Finite Group Databasebeta

← #10883573

Subgroups — #10883573

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
1C3×#2948C211yesmaximal, characteristic, complemented
2C2×#5160C211yesmaximal, characteristic, complemented
3#83836C211yesmaximal, characteristic, complemented
4#78579C211yesmaximal, characteristic, complemented
5#71652C212yesmaximal, complemented
6#71651C212yesmaximal, complemented
7#70665C211yesmaximal, characteristic, complemented
8#70664C211yesmaximal, characteristic, complemented
9#70331C212yesmaximal, complemented
10#70321C212yesmaximal, complemented
11#69896C211yesmaximal, characteristic, complemented
12#28183—31nomaximal, Sylow
13C22×C3×D8C2211yescharacteristic
14C3×#518C2212yes—
15#5805C2212yes—
16C2×#932C2212yes—
17#5748C2211yescharacteristic
18C2×#901C2211yescharacteristic
19C2×#901C2211yescharacteristic
20C3×#442—22no—
21C3×C4×D8—22no—
22C3×#427C2211yescharacteristic
23C3×#418C2212yes—
24C3×#405C2211yescharacteristic
25C2×#833C2212yes—
26#5215C2212yes—
27#5160—22no—
28#5160—22no—
29#5159—22no—
30#5157—22no—
31#5125C2211yescharacteristic
32C2×#819C2211yescharacteristic
33#5106C2211yescharacteristic
34#5006C2212yes—
35#5005C2211yescharacteristic
36#5005C2211yescharacteristic
37#5005C2212yes—
38#4717—22no—
39#4700—22no—
40#4556C2212yes—
41#4533C2212yes—
42#4531C2212yes—
43#4531C2212yes—
44C2×#472—31no—
45#3058—31no—
46#2955—31no—
47#2948D311yescharacteristic, complemented
48#2081—32no—
49#2008—32no—
50#1911—32no—
51#1904—32no—
52#1833—31no—
53#1814—31no—
54#1812—31no—
55C22×C3×D4C2312yes—
56C2×#235C2311yescharacteristic
57C2×#235C2311yescharacteristic
58C2×C3×D8—21no—
59C2×C3×D8—21no—
60C2×C3×D8—21no—
61C2×C3×D8—21no—
62C2×C3×D8D412yes—
63C2×C3×D8D412yes—
64C2×C3×M16C2311yescharacteristic
65C22×C3×C8C2311yescharacteristic
66C3×C4×D4—24no—
67C3×C4×D4—42no—
68C3×#117C2312yes—
69C3×#117C2311yescharacteristic
70C2×C3×C22⋊C4—42no—
71#953—42no—
72#940—22no—
73#938—24no—
74#936—22no—
75C2×#214—22no—
76#932—24no—
77#932—42no—
78C2×#213—22no—
79#926C2312yes—
80C2×#208C2311yescharacteristic
81C2×#208C2311yescharacteristic
82C2×#208C2312yes—
83#909—21no—
84#909—21no—
85#908—21no—
86#908—21no—
87#904—22no—
88#901—21no—
89#901—21no—
90#901—21no—
91#901—21no—
92#901—21no—
93#901—21no—
94#901—21no—
95#901—21no—
96C2×ΓL(1,𝔽25)—21no—
97C2×ΓL(1,𝔽25)—21no—
98C2×D3×C8—21no—
99C2×D3×C8—21no—
100C3×#107—21no—
101C3×#106—21no—
102C3×#102—24no—
103C3×#97D412yescomplemented
104C3×C4×C8D412yescomplemented
105#833—24no—
106#832—24no—
107#819—22no—
108#819—21no—
109#819—21no—
110#818—21no—
111#817—21no—
112#806—42no—
113#805—22no—
114#805—22no—
115#805—42no—
116#575—31no—
117C2×AGL(1,ℤ/8)—32no—
118C22×D8C2×D311yescharacteristic
119C2×#131—31no—
120C2×#131—31no—
121#535—32no—
122#518C2×D312yes—
123#472—62no—
124#472—62no—
125#469—62no—
126#466—62no—
127#444—62no—
128#443—62no—
129#442—22no—
130C4×D8—22no—
131#430—31no—
132C2×#108—31no—
133#427—31no—
134#427C2×D311yescharacteristic
135#421—31no—
136#421—31no—
137#421—32no—
138#418—32no—
139#418C2×D312yes—
140C2×#100—32no—
141#410—32no—
142#405C2×D311yescharacteristic
143#344—32no—
144#344—32no—
145H(3,ℤ/4)—32no—
146H(3,ℤ/4)—32no—
147C24×C3—22no—
148C2×C3×D4—22no—
149C2×C3×D4—22no—
150C2×C3×D4—22no—
151C2×C3×D4—22no—
152C2×C3×D4—22no—
153C2×C3×D4C2×D414yes—
154C22×C3×C4—44no—
155C22×C3×C4C2411yescharacteristic
156C2×#67—21no—
157C2×#67—21no—
158C2×#67—44no—
159#235—21no—
160#235—21no—
161#235—21no—
162#235—21no—
163#235—21no—
164#235—21no—
165#235—21no—
166#235—21no—
167C2×D12#4111yescharacteristic
168C2×D12#4111yescharacteristic
169C2×C4×D3—21no—
170C2×C4×D3—21no—
171C2×Dic6#4111yescharacteristic
172C2×Dic6#4111yescharacteristic
173C3×D8—22no—
174C3×D8—22no—
175C3×D8—22no—
176C3×D8—22no—
177C3×M16—21no—
178C3×M16—21no—
179C2×C3×C8C2×D411yescharacteristic
180C2×C3×C8C2×D411yescharacteristic
181C2×C3×C8C2×D412yes—
182C2×C3×C8C2×D411yescharacteristic
183C2×C3×C8C2×D411yescharacteristic
184C2×C3×C8#4111yescharacteristic
185C2×C3×C8#4111yescharacteristic
186C3×C4⋊C4—21no—
187C3×C4⋊C4—22no—
188C3×C22⋊C4—21no—
189C3×C22⋊C4—22no—
190C3×C22⋊C4—44no—
191C3×C42—22no—
192C3×C42C2×D412yes—
193#217—22no—
194#217—44no—
195#214—44no—
196#213—44no—
197C4⋊Dic3—22no—
198C4×Dic3—22no—
199#208—21no—
200#208—21no—
201#208—22no—
202#208—21no—
203#208—21no—
204#208—22no—
205C2×C3⋊C8—21no—
206C2×C3⋊C8—21no—
207C2×C3⋊C8—22no—
208Dic12—22no—
209D24—22no—
210#204—22no—
211#204—22no—
212ΓL(1,𝔽25)—41no—
213ΓL(1,𝔽25)—41no—
214ΓL(1,𝔽25)—41no—
215ΓL(1,𝔽25)—41no—
216D3×C8—41no—
217D3×C8—41no—
218D3×C8—41no—
219D3×C8—41no—
220C2×#41—31no—
221C2×#41—31no—
222C22×D4C22×D312yes—
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—21no—
232C2×D8—21no—
233C2×D8—21no—
234C2×D8—21no—
235C2×D8—62no—
236C2×D8D1212yes—
237C2×D8D1212yes—
238#131—61no—
239#131—61no—
240#131—61no—
241#131—61no—
242#131—61no—
243#131—61no—
244#131—61no—
245#131—61no—
246C2×M16—31no—
247C2×M16—31no—
248C2×M16—32no—
249C2×M16—61no—
250C2×M16—61no—
251C2×M16C22×D311yescharacteristic
252C22×C8—61no—
253C22×C8—61no—
254C22×C8C22×D311yescharacteristic
255#127—62no—
256#124—62no—
257#121—64no—
258#120—122no—
259C4×D4—24no—
260C4×D4—42no—
261#117—32no—
262#117C22×D312yes—
263#117C22×D311yescharacteristic
264C2×C22⋊C4—42no—
265#108—61no—
266#108—61no—
267#108—62no—
268#107—21no—
269#107—61no—
270#106—21no—
271#106—61no—
272#104—62no—
273#104—62no—
274#104—122no—
275#102—24no—
276#102—64no—
277#100—64no—
278#99—122no—
279#97#6712yescomplemented
280C4×C8#6712yescomplemented
281C23×C3—24no—
282C23×C3—42no—
283C23×C3—42no—
284C23×C3—162no—
285C23×C3H(5,𝔽2)11yescharacteristic
286C22×D3—21no—
287C22×D3—21no—
288C3×D4—24no—
289C3×D4—24no—
290C3×D4—42no—
291C3×D4—42no—
292C2×C3×C4—22no—
293C2×C3×C4—42no—
294C2×C3×C4—164no—
295C2×C3×C4C22×D411yescharacteristic
296C2×C3×C4C22×D411yescharacteristic
297C2×C3×C4C2×#4111yescharacteristic
298C2×C3×C4C2×#4111yescharacteristic
299C2×C3×C4C2×#4111yescharacteristic
300C2×C3×C4H(5,𝔽2)11yescharacteristic
301#67—41no—
302#67—41no—
303#67—41no—
304#67—41no—
305#67—164no—
306C2×Dic3—21no—
307C2×Dic3—21no—
308C2×Dic3—42no—
309D12—21no—
310D12—21no—
311D12—21no—
312D12—21no—
313C4×D3—41no—
314C4×D3—41no—
315C4×D3—41no—
316C4×D3—41no—
317Dic6—21no—
318Dic6—21no—
319Dic6—21no—
320Dic6—21no—
321C3×C8—22no—
322C3×C8#12111yescharacteristic
323C3×C8#12111yescharacteristic
324C3×C8#12111yescharacteristic
325C3×C8#12111yescharacteristic
326C3⋊C8—41no—
327C3⋊C8—41no—
328C3⋊C8—42no—
329C24—22no—
330#41—61no—
331#41—61no—
332#41—61no—
333#41—61no—
334#41—61no—
335#41—61no—
336#41—61no—
337#41—61no—
338C2×Q8—31no—
339C2×Q8—31no—
340C2×D4—22no—
341C2×D4—22no—
342C2×D4—22no—
343C2×D4—22no—
344C2×D4—22no—
345C2×D4—31no—
346C2×D4—31no—
347C2×D4—61no—
348C2×D4—61no—
349C2×D4—124no—
350C2×D4C2×D1214yes—
351C22×C4—44no—
352C22×C4—61no—
353C22×C4—61no—
354C22×C4C23×D311yescharacteristic
355Q16—62no—
356SD16—62no—
357SD16—62no—
358SD16—124no—
359D8—22no—
360D8—22no—
361D8—22no—
362D8—22no—
363D8—62no—
364D8—124no—
365M16—21no—
366M16—21no—
367M16—61no—
368M16—62no—
369M16—61no—
370M16—62no—
371M16—61no—
372M16—61no—
373M16—121no—
374M16—121no—
375M16—121no—
376M16—121no—
377C2×C8—61no—
378C2×C8—61no—
379C2×C8—62no—
380C2×C8—121no—
381C2×C8—121no—
382C2×C8—121no—
383C2×C8—121no—
384C2×C8C2×D1211yescharacteristic
385C2×C8C2×D1211yescharacteristic
386C2×C8#23511yescharacteristic
387C2×C8#23511yescharacteristic
388C2×C8C2×#6712yes—
389C2×C8C2×#6711yescharacteristic
390C2×C8C2×#6711yescharacteristic
391C4⋊C4—21no—
392C4⋊C4—22no—
393C4⋊C4—62no—
394C22⋊C4—21no—
395C22⋊C4—22no—
396C22⋊C4—44no—
397C22⋊C4—62no—
398C22⋊C4—124no—
399C42—22no—
400C42—62no—
401C42C2×#6712yes—
402C22×C3—41no—
403C22×C3—44no—
404C22×C3—84no—
405C22×C3—84no—
406C22×C3—162no—
407C22×C3#53511yescharacteristic
408C22×C3#54611yescharacteristic
409C22×C3#55311yescharacteristic
410C2×D3—42no—
411C2×D3—41no—
412C2×D3—41no—
413C2×D3—81no—
414C3×C4—42no—
415C3×C4—82no—
416C3×C4C2×#12111yescharacteristic
417C3×C4#52511yescharacteristic
418C3×C4#54611yescharacteristic
419C3×C4#55311yescharacteristic
420Dic3—41no—
421Dic3—41no—
422Dic3—82no—
423C23—24no—
424C23—42no—
425C23—42no—
426C23—61no—
427C23—61no—
428C23—162no—
429C23#100411yescharacteristic
430Q8—61no—
431Q8—61no—
432Q8—61no—
433Q8—61no—
434D4—24no—
435D4—24no—
436D4—42no—
437D4—42no—
438D4—61no—
439D4—61no—
440D4—61no—
441D4—61no—
442D4—121no—
443D4—121no—
444D4—121no—
445D4—121no—
446D4—484no—
447C2×C4—22no—
448C2×C4—42no—
449C2×C4—61no—
450C2×C4—61no—
451C2×C4—121no—
452C2×C4—121no—
453C2×C4—121no—
454C2×C4—121no—
455C2×C4—122no—
456C2×C4—164no—
457C2×C4C22×D1211yescharacteristic
458C2×C4C2×#23511yescharacteristic
459C2×C4#100411yescharacteristic
460C2×C4D3×#4111yescharacteristic
461C2×C4D3×#4111yescharacteristic
462C2×C4C22×#6711yescharacteristic
463C8—22no—
464C8—121no—
465C8—121no—
466C8—122no—
467C8#93011yescharacteristic
468C8#93011yescharacteristic
469C8#93011yescharacteristic
470C8#93011yescharacteristic
471C2×C3—21no—
472C2×C3—21no—
473C2×C3—21no—
474C2×C3—84no—
475C2×C3#253511yescharacteristic
476D3—81no—
477D3—81no—
478C22—41no—
479C22—44no—
480C22—84no—
481C22—84no—
482C22—122no—
483C22—121no—
484C22—121no—
485C22—162no—
486C22—241no—
487C22#555411yescharacteristic
488C22#555811yescharacteristic
489C22#580511yescharacteristic
490C4—42no—
491C4—82no—
492C4—121no—
493C4—121no—
494C4—242no—
495C4#555411yescharacteristic
496C4#555811yescharacteristic
497C4C2×#93011yescharacteristic
498C4#582811yescharacteristic
499C3#2818311yescharacteristic, complemented, Sylow
500C2—21no—
501C2—21no—
502C2—21no—
503C2—84no—
504C2—241no—
505C2—241no—
506C2#8241611yescharacteristic
subgroupG/H
centerC2#82416
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2948C2
socleC2×C3#2535
hypercenter#2948D3
hypocenterC3#28183
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C22×C4 ◁ #2948

Complements

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

NH ≅ G/Ntype
1. C3×#2948C2semidirect
2. C2×#5160C2semidirect
3. #83836C2semidirect
4. #78579C2semidirect
5. #71652C2semidirect
6. #71651C2semidirect
7. #70665C2semidirect
8. #70664C2semidirect
9. #70331C2semidirect
10. #70321C2semidirect
11. #69896C2semidirect
47. #2948D3semidirect
103. C3×#97D4semidirect
104. C3×C4×C8D4semidirect
279. #97#67semidirect
280. C4×C8#67semidirect
499. C3#28183semidirect

Maximal subgroups

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

subgroupindex
C3×#29482
C2×#51602
#838362
#785792
#716522
#716512
#706652
#706642
#703312
#703212
#698962
#281833