Finite Group Databasebeta

← #10822439

Subgroups — #10822439

Order 768.

2418 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5804C212yesmaximal, complemented
2C2×#5493C211yesmaximal, characteristic, complemented
3C3×#2308C211yesmaximal, characteristic, complemented
4C22×#806C212yesmaximal, complemented
5C2×#4526C211yesmaximal, characteristic, complemented
6#69539C214yesmaximal, complemented
7#69500C214yesmaximal, complemented
8#23444—31nomaximal, Sylow
9C23×C3×D4C2211yescharacteristic
10C23×#67C2212yes—
11C22×C3×C22⋊C4C2211yescharacteristic
12C22×C3×C22⋊C4C2211yescharacteristic
13C2×#953C2214yescomplemented
14C22×#217C2212yes—
15#5804—28no—
16#5804—28no—
17C2×#939—24no—
18C2×#937—22no—
19C2×#935—22no—
20C2×#930—22no—
21C2×#929C2214yescomplemented
22#5493—24no—
23#5493—22no—
24#5493C2214yes—
25C2×#886—22no—
26C2×#882—22no—
27C3×#386—22no—
28C3×#386—22no—
29C3×#381C2212yescomplemented
30C3×#379C2212yescomplemented
31C2×#806—28no—
32C2×#806C2218yescomplemented
33#4526C2214yescomplemented
34C2×#534—32no—
35C2×#534—31no—
36C22×#99—32no—
37#2308D311yescharacteristic, complemented
38#2022—34no—
39#2021—34no—
40C2×#342—31no—
41C25×C3C2311yescharacteristic
42C25×C3C2311yescharacteristic
43C24×D3D412yescomplemented
44C22×C3×D4—21no—
45C22×C3×D4—21no—
46C22×C3×D4—22no—
47C22×C3×D4—22no—
48C22×C3×D4—24no—
49C22×C3×D4—22no—
50C22×C3×D4D414yescomplemented
51C23×C3×C4—21no—
52C23×C3×C4D412yescomplemented
53C23×C3×C4C2311yescharacteristic
54C22×#67—24no—
55C22×#67—22no—
56C22×#67—22no—
57C22×#67—22no—
58C22×#67—22no—
59C22×#67—24no—
60C22×#67—24no—
61C22×#67—24no—
62C22×#67—24no—
63C22×#67D414yescomplemented
64C22×#67C2314yes—
65C23×Dic3—22no—
66C23×Dic3D412yescomplemented
67C22×D12—22no—
68C2×C3×C22⋊C4—22no—
69C2×C3×C22⋊C4—22no—
70C2×C3×C22⋊C4—24no—
71C2×C3×C22⋊C4—21no—
72C2×C3×C22⋊C4—24no—
73C2×C3×C22⋊C4D412yescomplemented
74C2×C3×C22⋊C4C2312yes—
75C2×C3×C22⋊C4C2312yes—
76#953—28no—
77#953—28no—
78C2×#217—22no—
79C2×#217—24no—
80C2×#217—24no—
81C2×#217D418yescomplemented
82C2×#217C2314yes—
83#939—232no—
84#937—216no—
85#935—216no—
86#930—28no—
87#930—28no—
88#929—28no—
89#929—28no—
90C2×(C2×D3)⋊C4—24no—
91C2×(C2×D3)⋊C4—24no—
92C2×C4⋊Dic3—22no—
93C2×Dic3⋊C4—24no—
94#886—28no—
95#886—28no—
96#882—28no—
97#882—28no—
98C3×#95—24no—
99#806—216no—
100C23×D4—32no—
101C23×D4C2×D311yescharacteristic
102#534—34no—
103#534—64no—
104#534—68no—
105#534—68no—
106#534—62no—
107C2×#123—34no—
108C2×#123—62no—
109C2×#123—62no—
110C2×#121—64no—
111C2×#121—62no—
112C2×#120—34no—
113C2×#120—62no—
114C2×#120—62no—
115C22×C22⋊C4—32no—
116C22×C22⋊C4C2×D311yescharacteristic
117C22×C22⋊C4C2×D311yescharacteristic
118C2×#99—38no—
119C2×#99—68no—
120#386—22no—
121#386—22no—
122#381C2×D312yescomplemented
123#379C2×D312yescomplemented
124#342—34no—
125C24×C3—22no—
126C24×C3—21no—
127C24×C3—22no—
128C24×C3—21no—
129C24×C3—82no—
130C24×C3—82no—
131C24×C3C2×D412yes—
132C24×C3C2×D412yes—
133C24×C3C2×D412yes—
134C24×C3C2×D414yes—
135C24×C3C2×D411yescharacteristic
136C24×C3C2×D411yescharacteristic
137C24×C3C2×D412yescomplemented
138C24×C3#4112yes—
139C24×C3C2411yescharacteristic
140C23×D3—22no—
141C23×D3—22no—
142C23×D3—24no—
143C23×D3—24no—
144C23×D3—84no—
145C23×D3C2×D414yes—
146C2×C3×D4—28no—
147C2×C3×D4—28no—
148C2×C3×D4—24no—
149C2×C3×D4—24no—
150C2×C3×D4—42no—
151C2×C3×D4—41no—
152C2×C3×D4—41no—
153C2×C3×D4—44no—
154C2×C3×D4—44no—
155C2×C3×D4—42no—
156C2×C3×D4—42no—
157C22×C3×C4—22no—
158C22×C3×C4—21no—
159C22×C3×C4—22no—
160C22×C3×C4—22no—
161C22×C3×C4—21no—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—42no—
165C22×C3×C4—82no—
166C22×C3×C4—82no—
167C22×C3×C4C2×D414yes—
168C22×C3×C4C2×D412yescomplemented
169C2×#67—24no—
170C2×#67—24no—
171C2×#67—28no—
172C2×#67—28no—
173C2×#67—24no—
174C2×#67—24no—
175C2×#67—24no—
176C2×#67—24no—
177C2×#67—24no—
178C2×#67—24no—
179C2×#67—216no—
180C2×#67—44no—
181C2×#67—44no—
182C2×#67—44no—
183C2×#67—44no—
184C2×#67—44no—
185C2×#67—44no—
186C2×#67—44no—
187C2×#67—44no—
188C2×#67—88no—
189C22×Dic3—22no—
190C22×Dic3—22no—
191C22×Dic3—24no—
192C22×Dic3—24no—
193C22×Dic3—24no—
194C22×Dic3—28no—
195C22×Dic3—84no—
196C22×Dic3C2×D414yes—
197C2×D12—28no—
198C2×D12—84no—
199C3×C22⋊C4—24no—
200C3×C22⋊C4—24no—
201C3×C22⋊C4—24no—
202C3×C22⋊C4—42no—
203C3×C22⋊C4—84no—
204#217—216no—
205#217—28no—
206#217—24no—
207#217C2×D418yescomplemented
208(C2×D3)⋊C4—216no—
209(C2×D3)⋊C4—216no—
210C4⋊Dic3—28no—
211Dic3⋊C4—216no—
212C25—32no—
213C25C22×D311yescharacteristic
214C25C22×D311yescharacteristic
215C22×D4—22no—
216C22×D4—22no—
217C22×D4—22no—
218C22×D4—24no—
219C22×D4—21no—
220C22×D4—21no—
221C22×D4—34no—
222C22×D4—34no—
223C22×D4—62no—
224C22×D4—64no—
225C22×D4—64no—
226C22×D4—62no—
227C22×D4—64no—
228C22×D4—62no—
229C22×D4—64no—
230C22×D4—62no—
231C22×D4—62no—
232C22×D4—64no—
233C22×D4D1214yescomplemented
234C23×C4—21no—
235C23×C4—32no—
236C23×C4—62no—
237C23×C4#6712yescomplemented
238C23×C4C22×D311yescharacteristic
239#123—68no—
240#123—68no—
241#123—68no—
242#123—616no—
243#123—68no—
244#121—68no—
245#121—632no—
246#121—68no—
247#120—68no—
248#120—68no—
249#120—68no—
250#120—616no—
251#120—68no—
252C2×C4⋊C4—64no—
253C2×C4⋊C4—62no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—38no—
260C2×C22⋊C4—34no—
261C2×C22⋊C4—64no—
262C2×C22⋊C4—64no—
263C2×C22⋊C4—64no—
264C2×C22⋊C4—64no—
265C2×C22⋊C4—62no—
266C2×C22⋊C4#6712yescomplemented
267C2×C22⋊C4C22×D312yes—
268C2×C22⋊C4C22×D312yes—
269#99—616no—
270#95—24no—
271C23×C3—24no—
272C23×C3—22no—
273C23×C3—44no—
274C23×C3—42no—
275C23×C3—42no—
276C23×C3—44no—
277C23×C3—42no—
278C23×C3—44no—
279C23×C3—48no—
280C23×C3—41no—
281C23×C3—42no—
282C23×C3—41no—
283C23×C3—82no—
284C23×C3—84no—
285C23×C3—82no—
286C23×C3—82no—
287C23×C3—84no—
288C23×C3—82no—
289C23×C3#12014yes—
290C23×C3#12014yes—
291C23×C3#12012yes—
292C23×C3#12011yescharacteristic
293C23×C3#12011yescharacteristic
294C23×C3#12114yes—
295C23×C3#12114yes—
296C23×C3#12112yes—
297C23×C3#12112yes—
298C23×C3#12712yes—
299C23×C3#12712yes—
300C23×C3C22×D411yescharacteristic
301C23×C3C22×D412yes—
302C23×C3C22×D412yes—
303C23×C3C22×D411yescharacteristic
304C23×C3C2×#4111yescharacteristic
305C22×D3—24no—
306C22×D3—24no—
307C22×D3—24no—
308C22×D3—48no—
309C22×D3—48no—
310C22×D3—44no—
311C22×D3—44no—
312C22×D3—84no—
313C22×D3—88no—
314C22×D3—88no—
315C3×D4—44no—
316C3×D4—48no—
317C3×D4—44no—
318C2×C3×C4—24no—
319C2×C3×C4—24no—
320C2×C3×C4—22no—
321C2×C3×C4—44no—
322C2×C3×C4—41no—
323C2×C3×C4—42no—
324C2×C3×C4—41no—
325C2×C3×C4—44no—
326C2×C3×C4—44no—
327C2×C3×C4—42no—
328C2×C3×C4—82no—
329C2×C3×C4—84no—
330#67—44no—
331#67—44no—
332#67—44no—
333#67—44no—
334#67—48no—
335#67—48no—
336#67—816no—
337C2×Dic3—24no—
338C2×Dic3—28no—
339C2×Dic3—24no—
340C2×Dic3—24no—
341C2×Dic3—48no—
342C2×Dic3—44no—
343C2×Dic3—44no—
344C2×Dic3—88no—
345D12—88no—
346C24—22no—
347C24—21no—
348C24—21no—
349C24—22no—
350C24—34no—
351C24—62no—
352C24—64no—
353C24—64no—
354C24—62no—
355C24—82no—
356C24—82no—
357C24—244no—
358C24C2×D1212yes—
359C24C2×D1212yescomplemented
360C24#23512yes—
361C24D3×D412yes—
362C24D3×D412yes—
363C24D3×D414yes—
364C24C2×#6711yescharacteristic
365C24C2×#6711yescharacteristic
366C24C23×D311yescharacteristic
367C2×D4—28no—
368C2×D4—28no—
369C2×D4—24no—
370C2×D4—24no—
371C2×D4—44no—
372C2×D4—44no—
373C2×D4—42no—
374C2×D4—41no—
375C2×D4—41no—
376C2×D4—42no—
377C2×D4—42no—
378C2×D4—68no—
379C2×D4—68no—
380C2×D4—68no—
381C2×D4—64no—
382C2×D4—64no—
383C2×D4—64no—
384C2×D4—64no—
385C2×D4—64no—
386C2×D4—64no—
387C2×D4—64no—
388C2×D4—64no—
389C2×D4—616no—
390C2×D4—124no—
391C2×D4—124no—
392C2×D4—124no—
393C2×D4—124no—
394C2×D4—124no—
395C2×D4—124no—
396C2×D4—124no—
397C2×D4—124no—
398C2×D4—244no—
399C2×D4—248no—
400C22×C4—22no—
401C22×C4—21no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—34no—
408C22×C4—42no—
409C22×C4—64no—
410C22×C4—64no—
411C22×C4—62no—
412C22×C4—62no—
413C22×C4—64no—
414C22×C4—68no—
415C22×C4—82no—
416C22×C4—82no—
417C22×C4—244no—
418C22×C4C2×D1212yescomplemented
419C22×C4C2×#6714yes—
420C4⋊C4—616no—
421C4⋊C4—68no—
422C22⋊C4—24no—
423C22⋊C4—24no—
424C22⋊C4—24no—
425C22⋊C4—38no—
426C22⋊C4—42no—
427C22⋊C4—616no—
428C22⋊C4—616no—
429C22⋊C4—616no—
430C22⋊C4—68no—
431C22⋊C4—64no—
432C22⋊C4—84no—
433C22×C3—24no—
434C22×C3—22no—
435C22×C3—22no—
436C22×C3—22no—
437C22×C3—24no—
438C22×C3—44no—
439C22×C3—42no—
440C22×C3—44no—
441C22×C3—44no—
442C22×C3—41no—
443C22×C3—42no—
444C22×C3—41no—
445C22×C3—44no—
446C22×C3—42no—
447C22×C3—42no—
448C22×C3—42no—
449C22×C3—48no—
450C22×C3—44no—
451C22×C3—42no—
452C22×C3—82no—
453C22×C3—84no—
454C22×C3—82no—
455C22×C3#39212yes—
456C22×C3#39214yes—
457C22×C3#39212yes—
458C22×C3C2×#12012yes—
459C22×C3C2×#12011yescharacteristic
460C22×C3C2×#12112yes—
461C22×C3C2×#12111yescharacteristic
462C22×C3C2×#12711yescharacteristic
463C2×D3—48no—
464C2×D3—44no—
465C2×D3—48no—
466C2×D3—88no—
467C2×D3—88no—
468C2×D3—84no—
469C3×C4—42no—
470C3×C4—44no—
471C3×C4—82no—
472Dic3—48no—
473Dic3—44no—
474C23—24no—
475C23—22no—
476C23—42no—
477C23—42no—
478C23—44no—
479C23—42no—
480C23—44no—
481C23—44no—
482C23—48no—
483C23—42no—
484C23—41no—
485C23—41no—
486C23—64no—
487C23—64no—
488C23—64no—
489C23—82no—
490C23—82no—
491C23—82no—
492C23—84no—
493C23—82no—
494C23—84no—
495C23—128no—
496C23—128no—
497C23—124no—
498C23—124no—
499C23—244no—
500C23—248no—
501C23—248no—
502C23#88214yes—
503C23#88214yes—
504C23#88414yes—
505C23#88414yes—
506C23#93012yes—
507C23#93012yes—
508C23#93712yes—
509C23#93711yescharacteristic
510C23#93711yescharacteristic
511C23#94012yes—
512C23#94012yes—
513C23C22×D1211yescharacteristic
514C23C2×#23511yescharacteristic
515C23C2×D3×D412yes—
516C23C2×D3×D412yes—
517C23C22×#6711yescharacteristic
518D4—44no—
519D4—48no—
520D4—44no—
521D4—128no—
522D4—128no—
523D4—124no—
524D4—124no—
525D4—124no—
526D4—124no—
527D4—248no—
528D4—2416no—
529C2×C4—24no—
530C2×C4—22no—
531C2×C4—24no—
532C2×C4—44no—
533C2×C4—42no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—42no—
539C2×C4—68no—
540C2×C4—64no—
541C2×C4—64no—
542C2×C4—64no—
543C2×C4—82no—
544C2×C4—84no—
545C2×C4—128no—
546C2×C4—124no—
547C2×C4—124no—
548C2×C4—248no—
549C2×C3—22no—
550C2×C3—21no—
551C2×C3—22no—
552C2×C3—22no—
553C2×C3—24no—
554C2×C3—21no—
555C2×C3—22no—
556C2×C3—82no—
557C2×C3C2×#39211yescharacteristic
558C2×C3#302612yes—
559D3—84no—
560C22—22no—
561C22—22no—
562C22—22no—
563C22—24no—
564C22—24no—
565C22—42no—
566C22—44no—
567C22—41no—
568C22—41no—
569C22—42no—
570C22—44no—
571C22—44no—
572C22—44no—
573C22—42no—
574C22—42no—
575C22—42no—
576C22—48no—
577C22—44no—
578C22—42no—
579C22—82no—
580C22—84no—
581C22—82no—
582C22—128no—
583C22—128no—
584C22—124no—
585C22—244no—
586C22—248no—
587C22—248no—
588C22#496012yes—
589C22#496014yes—
590C22#496012yes—
591C22C2×#88212yes—
592C22C2×#88412yes—
593C22C2×#93011yescharacteristic
594C22C2×#93711yescharacteristic
595C22C2×#94011yescharacteristic
596C4—44no—
597C4—42no—
598C4—82no—
599C4—128no—
600C4—124no—
601C3#2344411yescharacteristic, complemented, Sylow
602C2—21no—
603C2—22no—
604C2—21no—
605C2—22no—
606C2—22no—
607C2—24no—
608C2—22no—
609C2—82no—
610C2—244no—
611C2C2×#496011yescharacteristic
612C2#8337512yes—
subgroupG/H
centerC22C2×#937
commutator, cosocleC24×C3C24
FrattiniC24C23×D3
FittingC3×#2308C2
socleC22×C3C2×#120
hypercenter#2308D3
hypocenterC3#23444
LCSG ▷ C24×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C25 ◁ #2308

Complements

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

NH ≅ G/Ntype
1. C2×#5804C2semidirect
2. C2×#5493C2semidirect
3. C3×#2308C2semidirect
4. C22×#806C2semidirect
5. C2×#4526C2semidirect
6. #69539C2semidirect
7. #69500C2semidirect
13. C2×#953C22semidirect
21. C2×#929C22semidirect
29. C3×#381C22semidirect
30. C3×#379C22semidirect
32. C2×#806C22semidirect
33. #4526C22semidirect
37. #2308D3semidirect
43. C24×D3D4semidirect
50. C22×C3×D4D4semidirect
52. C23×C3×C4D4semidirect
63. C22×#67D4semidirect
66. C23×Dic3D4semidirect
73. C2×C3×C22⋊C4D4semidirect
81. C2×#217D4semidirect
122. #381C2×D3semidirect
123. #379C2×D3semidirect
137. C24×C3C2×D4semidirect
168. C22×C3×C4C2×D4semidirect
207. #217C2×D4semidirect
233. C22×D4D12semidirect
237. C23×C4#67semidirect
266. C2×C22⋊C4#67semidirect
359. C24C2×D12semidirect
418. C22×C4C2×D12semidirect
601. C3#23444semidirect

Maximal subgroups

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

subgroupindex
C2×#58042
C2×#54932
C3×#23082
C22×#8062
C2×#45262
#695392
#695002
#234443