Finite Group Databasebeta

← #10826888

Subgroups — #10826888

Order 768.

1898 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 1898 conjugacy classes of subgroups — too many to draw. Only its 203 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#87273C211yesmaximal, characteristic, complemented
2C2×#5830C211yesmaximal, characteristic, complemented
3C2×#4747C212yesmaximal, complemented
4#78637C212yesmaximal, complemented
5C3×#1785C211yesmaximal, characteristic, complemented
6#70116C214yesmaximal, complemented
7#70070C212yesmaximal, complemented
8#69966C212yesmaximal, complemented
9#23913—31nomaximal, Sylow
10C22×C3×#41C2211yescharacteristic
11C2×#1004C2211yescharacteristic, complemented
12C23×D12C2211yescharacteristic, complemented
13#5830—28no—
14#5829—24no—
15#5824—24no—
16#5816C2212yescomplemented
17C2×#947C2212yescomplemented
18#5804—22no—
19C2×#940C2212yes—
20C2×#937—22no—
21#5798C2211yescharacteristic, complemented
22#5791C2212yes—
23C2×#930—22no—
24C2×#929—22no—
25C2×#926C2211yescharacteristic, complemented
26C2×#908C2214yes—
27C2×C3×#101C2212yescomplemented
28C2×C3×#100C2212yescomplemented
29C3×#407C2212yescomplemented
30C2×#825C2214yescomplemented
31#5131C2214yescomplemented
32#4747—28no—
33#4732—24no—
34#4731—24no—
35#4555C2214yescomplemented
36#3485—31no—
37C2×#535—31no—
38C2×#453—32no—
39#3003—32no—
40#2003—34no—
41#1897—32no—
42#1887—32no—
43#1785D311yescharacteristic, complemented
44C24×D3—21no—
45C2×C3×#41—24no—
46C2×C3×#41—24no—
47C2×C3×#41—42no—
48C22×C3×Q8C2311yescharacteristic
49C22×C3×D4D412yes—
50C22×C3×D4C2311yescharacteristic
51C23×C3×C4D412yes—
52C23×C3×C4C2311yescharacteristic
53C22×#67—22no—
54C22×#67—24no—
55#1004—24no—
56#1004—42no—
57C2×#237—22no—
58C2×D3×D4—22no—
59C2×#235C2312yes—
60C22×D12—21no—
61C22×D12—24no—
62C22×D12—24no—
63C22×D12—42no—
64C22×D12C2312yes—
65C2×C3×M16C2314yes—
66#953—22no—
67C2×#217—21no—
68C2×#217—22no—
69#947—24no—
70#947—22no—
71#947—24no—
72#946—24no—
73#944—22no—
74#940—24no—
75#940D418yescomplemented
76#939—28no—
77#937—216no—
78#937—24no—
79#935—24no—
80D4×Dic3—24no—
81#930—216no—
82#930D414yescomplemented
83#929—28no—
84#929—22no—
85#929—24no—
86C4×#67—24no—
87C2×(C2×D3)⋊C4—24no—
88C2×(C2×D3)⋊C4—24no—
89#926—24no—
90#926C2312yes—
91C2×C4⋊Dic3—21no—
92#924D414yescomplemented
93C2×Dic3⋊C4—24no—
94C2×C4×Dic3C2312yes—
95#908—28no—
96#908—44no—
97C2×D24—24no—
98C2×#204—24no—
99C3×#101—42no—
100C3×#100—24no—
101C3×C22⋊C8—24no—
102#825—28no—
103#821—28no—
104C2×H(5,𝔽2)—31no—
105C22×#41C2×D311yescharacteristic
106C23×D4—31no—
107C2×AGL(1,ℤ/8)—34no—
108#547—64no—
109#546—64no—
110#536—32no—
111#535—68no—
112#534—62no—
113C2×#127—32no—
114C2×#124—32no—
115#525—32no—
116C2×#123—62no—
117C2×#121—62no—
118C2×#120—62no—
119#518—31no—
120C2×#117—31no—
121#453—68no—
122#450—64no—
123#447—64no—
124C2×#104—34no—
125#418—34no—
126C2×#101C2×D312yescomplemented
127C2×#100C2×D312yescomplemented
128#407C2×D312yescomplemented
129#339—34no—
130C24×C3C2×D411yescharacteristic
131C24×C3C2×D412yes—
132C23×D3—22no—
133C23×D3—22no—
134C23×D3—24no—
135C23×D3—42no—
136C23×D3—42no—
137C23×D3—84no—
138C3×#41—416no—
139C2×C3×Q8—22no—
140C2×C3×Q8—22no—
141C2×C3×Q8—22no—
142C2×C3×D4—24no—
143C2×C3×D4—24no—
144C2×C3×D4—22no—
145C2×C3×D4—22no—
146C2×C3×D4—42no—
147C2×C3×D4C2×D414yes—
148C22×C3×C4—22no—
149C22×C3×C4—22no—
150C22×C3×C4—24no—
151C22×C3×C4—24no—
152C22×C3×C4—42no—
153C22×C3×C4C2×D412yes—
154C22×C3×C4C2×D411yescharacteristic
155C22×C3×C4C2×D414yes—
156C22×C3×C4C2411yescharacteristic
157C2×#67—24no—
158C2×#67—22no—
159C2×#67—24no—
160C2×#67—216no—
161C2×#67—44no—
162C2×#67—44no—
163C2×#67—88no—
164C22×Dic3—22no—
165C22×Dic3—22no—
166C22×Dic3—24no—
167#237—48no—
168D3×D4—48no—
169#235—24no—
170#235—24no—
171C2×D12—24no—
172C2×D12—24no—
173C2×D12—24no—
174C2×D12—22no—
175C2×D12—416no—
176C2×D12—42no—
177C2×D12—42no—
178C2×D12C2×D412yes—
179C2×D12C2×D414yes—
180C2×C4×D3—22no—
181C2×Dic6C2×D412yes—
182C3×M16—24no—
183C3×M16—24no—
184C2×C3×C8—24no—
185#217—24no—
186#217—24no—
187#217—24no—
188#217—22no—
189(C2×D3)⋊C4—28no—
190(C2×D3)⋊C4—216no—
191(C2×D3)⋊C4—24no—
192C4⋊Dic3—24no—
193C4⋊Dic3C2×D414yes—
194Dic3⋊C4—28no—
195Dic3⋊C4—24no—
196C4×Dic3—22no—
197C4×Dic3—22no—
198C4×Dic3C2×D414yes—
199D24—48no—
200#204—48no—
201C25—61no—
202H(5,𝔽2)—64no—
203H(5,𝔽2)—122no—
204C2×#41—24no—
205C2×#41—24no—
206C2×#41—32no—
207C2×#41—42no—
208C2×#41—62no—
209C22×Q8C22×D311yescharacteristic
210C22×D4—32no—
211C22×D4—64no—
212C22×D4—62no—
213C22×D4—62no—
214C22×D4—64no—
215C22×D4—61no—
216C22×D4—64no—
217C22×D4—122no—
218C22×D4D1212yes—
219C22×D4C22×D311yescharacteristic
220C23×C4D1212yes—
221C23×C4C22×D311yescharacteristic
222AGL(1,ℤ/8)—68no—
223AGL(1,ℤ/8)—124no—
224C2×SD16—64no—
225C2×D8—64no—
226C2×M16C22×D314yes—
227C4⋊Q8—62no—
228#127—38no—
229#127—64no—
230#124—64no—
231#124—64no—
232#124—62no—
233#123—68no—
234#123—64no—
235#123—62no—
236#123—64no—
237C22⋊Q8—34no—
238C22⋊Q8—64no—
239#121—34no—
240#121—616no—
241#121—68no—
242#120—616no—
243#120—64no—
244#120—62no—
245C4×D4—64no—
246C4×D4—64no—
247#117—32no—
248#117—64no—
249C2×C4⋊C4—64no—
250C2×C4⋊C4—61no—
251C2×C22⋊C4—64no—
252C2×C22⋊C4—61no—
253C2×C22⋊C4—62no—
254C2×C22⋊C4—64no—
255C2×C42—32no—
256#104—68no—
257#102—68no—
258#101—42no—
259#100—24no—
260C22⋊C8—24no—
261C23×C3—21no—
262C23×C3—22no—
263C23×C3—21no—
264C23×C3—22no—
265C23×C3—42no—
266C23×C3—44no—
267C23×C3#12012yes—
268C23×C3#12014yes—
269C23×C3C22×D411yescharacteristic
270C22×D3—22no—
271C22×D3—24no—
272C22×D3—48no—
273C22×D3—48no—
274C22×D3—42no—
275C22×D3—42no—
276C22×D3—44no—
277C22×D3—88no—
278C22×D3—84no—
279C22×D3—88no—
280C22×D3—88no—
281C3×Q8—44no—
282C3×D4—28no—
283C3×D4—48no—
284C2×C3×C4—21no—
285C2×C3×C4—21no—
286C2×C3×C4—28no—
287C2×C3×C4—22no—
288C2×C3×C4—22no—
289C2×C3×C4—22no—
290C2×C3×C4—48no—
291C2×C3×C4#12012yes—
292C2×C3×C4C22×D412yes—
293C2×C3×C4C22×D412yes—
294C2×C3×C4H(5,𝔽2)12yes—
295#67—48no—
296#67—42no—
297#67—42no—
298#67—84no—
299#67—816no—
300C2×Dic3—24no—
301C2×Dic3—24no—
302C2×Dic3—22no—
303C2×Dic3—44no—
304C2×Dic3—44no—
305C2×Dic3—84no—
306D12—28no—
307D12—24no—
308D12—48no—
309D12—44no—
310C4×D3—44no—
311Dic6—24no—
312C3×C8—44no—
313C24—62no—
314C24—64no—
315C24—62no—
316C24—122no—
317C24—122no—
318C24—244no—
319C24C2×D1211yescharacteristic
320C24C2×D1212yes—
321#41—416no—
322#41—64no—
323#41—64no—
324#41—128no—
325C2×Q8—22no—
326C2×Q8—22no—
327C2×Q8—22no—
328C2×Q8—32no—
329C2×D4—24no—
330C2×D4—24no—
331C2×D4—22no—
332C2×D4—22no—
333C2×D4—32no—
334C2×D4—34no—
335C2×D4—42no—
336C2×D4—64no—
337C2×D4—64no—
338C2×D4—62no—
339C2×D4—64no—
340C2×D4—616no—
341C2×D4—64no—
342C2×D4—62no—
343C2×D4—64no—
344C2×D4—1216no—
345C2×D4—122no—
346C2×D4—128no—
347C2×D4—124no—
348C2×D4—124no—
349C2×D4—122no—
350C2×D4—248no—
351C2×D4D3×D414yes—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—42no—
357C22×C4—62no—
358C22×C4—62no—
359C22×C4—64no—
360C22×C4—62no—
361C22×C4C2×D1212yes—
362C22×C4C2×D1211yescharacteristic
363C22×C4D3×D414yes—
364C22×C4C23×D311yescharacteristic
365SD16—128no—
366D8—128no—
367M16—24no—
368M16—24no—
369C2×C8—24no—
370C4⋊C4—34no—
371C4⋊C4—64no—
372C4⋊C4—68no—
373C4⋊C4—64no—
374C22⋊C4—616no—
375C22⋊C4—68no—
376C22⋊C4—64no—
377C22⋊C4—64no—
378C22⋊C4—64no—
379C22⋊C4—62no—
380C22⋊C4—64no—
381C42—34no—
382C42—62no—
383C42—62no—
384C22×C3—22no—
385C22×C3—24no—
386C22×C3—22no—
387C22×C3—24no—
388C22×C3—44no—
389C22×C3—42no—
390C22×C3—42no—
391C22×C3—44no—
392C22×C3—82no—
393C22×C3C2×#12011yescharacteristic
394C22×C3C2×#12011yescharacteristic
395C22×C3#53411yescharacteristic
396C2×D3—48no—
397C2×D3—42no—
398C2×D3—44no—
399C2×D3—44no—
400C2×D3—84no—
401C2×D3—816no—
402C2×D3—84no—
403C2×D3—164no—
404C3×C4—22no—
405C3×C4—42no—
406C3×C4D4214yes—
407Dic3—44no—
408Dic3—42no—
409Dic3—44no—
410C23—21no—
411C23—22no—
412C23—21no—
413C23—22no—
414C23—42no—
415C23—44no—
416C23—64no—
417C23—62no—
418C23—128no—
419C23—128no—
420C23—124no—
421C23—122no—
422C23—122no—
423C23—248no—
424C23—248no—
425C23—244no—
426C23—248no—
427C23#88212yes—
428C23#88214yes—
429C23C22×D1211yescharacteristic
430Q8—44no—
431Q8—64no—
432D4—28no—
433D4—48no—
434D4—64no—
435D4—68no—
436D4—128no—
437D4—128no—
438D4—122no—
439D4—122no—
440D4—124no—
441D4—244no—
442D4—2416no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—28no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—48no—
450C2×C4—64no—
451C2×C4—64no—
452C2×C4—62no—
453C2×C4—124no—
454C2×C4—124no—
455C2×C4—124no—
456C2×C4—244no—
457C2×C4#88212yes—
458C2×C4C2×D3×D412yes—
459C2×C4C2×D3×D412yes—
460C2×C4#100412yes—
461C8—44no—
462C2×C3—21no—
463C2×C3—21no—
464C2×C3—42no—
465C2×C3—44no—
466C2×C3#240311yescharacteristic
467C2×C3#302112yes—
468D3—84no—
469D3—82no—
470C22—22no—
471C22—22no—
472C22—24no—
473C22—24no—
474C22—44no—
475C22—42no—
476C22—42no—
477C22—44no—
478C22—82no—
479C22—128no—
480C22—122no—
481C22—124no—
482C22—124no—
483C22—244no—
484C22—2416no—
485C22—244no—
486C22—484no—
487C22C2×#88211yescharacteristic
488C22C2×#88211yescharacteristic
489C22#549311yescharacteristic
490C4—22no—
491C4—42no—
492C4—124no—
493C4—122no—
494C4—124no—
495C4#559014yes—
496C3#2391311yescharacteristic, complemented, Sylow
497C2—21no—
498C2—21no—
499C2—42no—
500C2—44no—
501C2—244no—
502C2—242no—
503C2#7385211yescharacteristic
504C2#7871712yes—
subgroupG/H
centerC22C2×#882
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1785C2
socleC22×C3C2×#120
hypercenter#1785D3
hypocenterC3#23913
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1785

Complements

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

NH ≅ G/Ntype
1. #87273C2semidirect
2. C2×#5830C2semidirect
3. C2×#4747C2semidirect
4. #78637C2semidirect
5. C3×#1785C2semidirect
6. #70116C2semidirect
7. #70070C2semidirect
8. #69966C2semidirect
11. C2×#1004C22semidirect
12. C23×D12C22semidirect
16. #5816C22semidirect
17. C2×#947C22semidirect
21. #5798C22semidirect
25. C2×#926C22semidirect
27. C2×C3×#101C22semidirect
28. C2×C3×#100C22semidirect
29. C3×#407C22semidirect
30. C2×#825C22semidirect
31. #5131C22semidirect
35. #4555C22semidirect
43. #1785D3semidirect
75. #940D4semidirect
82. #930D4semidirect
92. #924D4semidirect
126. C2×#101C2×D3semidirect
127. C2×#100C2×D3semidirect
128. #407C2×D3semidirect
496. C3#23913semidirect

Maximal subgroups

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

subgroupindex
#872732
C2×#58302
C2×#47472
#786372
C3×#17852
#701162
#700702
#699662
#239133