Finite Group Databasebeta

← #10831130

Subgroups — #10831130

Order 768.

1290 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#87163C211yesmaximal, characteristic, complemented
2C3×#2932C211yesmaximal, characteristic, complemented
3#83805C212yesmaximal, complemented
4#83322C212yesmaximal, complemented
5#72014C211yesmaximal, characteristic, complemented
6#70531C212yesmaximal, complemented
7#70013C212yesmaximal, complemented
8#70009C212yesmaximal, complemented
9#69845C212yesmaximal, complemented
10#24288—31nomaximal, Sylow
11#5964C2212yes—
12C2×C3×#131C2211yescharacteristic
13C2×C3×#131C2211yescharacteristic, complemented
14C3×#532C2211yescharacteristic
15#5822C2214yescomplemented
16#5753—22no—
17#5752—22no—
18#5751—24no—
19#5748—22no—
20#5748C2212yes—
21C2×#904—22no—
22C2×#904C2212yes—
23C3×#435—22no—
24C3×#435—22no—
25C3×#423C2211yescharacteristic, complemented
26C3×#408C2211yescharacteristic
27C3×#408C2211yescharacteristic, complemented
28C2×C3×#97C2211yescharacteristic, complemented
29#5237C2212yes—
30#5143—22no—
31#5142—22no—
32#5141—24no—
33#5132C2214yescomplemented
34#5110C2214yescomplemented
35C2×#819C2212yescomplemented
36#5095—22no—
37#5077—22no—
38#5062—22no—
39#5061—22no—
40C2×#805C2214yescomplemented
41C3⋊#338C2212yescomplemented
42#3584—31no—
43#3067—32no—
44#3018—32no—
45#2932D311yescharacteristic, complemented
46#2050—32no—
47#2017—32no—
48#1975—32no—
49#1813—31no—
50#1791—32no—
51C2×C3×#41C2311yescharacteristic
52C2×C3×#41C2311yescharacteristic
53#1010—24no—
54#1009—24no—
55D3×#41—44no—
56#1007—22no—
57#1004—22no—
58#1004—24no—
59C2×#235—24no—
60C2×#235C2314yes—
61C3×#131—22no—
62C3×#131—22no—
63C3×#131—22no—
64C3×#131—22no—
65C2×C3×M16—21no—
66C2×C3×M16D412yes—
67C2×C3×M16C2311yescharacteristic
68C2×C3×M16C2311yescharacteristic
69C22×C3×C8—21no—
70C22×C3×C8D412yes—
71C22×C3×C8C2311yescharacteristic
72C22×C3×C8C2311yescharacteristic
73C3×C4⋊Q8D411yescharacteristic
74C3×#127D411yescharacteristic
75C3×#124D411yescharacteristic
76C3×#124D411yescharacteristic
77C3×#121—22no—
78C3×C4×D4—22no—
79C2×C3×C42C2311yescharacteristic
80#951—24no—
81#950—44no—
82#949—24no—
83#942—24no—
84#932—24no—
85C2×#208C2314yes—
86#909—28no—
87#909—24no—
88#908—28no—
89#908—24no—
90C2×Dic12—22no—
91C2×Dic12D412yescomplemented
92#904—24no—
93#904—24no—
94#904—22no—
95#904—24no—
96#904—22no—
97#904—22no—
98#904—22no—
99C2×D24—22no—
100C2×D24D412yescomplemented
101C2×#204—24no—
102C2×#204D414yescomplemented
103C3×C4⋊C8—22no—
104C3×C22⋊C8—22no—
105C3×C22⋊C8—22no—
106C3×#97—21no—
107C3×#97—21no—
108#836—22no—
109#833—22no—
110#824—24no—
111#823—24no—
112(C2×D3)⋊C8—44no—
113#819—22no—
114#819—22no—
115#805—24no—
116#805—24no—
117#585—32no—
118#578—62no—
119#577—64no—
120#576—62no—
121#575—34no—
122#575—32no—
123#575—62no—
124C2×#135—32no—
125C2×#135—62no—
126C2×#131C2×D311yescharacteristic
127C2×#131C2×D311yescharacteristic, complemented
128#532C2×D311yescharacteristic
129#473—62no—
130#472—62no—
131#471—64no—
132#456—62no—
133#455—62no—
134#454—62no—
135#453—62no—
136#435—22no—
137#435—22no—
138C2×#108—32no—
139#423C2×D311yescharacteristic, complemented
140C2×#104—34no—
141#413—32no—
142#413—34no—
143#408—34no—
144#408C2×D311yescharacteristic
145#408C2×D311yescharacteristic, complemented
146C2×#97C2×D311yescharacteristic, complemented
147#338—32no—
148C3×#41—22no—
149C3×#41—22no—
150C3×#41—22no—
151C3×#41—22no—
152C2×C3×Q8C2×D411yescharacteristic
153C2×C3×Q8C2×D411yescharacteristic
154C2×C3×D4—21no—
155C2×C3×D4—22no—
156C2×C3×D4C2×D411yescharacteristic
157C2×C3×D4C2×D411yescharacteristic
158C2×C3×D4#4112yes—
159C22×C3×C4—21no—
160C22×C3×C4—21no—
161C22×C3×C4C2×D412yes—
162C22×C3×C4C2411yescharacteristic
163C2×#67—24no—
164C2×#67—44no—
165#239—44no—
166D3×Q8—44no—
167#237—44no—
168#237—48no—
169D3×D4—44no—
170D3×D4—48no—
171#235—28no—
172#235—24no—
173#235—24no—
174#235—24no—
175#235—24no—
176#235—44no—
177#235—44no—
178C2×D12—24no—
179C2×D12C2×D414yes—
180C2×C4×D3—24no—
181C2×C4×D3—44no—
182C2×Dic6—24no—
183C2×Dic6C2×D414yes—
184C3×M16—22no—
185C3×M16—24no—
186C3×M16—22no—
187C3×M16—41no—
188C3×M16—41no—
189C2×C3×C8—22no—
190C2×C3×C8—22no—
191C2×C3×C8—21no—
192C2×C3×C8—21no—
193C2×C3×C8—22no—
194C2×C3×C8—42no—
195C2×C3×C8C2×D412yes—
196C2×C3×C8C2×D412yes—
197C2×C3×C8C2×D412yes—
198C2×C3×C8C2×D411yescharacteristic
199C2×C3×C8C2×D411yescharacteristic
200C3×C4⋊C4—22no—
201C3×C22⋊C4—22no—
202C3×C22⋊C4—22no—
203C3×C42—21no—
204C3×C42C2×D411yescharacteristic
205C3×C42C2×D411yescharacteristic
206#216—44no—
207#215—44no—
208#214—44no—
209#213—44no—
210#208—24no—
211#208—24no—
212C2×C3⋊C8—24no—
213Dic12—24no—
214Dic12—22no—
215Dic12—22no—
216D24—24no—
217D24—22no—
218D24—22no—
219#204—28no—
220#204—24no—
221#204—24no—
2222−1+4—64no—
2232−1+4—62no—
224H(5,𝔽2)—64no—
225H(5,𝔽2)—64no—
226H(5,𝔽2)—62no—
227C2×#41—34no—
228C2×#41—64no—
229C2×#41—124no—
230C2×#41C22×D311yescharacteristic
231C2×#41C22×D311yescharacteristic
232#137—64no—
233#137—68no—
234#137—64no—
235#137—64no—
236AGL(1,ℤ/8)—64no—
237AGL(1,ℤ/8)—68no—
238AGL(1,ℤ/8)—64no—
239AGL(1,ℤ/8)—64no—
240#135—64no—
241#135—64no—
242#135—62no—
243#135—62no—
244#135—62no—
245#135—62no—
246#135—64no—
247#135—124no—
248C2×Q16—32no—
249C2×Q16—62no—
250C2×SD16—34no—
251C2×SD16—64no—
252C2×D8—32no—
253C2×D8—62no—
254#131—22no—
255#131—22no—
256#131—22no—
257#131—22no—
258C2×M16—21no—
259C2×M16—34no—
260C2×M16#6712yes—
261C2×M16C22×D311yescharacteristic
262C2×M16C22×D311yescharacteristic
263C22×C8—21no—
264C22×C8#6712yes—
265C22×C8C22×D311yescharacteristic
266C22×C8C22×D311yescharacteristic
267C4⋊Q8D1211yescharacteristic
268#127D1211yescharacteristic
269#124D1211yescharacteristic
270#124D1211yescharacteristic
271#121—22no—
272C4×D4—22no—
273C2×C42C22×D311yescharacteristic
274#108—62no—
275#108—62no—
276C4⋊C8—22no—
277#104—64no—
278#104—64no—
279#101—64no—
280#101—62no—
281#100—64no—
282#100—62no—
283C22⋊C8—22no—
284C22⋊C8—22no—
285C22⋊C8—124no—
286#97—21no—
287#97—21no—
288C23×C3—21no—
289C23×C3#12112yes—
290C23×C3C2×#4111yescharacteristic
291C22×D3—24no—
292C22×D3—44no—
293C3×Q8—22no—
294C3×Q8—22no—
295C3×D4—24no—
296C3×D4—22no—
297C3×D4—22no—
298C3×D4—42no—
299C3×D4—82no—
300C2×C3×C4—22no—
301C2×C3×C4—22no—
302C2×C3×C4—21no—
303C2×C3×C4—21no—
304C2×C3×C4—21no—
305C2×C3×C4—41no—
306C2×C3×C4—41no—
307C2×C3×C4—81no—
308C2×C3×C4#12112yes—
309C2×C3×C4C22×D411yescharacteristic
310C2×C3×C4C22×D412yes—
311C2×C3×C4C22×D411yescharacteristic
312C2×C3×C4H(5,𝔽2)12yes—
313#67—48no—
314#67—44no—
315#67—44no—
316C2×Dic3—24no—
317C2×Dic3—44no—
318D12—24no—
319D12—24no—
320D12—44no—
321C4×D3—48no—
322C4×D3—44no—
323C4×D3—44no—
324Dic6—24no—
325Dic6—24no—
326Dic6—44no—
327C3×C8—22no—
328C3×C8—41no—
329C3×C8—41no—
330C3×C8#12011yescharacteristic
331C3×C8#12012yes—
332C3×C8#12011yescharacteristic
333C3⋊C8—44no—
334#41—22no—
335#41—22no—
336#41—22no—
337#41—22no—
338#41—68no—
339#41—64no—
340#41—64no—
341#41—64no—
342#41—64no—
343#41—124no—
344#41—128no—
345#41—124no—
346#41—124no—
347#41—124no—
348C2×Q8—34no—
349C2×Q8—64no—
350C2×Q8—124no—
351C2×Q8C2×D1211yescharacteristic
352C2×Q8C2×D1211yescharacteristic
353C2×D4—21no—
354C2×D4—22no—
355C2×D4—34no—
356C2×D4—64no—
357C2×D4—64no—
358C2×D4—124no—
359C2×D4—128no—
360C2×D4—124no—
361C2×D4C2×D1211yescharacteristic
362C2×D4C2×D1211yescharacteristic
363C2×D4#23512yes—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—64no—
367C22×C4—124no—
368C22×C4C2×#6712yes—
369C22×C4C23×D311yescharacteristic
370Q16—64no—
371Q16—62no—
372Q16—62no—
373Q16—124no—
374SD16—68no—
375SD16—64no—
376SD16—64no—
377SD16—124no—
378SD16—124no—
379D8—64no—
380D8—62no—
381D8—62no—
382D8—124no—
383M16—22no—
384M16—24no—
385M16—22no—
386M16—41no—
387M16—41no—
388M16—64no—
389M16—64no—
390C2×C8—22no—
391C2×C8—22no—
392C2×C8—22no—
393C2×C8—21no—
394C2×C8—21no—
395C2×C8—42no—
396C2×C8—64no—
397C2×C8D3×D412yes—
398C2×C8D3×D412yes—
399C2×C8C2×#6712yes—
400C2×C8C2×#6711yescharacteristic
401C2×C8C2×#6711yescharacteristic
402C4⋊C4—22no—
403C22⋊C4—22no—
404C22⋊C4—22no—
405C42—21no—
406C42C2×D1211yescharacteristic
407C42C2×D1211yescharacteristic
408C22×C3—22no—
409C22×C3—41no—
410C22×C3—42no—
411C22×C3—42no—
412C22×C3—41no—
413C22×C3C2×#12111yescharacteristic
414C22×C3#54612yes—
415C2×D3—44no—
416C2×D3—84no—
417C2×D3—164no—
418C3×C4—22no—
419C3×C4—41no—
420C3×C4—41no—
421C3×C4—41no—
422C3×C4C2×#12011yescharacteristic
423C3×C4#53511yescharacteristic
424C3×C4D4212yes—
425Dic3—44no—
426C23—21no—
427C23—64no—
428C23—124no—
429C23#93012yes—
430C23C2×#23511yescharacteristic
431Q8—22no—
432Q8—22no—
433Q8—64no—
434Q8—64no—
435Q8—124no—
436D4—24no—
437D4—22no—
438D4—22no—
439D4—42no—
440D4—64no—
441D4—64no—
442D4—82no—
443D4—124no—
444D4—128no—
445D4—124no—
446D4—124no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—64no—
455C2×C4—81no—
456C2×C4—128no—
457C2×C4—124no—
458C2×C4—124no—
459C2×C4—124no—
460C2×C4#93012yes—
461C2×C4C22×D1211yescharacteristic
462C2×C4C2×D3×D412yes—
463C2×C4#100912yes—
464C2×C4C22×#6711yescharacteristic
465C8—22no—
466C8—41no—
467C8—41no—
468C8—124no—
469C8#93711yescharacteristic
470C8#93711yescharacteristic
471C8#93712yes—
472C2×C3—21no—
473C2×C3—22no—
474C2×C3—42no—
475C2×C3—81no—
476C2×C3#240811yescharacteristic
477D3—84no—
478C22—22no—
479C22—41no—
480C22—42no—
481C22—42no—
482C22—41no—
483C22—124no—
484C22—244no—
485C22—484no—
486C22#555012yes—
487C22C2×#93011yescharacteristic
488C4—22no—
489C4—41no—
490C4—41no—
491C4—41no—
492C4—124no—
493C4D4×D1212yes—
494C4C2×#93711yescharacteristic
495C4#583011yescharacteristic
496C3#2428811yescharacteristic, complemented, Sylow
497C2—21no—
498C2—22no—
499C2—42no—
500C2—81no—
501C2—244no—
502C2#8240311yescharacteristic
subgroupG/H
centerC4#5830
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2932C2
socleC2×C3#2408
hypercenter#2932D3
hypocenterC3#24288
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C22×C4 ◁ #2932

Complements

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

NH ≅ G/Ntype
1. #87163C2semidirect
2. C3×#2932C2semidirect
3. #83805C2semidirect
4. #83322C2semidirect
5. #72014C2semidirect
6. #70531C2semidirect
7. #70013C2semidirect
8. #70009C2semidirect
9. #69845C2semidirect
13. C2×C3×#131C22semidirect
15. #5822C22semidirect
25. C3×#423C22semidirect
27. C3×#408C22semidirect
28. C2×C3×#97C22semidirect
33. #5132C22semidirect
34. #5110C22semidirect
35. C2×#819C22semidirect
40. C2×#805C22semidirect
41. C3⋊#338C22semidirect
45. #2932D3semidirect
91. C2×Dic12D4semidirect
100. C2×D24D4semidirect
102. C2×#204D4semidirect
127. C2×#131C2×D3semidirect
139. #423C2×D3semidirect
145. #408C2×D3semidirect
146. C2×#97C2×D3semidirect
496. C3#24288semidirect

Maximal subgroups

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

subgroupindex
#871632
C3×#29322
#838052
#833222
#720142
#705312
#700132
#700092
#698452
#242883