Finite Group Databasebeta

← #10883582

Subgroups — #10883582

Order 768.

812 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5142C211yesmaximal, characteristic, complemented
2#83792C211yesmaximal, characteristic, complemented
3#79183C211yesmaximal, characteristic, complemented
4#79167C211yesmaximal, characteristic, complemented
5C3×#1812C211yesmaximal, characteristic, complemented
6#70665C211yesmaximal, characteristic, complemented
7#70657C211yesmaximal, characteristic, complemented
8#70050C212yesmaximal, complemented
9#70049C212yesmaximal, complemented
10#68706C212yesmaximal, complemented
11#68705C212yesmaximal, complemented
12#28183—31nomaximal, Sylow
13C2×C3×#131C2211yescharacteristic
14#5822C2211yescharacteristic
15C2×#949C2212yes—
16C22×D24C2211yescharacteristic
17C2×#901C2211yescharacteristic
18#5525C2212yes—
19#5521C2212yes—
20C3×#427C2211yescharacteristic
21C3×#427C2211yescharacteristic
22C3×#410C2212yes—
23#5236C2211yescharacteristic
24#5145—22no—
25#5144—22no—
26#5142—22no—
27#5142—22no—
28#5139C2211yescharacteristic
29C2×#823C2212yes—
30#5124C2211yescharacteristic
31C2×#819C2211yescharacteristic
32#5101C2211yescharacteristic
33#5005C2212yes—
34#5001C2212yes—
35#5001C2212yes—
36#4883—22no—
37#4872—22no—
38#4869—22no—
39#4865—22no—
40C3×H(3,ℤ/4)C2212yes—
41#4533C2212yes—
42#4533C2212yes—
43#4531C2212yes—
44C2×#472—31no—
45#3058—31no—
46#2955—31no—
47#2948—31no—
48#2081—32no—
49#2008—32no—
50#1911—32no—
51#1904—32no—
52#1833—31no—
53#1814—31no—
54#1812D311yescharacteristic, complemented
55C2×C3×#41C2311yescharacteristic
56C2×#235C2311yescharacteristic
57C22×D12C2312yes—
58C3×#131—21no—
59C3×#131—21no—
60C3×#131—21no—
61C3×#131—21no—
62C2×C3×M16—21no—
63C2×C3×M16C2311yescharacteristic
64C22×C3×C8—21no—
65C22×C3×C8C2311yescharacteristic
66C3×#117C2312yes—
67C3×#117C2312yes—
68#951—21no—
69#950—22no—
70#949—21no—
71#949—24no—
72#949—42no—
73#942—21no—
74C2×#215—22no—
75#932—21no—
76C2×#213—22no—
77#926C2311yescharacteristic
78C2×#208C2311yescharacteristic
79C2×#208C2312yes—
80C2×#208C2311yescharacteristic
81C2×D24—21no—
82C2×D24—21no—
83C2×D24—21no—
84C2×D24—21no—
85C2×D24D412yes—
86C2×D24D412yes—
87#901—21no—
88#901—21no—
89#901—21no—
90#901—21no—
91C2×ΓL(1,𝔽25)—21no—
92C2×D3×C8—21no—
93#895—24no—
94#893—24no—
95#882—42no—
96D3×C22⋊C4—42no—
97#875—22no—
98#874—22no—
99C4×D12—42no—
100C3×#107—21no—
101C3×#107—21no—
102C3×#106—21no—
103C3×#106—21no—
104C3×#99—42no—
105#837—22no—
106#825—22no—
107#823—24no—
108#822—22no—
109#819—21no—
110#819—21no—
111#815D412yescomplemented
112C8×Dic3D412yescomplemented
113#809—24no—
114#809—24no—
115#805—42no—
116#575—31no—
117C2×AGL(1,ℤ/8)—32no—
118C22×D8—31no—
119C2×#131—31no—
120C2×#131C2×D311yescharacteristic
121#535—32no—
122#518—32no—
123#472—62no—
124#472—62no—
125#469—62no—
126#466—62no—
127#444—62no—
128#443—62no—
129#442—62no—
130C4×D8—62no—
131#430—31no—
132C2×#108—31no—
133#427C2×D311yescharacteristic
134#427C2×D311yescharacteristic
135#421—31no—
136#421—31no—
137#421—32no—
138#418—32no—
139#418—32no—
140C2×#100—32no—
141#410C2×D312yes—
142#405—31no—
143#344—32no—
144#344—32no—
145H(3,ℤ/4)—32no—
146H(3,ℤ/4)C2×D312yes—
147C23×D3—22no—
148C3×#41—21no—
149C3×#41—21no—
150C3×#41—21no—
151C3×#41—21no—
152C2×C3×Q8#4111yescharacteristic
153C2×C3×D4—21no—
154C2×C3×D4#4111yescharacteristic
155C22×C3×C4—21no—
156C22×C3×C4C2411yescharacteristic
157C2×#67—21no—
158#235—21no—
159#235—21no—
160#235—21no—
161#235—21no—
162C2×D12—22no—
163C2×D12—22no—
164C2×D12—22no—
165C2×D12—22no—
166C2×D12—22no—
167C2×D12—44no—
168C2×D12C2×D414yes—
169C2×D12#4111yescharacteristic
170C2×C4×D3—21no—
171C2×C4×D3—44no—
172C2×Dic6#4111yescharacteristic
173C3×M16—21no—
174C3×M16—21no—
175C3×M16—41no—
176C3×M16—41no—
177C2×C3×C8—21no—
178C2×C3×C8—41no—
179C2×C3×C8—41no—
180C2×C3×C8C2×D411yescharacteristic
181C2×C3×C8C2×D411yescharacteristic
182C2×C3×C8C2×D411yescharacteristic
183C2×C3×C8C2×D411yescharacteristic
184C2×C3×C8#4111yescharacteristic
185C2×C3×C8#4111yescharacteristic
186C3×C4⋊C4—22no—
187C3×C4⋊C4—22no—
188C3×C22⋊C4—22no—
189C3×C22⋊C4—22no—
190C3×C42—22no—
191C3×C42—22no—
192#217—21no—
193#216—22no—
194#215—22no—
195#215—44no—
196#214—22no—
197#213—22no—
198#213—44no—
199(C2×D3)⋊C4—44no—
200(C2×D3)⋊C4—44no—
201C4⋊Dic3—21no—
202C4×Dic3C2×D412yes—
203#208—21no—
204#208—21no—
205#208—22no—
206#208—21no—
207#208—21no—
208#208—22no—
209C2×C3⋊C8—21no—
210C2×C3⋊C8—22no—
211C2×C3⋊C8C2×D412yes—
212D24—22no—
213D24—22no—
214D24—22no—
215D24—22no—
216ΓL(1,𝔽25)—41no—
217ΓL(1,𝔽25)—41no—
218D3×C8—41no—
219D3×C8—41no—
220C2×#41—31no—
221C2×#41C22×D311yescharacteristic
222C22×D4—32no—
223#137—61no—
224#137—61no—
225AGL(1,ℤ/8)—61no—
226AGL(1,ℤ/8)—61no—
227AGL(1,ℤ/8)—64no—
228AGL(1,ℤ/8)—122no—
229#135—62no—
230C2×SD16—62no—
231C2×D8—32no—
232C2×D8—32no—
233C2×D8—62no—
234C2×D8—61no—
235C2×D8—61no—
236C2×D8—61no—
237C2×D8—61no—
238#131—21no—
239#131—21no—
240#131—21no—
241#131—21no—
242#131—61no—
243#131—61no—
244#131—61no—
245#131—61no—
246C2×M16—21no—
247C2×M16—31no—
248C2×M16—31no—
249C2×M16—32no—
250C2×M16—61no—
251C2×M16C22×D311yescharacteristic
252C22×C8—21no—
253C22×C8—61no—
254C22×C8C22×D311yescharacteristic
255#127—62no—
256#124—62no—
257#121—64no—
258#120—122no—
259C4×D4—64no—
260C4×D4—122no—
261#117—31no—
262#117C22×D312yes—
263#117C22×D312yes—
264C2×C22⋊C4—122no—
265#108—62no—
266#108—61no—
267#108—61no—
268#107—21no—
269#107—21no—
270#106—21no—
271#106—21no—
272#104—62no—
273#104—62no—
274#104—122no—
275#102—64no—
276#102—64no—
277#100—64no—
278#99—42no—
279#97—32no—
280C4×C8—32no—
281C23×C3—21no—
282C23×C3H(5,𝔽2)11yescharacteristic
283C22×D3—21no—
284C22×D3—24no—
285C22×D3—42no—
286C22×D3—42no—
287C22×D3—162no—
288C3×Q8—21no—
289C3×Q8—21no—
290C3×D4—21no—
291C3×D4—21no—
292C3×D4—41no—
293C3×D4—41no—
294C2×C3×C4—21no—
295C2×C3×C4—41no—
296C2×C3×C4—41no—
297C2×C3×C4—42no—
298C2×C3×C4—42no—
299C2×C3×C4C22×D411yescharacteristic
300C2×C3×C4C22×D411yescharacteristic
301C2×C3×C4C2×#4111yescharacteristic
302C2×C3×C4C2×#4111yescharacteristic
303C2×C3×C4C2×#4111yescharacteristic
304C2×C3×C4H(5,𝔽2)11yescharacteristic
305#67—41no—
306#67—41no—
307C2×Dic3—22no—
308C2×Dic3—21no—
309D12—24no—
310D12—21no—
311D12—24no—
312D12—21no—
313D12—42no—
314D12—42no—
315D12—164no—
316C4×D3—41no—
317C4×D3—41no—
318C4×D3—164no—
319Dic6—21no—
320Dic6—21no—
321C3×C8—41no—
322C3×C8#12111yescharacteristic
323C3×C8#12111yescharacteristic
324C3×C8#12111yescharacteristic
325C3×C8#12111yescharacteristic
326C3⋊C8—22no—
327C3⋊C8—41no—
328C3⋊C8—42no—
329C24—62no—
330#41—21no—
331#41—21no—
332#41—21no—
333#41—21no—
334#41—61no—
335#41—61no—
336#41—61no—
337#41—61no—
338C2×Q8—31no—
339C2×Q8#23911yescharacteristic
340C2×D4—21no—
341C2×D4—34no—
342C2×D4—31no—
343C2×D4—61no—
344C2×D4—62no—
345C2×D4—62no—
346C2×D4—62no—
347C2×D4—62no—
348C2×D4—62no—
349C2×D4—124no—
350C2×D4#23911yescharacteristic
351C22×C4—21no—
352C22×C4—61no—
353C22×C4—124no—
354C22×C4C23×D311yescharacteristic
355Q16—62no—
356SD16—62no—
357SD16—62no—
358SD16—124no—
359D8—62no—
360D8—62no—
361D8—62no—
362D8—62no—
363D8—62no—
364D8—124no—
365M16—21no—
366M16—21no—
367M16—41no—
368M16—41no—
369M16—61no—
370M16—61no—
371M16—61no—
372M16—62no—
373M16—61no—
374M16—62no—
375M16—121no—
376M16—121no—
377C2×C8—21no—
378C2×C8—32no—
379C2×C8—41no—
380C2×C8—41no—
381C2×C8—61no—
382C2×C8—62no—
383C2×C8—121no—
384C2×C8—121no—
385C2×C8#23511yescharacteristic
386C2×C8#23511yescharacteristic
387C2×C8D3×D411yescharacteristic
388C2×C8D3×D411yescharacteristic
389C2×C8D3×D411yescharacteristic
390C2×C8D3×D411yescharacteristic
391C4⋊C4—22no—
392C4⋊C4—22no—
393C4⋊C4—61no—
394C22⋊C4—22no—
395C22⋊C4—22no—
396C22⋊C4—61no—
397C22⋊C4—124no—
398C22⋊C4—124no—
399C42—22no—
400C42—22no—
401C42—32no—
402C22×C3—41no—
403C22×C3—41no—
404C22×C3—81no—
405C22×C3#53511yescharacteristic
406C22×C3#54611yescharacteristic
407C22×C3#55311yescharacteristic
408C2×D3—42no—
409C2×D3—41no—
410C2×D3—44no—
411C2×D3—84no—
412C2×D3—84no—
413C2×D3—162no—
414C3×C4—41no—
415C3×C4—82no—
416C3×C4—82no—
417C3×C4C2×#12111yescharacteristic
418C3×C4#52511yescharacteristic
419C3×C4#54611yescharacteristic
420C3×C4#55311yescharacteristic
421Dic3—42no—
422Dic3—41no—
423C23—21no—
424C23—61no—
425C23—64no—
426C23—122no—
427C23—122no—
428C23—482no—
429C23#100411yescharacteristic
430Q8—21no—
431Q8—21no—
432Q8—61no—
433Q8—61no—
434D4—21no—
435D4—21no—
436D4—41no—
437D4—41no—
438D4—64no—
439D4—61no—
440D4—61no—
441D4—64no—
442D4—121no—
443D4—121no—
444D4—122no—
445D4—122no—
446D4—484no—
447C2×C4—21no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—62no—
453C2×C4—61no—
454C2×C4—121no—
455C2×C4—121no—
456C2×C4—484no—
457C2×C4C2×#23511yescharacteristic
458C2×C4C2×D3×D411yescharacteristic
459C2×C4C2×D3×D411yescharacteristic
460C2×C4C2×#23911yescharacteristic
461C2×C4D3×#4111yescharacteristic
462C2×C4#100911yescharacteristic
463C8—41no—
464C8—62no—
465C8—121no—
466C8—122no—
467C8#88411yescharacteristic
468C8#88411yescharacteristic
469C8#88411yescharacteristic
470C8#88411yescharacteristic
471C2×C3—21no—
472C2×C3—21no—
473C2×C3—21no—
474C2×C3—81no—
475C2×C3#253511yescharacteristic
476D3—81no—
477D3—84no—
478C22—41no—
479C22—41no—
480C22—81no—
481C22—122no—
482C22—121no—
483C22—124no—
484C22—244no—
485C22—244no—
486C22—482no—
487C22#551411yescharacteristic
488C22#560711yescharacteristic
489C22#561211yescharacteristic
490C4—41no—
491C4—82no—
492C4—82no—
493C4—122no—
494C4—121no—
495C4C2×#88411yescharacteristic
496C4#553011yescharacteristic
497C4#562711yescharacteristic
498C4#563011yescharacteristic
499C3#2818311yescharacteristic, complemented, Sylow
500C2—21no—
501C2—21no—
502C2—21no—
503C2—81no—
504C2—241no—
505C2—244no—
506C2#7393311yescharacteristic
subgroupG/H
centerC2#73933
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1812C2
socleC2×C3#2535
hypercenter#1812D3
hypocenterC3#28183
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C22×C4 ◁ #1812

Complements

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

NH ≅ G/Ntype
1. C2×#5142C2semidirect
2. #83792C2semidirect
3. #79183C2semidirect
4. #79167C2semidirect
5. C3×#1812C2semidirect
6. #70665C2semidirect
7. #70657C2semidirect
8. #70050C2semidirect
9. #70049C2semidirect
10. #68706C2semidirect
11. #68705C2semidirect
54. #1812D3semidirect
111. #815D4semidirect
112. C8×Dic3D4semidirect
499. C3#28183semidirect

Maximal subgroups

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

subgroupindex
C2×#51422
#837922
#791832
#791672
C3×#18122
#706652
#706572
#700502
#700492
#687062
#687052
#281833