Finite Group Databasebeta

← #10831103

Subgroups — #10831103

Order 768.

1336 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#87163C211yesmaximal, characteristic, complemented
2C3×#2929C211yesmaximal, characteristic, complemented
3#83742C212yesmaximal, complemented
4#82936C212yesmaximal, complemented
5#72012C211yesmaximal, characteristic, complemented
6#70547C212yesmaximal, complemented
7#70010C212yesmaximal, complemented
8#69993C212yesmaximal, complemented
9#69897C212yesmaximal, complemented
10#24285—31nomaximal, Sylow
11#5964C2212yes—
12C2×C3×#131C2211yescharacteristic
13C23×C3×C8C2211yescharacteristic, complemented
14C3×#525C2211yescharacteristic
15#5791C2214yescomplemented
16#5753—22no—
17#5752—22no—
18#5751—24no—
19#5748—22no—
20#5748C2212yes—
21C2×#904—22no—
22C2×#904C2212yes—
23C3×#435—22no—
24C3×C8×D4—22no—
25C3×#424C2211yescharacteristic, complemented
26C3×#408C2211yescharacteristic
27C3×#407C2211yescharacteristic, complemented
28C3×#405C2211yescharacteristic, complemented
29#5240C2212yes—
30#5139C2214yescomplemented
31#5116—22no—
32#5115—22no—
33#5114—24no—
34#5113C2214yescomplemented
35#5106C2212yescomplemented
36#5053—24no—
37#5032—24no—
38#5006C2214yescomplemented
39#4557C2212yescomplemented
40#3584—31no—
41#3050—32no—
42#3025—32no—
43#2929D311yescharacteristic, complemented
44#2053—32no—
45#2010—32no—
46#1976—32no—
47#1812—31no—
48#1794—32no—
49C2×C3×#41C2311yescharacteristic
50C23×C3×C4C2311yescharacteristic
51#1010—24no—
52#1009—24no—
53D3×#41—44no—
54#1007—22no—
55#1004—22no—
56#1004—24no—
57C2×#235—24no—
58C2×#235C2314yes—
59C3×#131—22no—
60C3×#131—22no—
61C2×C3×M16D412yescomplemented
62C2×C3×M16C2311yescharacteristic
63C2×C3×M16C2311yescharacteristic
64C22×C3×C8—21no—
65C22×C3×C8—21no—
66C22×C3×C8—22no—
67C22×C3×C8—22no—
68C22×C3×C8D412yescomplemented
69C22×C3×C8C2311yescharacteristic
70C22×C3×C8C2311yescharacteristic
71C3×#123D412yes—
72C3×C22⋊Q8—21no—
73C3×#121—21no—
74C3×#120D412yes—
75C3×C4×D4—22no—
76C3×#117C2311yescharacteristic
77#953—24no—
78#930—24no—
79#929—24no—
80C4×#67—44no—
81#926C2314yes—
82#924—24no—
83#909—28no—
84#909—24no—
85#908—28no—
86#908—24no—
87C2×Dic12—22no—
88C2×Dic12D412yescomplemented
89#904—24no—
90#904—24no—
91#904—22no—
92#904—24no—
93#904—22no—
94#904—22no—
95#904—22no—
96C2×D24—22no—
97C2×D24D412yescomplemented
98C2×#204—24no—
99C2×#204D414yescomplemented
100C3×C4⋊C8—22no—
101C3×C22⋊C8—22no—
102C3×C22⋊C8—22no—
103C3×#97—21no—
104C3×C4×C8—21no—
105#834—24no—
106#825—44no—
107#821—24no—
108#818—22no—
109#817—22no—
110#816—24no—
111#806—28no—
112#585—32no—
113#578—62no—
114#577—64no—
115#576—62no—
116#575—32no—
117#575—62no—
118C2×#135—32no—
119C2×#135—62no—
120C2×#131C2×D311yescharacteristic
121C23×C8C2×D311yescharacteristic, complemented
122#525—34no—
123#525C2×D311yescharacteristic
124#467—62no—
125#466—62no—
126#465—64no—
127#458—64no—
128B(4,𝔽2)—64no—
129#435—22no—
130C8×D4—22no—
131#427—32no—
132#424C2×D311yescharacteristic, complemented
133#421—34no—
134#416—34no—
135#410—32no—
136#410—34no—
137#408C2×D311yescharacteristic
138#407C2×D311yescharacteristic, complemented
139#405C2×D311yescharacteristic, complemented
140H(3,ℤ/4)—32no—
141C24×C3C2×D411yescharacteristic
142C3×#41—22no—
143C3×#41—22no—
144C2×C3×Q8#4111yescharacteristic
145C2×C3×D4—22no—
146C2×C3×D4C2×D412yes—
147C2×C3×D4#4111yescharacteristic
148C22×C3×C4—21no—
149C22×C3×C4—22no—
150C22×C3×C4—21no—
151C22×C3×C4—22no—
152C22×C3×C4C2×D411yescharacteristic
153C22×C3×C4C2×D412yes—
154C22×C3×C4C2411yescharacteristic
155C2×#67—24no—
156C2×#67—44no—
157C2×#67—44no—
158#239—44no—
159D3×Q8—44no—
160#237—44no—
161#237—48no—
162D3×D4—44no—
163D3×D4—48no—
164#235—28no—
165#235—24no—
166#235—24no—
167#235—24no—
168#235—24no—
169#235—44no—
170#235—44no—
171C2×D12—24no—
172C2×D12C2×D414yes—
173C2×C4×D3—24no—
174C2×C4×D3—44no—
175C2×Dic6—24no—
176C2×Dic6C2×D414yes—
177C3×M16—22no—
178C3×M16—24no—
179C3×M16—22no—
180C2×C3×C8—22no—
181C2×C3×C8—22no—
182C2×C3×C8—22no—
183C2×C3×C8—21no—
184C2×C3×C8—21no—
185C2×C3×C8—41no—
186C2×C3×C8—42no—
187C2×C3×C8—41no—
188C2×C3×C8C2×D412yes—
189C2×C3×C8C2×D412yes—
190C2×C3×C8C2×D412yes—
191C2×C3×C8C2×D411yescharacteristic
192C2×C3×C8C2×D411yescharacteristic
193C3×C4⋊C4—22no—
194C3×C4⋊C4—21no—
195C3×C22⋊C4—22no—
196C3×C22⋊C4—22no—
197C3×C22⋊C4C2×D412yes—
198C3×C42—21no—
199#217—24no—
200#217—44no—
201(C2×D3)⋊C4—44no—
202C4⋊Dic3—24no—
203Dic3⋊C4—44no—
204C4×Dic3—24no—
205Dic12—24no—
206Dic12—22no—
207Dic12—22no—
208D24—24no—
209D24—22no—
210D24—22no—
211#204—28no—
212#204—24no—
213#204—24no—
2142−1+4—64no—
2152−1+4—62no—
216H(5,𝔽2)—64no—
217H(5,𝔽2)—64no—
218H(5,𝔽2)—62no—
219C2×#41—34no—
220C2×#41—64no—
221C2×#41—124no—
222C2×#41C22×D311yescharacteristic
223C23×C4C22×D311yescharacteristic
224#137—64no—
225#137—68no—
226AGL(1,ℤ/8)—64no—
227AGL(1,ℤ/8)—68no—
228#135—64no—
229#135—64no—
230#135—62no—
231#135—62no—
232#135—62no—
233#135—62no—
234#135—64no—
235C2×Q16—32no—
236C2×Q16—62no—
237C2×SD16—34no—
238C2×SD16—64no—
239C2×D8—32no—
240C2×D8—62no—
241#131—22no—
242#131—22no—
243C2×M16#6712yescomplemented
244C2×M16C22×D311yescharacteristic
245C2×M16C22×D311yescharacteristic
246C22×C8—22no—
247C22×C8—22no—
248C22×C8—21no—
249C22×C8—21no—
250C22×C8#6712yescomplemented
251C22×C8C22×D311yescharacteristic
252C22×C8C22×D311yescharacteristic
253#123—64no—
254#123D1212yes—
255C22⋊Q8—21no—
256C22⋊Q8—64no—
257#121—21no—
258#121—64no—
259#120—64no—
260#120D1212yes—
261C4×D4—22no—
262C4×D4—124no—
263#117—34no—
264#117C22×D311yescharacteristic
265#107—62no—
266#106—62no—
267C4⋊C8—22no—
268#104—124no—
269#103—64no—
270#102—64no—
271#99—64no—
272#99—68no—
273C22⋊C8—22no—
274C22⋊C8—22no—
275#97—21no—
276C4×C8—21no—
277C23×C3—21no—
278C23×C3—22no—
279C23×C3—42no—
280C23×C3#12112yes—
281C23×C3C22×D411yescharacteristic
282C22×D3—24no—
283C22×D3—44no—
284C3×Q8—22no—
285C3×D4—24no—
286C3×D4—22no—
287C3×D4—44no—
288C2×C3×C4—22no—
289C2×C3×C4—22no—
290C2×C3×C4—22no—
291C2×C3×C4—21no—
292C2×C3×C4—22no—
293C2×C3×C4—41no—
294C2×C3×C4—41no—
295C2×C3×C4—42no—
296C2×C3×C4#12112yes—
297C2×C3×C4C22×D412yes—
298C2×C3×C4C22×D411yescharacteristic
299C2×C3×C4C2×#4111yescharacteristic
300C2×C3×C4H(5,𝔽2)12yes—
301#67—48no—
302#67—44no—
303#67—44no—
304#67—88no—
305C2×Dic3—24no—
306C2×Dic3—44no—
307C2×Dic3—44no—
308D12—24no—
309D12—24no—
310D12—44no—
311C4×D3—48no—
312C4×D3—44no—
313C4×D3—44no—
314Dic6—24no—
315Dic6—24no—
316Dic6—44no—
317C3×C8—22no—
318C3×C8—41no—
319C3×C8—41no—
320C3×C8#12011yescharacteristic
321C3×C8#12012yes—
322C3×C8#12011yescharacteristic
323C24C2×D1211yescharacteristic
324#41—22no—
325#41—22no—
326#41—68no—
327#41—64no—
328#41—64no—
329#41—64no—
330#41—64no—
331#41—124no—
332#41—128no—
333#41—124no—
334#41—124no—
335#41—124no—
336C2×Q8—34no—
337C2×Q8—64no—
338C2×Q8—124no—
339C2×Q8#23511yescharacteristic
340C2×D4—22no—
341C2×D4—34no—
342C2×D4—64no—
343C2×D4—64no—
344C2×D4—124no—
345C2×D4—128no—
346C2×D4—124no—
347C2×D4—124no—
348C2×D4C2×D1212yes—
349C2×D4#23511yescharacteristic
350C22×C4—21no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—21no—
354C22×C4—64no—
355C22×C4—124no—
356C22×C4C2×D1211yescharacteristic
357C22×C4C2×#6712yes—
358C22×C4C23×D311yescharacteristic
359Q16—64no—
360Q16—62no—
361Q16—62no—
362SD16—68no—
363SD16—64no—
364SD16—64no—
365D8—64no—
366D8—62no—
367D8—62no—
368M16—22no—
369M16—24no—
370M16—22no—
371C2×C8—22no—
372C2×C8—22no—
373C2×C8—22no—
374C2×C8—21no—
375C2×C8—21no—
376C2×C8—41no—
377C2×C8—41no—
378C2×C8—42no—
379C2×C8D3×D412yes—
380C2×C8D3×D412yes—
381C2×C8C2×#6712yes—
382C2×C8C2×#6711yescharacteristic
383C2×C8C2×#6711yescharacteristic
384C4⋊C4—22no—
385C4⋊C4—21no—
386C4⋊C4—64no—
387C4⋊C4—124no—
388C22⋊C4—22no—
389C22⋊C4—22no—
390C22⋊C4—64no—
391C22⋊C4—124no—
392C22⋊C4—124no—
393C22⋊C4C2×D1212yes—
394C42—21no—
395C42—64no—
396C22×C3—24no—
397C22×C3—22no—
398C22×C3—22no—
399C22×C3—41no—
400C22×C3—42no—
401C22×C3—42no—
402C22×C3—41no—
403C22×C3—41no—
404C22×C3C2×#12111yescharacteristic
405C22×C3D4212yes—
406C2×D3—44no—
407C2×D3—84no—
408C2×D3—164no—
409C3×C4—22no—
410C3×C4—42no—
411C3×C4—41no—
412C3×C4C2×#12011yescharacteristic
413C3×C4#53511yescharacteristic
414C3×C4#54612yes—
415Dic3—44no—
416Dic3—84no—
417C23—21no—
418C23—22no—
419C23—42no—
420C23—64no—
421C23—124no—
422C23#93012yes—
423C23C22×D1211yescharacteristic
424Q8—22no—
425Q8—64no—
426Q8—64no—
427Q8—124no—
428D4—24no—
429D4—22no—
430D4—44no—
431D4—64no—
432D4—64no—
433D4—124no—
434D4—128no—
435D4—124no—
436D4—124no—
437D4—248no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—21no—
442C2×C4—22no—
443C2×C4—41no—
444C2×C4—42no—
445C2×C4—41no—
446C2×C4—64no—
447C2×C4—128no—
448C2×C4—124no—
449C2×C4—124no—
450C2×C4—124no—
451C2×C4—124no—
452C2×C4#93012yes—
453C2×C4C2×#23511yescharacteristic
454C2×C4C2×D3×D412yes—
455C2×C4#100912yes—
456C2×C4C22×#6711yescharacteristic
457C8—22no—
458C8—41no—
459C8—41no—
460C8#93711yescharacteristic
461C8#93711yescharacteristic
462C8#93712yes—
463C2×C3—22no—
464C2×C3—21no—
465C2×C3—42no—
466C2×C3—42no—
467C2×C3#240811yescharacteristic
468D3—84no—
469C22—22no—
470C22—24no—
471C22—22no—
472C22—41no—
473C22—42no—
474C22—41no—
475C22—42no—
476C22—41no—
477C22—124no—
478C22—244no—
479C22—484no—
480C22D4×D1212yes—
481C22C2×#93011yescharacteristic
482C4—22no—
483C4—42no—
484C4—41no—
485C4—124no—
486C4—244no—
487C4#555012yes—
488C4C2×#93711yescharacteristic
489C4#583011yescharacteristic
490C3#2428511yescharacteristic, complemented, Sylow
491C2—21no—
492C2—22no—
493C2—42no—
494C2—42no—
495C2—244no—
496C2#8240311yescharacteristic
subgroupG/H
centerC4#5830
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2929C2
socleC2×C3#2408
hypercenter#2929D3
hypocenterC3#24285
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C22×C4 ◁ #2929

Complements

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

NH ≅ G/Ntype
1. #87163C2semidirect
2. C3×#2929C2semidirect
3. #83742C2semidirect
4. #82936C2semidirect
5. #72012C2semidirect
6. #70547C2semidirect
7. #70010C2semidirect
8. #69993C2semidirect
9. #69897C2semidirect
13. C23×C3×C8C22semidirect
15. #5791C22semidirect
25. C3×#424C22semidirect
27. C3×#407C22semidirect
28. C3×#405C22semidirect
30. #5139C22semidirect
34. #5113C22semidirect
35. #5106C22semidirect
38. #5006C22semidirect
39. #4557C22semidirect
43. #2929D3semidirect
61. C2×C3×M16D4semidirect
68. C22×C3×C8D4semidirect
88. C2×Dic12D4semidirect
97. C2×D24D4semidirect
99. C2×#204D4semidirect
121. C23×C8C2×D3semidirect
132. #424C2×D3semidirect
138. #407C2×D3semidirect
139. #405C2×D3semidirect
243. C2×M16#67semidirect
250. C22×C8#67semidirect
490. C3#24285semidirect

Maximal subgroups

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

subgroupindex
#871632
C3×#29292
#837422
#829362
#720122
#705472
#700102
#699932
#698972
#242853