no—
281#102—62no—
282#102—64no—
283C22⋊C8—64no—
284C4×C8—22no—
285#95—62no—
286#95—61no—
287C23×C3—42no—
288C23×C3—42no—
289C23×C3—42no—
290C23×C3—42no—
291C23×C3—42no—
292C23×C3—42no—
293C23×C3—42no—
294C23×C3C22×D411yescharacteristic
295C22×D3—22no—
296C22×D3—22no—
297C22×D3—22no—
298C22×D3—22no—
299C22×D3—22no—
300C22×D3—22no—
301C22×D3—22no—
302C22×D3—22no—
303C22×D3—41no—
304C22×D3—41no—
305C22×D3—41no—
306C22×D3—41no—
307C22×D3—41no—
308C22×D3—41no—
309C3×D4—42no—
310C3×D4—42no—
311C3×D4—42no—
312C3×D4—42no—
313C3×D4—84no—
314C3×D4—84no—
315C2×C3×C4—21no—
316C2×C3×C4—21no—
317C2×C3×C4—42no—
318C2×C3×C4—42no—
319C2×C3×C4—42no—
320C2×C3×C4#12112yes—
321C2×C3×C4#12112yes—
322C2×C3×C4AGL(1,ℤ/8)12yes—
323C2×C3×C4AGL(1,ℤ/8)12yes—
324C2×C3×C4C22×D411yescharacteristic
325C2×C3×C4C22×D411yescharacteristic
326C2×C3×C4C2×#4111yescharacteristic
327C2×C3×C4C2×#4111yescharacteristic
328C2×C3×C4H(5,𝔽2)11yescharacteristic
329C2×C3×C4H(5,𝔽2)11yescharacteristic
330#67—88no—
331#67—88no—
332C2×Dic3—22no—
333C2×Dic3—22no—
334C2×Dic3—22no—
335C2×Dic3—22no—
336C2×Dic3—22no—
337C2×Dic3—41no—
338C2×Dic3—41no—
339C2×Dic3—41no—
340C2×Dic3—41no—
341C2×Dic3—41no—
342C2×Dic3—41no—
343C2×Dic3—41no—
344D12—22no—
345D12—22no—
346D12—41no—
347D12—41no—
348D12—44no—
349D12—88no—
350C4×D3—42no—
351C4×D3—42no—
352C4×D3—42no—
353C4×D3—42no—
354C4×D3—84no—
355C3×C8—22no—
356C3×C8—41no—
357C3⋊C8—24no—
358C24—22no—
359C24—32no—
360C24—61no—
361C2×D4—22no—
362C2×D4—22no—
363C2×D4—22no—
364C2×D4—22no—
365C2×D4—22no—
366C2×D4—22no—
367C2×D4—32no—
368C2×D4—42no—
369C2×D4—42no—
370C2×D4—42no—
371C2×D4—42no—
372C2×D4—62no—
373C2×D4—62no—
374C2×D4—62no—
375C2×D4—61no—
376C2×D4—61no—
377C2×D4—61no—
378C2×D4—61no—
379C2×D4—61no—
380C2×D4—81no—
381C2×D4—81no—
382C2×D4—124no—
383C2×D4—124no—
384C2×D4—124no—
385C2×D4—124no—
386C2×D4—124no—
387C2×D4—124no—
388C2×D4—242no—
389C2×D4—242no—
390C2×D4—242no—
391C22×C4—22no—
392C22×C4—32no—
393C22×C4—61no—
394C22×C4—61no—
395C22×C4—62no—
396C22×C4—62no—
397C22×C4—62no—
398C22×C4—62no—
399C22×C4—62no—
400C22×C4—62no—
401C22×C4—61no—
402C22×C4—122no—
403C22×C4—122no—
404C22×C4—122no—
405C22×C4D3×D411yescharacteristic
406C22×C4D3×D411yescharacteristic
407C22×C4C23×D311yescharacteristic
408D8—128no—
409M16—128no—
410C2×C8—21no—
411C2×C8—21no—
412C2×C8—21no—
413C2×C8—21no—
414C2×C8—21no—
415C2×C8—34no—
416C2×C8—62no—
417C2×C8—62no—
418C2×C8D3×D412yes—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—32no—
422C4⋊C4—32no—
423C4⋊C4—42no—
424C4⋊C4—62no—
425C4⋊C4—61no—
426C4⋊C4—61no—
427C4⋊C4—61no—
428C4⋊C4—61no—
429C4⋊C4—81no—
430C22⋊C4—64no—
431C22⋊C4—124no—
432C22⋊C4—124no—
433C22⋊C4—124no—
434C22⋊C4—124no—
435C22⋊C4—122no—
436C42—122no—
437C42#23512yes—
438C42D3×D412yes—
439C22×C3—82no—
440C22×C3—82no—
441C22×C3—82no—
442C22×C3—82no—
443C22×C3—82no—
444C22×C3—82no—
445C22×C3—82no—
446C22×C3C2×#12111yescharacteristic
447C22×C3D4211yescharacteristic
448C22×C3#54611yescharacteristic
449C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
450C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
451C22×C3#57611yescharacteristic
452C22×C3#57711yescharacteristic
453C2×D3—42no—
454C2×D3—42no—
455C2×D3—42no—
456C2×D3—42no—
457C2×D3—42no—
458C2×D3—42no—
459C2×D3—42no—
460C2×D3—42no—
461C2×D3—42no—
462C2×D3—42no—
463C2×D3—81no—
464C2×D3—81no—
465C2×D3—81no—
466C2×D3—81no—
467C3×C4—22no—
468C3×C4—82no—
469C3×C4#46914yescomplemented
470C3×C4#53211yescharacteristic
471C3×C4#54011yescharacteristic
472C3×C4D4211yescharacteristic
473C3×C4#55011yescharacteristic
474Dic3—42no—
475Dic3—42no—
476Dic3—42no—
477Dic3—81no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—62no—
486C23—62no—
487C23—62no—
488C23—62no—
489C23—62no—
490C23—62no—
491C23—62no—
492C23—62no—
493C23—121no—
494C23—121no—
495C23—121no—
496C23—121no—
497C23—121no—
498C23—121no—
499C23C2×D3×D411yescharacteristic
500D4—42no—
501D4—42no—
502D4—42no—
503D4—42no—
504D4—62no—
505D4—62no—
506D4—84no—
507D4—84no—
508D4—124no—
509D4—121no—
510D4—121no—
511D4—248no—
512D4—248no—
513D4—248no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—62no—
520C2×C4—62no—
521C2×C4—62no—
522C2×C4—62no—
523C2×C4—62no—
524C2×C4—121no—
525C2×C4—121no—
526C2×C4—121no—
527C2×C4—121no—
528C2×C4—122no—
529C2×C4—122no—
530C2×C4—122no—
531C2×C4—122no—
532C2×C4—121no—
533C2×C4—121no—
534C2×C4—121no—
535C2×C4—244no—
536C2×C4#88412yes—
537C2×C4#88412yes—
538C2×C4#91112yes—
539C2×C4#91412yes—
540C2×C4C2×#23511yescharacteristic
541C2×C4C2×D3×D411yescharacteristic
542C2×C4C2×D3×D411yescharacteristic
543C2×C4#100411yescharacteristic
544C2×C4D3×#4111yescharacteristic
545C2×C4#100911yescharacteristic
546C8—22no—
547C8—41no—
548C8—64no—
549C2×C3—162no—
550C2×C3#242811yescharacteristic
551C2×C3C2×#46911yescharacteristic
552C2×C3#306211yescharacteristic
553C2×C3#327211yescharacteristic
554C2×C3#327411yescharacteristic
555C2×C3#332511yescharacteristic
556C2×C3#333211yescharacteristic
557D3—82no—
558D3—82no—
559C22—82no—
560C22—82no—
561C22—82no—
562C22—82no—
563C22—82no—
564C22—82no—
565C22—82no—
566C22—122no—
567C22—122no—
568C22—122no—
569C22—122no—
570C22—122no—
571C22—122no—
572C22—122no—
573C22—122no—
574C22—122no—
575C22—122no—
576C22—241no—
577C22—241no—
578C22—241no—
579C22—241no—
580C22C2×#88411yescharacteristic
581C22#567611yescharacteristic
582C22#571711yescharacteristic
583C22C2×#91111yescharacteristic
584C22#575711yescharacteristic
585C22C2×#91411yescharacteristic
586C22#576211yescharacteristic
587C4—22no—
588C4—82no—
589C4—122no—
590C4—122no—
591C4—122no—
592C4—241no—
593C4#478214yescomplemented
594C4#554811yescharacteristic
595C4#556211yescharacteristic
596C4#559911yescharacteristic
597C4#560911yescharacteristic
598C3#2539211yescharacteristic, complemented, Sylow
599C2—162no—
600C2—242no—
601C2—242no—
602C2#7644411yescharacteristic
603C2C2×#478211yescharacteristic
604C2#7875211yescharacteristic
605C2#8158111yescharacteristic
606C2#8158211yescharacteristic
607C2#8177011yescharacteristic
608C2#8178811yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1957C2
socleC23×C3C22×D4
hypercenter#1957D3
hypocenterC3#25392
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1957

Complements

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

NH ≅ G/Ntype
1. #83188C2semidirect
2. C2×#5072C2semidirect
3. C2×#4782C2semidirect
4. C2×#4688C2semidirect
5. #74368C2semidirect
6. C3×#1957C2semidirect
7. #70207C2semidirect
8. #70204C2semidirect
9. #70203C2semidirect
10. C4×#821C2semidirect
11. #69568C2semidirect
52. #1957D3semidirect
111. #821D4semidirect
114. C4⋊(C3⋊C8)D4semidirect
469. C3×C4#469semidirect
593. C4#4782semidirect
598. C3#25392semidirect

Maximal subgroups

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

subgroupindex
#831882
C2×#50722
C2×#47822
C2×#46882
#743682
C3×#19572
#702072
#702042
#702032
C4×#8212
#695682
#253923