Finite Group Databasebeta

← #10846006

Subgroups — #10846006

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
1C2×C3×#436C211yesmaximal, characteristic, complemented
2C2×#5135C212yesmaximal, complemented
3#82417C211yesmaximal, characteristic, complemented
4#82378C211yesmaximal, characteristic, complemented
5C2×#4843C211yesmaximal, characteristic, complemented
6#71990C212yesmaximal, complemented
7#70454C211yesmaximal, characteristic, complemented
8#69961C212yesmaximal, complemented
9#69935C212yesmaximal, complemented
10#69772C211yesmaximal, characteristic, complemented
11C4×#821C211yesmaximal, characteristic, complemented
12#25392—31nomaximal, Sylow
13C22×C3×M16C2212yes—
14C2×C3×C4×D4C2211yescharacteristic
15C2×#940—21no—
16C2×#938—21no—
17C2×#930C2212yes—
18C2×#930C2212yes—
19C2×#930C2212yes—
20C22×D24C2212yes—
21C2×#874C2211yescharacteristic
22C2×C4×D12C2211yescharacteristic
23C3×#436—24no—
24C2×C3×C4⋊C8C2211yescharacteristic
25C2×C3×C22⋊C8C2212yes—
26C2×C3×C4×C8C2211yescharacteristic
27#5212C2212yes—
28#5135—28no—
29C2×#821C2212yes—
30C2×#821C2212yes—
31C2×#821C2211yescharacteristic
32C2×#821C2212yes—
33C2×#821C2211yescharacteristic
34C2×#817C2212yes—
35#4988—21no—
36#4987—21no—
37#4961—22no—
38#4960—22no—
39#4935C2211yescharacteristic
40C4×C4⋊Dic3C2211yescharacteristic
41#4843—24no—
42#4551C2211yescharacteristic
43#4551C2212yes—
44#4551C2211yescharacteristic
45C2×#469—32no—
46C2×#459—31no—
47C2×#436D311yescharacteristic, complemented
48#2541—31no—
49#2537—31no—
50#2034—32no—
51#1999—32no—
52#1957—31no—
53#1948—31no—
54#1895—32no—
55C4×#102—31no—
56C22×C3×D4D411yescharacteristic
57C23×C3×C4C2312yes—
58C22×#67—22no—
59C22×#67—22no—
60C22×#67—22no—
61C22×D12—21no—
62C22×D12—21no—
63C22×D12C2312yes—
64C22×D12C2311yescharacteristic
65C22×C4×D3—21no—
66C2×C3×M16—24no—
67C2×C3×M16—24no—
68C2×C3×M16—42no—
69C22×C3×C8C2311yescharacteristic
70C22×C3×C8C2311yescharacteristic
71C22×C3×C8C2312yes—
72C3×C4×D4—22no—
73C3×C4×D4—22no—
74C2×C3×C4⋊C4D411yescharacteristic
75C2×C3×C22⋊C4D412yes—
76C2×C3×C42C2311yescharacteristic
77C2×#217—22no—
78#940—44no—
79#938—44no—
80#930—24no—
81#930—24no—
82#930—24no—
83#930—42no—
84#930—42no—
85#930—42no—
86C2×(C2×D3)⋊C4—22no—
87C2×(C2×D3)⋊C4—22no—
88C2×(C2×D3)⋊C4—21no—
89C2×(C2×D3)⋊C4—22no—
90C2×C4⋊Dic3—21no—
91C2×C4⋊Dic3—21no—
92C2×C4⋊Dic3C2312yes—
93C2×C4⋊Dic3C2311yescharacteristic
94C2×C4⋊Dic3C2311yescharacteristic
95C2×C4⋊Dic3C2311yescharacteristic
96C2×C4×Dic3—21no—
97C2×D24—24no—
98C2×D24—24no—
99C2×D24—42no—
100#874—22no—
101#874—22no—
102C4×D12—22no—
103C4×D12—22no—
104C3×C4⋊C8D414yescomplemented
105C3×C22⋊C8—24no—
106C3×C4×C8—22no—
107C3⋊#95—21no—
108C3⋊#95—22no—
109#821—24no—
110#821—24no—
111#821—22no—
112#821—42no—
113#821D414yescomplemented
114#817—24no—
115C22×D8—32no—
116C22×M16C2×D312yes—
117C2×#127—31no—
118C2×#127—61no—
119C2×#121—32no—
120C2×#121—32no—
121C2×#121—32no—
122C2×#121—61no—
123C2×C4×D4—31no—
124C2×C4×D4C2×D311yescharacteristic
125#469—68no—
126#459—64no—
127#436—24no—
128C2×#106—32no—
129C2×C4⋊C8C2×D311yescharacteristic
130C2×#102—31no—
131C2×#102—32no—
132C2×#102—32no—
133C2×#102—31no—
134C2×#102—32no—
135C2×C22⋊C8C2×D312yes—
136C2×C4×C8C2×D311yescharacteristic
137#394—62no—
138#392—62no—
139#390—61no—
140#384—31no—
141#380—32no—
142#380—61no—
143C4×C4⋊C4—31no—
144#340—31no—
145#340—32no—
146#340—31no—
147C24×C3C2×D412yes—
148C23×D3—21no—
149C23×D3—22no—
150C2×C3×D4—22no—
151C2×C3×D4—22no—
152C2×C3×D4—22no—
153C22×C3×C4—22no—
154C22×C3×C4—22no—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
159C22×C3×C4C2×D412yes—
160C22×C3×C4C2×D411yescharacteristic
161C22×C3×C4C2×D411yescharacteristic
162C22×C3×C4C2411yescharacteristic
163C2×#67—44no—
164C2×#67—44no—
165C2×#67—44no—
166C2×#67—44no—
167C2×#67—44no—
168C2×#67—44no—
169C2×#67—82no—
170C2×#67—82no—
171C2×#67—82no—
172C22×Dic3—21no—
173C22×Dic3—21no—
174C22×Dic3—21no—
175C22×Dic3—22no—
176C2×D12—21no—
177C2×D12—21no—
178C2×D12—21no—
179C2×D12—21no—
180C2×D12—22no—
181C2×D12—22no—
182C2×D12—22no—
183C2×D12—22no—
184C2×D12—21no—
185C2×D12—22no—
186C2×D12—22no—
187C2×D12—42no—
188C2×D12—42no—
189C2×D12—42no—
190C2×D12—42no—
191C2×D12—81no—
192C2×D12—81no—
193C2×D12C2×D412yes—
194C2×C4×D3—42no—
195C2×C4×D3—42no—
196C2×C4×D3—42no—
197C3×M16—48no—
198C2×C3×C8—22no—
199C2×C3×C8—21no—
200C2×C3×C8—22no—
201C2×C3×C8—21no—
202C2×C3×C8—21no—
203C2×C3×C8—21no—
204C2×C3×C8—21no—
205C2×C3×C8C2×D412yes—
206C2×C3×C8C2×D414yes—
207C3×C4⋊C4—22no—
208C3×C22⋊C4—24no—
209C3×C42C2×D412yes—
210C3×C42#4112yes—
211#217—44no—
212(C2×D3)⋊C4—44no—
213(C2×D3)⋊C4—44no—
214(C2×D3)⋊C4—44no—
215(C2×D3)⋊C4—42no—
216C4⋊Dic3—21no—
217C4⋊Dic3—21no—
218C4⋊Dic3—22no—
219C4⋊Dic3—21no—
220C4⋊Dic3—21no—
221C4⋊Dic3—22no—
222C4⋊Dic3—42no—
223C4⋊Dic3—81no—
224C4⋊Dic3C2×D412yes—
225C4⋊Dic3#4112yes—
226C4×Dic3—42no—
227D24—48no—
228C22×D4—31no—
229C22×D4—32no—
230C22×D4—62no—
231C22×D4—62no—
232C22×D4—62no—
233C22×D4—61no—
234C22×D4—61no—
235C22×D4D1211yescharacteristic
236C23×C4—61no—
237C23×C4C22×D312yes—
238C2×D8—64no—
239C2×D8—64no—
240C2×D8—122no—
241C2×M16—24no—
242C2×M16—24no—
243C2×M16—42no—
244C22×C8C22×D311yescharacteristic
245C22×C8C22×D311yescharacteristic
246C22×C8C22×D312yes—
247#127—62no—
248#127—62no—
249#127—124no—
250#121—64no—
251#121—64no—
252#121—64no—
253#121—122no—
254#121—122no—
255#121—122no—
256#121—124no—
257C4×D4—22no—
258C4×D4—22no—
259C4×D4—62no—
260C4×D4—62no—
261C2×C4⋊C4—31no—
262C2×C4⋊C4—31no—
263C2×C4⋊C4—31no—
264C2×C4⋊C4—32no—
265C2×C4⋊C4—61no—
266C2×C4⋊C4—61no—
267C2×C4⋊C4D1211yescharacteristic
268C2×C22⋊C4—62no—
269C2×C22⋊C4—61no—
270C2×C22⋊C4—62no—
271C2×C22⋊C4—62no—
272C2×C22⋊C4—62no—
273C2×C22⋊C4D1212yes—
274C2×C42—61no—
275C2×C42C22×D311yescharacteristic
276#106—64no—
277C4⋊C8#6714yescomplemented
278#102—34no—
279#102—62no—
280#102—64no—
281#102—64no—
282#102—122no—
283C22⋊C8—24no—
284C4×C8—22no—
285#95—61no—
286#95—62no—
287C23×C3—22no—
288C23×C3—22no—
289C23×C3—22no—
290C23×C3—22no—
291C23×C3—22no—
292C23×C3—22no—
293C23×C3—22no—
294C23×C3C22×D411yescharacteristic
295C22×D3—22no—
296C22×D3—41no—
297C22×D3—41no—
298C22×D3—41no—
299C22×D3—41no—
300C22×D3—41no—
301C22×D3—41no—
302C22×D3—42no—
303C22×D3—42no—
304C22×D3—42no—
305C22×D3—42no—
306C22×D3—42no—
307C22×D3—42no—
308C22×D3—42no—
309C3×D4—44no—
310C2×C3×C4—21no—
311C2×C3×C4—21no—
312C2×C3×C4—22no—
313C2×C3×C4—22no—
314C2×C3×C4—22no—
315C2×C3×C4—42no—
316C2×C3×C4—42no—
317C2×C3×C4—42no—
318C2×C3×C4—42no—
319C2×C3×C4#12112yes—
320C2×C3×C4#12112yes—
321C2×C3×C4AGL(1,ℤ/8)12yes—
322C2×C3×C4AGL(1,ℤ/8)12yes—
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—
331#67—88no—
332C2×Dic3—22no—
333C2×Dic3—22no—
334C2×Dic3—41no—
335C2×Dic3—41no—
336C2×Dic3—41no—
337C2×Dic3—41no—
338C2×Dic3—41no—
339C2×Dic3—42no—
340C2×Dic3—42no—
341C2×Dic3—41no—
342C2×Dic3—41no—
343C2×Dic3—42no—
344D12—22no—
345D12—22no—
346D12—42no—
347D12—41no—
348D12—42no—
349D12—42no—
350D12—41no—
351D12—42no—
352D12—84no—
353D12—84no—
354C4×D3—84no—
355C3×C8—24no—
356C3×C8—22no—
357C3×C8—41no—
358C24—61no—
359C24—62no—
360C24C2×D1212yes—
361C2×D4—22no—
362C2×D4—22no—
363C2×D4—22no—
364C2×D4—32no—
365C2×D4—61no—
366C2×D4—62no—
367C2×D4—62no—
368C2×D4—61no—
369C2×D4—62no—
370C2×D4—61no—
371C2×D4—62no—
372C2×D4—61no—
373C2×D4—62no—
374C2×D4—61no—
375C2×D4—62no—
376C2×D4—124no—
377C2×D4—124no—
378C2×D4—124no—
379C2×D4—124no—
380C2×D4—124no—
381C2×D4—124no—
382C2×D4—122no—
383C2×D4—122no—
384C2×D4—122no—
385C2×D4—122no—
386C2×D4—241no—
387C2×D4—241no—
388C2×D4—242no—
389C2×D4—242no—
390C2×D4—242no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—61no—
398C22×C4—61no—
399C22×C4—61no—
400C22×C4—62no—
401C22×C4—122no—
402C22×C4—122no—
403C22×C4—122no—
404C22×C4C2×D1212yes—
405C22×C4C2×D1211yescharacteristic
406C22×C4C2×D1211yescharacteristic
407C22×C4C23×D311yescharacteristic
408D8—128no—
409M16—48no—
410C2×C8—22no—
411C2×C8—21no—
412C2×C8—22no—
413C2×C8—21no—
414C2×C8—21no—
415C2×C8—21no—
416C2×C8—21no—
417C2×C8D3×D412yes—
418C2×C8C2×#6714yes—
419C4⋊C4—22no—
420C4⋊C4—32no—
421C4⋊C4—32no—
422C4⋊C4—61no—
423C4⋊C4—61no—
424C4⋊C4—61no—
425C4⋊C4—62no—
426C4⋊C4—62no—
427C4⋊C4—61no—
428C4⋊C4—122no—
429C4⋊C4—241no—
430C22⋊C4—24no—
431C22⋊C4—124no—
432C22⋊C4—124no—
433C22⋊C4—124no—
434C22⋊C4—124no—
435C22⋊C4—122no—
436C42—122no—
437C42#23512yes—
438C42C2×#6712yes—
439C22×C3—42no—
440C22×C3—42no—
441C22×C3—42no—
442C22×C3—42no—
443C22×C3—42no—
444C22×C3—42no—
445C22×C3—42no—
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—81no—
457C2×D3—81no—
458C2×D3—81no—
459C2×D3—81no—
460C2×D3—82no—
461C2×D3—82no—
462C2×D3—82no—
463C2×D3—82no—
464C2×D3—82no—
465C2×D3—82no—
466C2×D3—82no—
467C3×C4—22no—
468C3×C4—42no—
469C3×C4#46914yescomplemented
470C3×C4#53211yescharacteristic
471C3×C4#54011yescharacteristic
472C3×C4D4211yescharacteristic
473C3×C4#55011yescharacteristic
474Dic3—42no—
475Dic3—42no—
476Dic3—82no—
477Dic3—81no—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—22no—
482C23—22no—
483C23—22no—
484C23—22no—
485C23—62no—
486C23—121no—
487C23—121no—
488C23—121no—
489C23—121no—
490C23—121no—
491C23—121no—
492C23—122no—
493C23—122no—
494C23—122no—
495C23—122no—
496C23—122no—
497C23—122no—
498C23—122no—
499C23C22×D1211yescharacteristic
500D4—44no—
501D4—62no—
502D4—62no—
503D4—122no—
504D4—122no—
505D4—121no—
506D4—122no—
507D4—121no—
508D4—122no—
509D4—244no—
510D4—248no—
511D4—248no—
512D4—248no—
513D4—244no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—62no—
524C2×C4—62no—
525C2×C4—121no—
526C2×C4—121no—
527C2×C4—121no—
528C2×C4—121no—
529C2×C4—121no—
530C2×C4—122no—
531C2×C4—121no—
532C2×C4—122no—
533C2×C4—121no—
534C2×C4—122no—
535C2×C4—244no—
536C2×C4#90812yes—
537C2×C4#90812yes—
538C2×C4#93012yes—
539C2×C4#93012yes—
540C2×C4C2×#23511yescharacteristic
541C2×C4C2×D3×D411yescharacteristic
542C2×C4#100411yescharacteristic
543C2×C4D3×#4111yescharacteristic
544C2×C4#100911yescharacteristic
545C2×C4C22×#6711yescharacteristic
546C8—22no—
547C8—24no—
548C8—41no—
549C2×C3—82no—
550C2×C3#242811yescharacteristic
551C2×C3C2×#46911yescharacteristic
552C2×C3#306211yescharacteristic
553C2×C3#327211yescharacteristic
554C2×C3#327411yescharacteristic
555C2×C3#332511yescharacteristic
556C2×C3#333211yescharacteristic
557D3—82no—
558D3—162no—
559C22—42no—
560C22—42no—
561C22—42no—
562C22—42no—
563C22—42no—
564C22—42no—
565C22—42no—
566C22—122no—
567C22—122no—
568C22—122no—
569C22—241no—
570C22—241no—
571C22—241no—
572C22—241no—
573C22—242no—
574C22—242no—
575C22—242no—
576C22—242no—
577C22—242no—
578C22—242no—
579C22—242no—
580C22D4×D1211yescharacteristic
581C22#555411yescharacteristic
582C22C2×#90811yescharacteristic
583C22C2×#90811yescharacteristic
584C22#575111yescharacteristic
585C22#575211yescharacteristic
586C22C2×#93011yescharacteristic
587C4—22no—
588C4—42no—
589C4—122no—
590C4—122no—
591C4—241no—
592C4—242no—
593C4#513514yescomplemented
594C4#554811yescharacteristic
595C4#556211yescharacteristic
596C4D4×#6711yescharacteristic
597C4#583411yescharacteristic
598C3#2539211yescharacteristic, complemented, Sylow
599C2—82no—
600C2—242no—
601C2—482no—
602C2#7954811yescharacteristic
603C2#7955011yescharacteristic
604C2#7955611yescharacteristic
605C2#7956511yescharacteristic
606C2#8240911yescharacteristic
607C2C2×#513511yescharacteristic
608C2#8380211yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#436C2
socleC23×C3C22×D4
hypercenterC2×#436D3
hypocenterC3#25392
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#436

Complements

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

NH ≅ G/Ntype
1. C2×C3×#436C2semidirect
2. C2×#5135C2semidirect
3. #82417C2semidirect
4. #82378C2semidirect
5. C2×#4843C2semidirect
6. #71990C2semidirect
7. #70454C2semidirect
8. #69961C2semidirect
9. #69935C2semidirect
10. #69772C2semidirect
11. C4×#821C2semidirect
47. C2×#436D3semidirect
104. C3×C4⋊C8D4semidirect
113. #821D4semidirect
277. C4⋊C8#67semidirect
469. C3×C4#469semidirect
593. C4#5135semidirect
598. C3#25392semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4362
C2×#51352
#824172
#823782
C2×#48432
#719902
#704542
#699612
#699352
#697722
C4×#8212
#253923