Finite Group Databasebeta

← #72258813

Subgroups — #72258813

Order 1536.

742 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10813663C211yesmaximal, characteristic
2#10711285C211yesmaximal, characteristic
3C2×#67568C211yesmaximal, characteristic
4#10688953C211yesmaximal, characteristic, complemented
5#10682359C211yesmaximal, characteristic, complemented
6#10682127C211yesmaximal, characteristic, complemented
7C3×#13186C211yesmaximal, characteristic, complemented
8#136361—31nomaximal, Sylow
9#79178—21no—
10#78574—21no—
11C2×#4700C2211yescharacteristic
12C2×C3×#365—21no—
13C2×C3×#364C2211yescharacteristic
14C3×#1844C2211yescharacteristic
15C3×#1772C2211yescharacteristic
16#70720—21no—
17C2×#4488—21no—
18#69893—21no—
19#69764C2211yescharacteristic
20C2×#4463C2211yescharacteristic
21C2×C8×C3⋊C8C2211yescharacteristic
22#69576—22no—
23#68771—22no—
24C3×#1386—21no—
25#67720—21no—
26#67568C414yescomplemented
27#22689—31no—
28#13186D311yescharacteristic, complemented
29#12823—31no—
30C2×#1335—31no—
31#8813—31no—
32#7268—31no—
33#7179—31no—
34C2×#904D411yescharacteristic
35C2×#901—21no—
36#5523—22no—
37C2×C3×M32—21no—
38C2×C3×#108D411yescharacteristic
39C3×#427D411yescharacteristic
40C2×C3×#106—21no—
41C2×C3×#106—21no—
42C3×#405—21no—
43C2×C3×C4×C8C2311yescharacteristic
44C3×#384—21no—
45#5133—22no—
46C2×#819D411yescharacteristic
47C2×#817—21no—
48C2×#817—21no—
49C2×#812—21no—
50#4995—22no—
51C2×#808—44no—
52#4935D412yes—
53C2×#805—21no—
54C2×C4×C3⋊C8—21no—
55#4881—42no—
56#4860—41no—
57D3×#106—41no—
58#4716—41no—
59#4711—41no—
60#4700—21no—
61#4700—21no—
62#4700C2×C412yes—
63#4700Q812yes—
64C4×#204—42no—
65C3×#365—42no—
66C3×#364—21no—
67C3×#364C2×C412yescomplemented
68C3×#339—21no—
69#4554—22no—
70#4488—42no—
71#4463—21no—
72#4463—21no—
73C8×C3⋊C8—21no—
74C8×C3⋊C8C2×C412yescomplemented
75C2×#443—31no—
76#2952—61no—
77#2912—61no—
78C2×#365—21no—
79C2×#364—31no—
80C2×#364C2×D311yescharacteristic
81C2×#360—61no—
82#1951—61no—
83#1859—62no—
84#1844—31no—
85#1844—62no—
86#1844C2×D311yescharacteristic
87#1815—61no—
88#1772C2×D311yescharacteristic
89C2×C82—31no—
90#1386—21no—
91#1383—61no—
92#1335—34no—
93C2×#235—21no—
94C2×C3×M16—21no—
95C22×C3×C8C2×D411yescharacteristic
96C22×C3×C8C2×D411yescharacteristic
97C3×#117—21no—
98C2×C3×C4⋊C4—42no—
99C2×C3×C4⋊C4—41no—
100C2×C3×C42C2×D411yescharacteristic
101C2×C4⋊Dic3—22no—
102C2×C4⋊Dic3—42no—
103C2×#208—21no—
104C22×C3⋊C8—22no—
105C2×Dic12SD1611yescharacteristic
106#904—21no—
107#904—21no—
108#904C4⋊C412yes—
109#904SD1612yes—
110C2×D24SD1611yescharacteristic
111C2×#204—21no—
112#901—22no—
113#901—22no—
114#901—41no—
115#901—41no—
116C2×ΓL(1,𝔽25)—41no—
117C2×D3×C8—41no—
118#892—42no—
119D3×C4⋊C4—42no—
120#67⋊C4—82no—
121C4×D12—42no—
122C4×Dic6—42no—
123C3×M32—22no—
124C3×M32—41no—
125C2×C3×C16—41no—
126C3×#108—21no—
127C3×#108—21no—
128C3×#107SD1612yes—
129C3×#106—21no—
130C3×#106—41no—
131C3×#106—41no—
132C3×#106—41no—
133C3×#106—41no—
134C3×#97—41no—
135C3×C4×C8—41no—
136C3×C4×C8C22×C411yescharacteristic
137C3×C4×C8C2×Q811yescharacteristic
138C3×C4×C8#4111yescharacteristic
139C3×C4×C8#4111yescharacteristic
140#821—42no—
141#819—21no—
142#819C4⋊C412yes—
143#817—41no—
144#817—41no—
145#817—41no—
146#817—41no—
147#816—42no—
148#812—41no—
149#812—41no—
150C2×C3⋊C16—41no—
151#808—42no—
152#808—88no—
153#807—42no—
154#805—22no—
155#805—22no—
156C4×C3⋊C8—22no—
157C4×C3⋊C8—22no—
158C2×#135—31no—
159C2×#131—61no—
160#520—62no—
161C2×M32—21no—
162C2×M32—61no—
163#443—32no—
164#443—32no—
165#443—61no—
166#443—61no—
167#442—121no—
168#441—121no—
169C4×SD16—122no—
170C2×#108—31no—
171C2×#108#6711yescharacteristic
172#428—62no—
173#428—122no—
174#427—121no—
175#427#6711yescharacteristic
176C2×#106—21no—
177C2×#106—21no—
178C2×#106—61no—
179C2×#106—61no—
180C2×#106—121no—
181C2×#106—124no—
182C2×#104—61no—
183#417—62no—
184#405—21no—
185C2×C4×C8—61no—
186C2×C4×C8C22×D311yescharacteristic
187#384—21no—
188#384—32no—
189#365—42no—
190#364—21no—
191#364—61no—
192#364—61no—
193#364C4×D312yescomplemented
194#360—122no—
195#343—62no—
196#339—21no—
197C82—32no—
198C82—61no—
199C22×C3×C4—21no—
200C22×C3×C4—81no—
201C22×C3×C4#12011yescharacteristic
202C2×#67—41no—
203C22×Dic3—42no—
204#235—22no—
205#235—22no—
206#235—41no—
207#235—41no—
208C2×D12—21no—
209C2×C4×D3—41no—
210C2×C4×D3—82no—
211C2×Dic6—21no—
212C3×M16—41no—
213C3×M16—41no—
214C2×C3×C8—41no—
215C2×C3×C8C2×C4⋊C411yescharacteristic
216C2×C3×C8C4×D411yescharacteristic
217C2×C3×C8C22⋊Q811yescharacteristic
218C2×C3×C8#12311yescharacteristic
219C2×C3×C8C2×SD1611yescharacteristic
220C2×C3×C8C2×SD1611yescharacteristic
221C2×C3×C8C2×SD1611yescharacteristic
222C2×C3×C8#13511yescharacteristic
223C2×C3×C8AGL(1,ℤ/8)11yescharacteristic
224C2×C3×C8AGL(1,ℤ/8)11yescharacteristic
225C2×C3×C8AGL(1,ℤ/8)11yescharacteristic
226C2×C3×C8#13711yescharacteristic
227C3×C4⋊C4—22no—
228C3×C4⋊C4—84no—
229C3×C4⋊C4—81no—
230C3×C4⋊C4—81no—
231C3×C22⋊C4—41no—
232C3×C42—21no—
233C3×C42—41no—
234C3×C42C4×D411yescharacteristic
235C3×C42C22⋊Q811yescharacteristic
236(C2×D3)⋊C4—82no—
237C4⋊Dic3—42no—
238C4⋊Dic3—42no—
239C4⋊Dic3—84no—
240Dic3⋊C4—82no—
241C4×Dic3—82no—
242#208—22no—
243#208—41no—
244C2×C3⋊C8—24no—
245C2×C3⋊C8—28no—
246C2×C3⋊C8—41no—
247Dic12—21no—
248Dic12#10612yescomplemented
249D24—21no—
250D24#10612yescomplemented
251#204—22no—
252#204—22no—
253ΓL(1,𝔽25)—42no—
254ΓL(1,𝔽25)—42no—
255D3×C8—42no—
256D3×C8—42no—
257C3×C16—42no—
258C3⋊C16—81no—
259C2×#41—61no—
260#135—32no—
261#135—32no—
262#135—61no—
263#135—61no—
264C2×Q16—31no—
265C2×SD16—61no—
266C2×D8—31no—
267#131—62no—
268#131—62no—
269#131—121no—
270#131—121no—
271C2×M16—21no—
272C2×M16—61no—
273C2×M16—121no—
274C22×C8—62no—
275C22×C8—121no—
276C22×C8D3×D411yescharacteristic
277C22×C8C2×#6711yescharacteristic
278C4×Q8—122no—
279C4×D4—122no—
280C4×D4—242no—
281#117—21no—
282#117—122no—
283C2×C4⋊C4—42no—
284C2×C4⋊C4—41no—
285C2×C4⋊C4—62no—
286C2×C4⋊C4—122no—
287C2×C4⋊C4—122no—
288C2×C42D3×D411yescharacteristic
289M32—22no—
290M32—41no—
291M32—121no—
292M32—121no—
293C2×C16—41no—
294C2×C16—121no—
295#108—21no—
296#108—21no—
297#108—32no—
298#108—61no—
299#107—122no—
300#107#21512yes—
301#106—21no—
302#106—41no—
303#106—41no—
304#106—41no—
305#106—41no—
306#106—122no—
307#106—121no—
308#106—121no—
309#106—121no—
310#106—121no—
311#106—248no—
312#104—62no—
313#104—62no—
314#103—122no—
315#102—122no—
316#97—41no—
317C4×C8—41no—
318C4×C8—62no—
319C4×C8—62no—
320C4×C8C2×C4×D311yescharacteristic
321C4×C8#23511yescharacteristic
322C4×C8D3×Q811yescharacteristic
323C4×C8#23911yescharacteristic
324C23×C3#45311yescharacteristic
325C22×D3—41no—
326C2×C3×C4—24no—
327C2×C3×C4—21no—
328C2×C3×C4—21no—
329C2×C3×C4—42no—
330C2×C3×C4—162no—
331C2×C3×C4—161no—
332C2×C3×C4#38511yescharacteristic
333C2×C3×C4#44811yescharacteristic
334C2×C3×C4#44911yescharacteristic
335C2×C3×C4#45011yescharacteristic
336C2×C3×C4#45011yescharacteristic
337C2×C3×C4#45611yescharacteristic
338#67—42no—
339#67—42no—
340C2×Dic3—41no—
341C2×Dic3—82no—
342C2×Dic3—162no—
343D12—22no—
344D12—41no—
345C4×D3—42no—
346C4×D3—42no—
347C4×D3—322no—
348Dic6—22no—
349Dic6—41no—
350C3×C8—81no—
351C3×C8C2×#10611yescharacteristic
352C3×C8#42811yescharacteristic
353C3×C8C4×SD1611yescharacteristic
354C3×C8#44211yescharacteristic
355C3×C8#47611yescharacteristic
356C3×C8#47811yescharacteristic
357C3×C8#48111yescharacteristic
358C3×C8#48311yescharacteristic
359C3⋊C8—28no—
360C3⋊C8—42no—
361#41—62no—
362#41—62no—
363#41—121no—
364#41—121no—
365C2×Q8—61no—
366C2×D4—61no—
367C2×D4—121no—
368C22×C4—21no—
369C22×C4—81no—
370C22×C4—121no—
371C22×C4—122no—
372C22×C4—242no—
373C22×C4#93711yescharacteristic
374Q16—32no—
375Q16—61no—
376SD16—62no—
377SD16—62no—
378D8—32no—
379D8—61no—
380M16—41no—
381M16—41no—
382M16—62no—
383M16—121no—
384M16—122no—
385M16—122no—
386C2×C8—41no—
387C2×C8—64no—
388C2×C8—68no—
389C2×C8—121no—
390C2×C8—122no—
391C2×C8—122no—
392C2×C8D3×C4⋊C411yescharacteristic
393C2×C8#89411yescharacteristic
394C2×C8D3×SD1611yescharacteristic
395C2×C8D3×SD1611yescharacteristic
396C2×C8#91411yescharacteristic
397C2×C8#91511yescharacteristic
398C2×C8C4×#6711yescharacteristic
399C2×C8#93211yescharacteristic
400C2×C8C2×#21511yescharacteristic
401C2×C8#94611yescharacteristic
402C2×C8#94911yescharacteristic
403C2×C8#95011yescharacteristic
404C4⋊C4—22no—
405C4⋊C4—84no—
406C4⋊C4—81no—
407C4⋊C4—81no—
408C4⋊C4—122no—
409C4⋊C4—122no—
410C4⋊C4—244no—
411C4⋊C4—242no—
412C22⋊C4—41no—
413C22⋊C4—242no—
414C42—21no—
415C42—41no—
416C42—242no—
417C42#89311yescharacteristic
418C42#89611yescharacteristic
419C16—42no—
420C16—241no—
421C22×C3—22no—
422C22×C3#162211yescharacteristic
423C22×C3#162911yescharacteristic
424C22×C3#185911yescharacteristic
425C2×D3—42no—
426C2×D3—82no—
427C3×C4—22no—
428C3×C4—22no—
429C3×C4—82no—
430C3×C4—321no—
431C3×C4#165111yescharacteristic
432C3×C4#165411yescharacteristic
433C3×C4#185311yescharacteristic
434C3×C4#186411yescharacteristic
435Dic3—42no—
436Dic3—162no—
437C23—121no—
438C23#519811yescharacteristic
439Q8—62no—
440Q8—121no—
441D4—62no—
442D4—122no—
443D4—122no—
444D4—121no—
445C2×C4—24no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—42no—
449C2×C4—121no—
450C2×C4—122no—
451C2×C4—122no—
452C2×C4—162no—
453C2×C4—161no—
454C2×C4—242no—
455C2×C4—482no—
456C2×C4—962no—
457C2×C4#498911yescharacteristic
458C2×C4#503711yescharacteristic
459C2×C4#504611yescharacteristic
460C2×C4#516911yescharacteristic
461C2×C4#517011yescharacteristic
462C2×C4#520511yescharacteristic
463C8—68no—
464C8—81no—
465C8—122no—
466C8D3×#10611yescharacteristic
467C8#486111yescharacteristic
468C8#486211yescharacteristic
469C8#486311yescharacteristic
470C8#501411yescharacteristic
471C8C4×#21511yescharacteristic
472C8#506611yescharacteristic
473C8#508411yescharacteristic
474C2×C3—21no—
475C2×C3—21no—
476C2×C3#888611yescharacteristic
477C2×C3#1236412yes—
478D3—82no—
479C22—22no—
480C22—122no—
481C22—242no—
482C22#7068011yescharacteristic
483C22#7102811yescharacteristic
484C22#7114311yescharacteristic
485C4—22no—
486C4—22no—
487C4—82no—
488C4—122no—
489C4—321no—
490C4—482no—
491C4#6962611yescharacteristic
492C4#7057911yescharacteristic
493C4#7105111yescharacteristic
494C4#7112711yescharacteristic
495C3#13636111yescharacteristic, complemented, Sylow
496C2—21no—
497C2—21no—
498C2—242no—
499C2#1068937411yescharacteristic
500C2#1070894512yes—
subgroupG/H
centerC22#71143
commutatorC3×C4×C8C22×C4
FrattiniC2×C4×C8C22×D3
FittingC3×#13186C2
socleC22×C3#1629
prime radicalC2×#904D4
cosocleC2×C3×C4×C8C23
hypercenter#13186D3
hypocenterC3#136361
LCSG ▷ C3×C4×C8 ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ #13186

Complements

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

NH ≅ G/Ntype
4. #10688953C2semidirect
5. #10682359C2semidirect
6. #10682127C2semidirect
7. C3×#13186C2semidirect
26. #67568C4semidirect
28. #13186D3semidirect
67. C3×#364C2×C4semidirect
74. C8×C3⋊C8C2×C4semidirect
193. #364C4×D3semidirect
248. Dic12#106semidirect
250. D24#106semidirect
495. C3#136361semidirect

Maximal subgroups

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

subgroupindex
#108136632
#107112852
C2×#675682
#106889532
#106823592
#106821272
C3×#131862
#1363613