Finite Group Databasebeta

← #10830382

Subgroups — #10830382

Order 768.

1586 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#85786C211yesmaximal, characteristic, complemented
2C3×C4×#127C211yesmaximal, characteristic, complemented
3C2×#5077C212yesmaximal, complemented
4#82307C212yesmaximal, complemented
5#71617C211yesmaximal, characteristic, complemented
6#69664C212yesmaximal, complemented
7#69662C212yesmaximal, complemented
8#69593C212yesmaximal, complemented
9#69395C212yesmaximal, complemented
10#24222—31nomaximal, Sylow
11C2×#1004C2212yes—
12C2×C3×#127C2211yescharacteristic
13C2×C3×C4×D4—22no—
14C2×C3×C4×D4C2211yescharacteristic
15C2×C3×C4×D4C2211yescharacteristic, complemented
16C2×#932C2214yescomplemented
17C2×#930—22no—
18#5555—22no—
19#5554—24no—
20D4×D12—22no—
21#5510C2212yes—
22#5493—22no—
23#5477C2212yes—
24C3×#390—22no—
25C3×#390C2211yescharacteristic
26C3×#390C2211yescharacteristic, complemented
27C3×#384C2211yescharacteristic, complemented
28C3×C43C2211yescharacteristic, complemented
29C2×#833C2212yes—
30#5077—28no—
31#5019—22no—
32#5016—24no—
33#5015—22no—
34#4966C2214yescomplemented
35C2×#805C2214yescomplemented
36C2×#805C2214yescomplemented
37#4924C2212yescomplemented
38#4530C2212yescomplemented
39#3496—31no—
40#3036—32no—
41C2×#453—32no—
42C4×#127D311yescharacteristic, complemented
43#2039—32no—
44#2006—32no—
45#1962—32no—
46#1800—31no—
47#1792—32no—
48C22×C3×D4—22no—
49C22×C3×D4C2311yescharacteristic
50C22×C3×D4C2311yescharacteristic
51C23×C3×C4—21no—
52C23×C3×C4D412yes—
53C22×#67—24no—
54#1004—24no—
55#1004—24no—
56#1004—44no—
57C2×#237—24no—
58C2×D3×D4—24no—
59C2×#235C2314yes—
60C22×D12—22no—
61C3×#127—22no—
62C3×#127D414yescomplemented
63C3×C4×D4—22no—
64C3×C4×D4—22no—
65C3×C4×D4—22no—
66C3×C4×D4—22no—
67C3×C4×D4—48no—
68C2×C3×C4⋊C4—22no—
69C2×C3×C4⋊C4C2311yescharacteristic
70C2×C3×C4⋊C4C2311yescharacteristic
71C2×C3×C22⋊C4—22no—
72C2×C3×C22⋊C4—21no—
73C2×C3×C22⋊C4—22no—
74C2×C3×C22⋊C4D412yes—
75C2×C3×C42—21no—
76C2×C3×C42—21no—
77C2×C3×C42C2311yescharacteristic
78C2×C3×C42C2311yescharacteristic
79C2×C3×C42C2311yescharacteristic
80C2×#214—24no—
81#932—28no—
82#932—44no—
83C2×#213—24no—
84#930—24no—
85#930—24no—
86#930—22no—
87#930—22no—
88#930—24no—
89C2×(C2×D3)⋊C4—24no—
90C2×C4⋊Dic3—22no—
91C2×#208C2314yes—
92#897—24no—
93#895—24no—
94#886—28no—
95#882—28no—
96#875D414yescomplemented
97#874D412yescomplemented
98C4×D12—22no—
99C4×D12—22no—
100#869D412yescomplemented
101#833—24no—
102#810—24no—
103#809—24no—
104#805—28no—
105#805—44no—
106C4⋊(C3⋊C8)—22no—
107C4⋊(C3⋊C8)—22no—
108C2×H(5,𝔽2)—32no—
109C2×AGL(1,ℤ/8)—34no—
110#547—62no—
111#546—64no—
112D42—62no—
113#537—32no—
114#534—62no—
115#532—32no—
116C2×#127C2×D311yescharacteristic
117C2×#121—62no—
118C2×C4×D4—22no—
119C2×C4×D4C2×D311yescharacteristic
120C2×C4×D4C2×D311yescharacteristic, complemented
121#463—64no—
122#461—62no—
123#459—62no—
124#453—68no—
125#423—32no—
126C2×#104—34no—
127C2×#104—34no—
128#417—34no—
129C2×#100—32no—
130#390—22no—
131#390C2×D311yescharacteristic
132#390C2×D311yescharacteristic, complemented
133#384C2×D311yescharacteristic, complemented
134C43C2×D311yescharacteristic, complemented
135#343—32no—
136C24×C3—21no—
137C24×C3C2×D412yes—
138C23×D3—24no—
139C2×C3×D4—28no—
140C2×C3×D4—22no—
141C2×C3×D4—22no—
142C2×C3×D4—21no—
143C2×C3×D4—22no—
144C2×C3×D4—22no—
145C2×C3×D4—44no—
146C2×C3×D4—44no—
147C2×C3×D4C2×D412yes—
148C2×C3×D4C2×D412yescomplemented
149C2×C3×D4#4112yes—
150C22×C3×C4—21no—
151C22×C3×C4—21no—
152C22×C3×C4—22no—
153C22×C3×C4—22no—
154C22×C3×C4—21no—
155C22×C3×C4—44no—
156C22×C3×C4—44no—
157C22×C3×C4—41no—
158C22×C3×C4—41no—
159C22×C3×C4C2×D411yescharacteristic
160C22×C3×C4C2×D412yes—
161C22×C3×C4C2×D411yescharacteristic
162C22×C3×C4C2411yescharacteristic
163C2×#67—28no—
164C2×#67—24no—
165C2×#67—48no—
166C2×#67—48no—
167C22×Dic3—24no—
168#237—48no—
169#237—48no—
170D3×D4—48no—
171D3×D4—48no—
172#235—24no—
173#235—24no—
174#235—24no—
175#235—24no—
176C2×D12—22no—
177C2×D12—24no—
178C2×D12—22no—
179C2×D12—84no—
180C2×D12C2×D414yes—
181C2×C4×D3—24no—
182C2×Dic6C2×D414yes—
183C3×C4⋊C4—22no—
184C3×C4⋊C4—22no—
185C3×C4⋊C4—44no—
186C3×C22⋊C4—24no—
187C3×C22⋊C4—44no—
188C3×C22⋊C4—44no—
189C3×C22⋊C4—42no—
190C3×C42—24no—
191C3×C42—21no—
192C3×C42—22no—
193C3×C42—42no—
194C3×C42C2×D412yes—
195C3×C42C2×D412yes—
196C3×C42C2×D412yescomplemented
197#214—48no—
198#213—48no—
199(C2×D3)⋊C4—28no—
200(C2×D3)⋊C4—24no—
201(C2×D3)⋊C4—24no—
202C4⋊Dic3—24no—
203C4⋊Dic3—22no—
204C4⋊Dic3—22no—
205#208—24no—
206#208—24no—
207C2×C3⋊C8—24no—
208H(5,𝔽2)—64no—
209H(5,𝔽2)—64no—
210H(5,𝔽2)—124no—
211C2×#41—34no—
212C2×#41—64no—
213C22×D4—22no—
214C22×D4—62no—
215C22×D4—64no—
216C22×D4—64no—
217C22×D4C22×D311yescharacteristic
218C22×D4C22×D311yescharacteristic
219C23×C4—21no—
220C23×C4#6712yes—
221AGL(1,ℤ/8)—68no—
222AGL(1,ℤ/8)—124no—
223C2×SD16—64no—
224C2×D8—64no—
225C2×M16—34no—
226C4⋊Q8—32no—
227#127—22no—
228#127—32no—
229#127D1214yescomplemented
230#124—34no—
231#123—68no—
232C22⋊Q8—64no—
233#121—64no—
234#121—64no—
235#121—64no—
236#121—62no—
237#121—64no—
238#121—62no—
239#120—68no—
240C4×D4—22no—
241C4×D4—22no—
242C4×D4—22no—
243C4×D4—22no—
244C4×D4—48no—
245C4×D4—62no—
246C4×D4—62no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—62no—
249C2×C4⋊C4C22×D311yescharacteristic
250C2×C4⋊C4C22×D311yescharacteristic
251C2×C22⋊C4—22no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—64no—
255C2×C22⋊C4#6712yes—
256C2×C42—21no—
257C2×C42—21no—
258C2×C42C22×D311yescharacteristic
259C2×C42C22×D311yescharacteristic
260C2×C42C22×D311yescharacteristic
261C4⋊C8—62no—
262C4⋊C8—62no—
263#104—68no—
264#104—124no—
265#103—64no—
266#102—64no—
267#100—64no—
268C23×C3—22no—
269C23×C3—22no—
270C23×C3—22no—
271C23×C3—42no—
272C23×C3—42no—
273C23×C3—42no—
274C23×C3—42no—
275C23×C3—41no—
276C23×C3—41no—
277C23×C3#12114yes—
278C23×C3C22×D411yescharacteristic
279C22×D3—24no—
280C22×D3—48no—
281C22×D3—84no—
282C22×D3—84no—
283C3×D4—24no—
284C3×D4—24no—
285C3×D4—24no—
286C3×D4—416no—
287C3×D4—42no—
288C2×C3×C4—22no—
289C2×C3×C4—22no—
290C2×C3×C4—22no—
291C2×C3×C4—22no—
292C2×C3×C4—22no—
293C2×C3×C4—22no—
294C2×C3×C4—41no—
295C2×C3×C4—41no—
296C2×C3×C4—41no—
297C2×C3×C4—44no—
298C2×C3×C4—42no—
299C2×C3×C4—42no—
300C2×C3×C4—41no—
301C2×C3×C4—82no—
302C2×C3×C4—82no—
303C2×C3×C4—82no—
304C2×C3×C4#12012yes—
305C2×C3×C4#12011yescharacteristic
306C2×C3×C4#12011yescharacteristic
307C2×C3×C4C22×D411yescharacteristic
308C2×C3×C4C22×D412yes—
309C2×C3×C4C2×#4111yescharacteristic
310C2×C3×C4H(5,𝔽2)12yes—
311#67—416no—
312#67—44no—
313#67—44no—
314C2×Dic3—24no—
315C2×Dic3—48no—
316D12—24no—
317D12—24no—
318D12—44no—
319D12—82no—
320D12—84no—
321C4×D3—44no—
322C4×D3—44no—
323Dic6—24no—
324Dic6—24no—
325C3⋊C8—44no—
326C24—21no—
327C24—64no—
328C24C2×#6712yes—
329#41—64no—
330#41—64no—
331#41—64no—
332#41—64no—
333#41—128no—
334#41—128no—
335C2×Q8—34no—
336C2×D4—28no—
337C2×D4—21no—
338C2×D4—22no—
339C2×D4—22no—
340C2×D4—22no—
341C2×D4—22no—
342C2×D4—34no—
343C2×D4—44no—
344C2×D4—44no—
345C2×D4—64no—
346C2×D4—68no—
347C2×D4—64no—
348C2×D4—62no—
349C2×D4—62no—
350C2×D4—128no—
351C2×D4—128no—
352C2×D4—128no—
353C2×D4—128no—
354C2×D4—244no—
355C2×D4C2×D1212yes—
356C2×D4C2×D1212yescomplemented
357C2×D4#23512yes—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—21no—
363C22×C4—44no—
364C22×C4—44no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—64no—
368C22×C4—64no—
369C22×C4C2×#6711yescharacteristic
370C22×C4C2×#6712yes—
371C22×C4C2×#6711yescharacteristic
372C22×C4C23×D311yescharacteristic
373SD16—128no—
374D8—128no—
375M16—64no—
376M16—64no—
377C2×C8—64no—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C4⋊C4—44no—
381C4⋊C4—64no—
382C4⋊C4—62no—
383C4⋊C4—62no—
384C22⋊C4—24no—
385C22⋊C4—44no—
386C22⋊C4—44no—
387C22⋊C4—42no—
388C22⋊C4—68no—
389C22⋊C4—64no—
390C22⋊C4—64no—
391C42—24no—
392C42—21no—
393C42—22no—
394C42—42no—
395C42C2×D1212yescomplemented
396C42D3×D412yes—
397C42D3×D412yes—
398C22×C3—21no—
399C22×C3—21no—
400C22×C3—24no—
401C22×C3—44no—
402C22×C3—44no—
403C22×C3—42no—
404C22×C3—42no—
405C22×C3—42no—
406C22×C3—82no—
407C22×C3—82no—
408C22×C3—81no—
409C22×C3—81no—
410C22×C3C2×#12011yescharacteristic
411C22×C3C2×#12111yescharacteristic
412C22×C3#53511yescharacteristic
413C2×D3—44no—
414C2×D3—44no—
415C2×D3—44no—
416C2×D3—168no—
417C3×C4—24no—
418C3×C4—21no—
419C3×C4—22no—
420C3×C4—42no—
421C3×C4—42no—
422C3×C4—81no—
423C3×C4#45312yescomplemented
424C3×C4D4212yes—
425C3×C4#54612yes—
426Dic3—44no—
427C23—22no—
428C23—22no—
429C23—22no—
430C23—42no—
431C23—42no—
432C23—42no—
433C23—42no—
434C23—41no—
435C23—41no—
436C23—64no—
437C23—128no—
438C23—244no—
439C23—244no—
440C23#93014yes—
441C23C22×#6711yescharacteristic
442Q8—64no—
443Q8—64no—
444D4—24no—
445D4—24no—
446D4—24no—
447D4—416no—
448D4—42no—
449D4—64no—
450D4—64no—
451D4—1216no—
452D4—124no—
453D4—124no—
454D4—124no—
455D4—242no—
456D4—244no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—44no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—41no—
470C2×C4—64no—
471C2×C4—82no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4—128no—
475C2×C4—124no—
476C2×C4—124no—
477C2×C4#93712yes—
478C2×C4#93711yescharacteristic
479C2×C4#93711yescharacteristic
480C2×C4C22×D1211yescharacteristic
481C2×C4C2×#23511yescharacteristic
482C2×C4C2×D3×D412yes—
483C2×C4#100912yes—
484C8—124no—
485C2×C3—21no—
486C2×C3—21no—
487C2×C3—44no—
488C2×C3—82no—
489C2×C3#240811yescharacteristic
490C2×C3C2×#45311yescharacteristic
491C2×C3#301811yescharacteristic
492D3—84no—
493C22—21no—
494C22—21no—
495C22—24no—
496C22—44no—
497C22—44no—
498C22—42no—
499C22—42no—
500C22—42no—
501C22—82no—
502C22—82no—
503C22—81no—
504C22—81no—
505C22—124no—
506C22—124no—
507C22—124no—
508C22—488no—
509C22C2×#93011yescharacteristic
510C22C2×#93711yescharacteristic
511C22#583011yescharacteristic
512C4—24no—
513C4—22no—
514C4—21no—
515C4—42no—
516C4—42no—
517C4—81no—
518C4—124no—
519C4#507712yescomplemented
520C4D4×D1212yes—
521C4#555012yes—
522C3#2422211yescharacteristic, complemented, Sylow
523C2—21no—
524C2—21no—
525C2—44no—
526C2—82no—
527C2—244no—
528C2#8240311yescharacteristic
529C2C2×#507711yescharacteristic
530C2#8332211yescharacteristic
subgroupG/H
centerC22C2×#937
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×C4×#127C2
socleC22×C3C2×#120
hypercenterC4×#127D3
hypocenterC3#24222
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ C4×#127

Complements

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

NH ≅ G/Ntype
1. #85786C2semidirect
2. C3×C4×#127C2semidirect
3. C2×#5077C2semidirect
4. #82307C2semidirect
5. #71617C2semidirect
6. #69664C2semidirect
7. #69662C2semidirect
8. #69593C2semidirect
9. #69395C2semidirect
15. C2×C3×C4×D4C22semidirect
16. C2×#932C22semidirect
26. C3×#390C22semidirect
27. C3×#384C22semidirect
28. C3×C43C22semidirect
34. #4966C22semidirect
35. C2×#805C22semidirect
36. C2×#805C22semidirect
37. #4924C22semidirect
38. #4530C22semidirect
42. C4×#127D3semidirect
62. C3×#127D4semidirect
96. #875D4semidirect
97. #874D4semidirect
100. #869D4semidirect
120. C2×C4×D4C2×D3semidirect
132. #390C2×D3semidirect
133. #384C2×D3semidirect
134. C43C2×D3semidirect
148. C2×C3×D4C2×D4semidirect
196. C3×C42C2×D4semidirect
229. #127D12semidirect
356. C2×D4C2×D12semidirect
395. C42C2×D12semidirect
423. C3×C4#453semidirect
519. C4#5077semidirect
522. C3#24222semidirect

Maximal subgroups

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

subgroupindex
#857862
C3×C4×#1272
C2×#50772
#823072
#716172
#696642
#696622
#695932
#693952
#242223