Finite Group Databasebeta

← #10826899

Subgroups — #10826899

Order 768.

1886 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#535C211yesmaximal, characteristic, complemented
2#87274C211yesmaximal, characteristic, complemented
3#84038C212yesmaximal, complemented
4C2×#5062C212yesmaximal, complemented
5#72137C211yesmaximal, characteristic, complemented
6#71769C214yesmaximal, complemented
7#71724C212yesmaximal, complemented
8#71591C212yesmaximal, complemented
9#23915—31nomaximal, Sylow
10C22×C3×#41C2211yescharacteristic
11C23×C3×D4C2211yescharacteristic
12C2×#1007C2211yescharacteristic, complemented
13C3×#535—28no—
14C2×C3×#127C2212yes—
15C2×C3×#124C2212yes—
16C2×C3×#121—22no—
17C2×C3×#120—22no—
18C2×C3×#117C2211yescharacteristic
19#5834—24no—
20#5831—24no—
21#5812C2211yescharacteristic, complemented
22#5805—22no—
23#5805C2212yescomplemented
24C2×#932C2214yes—
25#5791—22no—
26#5791C2212yes—
27C2×#836C2212yescomplemented
28#5218—24no—
29#5217—24no—
30C2×#833C2212yescomplemented
31#5215C2214yes—
32#5207C2212yes—
33#5062—28no—
34C2×#805C2214yescomplemented
35#4523C2214yescomplemented
36#3487—31no—
37C2×#535D311yescharacteristic, complemented
38C2×#454—32no—
39#3003—32no—
40#2002—34no—
41#1897—32no—
42#1888—32no—
43#1785—31no—
44C25×C3—21no—
45C2×C3×#41—24no—
46C2×C3×#41—24no—
47C2×C3×#41—42no—
48C22×C3×Q8C2311yescharacteristic
49C22×C3×D4—24no—
50C22×C3×D4—24no—
51C22×C3×D4—21no—
52C22×C3×D4—24no—
53C22×C3×D4—42no—
54C22×C3×D4D412yes—
55C22×C3×D4C2311yescharacteristic
56C22×C3×D4C2312yes—
57C23×C3×C4D412yes—
58C23×C3×C4C2311yescharacteristic
59#1007—24no—
60#1007—42no—
61C2×#239—22no—
62C2×D3×Q8—22no—
63C2×#235—22no—
64C2×#235C2312yes—
65C3×#127—24no—
66C3×#127—24no—
67C3×#124—24no—
68C3×#124D418yescomplemented
69C3×#121—216no—
70C3×#120—216no—
71C3×#117—24no—
72C2×C3×C4⋊C4—21no—
73C2×C3×C22⋊C4—21no—
74C2×C3×C22⋊C4—24no—
75C2×C3×C42C2312yes—
76#953—24no—
77#953—22no—
78Q8×Dic3—24no—
79#940—24no—
80#940—22no—
81#938—28no—
82#938—24no—
83#936—24no—
84#936—22no—
85C2×#214—24no—
86#932—28no—
87#932—44no—
88C2×#213—24no—
89#930—24no—
90#930D414yescomplemented
91#929—22no—
92C4×#67—28no—
93C4×#67—24no—
94#926—22no—
95#926C2312yes—
96#924—24no—
97#924D414yescomplemented
98C2×#208C2314yes—
99#836—24no—
100#833—42no—
101#832—28no—
102#830—24no—
103#805—28no—
104C2×2−1+4—31no—
105C22×#41C2×D311yescharacteristic
106C23×D4C2×D311yescharacteristic
107C2×AGL(1,ℤ/8)—34no—
108#550—64no—
109#548—64no—
110#535—28no—
111#535—32no—
112#535—62no—
113C2×#127C2×D312yes—
114C2×#124C2×D312yes—
115#525—32no—
116#525—62no—
117C2×#121—22no—
118C2×#120—22no—
119#519—31no—
120C2×#117C2×D311yescharacteristic
121#454—68no—
122#450—64no—
123#447—64no—
124C2×#104—34no—
125#418—34no—
126C2×#101—32no—
127C2×#100—32no—
128#407—32no—
129#339—34no—
130C24×C3—24no—
131C24×C3—22no—
132C24×C3—42no—
133C24×C3—42no—
134C24×C3—84no—
135C24×C3C2×D412yes—
136C24×C3C2×D411yescharacteristic
137C3×#41—416no—
138C2×C3×Q8—22no—
139C2×C3×Q8—22no—
140C2×C3×Q8C2×D414yes—
141C2×C3×D4—24no—
142C2×C3×D4—24no—
143C2×C3×D4—216no—
144C2×C3×D4—24no—
145C2×C3×D4—24no—
146C2×C3×D4—22no—
147C2×C3×D4—22no—
148C2×C3×D4—22no—
149C2×C3×D4—24no—
150C2×C3×D4—22no—
151C2×C3×D4—416no—
152C2×C3×D4—42no—
153C2×C3×D4—42no—
154C2×C3×D4—42no—
155C2×C3×D4—88no—
156C2×C3×D4C2×D414yes—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
159C22×C3×C4—22no—
160C22×C3×C4—24no—
161C22×C3×C4—24no—
162C22×C3×C4—42no—
163C22×C3×C4C2×D414yes—
164C22×C3×C4C2×D412yes—
165C22×C3×C4C2×D411yescharacteristic
166C22×C3×C4C2411yescharacteristic
167C2×#67—22no—
168C2×#67—28no—
169C2×#67—24no—
170#239—48no—
171D3×Q8—48no—
172#235—24no—
173#235—24no—
174#235—44no—
175#235—44no—
176C2×D12—22no—
177C2×D12C2×D412yes—
178C2×C4×D3—24no—
179C2×C4×D3—22no—
180C2×Dic6—22no—
181C2×Dic6C2×D412yes—
182C3×C4⋊C4—24no—
183C3×C22⋊C4—24no—
184C3×C22⋊C4—216no—
185C3×C42—22no—
186C3×C42C2×D414yes—
187#217—28no—
188#217—24no—
189#217—22no—
190#214—48no—
191#213—48no—
192(C2×D3)⋊C4—24no—
193C4⋊Dic3—24no—
194C4⋊Dic3C2×D414yes—
195Dic3⋊C4—24no—
196C4×Dic3—24no—
197C4×Dic3—22no—
198#208—24no—
199#208—24no—
200C2×C3⋊C8—24no—
201C25—21no—
2022−1+4—64no—
2032−1+4—122no—
204C2×#41—24no—
205C2×#41—24no—
206C2×#41—32no—
207C2×#41—42no—
208C2×#41—62no—
209C2×#41—62no—
210C22×Q8—62no—
211C22×Q8C22×D311yescharacteristic
212C22×D4—24no—
213C22×D4—24no—
214C22×D4—24no—
215C22×D4—21no—
216C22×D4—42no—
217C22×D4#6712yes—
218C22×D4C22×D311yescharacteristic
219C22×D4C22×D312yes—
220C23×C4#6712yes—
221C23×C4C22×D311yescharacteristic
222AGL(1,ℤ/8)—68no—
223AGL(1,ℤ/8)—124no—
224C2×SD16—64no—
225C2×D8—64no—
226C2×M16—34no—
227#127—24no—
228#127—24no—
229#127—62no—
230#127—64no—
231#124—24no—
232#124—62no—
233#124—64no—
234#124#6718yescomplemented
235#123—62no—
236C22⋊Q8—34no—
237C22⋊Q8—64no—
238#121—216no—
239#121—34no—
240#121—64no—
241#121—64no—
242#121—68no—
243#120—216no—
244#120—62no—
245#120—64no—
246C4×Q8—64no—
247C4×D4—68no—
248C4×D4—64no—
249#117—24no—
250#117—32no—
251#117—62no—
252C2×C4⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—24no—
255C2×C42C22×D312yes—
256#104—68no—
257#102—68no—
258#101—64no—
259#100—122no—
260C22⋊C8—64no—
261C23×C3—21no—
262C23×C3—21no—
263C23×C3—22no—
264C23×C3—24no—
265C23×C3—22no—
266C23×C3—22no—
267C23×C3—42no—
268C23×C3—44no—
269C23×C3—48no—
270C23×C3—48no—
271C23×C3—42no—
272C23×C3—42no—
273C23×C3—88no—
274C23×C3—88no—
275C23×C3—88no—
276C23×C3#12012yes—
277C23×C3C22×D411yescharacteristic
278C22×D3—22no—
279C3×Q8—28no—
280C3×D4—28no—
281C3×D4—48no—
282C3×D4—48no—
283C3×D4—44no—
284C3×D4—44no—
285C3×D4—816no—
286C2×C3×C4—22no—
287C2×C3×C4—21no—
288C2×C3×C4—22no—
289C2×C3×C4—21no—
290C2×C3×C4—28no—
291C2×C3×C4—24no—
292C2×C3×C4—48no—
293C2×C3×C4—44no—
294C2×C3×C4#12012yes—
295C2×C3×C4#12014yes—
296C2×C3×C4C22×D412yes—
297C2×C3×C4C22×D412yes—
298C2×C3×C4H(5,𝔽2)12yes—
299#67—42no—
300#67—42no—
301#67—84no—
302#67—88no—
303C2×Dic3—24no—
304C2×Dic3—22no—
305D12—24no—
306D12—44no—
307C4×D3—48no—
308C4×D3—44no—
309Dic6—24no—
310Dic6—44no—
311C3⋊C8—44no—
312C24—24no—
313C24—22no—
314C24—42no—
315C24—42no—
316C24—84no—
317C24C2×#6712yes—
318C24C2×#6711yescharacteristic
319#41—416no—
320#41—64no—
321#41—64no—
322#41—124no—
323#41—128no—
324#41—124no—
325C2×Q8—22no—
326C2×Q8—22no—
327C2×Q8—32no—
328C2×Q8—62no—
329C2×Q8—128no—
330C2×Q8C2×#6714yes—
331C2×D4—24no—
332C2×D4—24no—
333C2×D4—216no—
334C2×D4—24no—
335C2×D4—24no—
336C2×D4—22no—
337C2×D4—24no—
338C2×D4—22no—
339C2×D4—22no—
340C2×D4—22no—
341C2×D4—32no—
342C2×D4—416no—
343C2×D4—42no—
344C2×D4—42no—
345C2×D4—42no—
346C2×D4—62no—
347C2×D4—62no—
348C2×D4—68no—
349C2×D4—64no—
350C2×D4—88no—
351C2×D4C2×#6714yes—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—42no—
358C22×C4—64no—
359C22×C4—62no—
360C22×C4D3×D414yes—
361C22×C4C2×#6711yescharacteristic
362C22×C4C2×#6712yes—
363C22×C4C23×D311yescharacteristic
364SD16—128no—
365D8—128no—
366M16—64no—
367M16—64no—
368C2×C8—64no—
369C4⋊C4—24no—
370C4⋊C4—34no—
371C4⋊C4—64no—
372C4⋊C4—64no—
373C22⋊C4—24no—
374C22⋊C4—216no—
375C22⋊C4—68no—
376C22⋊C4—64no—
377C22⋊C4—62no—
378C22⋊C4—64no—
379C42—22no—
380C42—64no—
381C42—62no—
382C42C2×#6714yes—
383C22×C3—22no—
384C22×C3—22no—
385C22×C3—24no—
386C22×C3—42no—
387C22×C3—42no—
388C22×C3—44no—
389C22×C3—48no—
390C22×C3—42no—
391C22×C3—44no—
392C22×C3—44no—
393C22×C3—82no—
394C22×C3—84no—
395C22×C3—816no—
396C22×C3—84no—
397C22×C3C2×#12011yescharacteristic
398C22×C3C2×#12011yescharacteristic
399C22×C3#53411yescharacteristic
400C2×D3—42no—
401C2×D3—44no—
402C3×C4—24no—
403C3×C4—22no—
404C3×C4—44no—
405C3×C4D4214yes—
406Dic3—42no—
407Dic3—44no—
408C23—21no—
409C23—21no—
410C23—22no—
411C23—24no—
412C23—22no—
413C23—22no—
414C23—42no—
415C23—44no—
416C23—48no—
417C23—48no—
418C23—42no—
419C23—42no—
420C23—62no—
421C23—88no—
422C23—88no—
423C23—88no—
424C23#93712yes—
425C23C22×#6711yescharacteristic
426Q8—28no—
427Q8—64no—
428Q8—124no—
429D4—28no—
430D4—48no—
431D4—48no—
432D4—44no—
433D4—44no—
434D4—64no—
435D4—816no—
436D4—124no—
437D4—122no—
438D4—122no—
439D4—244no—
440D4—248no—
441C2×C4—22no—
442C2×C4—21no—
443C2×C4—22no—
444C2×C4—21no—
445C2×C4—28no—
446C2×C4—24no—
447C2×C4—48no—
448C2×C4—44no—
449C2×C4—64no—
450C2×C4—62no—
451C2×C4—128no—
452C2×C4—124no—
453C2×C4#93712yes—
454C2×C4#95314yes—
455C2×C4C2×D3×D412yes—
456C2×C4#100412yes—
457C2×C4C22×#6712yes—
458C8—124no—
459C2×C3—21no—
460C2×C3—21no—
461C2×C3—42no—
462C2×C3—82no—
463C2×C3—84no—
464C2×C3#240311yescharacteristic
465C2×C3#302112yes—
466D3—82no—
467C22—22no—
468C22—22no—
469C22—24no—
470C22—42no—
471C22—42no—
472C22—44no—
473C22—48no—
474C22—42no—
475C22—44no—
476C22—44no—
477C22—82no—
478C22—84no—
479C22—816no—
480C22—84no—
481C22—122no—
482C22—124no—
483C22C2×#93711yescharacteristic
484C22#580411yescharacteristic
485C22C2×#95311yescharacteristic
486C4—22no—
487C4—24no—
488C4—44no—
489C4—122no—
490C4—124no—
491C4D4×#6714yes—
492C3#2391511yescharacteristic, complemented, Sylow
493C2—21no—
494C2—21no—
495C2—42no—
496C2—82no—
497C2—84no—
498C2—242no—
499C2#8339512yes—
500C2#8405211yescharacteristic
subgroupG/H
centerC22C2×#937
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#535C2
socleC22×C3C2×#120
hypercenterC2×#535D3
hypocenterC3#23915
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#535

Complements

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

NH ≅ G/Ntype
1. C2×C3×#535C2semidirect
2. #87274C2semidirect
3. #84038C2semidirect
4. C2×#5062C2semidirect
5. #72137C2semidirect
6. #71769C2semidirect
7. #71724C2semidirect
8. #71591C2semidirect
12. C2×#1007C22semidirect
21. #5812C22semidirect
23. #5805C22semidirect
27. C2×#836C22semidirect
30. C2×#833C22semidirect
34. C2×#805C22semidirect
35. #4523C22semidirect
37. C2×#535D3semidirect
68. C3×#124D4semidirect
90. #930D4semidirect
97. #924D4semidirect
234. #124#67semidirect
492. C3#23915semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5352
#872742
#840382
C2×#50622
#721372
#717692
#717242
#715912
#239153